Introduction

Atmospheric flight is governed by aerodynamics, so a sound understanding of aerodynamic principles is essential for the design of all flight vehicles. Aeronautical and astronautical engineers must understand aerodynamic behavior across a broad range of conditions. Air is a fluid; fluids can be liquids or gases. Air is also a gas. To understand the aerodynamic forces acting on flight vehicles, it is first necessary to become intimately familiar with the fundamental physical properties that describe fluid behavior and their relationships. This general field is called fluid mechanics^[1]. One must learn about fluid mechanics, in general, and aerodynamics, in particular, before a deeper understanding of the other characteristics of flight vehicles becomes possible.

In all branches of science and engineering, physical properties are defined to help describe how things behave in the physical world. For example, mass, weight, energy, work, power, and related quantities are essential properties relevant to physical phenomena and problem solving. The pertinent properties of fluids include pressure, density, temperature, viscosity, flow velocity, and the speed of sound. These are also point properties because their values can vary from one spatial location to another within the fluid and may vary over time at a given location. These are called macroscopic properties.^[2] In this regard, they apply to bulk matter or a finite group of molecules rather than to each molecule, the matter having net physical dimensions much greater than the mean free path^[3] between the molecules; this approach is known as a continuum assumption.

Furthermore, it must be recognized that these fluid properties are interdependent and are often mediated by temperature. Changing the value of one property may affect the values of other properties. For example, increasing air pressure by compressing it increases its density and may also raise its temperature, depending on the amount of heat transferred during compression. The relationships between pressure, temperature, and density can be established using thermodynamic principles and the ideal gas laws, which are formally encapsulated in an equation of state.

What is a Fluid?

Fluids are substances with mass and volume that have no predefined shape and can flow easily. They can be liquids (e.g., water) or gases (e.g., air). Unlike solids, fluids have relatively mobile molecules, as illustrated in Figure 1. In a solid, molecules are tightly packed in a lattice and are immobile, except for small vibrations about their fixed positions. Solids are primarily rigid and have shapes that are difficult to change under the action of external forces.

However, the molecules are farther apart and more mobile in fluids. This characteristic means that fluids are easily deformed and flow readily under external forces. The tendency of fluids to flow and continuously deform under applied forces makes them more difficult to understand. Of particular interest to aerospace engineers is the gas commonly called “air.” Like all gases, air is composed of molecules that are relatively far apart from one another. Air can be compressed relatively easily, which has numerous consequences for the aerodynamic characteristics of flight vehicles.

Concept of a Continuum

A molecular model is used to describe fluid behavior within the continuum framework. In a continuum model, it is assumed that the distance between individual fluid molecules, or more specifically, their mean free path, denoted by the length scale , is tiny compared to the scale and physical dimensions of the problem, , as suggested in Figure 2. In a continuum model, macroscopic fluid properties such as temperature, density, pressure, and flow velocity are treated as constant at each point in space and vary continuously across the problem domain. In a continuum, any local changes associated with individual molecular motion are irrelevant, which is easily justified in most practical cases of fluid flows.

One way to think about a continuum is to consider a measurement volume, , like a probe used in a wind tunnel to measure pressure. Figure 3 shows the measurement volume at different scales. When is very small, it contains only a few molecules, making it difficult to accurately define macroscopic properties such as density, temperature, and pressure. As increases, it encompasses more molecules, allowing the averaged properties to converge to well-defined macroscopic values. This is the essence of the continuum assumption: that at sufficiently large scales, the discrete nature of matter can be ignored, and the material can be treated as a continuous medium. Eventually, as becomes too large, the macroscopic properties can vary within the volume, making it challenging to make a valid measurement of “point” macroscopic properties.

A continuum model is the most common approach for describing fluids, and this assumption will be employed throughout this eBook. But why does this distinction of a continuum matter for flight vehicles? For a gas such as air in the lower atmosphere, the mean free path is on the order of m. The physical dimensions of a flight vehicle flying in the lower atmosphere will be many orders of magnitude greater than . Therefore, in this case, the fluid flow around the vehicle can be treated as a continuum, and all standard macroscopic fluid properties, such as pressure, density, and temperature, will apply.

However, consider a situation where becomes comparable to the length scale of the flight vehicle, such as at the edge of space, where the air density is very low, and satellites operate in low Earth orbit. At an altitude of 100 km, is on the order of centimeters to tenths of a meter, while at 300 km, can exceed several meters. In this context, air molecules are sufficiently separated that interactions with the vehicle occur infrequently. Consequently, the properties will not vary continuously from point to point, meaning the flow cannot be considered a continuum and will behave differently.

The Knudsen number, , is often used to quantify such low-density flows. The validity of the continuum assumption is inherently tied to the collision rate of gas molecules. Therefore, letting represent the intermolecular collision rate or frequency, i.e., collisions per unit time, and represent the characteristic flow time, the Knudsen number (which is a non-dimensional similarity parameter) can be defined as

(1)  

Alternatively, if is the mean molecular speed, is the mean free path (the average distance traveled by a molecule before encountering a collision), and represents a characteristic length scale, then

(2)  

Therefore, the ratio of to a characteristic length, , becomes a measure of the degree of departure from a continuum.

Usually, when is larger than 0.01, continuum assumptions become increasingly questionable. As approaches unity, the flow enters the transitional regime, in which the mean free path of the molecules becomes comparable to the characteristic length scale. For still larger values, i.e., , the flow approaches the free-molecular regime, where intermolecular collisions are infrequent compared to molecule-surface interactions. Consequently, gas behavior cannot be explained by macroscopic quantities; it must be described using a rarefied-gas model, in which the behavior of individual molecules must be accounted for, typically through statistical models.

However, the definition of a characteristic length scale may require qualification. Choosing a length scale, such as the flight vehicle’s length, provides a global measure of the extent to which the continuum assumption holds at a given flight condition. Another option is to use a length scale associated with the local flow characteristics, in which case the continuum assumption may become invalid across flight conditions, depending on the location within the flow.

Fluid Pressure

Remember that a fluid contains many relatively mobile molecules. In physical terms,^[4] the pressure can be thought of as the magnitude of the force produced in a direction normal to this elemental area from the average reaction force (i.e., the time rate of change of momentum) of the molecules per unit time that are impacting upon this surface, the concept being shown in Figure 4.

Therefore, the pressure at point B in a fluid can be defined as

(3)  

where is a measurable dimension compared to the mean distance between the fluid molecules, i.e., . This latter definition means that the pressure is the limiting form of the time-averaged force per unit area as the area shrinks to a point, but that “point” is still big enough to be described by a continuum model. Under the assumption of a continuum model, the area cannot shrink to zero.

Notice that pressure can be interpreted as a normal compressive force per unit area, which can be recognized as equivalent to a stress. The physical interpretation of pressure assumes it is caused by the time rate of change of momentum of fluid molecules, such as when they strike the walls of a surface or container. Higher or lower pressure would be associated with more or fewer molecules impacting a given surface area per unit of time. A large force is required to generate a high pressure over a given area. Alternatively, the same force must be exerted over a smaller area to achieve higher pressure.

Pressure is also a point property, meaning that its value can differ from one point to another throughout the fluid; for example, when flow moves around a shape such as an airfoil section, as shown in Figure 5. Well upstream of the airfoil “at infinity,” the flow velocity and pressure are constant, i.e., the velocity is , and the static pressure is . However, both the local velocity and pressure will change near the airfoil. Pressure is also related to other fluid properties, such as flow velocity , density, , and temperature, . Therefore, the pressure becomes a function of the Cartesian spatial coordinates, i.e., . Sometimes, the pressure value at a point may also change over time (the symbol denotes time); therefore, pressure can be written more generally as . It is also essential to recognize that pressure is a scalar quantity (it has magnitude but no direction). At a given point, the pressure has the same value in all directions, a principle known as Pascal’s Law.

Units of Pressure

Pressure is measured in units of N m (Pascals, Pa) in the SI system, or lb ft (pounds per square foot) in the U.S. Customary (USC) system. In practice, kilopascals (kPa) are commonly used because a Pascal is a small unit of pressure. Hectopascals are often used in barometric pressure measurements, where 1 hectopascal (hPa) equals 100 Pascals. One hPa equals one millibar; one bar is 100,000 Pa or 100 kPa. However, a bar is not an SI unit and should be avoided in most engineering applications unless specifically required. In the USC system, units of lb in (pounds per square inch) are also common; converting from units of lb in to units of lb ft requires a multiplication factor of 144, i.e. one lb in = 144 lb ft

Pressure as a Force

Sometimes, the effects of pressure may be interpreted as a force, i.e., as a quantity with magnitude and direction. However, pressure can be viewed as a force only when the area and orientation of the surface over which it acts are specified. Therefore, a line of action is needed to determine the force acting on the surface where the given pressure acts.

For example, if the small elemental surface had an outward-pointing unit normal vector , as shown in Figure 6, then the pressure force normal to the surface would be

(4)  

where the minus sign indicates that the pressure force will act inward in the opposite direction to the outward-pointing direction of .

Fluid Density

Another essential property to describe a fluid’s characteristics is its density, given the symbol or . Because the symbol used for pressure looks similar to , it is better to use for density to preserve clarity, mainly when and are used in the same equation, and the symbol will be used throughout this eBook.

Again, consider some point B in the fluid, as shown in Figure 7. Let be an elemental volume surrounding point B, and is the associated mass of fluid inside . In the continuum assumption, millions of molecules will remain within the small elemental volume. Density is defined as the mass of a fluid per unit volume and is a measure of the number of molecules per unit volume. Mass is usually given the symbol . Volume is given the symbol , i.e., a curly form of “V.” Notice: Do not confuse volume with velocity , the latter usually being written in vector form, i.e., .

The density of the fluid at point B is formally defined as

(5)  

where is a large linear dimension compared with the mean distance between the molecules . Therefore, flow density refers to the ratio of the mass of a small volume of fluid to the volume that contains it. Flow density is also a scalar quantity and, in general, can be written as .

Units of Fluid Density

Fluid density has units of kg/m (or more appropriately as kg m) in the SI system or slugs ft in the U.S. customary or USC system, where the slug is the base unit of mass.^[5] In dealing with aerodynamic problems, it is helpful to remember that air has a density of 1.225 kg m or 0.002378 slugs ft at sea level standard temperature and pressure. These values are at mean sea level (MSL) as defined in the International Standard Atmosphere (ISA) and are usually designated by the symbol .

Other types of fluid density measurements may be used in practice, particularly for liquids, which are often referenced to the density of water. These values include specific volume, specific weight, and specific gravity.

Specific Volume

The specific volume of a fluid is the reciprocal of its density and is given the symbol or and can be expressed as

(6)  

The units of specific volume are volume per unit mass, so m/kg (i.e., m kg) in the SI system, or ft/slug (i.e., ft slug) in the USC system.

Specific Weight

Specific weight is the weight per unit volume of a fluid. It is often denoted by the symbol or , i.e.,

(7)  

Specific weight has units of weight per unit volume, so its value depends on the acceleration under gravity or “g.” The units of specific weight are N/m (i.e., N m) in the SI system or lb/ft (i.e., lb ft) in the USC system.

Specific Gravity

When dealing with liquids, density is often measured relative to a reference fluid, called the specific gravity, . Usually, the reference is the density of water, so

(8)  

Notice that specific gravity is a non-dimensional or unitless quantity. is the most commonly used alternative measure of fluid density.

Fluid Temperature

The temperature of a fluid, denoted , is related to the average internal energy of its molecules at that point. Temperature affects fluid properties differently, depending on whether the fluid is a liquid or a gas and on its molecular composition. Temperature plays a vital role in aerodynamics, where air is compressible and aerodynamic heating from friction can occur.

Again, the molecular model can help explain the concept of temperature. This relationship is typically written as per molecule for a monatomic gas, where is Boltzmann’s constant, which serves as the conversion factor between energy and temperature. The Boltzmann constant is defined as 1.380649×10⁻²³ J K⁻¹ in SI units, the Joule (J) being the unit of energy. Therefore, as shown in Figure 8, a higher-temperature fluid is one in which the molecules move at relatively high speeds. In contrast, a lower-temperature fluid would have relatively low molecular speeds. Temperature is also a point scalar property. In general, the temperature in a fluid will vary from point to point; temperature may also vary with time at a given point, i.e., .

Units of Temperature

Two fixed points are taken to construct a temperature scale. The first fixed point is the freezing point of water, called the lower fixed point or . The second fixed point is the boiling point of water, which is called the upper fixed point or . Temperature is measured in units of Centigrade or Celsius C or Kelvin, K or K, in the SI system or Fahrenheit F or Rankine, R or R, in the USC system.

Celsius or Centigrade Scale

This scale was devised by Anders Celsius in 1742. The interval between and is 100 units, where each unit is called one degree Celsius (). In this scale, then the lower fixed point is = 0C (freezing point of water), and the upper fixed point is = 100C (boiling point of water).

Fahrenheit Scale

This scale was devised by Gabriel Fahrenheit in 1724. The interval between and is 180 units, where each unit is called one degree Fahrenheit (). In this scale, then the lower fixed point is = 32F and the upper fixed point is = 212F.

Kelvin Scale

This scale was devised by William Thomson (later Lord Kelvin) in 1848. The zero temperature is absolute zero^[6] on this scale. It is the thermodynamic scale for use in SI units. The interval between and is 100 units, where each unit is called one degree Kelvin ( or ). In this scale, then the lower fixed point is = 273.15K, and the upper fixed point is = 373.15K.^[7]

Rankine Scale

This scale was devised by William John Macquorn Rankine in 1859. On this scale, the zero temperature is also equivalent to absolute zero. It is the thermodynamic scale for use in USC units. The interval between and is 180 units, where each unit is called one degree Rankine (R or 1 R). In this scale, then = 491.67R, and = 671.67R.

Temperature Scale Conversions

Converting between temperature scales is straightforward because they are linearly related, as shown in Figure 9; this figure should not be used for numerical calculations. Converting to Centigrade or Celsius C from Fahrenheit F is performed using

(9)  

Converting to Fahrenheit from Centigrade or Celsius uses

(10)  

Converting to Kelvin K from Centigrade or Celsius uses

(11)  

Finally, converting to Rankine R from Fahrenheit uses

(12)  

Notice that it is often suggested that the degree symbol () not be used when citing temperature units (especially for the Kelvin and Rankine scales). However, many publications can be found with and without the degree symbol. Nevertheless, retaining the degree symbol in temperature units is entirely acceptable for students and others working on aerodynamic problems. Finally, it is helpful to remember the standard sea level values of temperature (based on the ISA model), usually given the symbol , which are 15C = 59F = 288.15K = 518.67R.

Using Temperatures in Engineering Problem-Solving

In engineering problem-solving, caution should be applied so that the correct absolute (engineering) units of temperature are used, i.e., units of Kelvin or Rankine, because these scales measure the temperature relative to absolute zero temperature, i.e., the temperature when the average internal energy and motion of the molecules becomes effectively zero. For example, for two temperatures and = 40, then the ratio is written correctly as

(13)  

but incorrectly as

(14)  

Equation of State

Having introduced the concepts of pressure, density, and temperature, it is essential to recognize that these quantities are interdependent; changes in one affect the others. In physics and chemistry, there are four fundamental gas laws: Boyle’s Law, Charles’s Law, Gay-Lussac’s Law, and Avogadro’s Law. These are all empirical gas laws because their relationships were derived from laboratory experiments with gases.

These laws are combined into a single equation, known as the equation of state, that is one of the most useful in thermodynamics and aerodynamics. The general form of the equation of state is

(15)  

where is the universal gas constant.

For many engineering applications, it is more useful to express the previous relation in terms of specific properties. Dividing through by mass gives the specific volume, , leading to

(16)  

Equation 16 is the engineering form of the equation of state, widely used in gas dynamics and aerodynamics. Here is the gas-specific constant, equal to the universal gas constant divided by the molar mass. These forms express the same ideal-gas relationship, but they use different measures of amount: total volume and number of moles in the first form, specific volume in the second form, and density in the third form, i.e.,

(17)  

If pressure and temperature are known, the density follows directly from the latter form of the equation of state, i.e.,

(18)  

which is one of the most common uses of the equation of state in aerospace applications. Therefore, it will be apparent that the equation of state reduces the number of independent thermodynamic variables from three to two. The relationships may also be visualized as a surface in the , , and space, where each point represents a unique equilibrium state, as shown in Figure 10. For air under standard and moderate flight conditions, the ideal gas law is an excellent approximation. Deviations occur mainly at very high pressures or very low temperatures, where so-called “real-gas” effects become significant.

Bulk Modulus

The bulk modulus, denoted by the symbol , measures a substance’s resistance to uniform compression under a change in pressure. The bulk modulus is equivalent to a compressive “stiffness” and is an essential property in fluid mechanics. In particular, its value is related to the speed of sound propagation in liquids, which plays a fundamental role in various scientific, industrial, and medical imaging^[8], and other technological applications^[9], where precise measurement and understanding of acoustic properties are crucial.

Definition of the Bulk Modulus

When a fluid of original volume is subjected to a change in external pressure , it undergoes a corresponding volume change . For a compression, the pressure increases, so is positive, but the volume decreases, so is negative. The relationship is written as

(19)  

where the coefficient is called the bulk modulus. The minus sign is needed because a positive change in applied pressure produces a negative volume change, as illustrated in Figure 11. In the limit when the changes become infinitesimally small, Eq. 19 becomes

(20)  

Therefore, the bulk modulus is defined as

(21)  

It is often useful to write the bulk modulus in terms of density rather than volume. For a fixed mass of fluid, conservation of mass requires

(22)  

Taking the differential of this product gives

(23)  

or

(24)  

Dividing by gives

(25)  

This result states that if the volume decreases during compression, the density must increase.

Substituting this relationship into the differential form of the bulk modulus gives

(26)  

Therefore,

(27)  

In other words, Eqs. 21 and 27 quantify the degree of compressibility of the fluid. A high bulk modulus indicates that the substance is difficult to compress, i.e., nearly incompressible. It will also be apparent that the bulk modulus has units of pressure.

Bulk Moduli of Fluids

Liquids have relatively high bulk moduli because their molecules are closely packed, resisting external forces that would otherwise cause compression. Indeed, for all practical purposes, liquids cannot be compressed significantly (their values of bulk moduli are in the GPa or Giga-Pascal range), so, with few exceptions, they can be considered incompressible fluids.

Gases have much lower bulk moduli because their molecules are farther apart, allowing them to be compressed more easily. In general, gases cannot be treated as incompressible fluids, and their bulk modulus varies with pressure and temperature. One type of bulk modulus used in practice is the isothermal (i.e., constant temperature) bulk modulus, , which can be written as

(28)  

The isothermal gas law (Boyle’s law) states that constant, so differentiating with respect to using the product rule gives

(29)  

so that

(30)  

Substituting this result into Eq. 28 gives

(31)  

This means that a gas’s isothermal bulk modulus is its pressure; therefore, its “stiffness” or resistance to compression is directly proportional to its pressure. As the gas’s pressure increases, its resistance to further compression rises proportionally.

If the compression is adiabatic, the gas’s temperature increases because the work done on it increases its internal energy. Conversely, the temperature decreases during adiabatic expansion as the gas does work on its surroundings, reducing its internal energy. In an adiabatic process, the relationship between pressure and density is

(32)  

Differentiating this equation with respect to using the product rule leads to

(33)  

and solving for gives

(34)  

Therefore, substituting this previous result into Eq. 27 for an adiabatic process, gives the bulk modulus as

(35)  

Connection to the Speed of Sound

The bulk modulus plays a fundamental role in determining how pressure disturbances propagate through a fluid. In particular, the compressibility of the medium governs the speed at which small pressure waves travel. For a general fluid, the speed of sound, , is given by

(36)  

where the derivative is evaluated under the appropriate thermodynamic conditions. For sound propagation in gases, the process is usually treated as adiabatic. For gases, sound propagation occurs rapidly enough that the process is essentially adiabatic, so that

(37)  

Using the result for an ideal gas undergoing an adiabatic process, , then gives the well-known expression

(38)  

Therefore, for a given ideal gas, the speed of sound depends primarily on the absolute temperature; across different gases, it also depends on and . This result is of central importance in aerodynamics, particularly in the study of compressible flows and Mach number effects. A further discussion of the speed of sound in fluid media is given later in this chapter.

Mach Number & Compressibility Effects

The speed of sound provides a natural reference velocity for characterizing fluid motion. The ratio of the local flow velocity, , to the local speed of sound, , is defined as the Mach number, i.e.,

(39)  

The Mach number is one of the most important non-dimensional parameters in aerodynamics because it determines the relative importance of compressibility effects in a flow.

When , the flow velocity is much smaller than the speed of sound, and compressibility effects are negligible; the flow may be treated as incompressible. As increases and approaches unity, compressibility effects become significant, and changes in pressure are accompanied by appreciable changes in density. At , the flow speed equals the speed of sound, and the flow is said to be sonic. For , the flow is supersonic, and pressure disturbances cannot propagate upstream, leading to the formation of shock waves and other nonlinear phenomena. Therefore, the concepts of bulk modulus, compressibility, and the speed of sound are fundamentally linked through the Mach number, which governs the aerodynamic behavior of flight vehicles over a wide range of operating conditions.

Viscosity of Fluids

Viscosity is the fluid’s resistance to shear when different parts of the fluid move relative to each other, i.e., its “internal friction” or resistance to deformation. Viscosity can also be viewed as a measure of fluidity, i.e., the higher the viscosity, the lower the fluidity. The symbol (i.e., the Greek symbol “mu”) is a constant known as the coefficient of dynamic viscosity, or more simply, just the fluid’s viscosity.

All fluids have viscosity to a lesser or greater degree, so for a fluid in relative motion, the property of viscosity causes shear forces to be produced within the fluid. Gases generally have much lower viscosities than liquids, as expected. Some liquids are very viscous, such as molasses, corn syrup, grease, and heavy oils. However, viscosity affects the behavior of all types of fluids.

Viscometer

One way to begin to understand the property of viscosity is to consider a demonstration using three columns of oil, as shown in Figure 12. Each oil has a different viscosity, ranging from SAE 20 (the thinnest and least viscous) to SAE 40 (the most viscous). SAE stands for the Society of Automotive Engineers. Suppose a heavy steel ball is dropped into the oil. In that case, it will descend at a velocity inversely proportional to its viscosity, as the oil exerts shear stress on the ball’s surface, thereby inducing viscous drag as it moves downward under gravity.

The balance of forces in equilibrium descent is such that the weight of the ball, , less any buoyancy force, , is equal to the viscous drag on the ball, . The weight will be density times volume times acceleration under gravity, i.e.,

(40)  

where is the density of the steel ball. The (upward) buoyancy force on the ball (Archimedes’ principle) will be equal to the weight of oil displaced by the ball, i.e.,

(41)  

Therefore, the equilibrium equation is

(42)  

The drag force on a sphere of radius dropping through a fluid of high viscosity at low speed (this is called a creeping flow) is given by Stokes’s law, i.e.,

(43)  

where is the viscosity of the fluid. Therefore, in equilibrium, then

(44)  

and rearranging to solve for gives

(45)  

For a ball of the same weight and size, Eq. 45 shows it will drop in the oil at a velocity that is inversely proportional to the oil’s viscosity, , i.e., the higher the viscosity, the slower the ball drops. The foregoing is the principle used in the falling-sphere viscometer. The time, , it takes for a steel sphere of known size (radius), , and weight (material density ), can be measured using two lines on the tube a distance apart, from which the ball’s velocity, , is determined. Stokes’s law (Eq. 43) can be used to determine the viscosity, , of the oil (or other liquid) from the resulting velocity by knowing the size and weight of the sphere as well as the density of the liquid, i.e.,

(46)  

Units of Viscosity

Viscosity has units of kg m^-1 s^-1, N s m^-2, or Pa s, i.e., the pascal-second, in the SI system, or slug ft^-1 s^-1 in the USC system. However, the unit of viscosity typically used in practice is the “Poise” (P), or gram cm^-1 s^-1. The unit of poise is 1 Poise = 0.1 Pa s, i.e., 1 Pa s = 10 Poise. The viscosity of liquids is typically low, so it is often reported in centipoise (cP). In contrast, the viscosity of gases, which are much less viscous than liquids, is usually reported in units of micropoise (P). In base units, at MSL ISA, then = 1.789 x 10^-5 kg m^-1 s^-1 = 1.789 x 10^-5 Pa s = 3.737 x 10^-7 slug ft^-1 s^-1.

Shear in a Fluid

In Newton’s Philosophiæ Naturalis Principia Mathematica of 1687, he uses the Latin word tenacitas to refer to what is now called viscosity. He then defines the concept of viscosity as “Resistentia, quae oritur ex inopia lubricitatis partium fluidi, caeteris paribus, proportionalis est velocitati, qua partes fluidi separantur ab invicem,” which can be translated as “The resistance which arises from the lack of slipperiness originating in a fluid, all other things being equal, is proportional to the velocity by which the parts of the fluid are being separated from each other.” This statement is the historical basis for what is now called Newton’s law of viscosity.

This idea is now commonly illustrated by the experiment shown in Figure 13, using a fluid of depth between a moving upper plate and a stationary lower plate, i.e., a flow containing a velocity gradient in one direction, with the velocity in the fluid increasing as it moves from one point to another. The upper (faster) layer draws the lower (slower) layer along by exerting a shear force on it, so a shear force must act between the layers. Simultaneously, the lower layer exerts an equal and opposite force on the upper layer (i.e., Newton’s Third Law).

The fluid adjacent to the bottom plate remains at rest, whereas the fluid in contact with the top plate is dragged along (by viscosity) with velocity . Consequently, a velocity gradient forms in the fluid between the two plates. Maintaining this gradient requires the application of a force, , where

(47)  

where is the area of the plate. Notice that the ratio is the slope of the velocity profile or the velocity gradient. In terms of force per unit area, which is a stress, , then

(48)  

where the constant of proportionality, , is the “resistance to shear,” i.e., the fluid’s viscosity.

In general, for the straight and parallel motion of a given fluid, the tangential stress produced between two adjacent fluid layers is proportional to the velocity gradient in a direction perpendicular to the layers, i.e.,

(49)  

where is the velocity at some distance . The quantity is the velocity gradient in the direction. The velocity gradient is equivalent to a strain rate, so this preceding equation is just a statement of a fluid’s linear stress/strain rate relationship. Equation 49 is called Newton’s law of viscosity. Remember that maintaining a velocity gradient and shear stresses in a fluid requires a continuous force; if the force stops, the fluid ceases to deform, and the shear stresses become zero.

To explain this latter point further, consider a fluid element as it flows in a fluid with a velocity gradient, as shown in the inset of Figure 13. If is positive, then the upper surface of the element will move faster than the lower surface, so over some time , the upper AC surface will travel further than the lower surface DE by a distance

(50)  

Consequently, a shear deformation or strain is produced in the fluid. This strain can be calculated from the deformation geometry shown in Figure 13. The strain, , which is the angle between the lines EB and EC, is

(51)  

to a small-angle approximation. Rearranging the equation gives

(52)  

And so the shear stress in the fluid is

(53)  

This latter result is another way of writing Newton’s law of viscosity. Notice that the shear stress depends on the strain rate, i.e., . Remember that the shear stress in a solid is proportional to strain, so a constantly applied strain will create constant deformation and stress. In a fluid, however, shear stress is only produced by a strain rate because the fluid continuously flows and deforms.

Velocity Gradients & Shear Stresses

Newton’s law of viscosity should be written more precisely using the partial derivative on the velocity gradient, i.e., it should be written as

(54)  

because the velocity in a fluid may vary in other directions as well, e.g., the flow is three-dimensional, so there could be velocity gradients in the and directions, i.e., , and . In general, these gradients can be written in the matrix (or tensor) form as

(55)  

The velocity gradient tensor is not, in general, symmetric. It can be decomposed into a symmetric part associated with deformation (strain rate) and an antisymmetric part associated with rigid-body rotation of the fluid (vorticity). For an incompressible Newtonian fluid with constant viscosity, the symmetric part of the velocity gradient tensor leads to the viscous stress tensor, which becomes

(56)  

and in component form, then

(57)  

This significant result is used to derive the momentum equation for a fluid, known as the Navier-Stokes equation. Notice that the diagonal terms in this viscous stress tensor are normal viscous stresses because they act perpendicular, or normally, to the respective surfaces. The off-diagonal terms represent shear stresses that act tangentially on the surfaces. The thermodynamic pressure contributes an additional isotropic normal stress, which is treated separately in the fluid’s full stress state.

Mechanisms of Viscosity

The viscous properties of a fluid arise from two sources: 1. Intermolecular momentum transfer between the molecules, and 2. Bonding between the molecules. Therefore, the viscosity of a fluid depends on whether it is a gas or a liquid; its properties primarily depend on the physics of the mean intermolecular spacing. The molecules in a liquid are relatively close together, but they are not as mobile as those in a gas. In this case, viscosity arises more from molecular bonding than from intermolecular momentum transfer. As shown in Figure 14, stronger bonding yields greater resistance to deformation (i.e., higher viscosity). In general, intermolecular bonding can be influenced by factors such as molecular size and weight, bond strength, and the liquid’s temperature.

For example, liquids with large, heavy molecules tend to have higher viscosities than those with small, light molecules because the larger molecules have more intermolecular bonds and are more resistant to flow. Liquids such as benzene, diethyl ether, gasoline, ethanol, and water flow readily because of their low viscosity. Others, such as honey, heavy oils, glycerin, motor oil, molasses, and maple syrup, flow very slowly and have a high viscosity. There is also a correlation between viscosity and molecular shape. Liquids of long, flexible molecules tend to have higher viscosities than those of more spherical or shorter-chain molecules. The longer the molecules, the more likely they are to become “tangled” with one another, increasing the liquid’s viscosity.

The intermolecular bonding is much weaker because gas-phase molecules are farther apart. The mechanism underlying viscosity arises from intermolecular momentum transfer as the relatively mobile molecules diffuse throughout the gas. Nevertheless, gases remain viscous and exhibit the characteristics associated with viscosity. This effect becomes more apparent when the gas has an initial velocity gradient, as shown in Figure 15.

The random motion of gas molecules between fluid layers means that collisions inevitably occur and momentum is exchanged, with slower molecules gaining momentum from faster ones. The consequence of intermolecular momentum transfer is a shear force between gas layers in regions of velocity gradient, which resists further deformation and manifests as viscosity. Remember that fluids must be continuously deformed to produce stresses, so in the absence of additional shear rates, the velocity gradients will diminish as momentum is balanced throughout the gas layers.

Newtonian Versus Non-Newtonian Fluids

A Newtonian fluid is a fluid for which, at a specified thermodynamic state, the shear stress is proportional to the strain rate. In this sense, is independent of the magnitude of the velocity gradient. In simple shear flow, the shear strain rate is defined as the time rate of change of the shear strain , i.e.,

(58)  

where is the velocity in the flow direction and is the coordinate normal to the flow. For a fluid element undergoing planar shear deformation, the shear strain is the angular deformation of the element. The velocity gradient determines the rate at which this deformation is produced, so for simple shear flow

(59)  

or, more generally,

(60)  

and for a Newtonian fluid, then

(61)  

where is the shear stress and is the dynamic viscosity; recall that Eq. 61 is called Newton’s law of viscosity. For a Newtonian fluid, the viscosity is a constant and a material property that is independent of the strain rate, i.e.,

(62)  

Many fluids, including air and water, behave as Newtonian fluids, i.e., they obey a linear stress-strain-rate relationship, as shown in Figure 16. Therefore, the value of can be assumed to be constant. Remember that in a fluid, the shear stresses produced by viscosity are directly related to the strain rate generated in the fluid by its deformation. However, as the figure shows, not all fluids behave linearly.

Many other fluids, such as oils, blood, inks, and most paints, exhibit non-Newtonian behavior, in which their effective viscosity depends on the strain rate. The shear stress now becomes a nonlinear function of the shear strain rate, i.e.,

(63)  

where is the apparent viscosity, defined as the local ratio of shear stress to strain rate, i.e.,

(64)  

This apparent viscosity, , is not a fluid property but a descriptive quantity that depends on the instantaneous strain rate. It is the nonlinear nature of the stress-strain relationship that distinguishes non-Newtonian fluids from Newtonian ones.

Notice from Figure 16 that the apparent viscosity may increase or decrease with an increasing strain rate, depending on the nature of the fluid. For example, a dilatant fluid exhibits shear thickening, while a pseudoplastic fluid displays shear thinning, meaning the fluid becomes less viscous when sheared more rapidly. In other cases, it becomes more resistant to increasing shear rates. Some materials do not flow until a critical stress threshold is exceeded, a phenomenon known as Bingham plasticity. Other types of non-Newtonian fluids include thixotropic and rheopectic fluids, whose viscosity changes not only with shear rate but also with the duration of the applied shear. Thixotropic fluids thin when sheared and regain their viscosity when at rest. In contrast, rheopectic fluids become thicker with sustained shear and are far less common.

The study of non-Newtonian fluids falls under the umbrella of a field called rheology, which generally encompasses the examination of any substance, fluid, or solid that exhibits a non-Newtonian response to applied stress. The behavior of non-Newtonian fluids is less well understood than that of Newtonian fluids, and their nonlinear characteristics make their behavior less predictable; however, they have numerous practical applications. For example, paints (by chemical formulation) are typically both pseudoplastic and thixotropic fluids. They thin under shear during application (shear-thinning) but also recover viscosity over time when at rest (thixotropy). This helps them resist dripping or sagging after application, preventing them from running off the surface.

Conventional methods cannot be used to measure the viscosity of a non-Newtonian fluid. Instead, it is necessary to measure the apparent viscosity, which considers the shear rate at which the viscosity measurement was made. Despite their molecular-level complexity, the macroscopic behavior of many non-Newtonian fluids can be modeled using relatively simple empirical relationships. One of the most commonly used is the power-law model, in which the shear stress is nonlinearly related to the shear rate by an expression of the form

(65)  

where is known as the consistency index and is the behavior index.  For a given fluid, both values can be determined from standard reference sources, including rheology books.

The parameters for less common non-Newtonian fluids may need to be determined experimentally. One accepted method is to use a rotational rheometer to measure the relationship between shear stress and shear rate . For the assumed power-law for a non-Newtonian fluid in Eq. 65, taking the logarithm of both sides yields a linear equation, i.e.,

(66)  

Plotting measured values of versus usually produces a straight line. The slope of this line gives , and the intercept gives . This is the most common method for extracting rheological parameters of a non-Newtonian fluid over the required range of shear rates.

Notice that the model in Eq. 65 reduces to the Newtonian case when , with , the dynamic viscosity. When , the material is shear-thinning (pseudoplastic), meaning the effective viscosity decreases as the shear rate increases. When , the fluid exhibits shear-thickening (dilatant) behavior, giving more resistance as it deforms more rapidly. Consequently, the effective viscosity in such a fluid is not a material constant but varies with shear rate and can be modeled using

(67)  

As shown in Figure 17 (which is plotted on a log-log scale because of the nonlinearity in Eq. 67), for a relatively viscous power-law fluid with a large consistency index , the effective viscosity decreases with increasing shear rate when . For shear-thickening materials with , the effective viscosity increases with increasing shear rate.

Although the power-law model is only an approximation and fails to capture behavior at very low or very high shear rates, it remains a practical and widely used model for analyzing slowly moving, or “creeping,” flows in viscous, non-Newtonian fluids. However, from a modeling perspective, adopting a power-law fluid introduces significant additional nonlinearity into the governing equations, making analytical and numerical solutions more challenging.

The ability to model and predict non-Newtonian fluid behavior is crucial for the production of various fluids and likewise depends on precise control of rheological properties. Even in additive manufacturing and 3-D printing, the extrusion of pastes and gels requires careful modeling of non-Newtonian flow to achieve accuracy and repeatability. However, for many canonical flows, such as between plates, in channels, or over inclined surfaces, exact or approximate solutions can still be obtained, allowing reference predictions of flow rates and stress distributions.

Effects of Temperature on Viscosity

Temperature significantly affects the viscosity of both gases and liquids. Consequently, the viscous characteristics of gases differ from those of liquids when subjected to changes in temperature, as shown in Figure 18. For example, the viscosity of a liquid generally decreases with increasing temperature. This effect occurs because the reduction in bonding forces, as molecular motion moves them farther apart, dominates over any increase in intermolecular momentum transfer. Consequently, liquids become less viscous and flow more easily as the temperature increases. However, gases, including air, typically exhibit higher viscosity with increasing temperature because of increased intermolecular momentum transfer, which increases the resistance to the fluid’s deformation.

Sutherland’s Law of Viscosity for Gases

The coefficient of dynamic viscosity, (or often referred to as the coefficient of viscosity or viscosity), for a gas can be calculated using Sutherland’s formula or Sutherland’s law. This empirical (i.e., experimentally derived) law was first published in 1893 and can be written as a function of absolute temperature, , as

(68)  

where reference values (subscript “ref”) are in appropriate SI or USC units. The parameter is known as Sutherland’s constant. A graphical interpretation of Sutherland’s law is shown in Figure 19.

For air in SI units, then = 273.15K and kg m s (also known as units of Pa~s), with a Sutherland constant of = 110.0K. In USC units at = 518.67R, then = 198.72R and slugs s ft. Coefficients for other gases are widely available in reference books and online data sources.

Sutherland’s law is widely used in various engineering fields to predict the viscosity of gases as a function of temperature. It operates over a wide temperature range for air and gases such as oxygen, nitrogen, and helium. However, it is not universally applicable to all gases. For example, Sutherland’s law does not apply to gases that exhibit significant deviations from ideal behavior, such as rarefied gases or gases containing large molecules. Additionally, Sutherland’s law does not apply to liquids, which exhibit a more complex relationship between viscosity and temperature because of the effects of intermolecular bonding. In general, Sutherland’s law should be used cautiously, and its applicability should be verified for each specific gas and temperature range of interest.

Temperature Effects on the Viscosity of Liquids

The Andrade equation, named after British physicist Edward Andrade, is a commonly used semi-empirical model to predict the effects of temperature on the viscosity of liquids. This equation relates the viscosity of a liquid to temperature based on an exponential relationship, i.e.,

(69)  

where is the viscosity of the liquid and is the absolute temperature, in Kelvin or Rankine, and the coefficients and depend on the liquid and the unit system used. Values of and for this two-parameter model have been widely reported for many liquids. This equation describes the behavior in which a liquid’s viscosity decreases as temperature increases.

There are also other semi-empirical models for the effects of temperature on the viscosity of liquids. The three-parameter model is

(70)  

and the four-parameter model is

(71)  

Again, the coefficients, A, B, C, and D for most liquids can be found in published sources.

Flow Velocity

In fluid mechanics, the primary focus is on fluids in motion, known as fluid dynamics. Hence, the velocity of the fluid is a significant quantity that must be defined carefully. By definition, a velocity is a vector quantity, so the velocity of any given fluid packet will have both a magnitude (a speed) and a direction. When the concept of the velocity of a fluid is considered, which will have relative motion between the fluid packets, then its velocity becomes more subtle to describe than for a solid body, where all the parts will move in unison.

For example, for a solid body in translational motion, it is evident that all points of the body will be traveling at the same velocity, i.e., with the same speed and direction. However, different parts of a fluid in motion will likely travel at different velocities, resulting in relative motion. This is just one reason why the motion of fluids is somewhat more challenging to describe, both physically and mathematically.

Tracking the movement of the fluid in space and time is essential for understanding flow problems, as it allows one to visualize the flow’s path as it passes around an airfoil or other object. Consider the flow about an airfoil at a steady angle of attack and follow the path of a small group of fluid particles initially upstream of the airfoil at point 1, as shown in Figure 20. This group is called a fluid element because it represents a small volume of elemental flow. The speed and direction of the fluid elements will change as they move downstream from point 1 to points 2 and 3, with point 3 being nearer to the nose of the airfoil. Point 3 at the nose is called a stagnation point because the air is brought to rest there and stagnates. Therefore, the flow velocity at 1, 2, or 3, or any other point in the field, is just the velocity of an infinitesimally small fluid element as it passes through that point. The flow passes around the airfoil and leaves it at point 4. In this case, the paths followed by the fluid elements are also referred to as streamlines, although the distinction between streamlines, pathlines, and streaklines remains to be clarified.

Velocity has magnitude and direction, but it is still a point property in that its value will change from point to point in the flow, and it can also change with respect to time, i.e., . Flow velocities are measured in units of m s in the SI system or ft s in the USC system.

Equation of a Streamline

Determining the equation of a streamline is straightforward. For a two-dimensional flow in the – plane, then , so the slope of a streamline is just , which is an ordinary differential equation. Therefore, this differential equation could be solved with the known velocity field to trace a streamline in a given plane. In three-dimensions, i.e., in , and space then . In this case, a direction vector, say , can be defined that points along the streamline, i.e., in a direction parallel to the streamline, as shown in Figure 21.

By definition, there is no flow across a streamline, so the equation of a streamline in three-dimensional space is just

(72)  

noting that is the zero vector. The meaning of this latter equation becomes clearer by expanding out the vector equation in terms of its scalar components, i.e.,

(73)  

In terms of the components, then

(74)  

which can be physically interpreted from Figure 22. Therefore, it will be apparent that the equation of a streamline is

(75)  

Streamlines, Pathlines & Streaklines

As previously discussed, by definition, a streamline is a line drawn tangential to the instantaneous local velocity vector field, i.e., there is no flow perpendicular to a streamline. Streamlines can be considered an instantaneous realization, or a “snapshot,” of the flow. A pathline is the trajectory or path a fluid element traces out in time, i.e., following the path of the fluid element as it moves through the flow. A streakline is the locus of fluid elements that have passed through the same point in the flow.

In general, streamlines are different from both pathlines and streaklines. In steady flow problems, when the flow properties do not change with respect to time, called a steady flow, then streamlines, pathlines, and streaklines are all the same. In an unsteady flow, the path followed by a fluid element, i.e., the pathline, is not the same as the streamline, as shown in Figure 23. At each instant, the airfoil experiences a new angle of attack relative to the flow, and the streamline pattern changes. The fluid element, however, moves along its own pathline as it traces its way through the flow. This is why streamlines are referred to as instantaneous realizations, or snapshots, of the flow. In contrast, the pathlines and the streaklines depend on the prior history of the velocity field.

Frames of Reference

An interesting issue in fluid dynamics and streamline calculations is the choice of reference frame. The question is: Is the body moving in a stationary flow, or is the flow moving past a stationary body? Are these two scenarios equivalent? The answer lies in the frame of reference.

One frame of reference is to consider a moving body in a stationary flow, as illustrated in Figure 24. In this frame of reference, the airfoil moves through the air, which is considered undisturbed and stationary. The other frame of reference, a stationary body in a moving flow, is a typical setup for wind tunnel testing, theoretical analyses, and numerical simulations. In this frame, the airfoil is stationary, and the air flows past it. This setup is also used because it simplifies the boundary conditions and problem setup. Note, however, that the streamline patterns will differ in each case. In the case of a stationary body, an observer will see the same streamline pattern for all time.  In the case of a moving body, an observer will see that the initially stationary flow develops a streamlined pattern and then returns to its undisturbed state; thus, the observer perceives the behavior as unsteady.

Mathematically, both frames of reference are equivalent for steady, non-accelerating motion because of Galilean relativity. This means the physical phenomena observed or measured, such as lift, drag, and pressure distribution, will be the same whether the airfoil moves through still air or the air moves past a stationary airfoil at the same relative speed. However, the choice of frame can affect the streamline pattern seen by an observer, as well as the complexity of the analysis and the implementation of numerical methods.

In a more general sense, the streamline equations are often solved using a marching parameter, say , having units of time. For a steady velocity field, this parameter may be interpreted as the travel time of a fluid element because pathlines and streamlines coincide. For an unsteady flow, however, the velocity field must be frozen at a given instant to compute streamlines. The streamline equations can then be written as

(76)  

To find the streamline, integrate each equation with respect to , i.e.,

(77)  

(78)  

(79)  

where is the starting point of the streamline at .

Numerical integration methods include Euler’s method, Runge-Kutta methods, and others. Second-order Runge-Kutta methods are commonly used to calculate streamlines. If the marching increment is sufficiently small, an explicit Euler method is sufficient to calculate streamlines in two dimensions. For example, for the component, a one-step explicit method for a given increment is of the form

and for the component

where represents the current marching location and represents the next location along the streamline. This algorithm is easily programmable, and if and are given as simple functions of and , then the streamlines can be solved. However, the value of must be small enough to prevent errors from accumulating in the values of and , and some trial and error may be involved. Some trial and error may also be necessary to find suitable initial points for integration, allowing the nature of the flow field to become apparent.

MATLAB provides various solvers for initial-value problems in ODEs and can be used to trace pathlines and streamlines in simple flows; an example is shown in Figure 25. However, the accuracy of streamline calculations depends strongly on the spatial quality of the velocity field used, whether computed or measured, rather than on the accuracy of the numerical method alone.

Streamtubes

A streamtube is a concept in fluid dynamics that represents a bundle of streamlines. These streamlines delineate the boundaries of a particular flow or control volume in the form of a tubular structure, as shown in Figure 26. Because the boundary of a streamtube is made from streamlines, there is no flow across its sides. For steady flow, the mass flow rate entering one end of the streamtube must equal the mass flow rate leaving the other end, as required by conservation of mass. A streamtube need not be axisymmetric; it may have any shape determined by the surrounding velocity field. However, in many engineering applications, a uniform or axisymmetric streamtube is used as a useful simplifying model.

Speed of Sound in Fluid Media

The speed of sound in any medium depends on how quickly vibrational energy can be transferred from molecule to molecule through the medium. All gases are compressible; therefore, pressure disturbances produced at one point will propagate to another at a finite speed. This propagation speed is called the speed of sound, given the symbol , and its value differs from gas to gas. Sound waves oscillate very rapidly, causing compressions and rarefactions in the medium. The timescale for these oscillations is usually so short that there is insufficient time for significant heat exchange with the surroundings, so sound propagation can be assumed adiabatic. For small-amplitude sound waves, dissipative effects are also negligible, allowing the disturbance to be treated as approximately reversible and hence isentropic.

Speed of Sound in a Gas

It can be shown, in general, that the speed of sound is related to changes in pressure and density of the fluid medium using

(80)  

where the subscript means that the derivative is evaluated at constant entropy, i.e., for an isentropic disturbance. For sound waves of small amplitude, this is usually an excellent approximation because the compressions and rarefactions occur so rapidly that there is insufficient time for significant heat transfer.
For an adiabatic process, then

(81)  

where is the ratio of specific heats, i.e., . Solving for gives

(82)  

so that

(83)  

using where is the specific gas constant. Therefore, the speed of sound in a gas depends on the gas type and the absolute temperature of that gas.

Because liquids and solids are very difficult to compress and to change their density, the speed of sound in such media is generally greater than in gases; for example, sound travels about four times faster in water than in air. Note that the values of and vary across gases; a useful table is provided below.

Thermodynamic properties of some gases.

Gas		( J kg K)	(ft-lb slug R)
Air	1.4	287.05	1717.0
Nitrogen	1.4	296.8	1775.0
Hydrogen	1.41	4124.2	24663.0
Helium	1.66	2077.1	12421.0
Oxygen	1.395	259.84	1554.0
Carbon Dioxide	1.289	188.92	1130.0
Carbon Monoxide	1.4	296.84	1775.0

Doppler Effect

Sound is a pressure disturbance, and the speed of sound propagation in any gas at a given temperature will be constant. Let the frequency of the sound source be , which will be the frequency of the sound heard by any listener if the sound source is stationary. However, the perceived frequency of sound propagation will change if the location of the sound source S and the listener locations (comparing locations L₁ and L₂) are in relative movement to each other, which is known as the Doppler effect, as illustrated in Figure 27.

The frequency , heard by the listener, is given by

(84)  

where is the emitted frequency of the sound source. In this form, is taken as positive when the listener moves toward the source, and is taken as positive when the source moves away from the listener. The same expression may be written in terms of Mach numbers as

(85)  

where and . If the listeners are stationary, then . A source moving toward a listener has , which decreases the denominator and increases the heard frequency. Therefore, as shown in Figure 27, the listener at , ahead of the sound source, will hear a higher frequency than the listener at , who is behind the sound source.

When the sound source moves at or above the speed of sound, the emitted sound waves cannot move ahead of the source. Instead, the sound energy becomes concentrated along a wavefront known as a Mach wave. In three dimensions, this wavefront forms a Mach cone. In this case, a listener ahead of a supersonic sound source would not hear it before the Mach cone reaches the listener. At that point, there would be a loud “bang” as it passes by, as shown in Figure 28.

In this case, the lower frequency heard by a stationary listener after the passage of the sound source traveling at a Mach number of will be

(86)  

The speed of sound is critical for all flight vehicles, which create pressure disturbances as they fly. An aircraft’s airspeed relative to the speed of sound affects the flow physics and the forces acting on the aircraft. This ratio is called the flight Mach number, . If the aircraft flies much slower than the speed of sound, the conditions are said to be subsonic, and compressibility effects are minor. However, if the aircraft moves faster and approaches or exceeds the speed of sound, known as supersonic, compressibility effects become essential, and the flow physics change. In this case, the issue of the “sonic boom” generated by the aircraft also becomes a consideration for people on the ground.

Speed of Sound in a Liquid

The speed of sound in a liquid varies depending on the type of liquid and its properties, including its density, , and its bulk modulus, . The speed of sound in a liquid can be calculated using

(87)  

where the bulk modulus of the liquid, which, as previously explained, is a measure of its incompressibility. Equation 87 is often referred to as the Newton-Laplace equation.

Sound travels much faster in media with a higher bulk modulus because they transmit pressure changes more readily. For example, the speed of sound in freshwater at average room temperature is approximately 1,482 m/s ( 4,850 ft/s). The speed of sound in seawater is generally higher because of the presence of salts and minerals, averaging around 1,533 m/s ( 5,029 ft/s) at similar temperatures. In general, the speed of sound can vary widely across different liquids; for example, in ethanol it is around 1,160 m/s ( 3,807 ft/s), and in mercury it is about 1,450 m/s ( 4,760 ft/s). For reference, some numerical values of the bulk modulus for liquids are given in the table below, and others are available online or at other authoritative sources. Notice that bulk modulus has units of pressure. However, the temperature and impurities in the liquid can also affect the speed of sound.

Numerical values of the bulk modulus of some common liquids.

Liquid	(Pa)	(lb/ft)	(kg/m)	(slugs/ft)	(m/s)	(ft/s)
Freshwater			1,000	1.94	~1,482	~4,850
Seawater			1,025	1.99	~1,533	~5,029
Ethanol			780	1.51	~1,160	~3,807
Mercury			13,600	26.4	~1,450	~4,760
Glycerol			1,260	2.45	~1,890	~6,200
Benzene			876	1.70	~1,105	~3,625
Methanol			792	1.54	~1,011	~3,318
Olive oil			920	1.79	~1,189	~3,900
Acetone			790	1.53	~1,210	~3,970
Carbon tetrachloride			1,590	3.09	~904	~2,966

Diffusion & Effusion

One notable characteristic of fluids is their ability to rapidly diffuse or mix at the molecular scale, even in the absence of turbulence or other forms of agitation. Gaseous molecules travel at relatively high speeds, so they collide frequently with other molecules as they travel in many directions. Understanding and predicting the diffusion of gases and other fluids, as well as aerosols^[10] or particulates suspended in fluids (colloids),^[11] is fundamental to many engineering applications.

Diffusion

Diffusion is a process in which fluid molecules move through a concentration gradient from an area of higher concentration to an area of lower concentration, until they are more evenly mixed, as shown in Figure 29. This movement is driven by molecular kinetic energy and natural molecular mixing and continues until a uniform equilibrium is reached. Diffusion is generally faster in gases because molecules are farther apart and move more freely, particularly in gases with low molecular weights. Diffusion at the molecular level is relatively slow, but its rate is considerably enhanced by turbulence or other forced processes such as convection.

The diffusion characteristics of fluids can be predicted using Fick’s laws, after the work of Adolf Fick. In the case where the concentration gradient is time-invariant, the diffusion flux, J, which is the amount of substance per unit area per unit time, is proportional to the negative gradient of the concentration, C, i.e.,

(88)  

where C is the concentration of the diffusing species, D is the diffusion coefficient, i.e., a measure of the diffusivity, and x is the spatial coordinate. Equation 88 is called Fick’s first law of diffusion. The negative sign in Eq. 88 indicates that the flux occurs in the direction of decreasing concentration. The diffusion coefficient, D, depends on the properties of the fluid, including its viscosity and temperature. Fick’s law is empirically derived and was developed by building on the work of Thomas Graham.

Fick’s first law of diffusion can also be expressed using the mass concentration, or partial density, of the diffusing species, denoted by . This quantity represents the mass of species per unit volume of the mixture. This alternative form of Fick’s first law is

(89)  

where is the diffusive mass flux of species through a unit area per unit time and is the corresponding diffusion coefficient. In this form, Fick’s first law states that the mass flux of a substance is proportional to the concentration gradient, with the flux directed opposite to the gradient. A related but distinct result, Graham’s law, expresses how the relative diffusion or effusion rates of gases depend on their molecular masses.

Fick’s second law applies to non-steady-state diffusion, where the concentration within the diffusion medium changes with time, which is expressed by

(90)  

For practical problems involving the diffusion of gases, Fick’s second law can also be used to estimate a characteristic diffusion time. The scaling depends on the geometry of the diffusion process. For one-dimensional diffusion over a characteristic distance , the diffusion time is often estimated as

(91)  

For three-dimensional spherical diffusion, the corresponding estimate is

(92)  

where is the characteristic diffusion distance and is the diffusion coefficient. These expressions are order-of-magnitude estimates, but they are useful for judging whether molecular diffusion alone is likely to be fast or slow in a given engineering problem.

In aerospace applications, diffusion is a fundamental mechanism for combustion processes. It is also an essential component of air-conditioning systems on flight vehicles, ensuring breathable air and removing carbon dioxide and other contaminants. In advanced materials, the diffusion of resin and curing agents affects the final mechanical properties and performance of composites.

Effusion

Effusion refers to the escape of molecules through a porous surface or a small hole; the underlying process is similar to diffusion, as shown in Figure 30. In this case, molecules move through the holes in a metered fashion at a rate related to their kinetic energy and molecular size, i.e., the faster the molecules move and the smaller they are, the more frequently they will pass through a given size of hole in a given time, and so move from one side of the porous surface to the other.

The behavior of effusion and diffusion can be predicted using Graham’s law. Graham’s law states that the rate of effusion or diffusion of a given gas is inversely proportional to the square root of its relative molecular mass, i.e.,

(93)  

where and are the rates of effusion or diffusion of gases 1 and 2, respectively, and and are the relative molecular masses of gases 1 and 2, respectively.

The proof of this result can be developed from the kinetic theory of gases, which states that the average kinetic energy of gas molecules is proportional to their absolute temperature, , i.e.,

(94)  

where is the mass of a gas molecule and is the average root-mean-square speed of the molecules. The average kinetic energy of one molecule of gas is also proportional to the absolute temperature, i.e.,

(95)  

where is Boltzmann’s constant. Therefore,

(96)  

where is the mass of one molecule and is the root-mean-square molecular speed. If is the molar mass of the gas, then , where is Avogadro’s number. Substituting gives

(97)  

and solving for gives

(98)  

Because , where is the universal gas constant, then

(99)  

showing that the root-mean-square speed, , of the molecules is inversely proportional to the square root of their molar mass, .

It can be deduced, therefore, that the rate of effusion, , of a gas is proportional to the root-mean-square speed of its molecules, i.e.,

(100)  

So, for two different gases (1 and 2) at the same temperature, then

(101)  

which is known as Graham’s law. Stated in words, Eq. 101 shows that the effusion rate is inversely proportional to the square root of its relative molecular mass.

Graham’s law also holds for diffusion, albeit only approximately, and it has been shown to perform well in practice. Likewise, the effusion or diffusion of a gas mixture can be predicted only approximately using Graham’s law, because interactions among the different gas molecules affect the combined rate. Note that Fick’s laws concern diffusion through a medium driven by concentration gradients. Graham’s law concerns the diffusion rate as a function of molar mass.

Thomas Graham’s original work pertained mainly to the effusion and diffusion of gases. For gases, Graham’s law gives a useful approximate relationship between diffusion or effusion rate and molecular mass. For liquids, however, diffusion is usually governed more strongly by viscosity, molecular size and shape, temperature, and intermolecular interactions. For this reason, liquid diffusion rates are normally predicted using measured diffusion coefficients or empirical correlations rather than Graham’s law.

Graham’s law of effusion is also known to work approximately for gases that exit through somewhat larger holes and orifices, in which case the law is often modified to

(102)  

where is an empirical constant related to the size and shape of the hole or opening and to the conditions under which the gas escapes. If different gases exit through the same hole size, then the ratio of their effusion rates (but not their rates per se) can be predicted using Eq. 101.

Surface Tension & Capillary Action

Surface tension is the tendency of a liquid-gas interface, or a free-surface between two immiscible fluids to behave like a stretched elastic membrane or “skin.” Capillary action is directly related to surface tension, which is usually understood as describing the behavior of a liquid-gas interface in narrow spaces, usually with the formation of a surface meniscus. Capillary action and surface tension are phenomena that arise from the relative effects of intermolecular forces in fluids. The surface-tension property is widely used in various applications, including lubrication, painting and coating technologies, inkjet printing, the production of emulsions and foams, bubble formation, microfluidics, and various manufacturing processes.

Surface Tension

There is a natural tendency for all liquids to minimize their net energy state by minimizing their surface area. For example, the often-observed behavior of water “beading” on a surface covered with a thin film of oil or wax occurs because the surface is poorly wetted by water, so cohesive forces within the water dominate over adhesive forces between the water and the surface.

Consider Figure 31, which shows beads of water sitting on a solid surface. The physics of surface tension can be explained by the cohesive forces among the liquid molecules. Away from the surface, the liquid molecules are pulled approximately uniformly in every direction by neighboring molecules, giving no net force. As the liquid-gas interface or free surface is approached, however, the intermolecular forces are no longer balanced because there are fewer liquid molecules above the surface. The molecules at the interface experience a net inward attraction toward the liquid, causing the surface to behave as though it were under tension and tending to minimize its area. The interaction between the liquid and the solid surface determines the degree of wetting and the resulting shape of the droplet.

For small water droplets sitting on a surface, as shown in Figure 31, it will be noticed that the droplet is almost perfectly spherical because, in this configuration, there is the least surface area for a given volume. For larger droplets, the shape becomes somewhat flatter and bulges because of the increasingly significant effects of gravity, which are roughly proportional to the droplet’s weight, i.e., proportional to , where is the approximate radius of the droplet.

This latter situation is analogous to that of a water-filled balloon: the water’s weight accounts for the gravitational effect, and the balloon’s skin stretches as the water volume increases, mimicking the effects of surface tension. Because the surface area is proportional to , the gravitational to surface tension ratio depends on the ratio ; this latter effect becomes increasingly important for larger droplets.

Surface tension, typically denoted by the symbol (though is sometimes used, resulting in a symbol conflict), has units of force per unit length. The base units of surface tension are Newtons per meter (N/m) in SI units or pounds per foot (lb/ft) in USC units. The Young-Laplace equation describes the pressure difference across a curved liquid-gas interface resulting from surface tension.

Consider a small, curved element of the liquid surface, i.e., part of a spherical or cylindrical cap defined by two principal radii of curvature, and , as shown in Figure 32. Although signs for these values can vary, sign convention usually dictates positive curvature when it is convex and negative when it is concave. The curvature of the surface affects how surface tension acts.

The surface tension acts tangentially along the boundary of the patch. For an infinitesimal short boundary length , the force from surface tension is

(103)  

The pressure difference across the surface, , which, because it is a liquid/gas interface, is usually referred to as the capillary pressure or the Laplace pressure, creates a normal force on the patch, . The surface-tension force must balance the capillary-pressure force to maintain equilibrium. The net inward force from surface tension on the patch can be approximated by considering the contributions from both radii of curvature, and .

It can be shown that the force from surface tension in one direction is proportional to and inversely proportional to . This outcome occurs because a smaller radius of curvature (a sharper curve) results in a larger force. Similarly, the other radius of curvature, , also contributes an inward force proportional to but inversely proportional to .

Therefore, the total force from the surface curvature must balance the capillary pressure, i.e.,

(104)  

and so leading to

(105)  

This latter relationship is called the Young-Laplace equation. Notice that in the case of spherical shapes such as bubbles or droplets of radius , then = = and so

(106)  

For cylindrical cases, then = and , so and

(107)  

In the general case, for a free surface where gravity is also important, the pressure jump across the gas/liquid interface must be balanced by both hydrostatic pressure variations and surface-tension effects. The precise sign of each term depends on the chosen coordinate direction and curvature convention. For this reason, the Young-Laplace relation is usually applied locally as

(108)  

with any hydrostatic pressure variation added separately using the usual relation

(109)  

after the coordinate system has been defined.

It is worth noting that the magnitude of surface tension depends on the type of liquid and gas, as well as their respective temperatures. Generally, liquids with stronger intermolecular forces exhibit higher values of surface tension. Water, for example, has a relatively high surface tension from the strong cohesive interactions among its hydrogen-bonded molecules. Surface tension is an essential physical property of most liquids. It is crucial in various engineering and scientific problems, including droplet formation, capillarity and meniscus formation, bubble dynamics, surface coatings, and biological and medical applications.

Values of surface tension for a liquid-air interface at 25^oC.

Fluid	(N/m)	(lb/ft)
Water	0.0728	0.00498
Soapy water	0.025–0.040	0.00171–0.00274
Ethanol	0.0223	0.00153
Methanol	0.0226	0.00154
Glycerol	0.0631	0.00432
Mercury	0.485	0.0332
Benzene	0.0289	0.00198
Acetone	0.0237	0.00162
Olive Oil	0.032	0.00219

Blowing Bubbles!

The formation of bubbles, whether gas bubbles in liquids (e.g., carbon dioxide bubbles in a soda or other fizzy drink) or soap bubbles, involves surface-tension effects. Gas bubbles in a liquid rise quickly because of their buoyancy, which is proportional to the density difference between the gas inside the bubble and the surrounding liquid. Surface tension plays a critical role in the stability and shape of the bubble. It tends to minimize surface energy and, hence, the bubble’s surface area, thereby forming a spherical shape in the absence of external forces.

For any bubble to be in equilibrium with itself in ambient air, the internal pressure of the gas inside the bubble must be greater than the atmospheric pressure by an amount exactly equal to the pressure difference caused by the bubble’s surface tension, as shown in Figure 33. This means that at equilibrium, the external pressure exerts an inward force that balances the internal pressure minus the surface-tension force. Therefore, a bubble will maintain its shape and size if these forces are precisely balanced.

A soap bubble consists of a thin film of water sandwiched between a small amount of soap and enclosing a volume of gas, often air or helium. The total thickness of the film is typically very small, ranging from a few micrometers to hundreds of nanometers as the film drains and thins. Note that there are inner and outer soap film surfaces, both of which exhibit surface tension.  The atmospheric pressure and the pressure from surface tension balance the internal pressure of the gas inside the bubble. This balance is described by the Young-Laplace equation, i.e., for the case of a sphere, given by Eq. 106, then

(110)  

where is the pressure difference across the bubble’s surface, is the surface tension, and is the radius of the bubble. Because the film has inner and outer soap surfaces, this result must be multiplied by two, i.e.,

(111)  

Therefore, for bubble equilibrium, then

(112)  

Capillary Action

Narrow-diameter cylindrical tubes are called capillary tubes. If one end of these tubes is dipped into a liquid, as shown in Figure 34, the liquid in the capillary “wets” the inside of the tube and either rises or falls relative to the surrounding liquid level. This phenomenon is called capillary action. Capillary action occurs at a molecular level because of the intermolecular cohesive attraction between the molecules of the liquid and the adhesive attraction between the capillary walls. Leonardo da Vinci first observed capillary action with water in the late 15th century; he was fascinated by water’s behavior and made numerous observations, though he did not develop any physical laws governing this phenomenon. Today, the physics of capillary action is applied in many engineering and scientific fields, including microfluidics.

In capillary action, three main forces are in balance: adhesion, cohesion, and surface tension. Adhesion happens when different molecules are attracted to each other. Cohesion is when molecules that are the same cling to each other. If the liquid wets the tube sufficiently for wall adhesion to dominate, it will rise in the capillary tube. If the liquid does not wet the tube, such as mercury in glass, the liquid level will be depressed. Furthermore, the narrower the tube, the larger the magnitude of the capillary rise or depression.

Another interesting feature of capillary tubes is the meniscus formation at the liquid-gas interface. A concave meniscus occurs when the attraction between the particles of the liquid and the container (adhesion) is greater than the attraction of the particles of the liquid to each other (cohesion), causing the liquid to climb the walls of the container. This behavior occurs at the water-glass interface; water-based fluids will form a concave meniscus in glass or other wettable containers. A convex meniscus occurs when the adhesion energy is less than the cohesion energy. Convex menisci arise, for example, between mercury and glass, which was apparent in the old mercury thermometers.

Jurin’s law, named after the 18th-century physicist James Jurin, mathematically describes the capillary action observed when a liquid rises or falls in a narrow tube. The physics involves the balance between adhesive forces (attraction between the liquid and the walls) and cohesive forces (attraction between liquid molecules). When a narrow tube is inserted vertically into a liquid, the liquid either rises or falls. If the liquid wets the tube (e.g., water in a glass tube), it will rise; if it does not wet the tube (e.g., mercury in a glass tube), it will fall. According to Jurin’s law, the height to which the liquid rises or falls is given by

(113)  

where is the surface tension of the liquid, is the contact angle between the liquid and the tube, is the density of the liquid, is the acceleration under gravity, and is the diameter of the tube.

Surface tension, , is the force per unit length at the surface of a liquid because of molecular attractions, causing the liquid surface to behave like a stretched elastic membrane. The contact angle, , is the angle at which the liquid interface meets the solid surface, being less than 90^o for a liquid that wets the surface and greater than 90^o for one that does not. The smaller the diameter of the tube, the higher the liquid will rise or fall because capillary forces exert a more substantial effect in narrower tubes. For a concave meniscus, which occurs when the liquid wets the tube walls (e.g., water in a glass tube), the height is measured from the bottom of the meniscus to the free surface of the liquid outside the tube.  For a convex meniscus, which occurs when the liquid does not wet the tube walls (e.g., mercury in a glass tube), the height is measured from the top of the meniscus to the free surface of the liquid outside the tube.

Summary of Properties of Air at MSL Standard

Values of the properties of air at standard mean sea level (MSL) conditions are helpful to have on hand and are given in the table below. They are often referred to as MSL ISA conditions, where “MSL” denotes Mean Sea Level and “ISA” denotes the International Standard Atmosphere. Air properties at MSL ISA conditions are often used as a standard reference.

Table of ISA Properties at MSL.

Property	Symbol	SI units	USC units
Pressure		1.0132510 Nm (Pa)	2116.4 lb/ft (29.92 inches of Hg)
Density		1.225 kg m	0.002378 slugs ft
Temperature		288.15 K	518.67 R
Dynamic viscosity		1.78910 kg m s	3.73710 slugs ft s
Speed of sound		340.3 m s	1116.47 ft s
Gas constant		287.057 J kg K	1716.49 ft-lb slugR