Viscous-Dominated Flows

J. Gordon Leishman

29 Viscous-Dominated Flows

Introduction

Viscous-dominated flows, which are usually called “creeping” or “Stokes” flows, occur when viscous forces rather than inertial forces predominantly govern fluid motion. These flows are characterized by smooth (laminar), steady, and highly predictable fluid motion. This flow regime is characterized by Reynolds numbers ( $Re$ ) based on a length dimension, $L$ , of much less than one ( $Re \ll 1$ ), where the effects of inertia in the flow are negligible compared to viscous forces. Such conditions often arise in scenarios involving very slow flow velocities, small characteristic length scales, or highly viscous fluids.

Recall that the Reynolds number assesses the relative significance of inertial to viscous forces in a fluid. The Reynolds number is expressed as

(1) $\begin{equation*} Re = \frac{\varrho V L}{\mu} = \frac{\varrho V L (V/L)}{\mu (V/L)} = \frac{\varrho V^2}{\mu (V/L)} \equiv \dfrac{\text{Inertial effects in a flow}}{\text{Viscous shear effects in a flow}} \end{equation*}$

where $\varrho$ is the fluid density, $V$ is a characteristic velocity, $L$ is a characteristic length scale, and $\mu$ is the fluid’s viscosity. At low values of the Reynolds numbers, the inertial term in the Navier-Stokes equation for fluid motion becomes negligible, and the equations simplify to what is known as the Stokes equation, i.e.,

(2) $\begin{equation*} \nabla p = \mu \nabla^2 \vec{V} \end{equation*}$

where $p$ is the pressure, $\vec{V}$ is the velocity, and the term $\nabla^2 \vec{V}$ represents the viscous terms from the velocity gradients. The Stokes equation defines a balance between the flow’s pressure forces and viscous forces. The absence of the nonlinear inertial terms implies that the flow is steady and linear, with any external forces being immediately transmitted through the entire fluid without inducing any transient or time-history effects.

Mathematical models in this flow regime can be used to find closed-form solutions in some cases and also allow for easier^[1] numerical simulations. One of the defining features of creeping flows is their reversibility. In the absence of inertial effects, reversing the driving force or boundary motion results in the fluid flow that initially developed and then retracing its path exactly. This particular flow behavior contrasts distinctly with higher Reynolds-number flows, where turbulence introduces completely irreversible flow dynamics. Observations and measurements of creeping flows abound in the published scientific literature,^[2] highlighting the importance of this type of flow within the framework of fluid mechanics. An example of a precise flow visualization of the Stokes flow about the cross-section of a circular cylinder between two glass plates (called a Hele-Shaw flow) is shown below.

Flow visualization of a form of Stokes flow over a circular cylinder sandwiched between glass plates. The dye marks streamlines in water flowing at 1 mm/second. (Photograph by D. H. Peregrine.)

In engineering, creeping flows play a critical role in the study of lubrication within thin films, particularly in machinery. In aerospace technologies, microfluidic systems are increasingly used for thermal management of sensitive electronics, where the dominance of viscous forces ensures precise and stable fluid motion. Bearings and hydraulic systems rely on these flows to ensure smooth and reliable operation. Highly viscous or creeping flows are also prevalent in natural, biological, and engineering systems. The principles of Stokes flow are also essential in analyzing particle behavior in colloidal suspensions, with applications in pharmaceuticals, food processing, and other industries. Microfluidics represents a modern application of creeping flow, where small fluid volumes are manipulated for various applications, including chips for control devices. Similarly, biomedical engineering relies on understanding highly viscous flows to analyze blood flow in capillaries and mucus transport in the respiratory system. Similar creeping flow principles govern various geophysical phenomena, such as the slow movement of glaciers and lava flows.

Learning Objectives

Develop an understanding of highly viscous creeping flows, including the conditions under which they occur.
Be able to reduce the Navier-Stokes equation to the Stokes equation for viscous-dominated flows.
Derive and analyze analytical solutions to the Stokes equations for simple viscous-dominated flow problems.

Stokes Flow Around a Sphere

It is instructive to introduce the characteristics of viscous-dominated Stokes flows or creeping flows by considering the Reynolds number-dependent flow about a sphere. The creeping flow about a sphere can also be analyzed using pure mathematics, providing a basis for a general understanding of creeping flows. For a sphere, the characteristic length scale $L$ is taken as the sphere’s diameter $D$ , which is a key parameter in determining the flow regime around the object.

At very low Reynolds numbers ( $Re \ll 1$ ), the flow is dominated by viscous forces where the flow field around the sphere becomes smooth (laminar) and steady, as shown in the figure below. Both spheres and circular cylinders in the low-Reynolds-number regime ( $Re < 5$ ) represent a unique flow regime where viscous forces rather than pressure forces dictate the flow behavior, resulting in laminar, predictable flow patterns. In this regime, the flow is characterized by symmetric streamlines around the sphere, with no wake formation or flow separation. At low Reynolds numbers, even as “high” as ( $Re = 5$ ), the flow retains Stokes-like characteristics, remaining laminar, with the viscous forces shaping the flow patterns and with streamlines exhibiting smooth and symmetric curvature around the sphere. In this range, viscous effects dominate the drag experienced by the sphere, which can be calculated theoretically, leading to what is known as Stokes’ drag law.

Flow patterns around a sphere at low Reynolds numbers. Stokes flows exist for Reynolds numbers of less than unity, although they may persist to slightly higher Reynolds numbers.

As the Reynolds number increases, the balance of forces shifts, and inertial effects begin to influence the flow. For spheres and circular cylinders, the onset of flow separation occurs at moderate Reynolds numbers ( $Re \ge 5$ ), where a small symmetric recirculating wake forms behind the sphere, breaking the fore-and-aft symmetry observed in the Stokes regime. This flow becomes increasingly unstable with periodic vortex shedding at even modestly higher Reynolds numbers, eventually transitioning to a more turbulent flow beyond a critical Reynolds number, typically around $Re \approx 40$ , depending on the specific conditions. The distinction between spheres and cylinders lies in their three-dimensional versus two-dimensional geometry, which subtly alters the drag forces and flow patterns but follows the same fundamental principles dictated by the magnitude of the Reynolds number.

Who was George Stokes?

George Gabriel Stokes (1819–1903) was an influential mathematician and physicist who made foundational contributions to fluid dynamics and other fields of science. In 1851, he derived a fluid law, now known as Stokes’ Law, which describes the drag force experienced by a spherical particle moving through a viscous fluid at low Reynolds numbers. Stokes’ contributions laid the groundwork for understanding lubrication, sedimentation, microfluidics, and bioengineering. Stokes held the Lucasian Professor of Mathematics position at Cambridge University, significantly contributing to the field of fluid mechanics.

Stokes’ Equation

The Stokes equation is determined by reducing the Navier-Stokes equation without the inertia and time-dependent terms. The Navier-Stokes equation for incompressible flow is given in vector form as

(3) $\begin{equation*} \varrho \frac{D \vec{V}}{Dt} = \varrho \left( \frac{\partial \vec{V}}{\partial t} + \vec{V} \bigcdot \nabla \vec{V} \right) = - \nabla p + \mu \nabla^2 \vec{V} + \vec{f_b} \end{equation*}$

where $\varrho$ is the fluid density, $\vec{V}$ is the velocity vector of the fluid, $p$ is the pressure field, $\mu$ is the viscosity of the fluid, and $\vec{f_b}$ accounts for external body forces such as gravity. The term $\dfrac{\partial \vec{V}}{\partial t}$ represents the unsteady or time-dependent change in velocity. In contrast, $\vec{V} \bigcdot \nabla \vec{V}$ is the nonlinear convective term describing the effects of fluid inertia. The term $-\nabla p$ corresponds to pressure-driven acceleration, $\mu \nabla^2 \vec{V}$ represents the viscous forces, and $\vec{f_b}$ introduces any additional external forces acting per unit volume. In addition, For incompressible flows, the density of the fluid remains constant, leading to the incompressible form of the continuity equation, i.e.,

(4) $\begin{equation*} \nabla \bigcdot \vec{V} = 0 \end{equation*}$

This form of the Navier-Stokes and continuity equations form the pair of governing equations for the motion of incompressible, viscous fluids.

For Stokes flow, the Navier-Stokes equation can be simplified. The first fundamental assumption is that the flow is steady. This condition mathematically translates to $\dfrac{\partial \vec{V}}{\partial t} = 0$ , eliminating the unsteady terms from the Navier-Stokes equation. The second assumption involves neglecting inertial terms, which is valid in the low Reynolds number regime. At such small Reynolds numbers, the inertial terms $\varrho (\vec{V} \bigcdot \nabla) \vec{V}$ become insignificant compared to viscous forces, effectively reducing this term to zero.

Under these two assumptions, the Navier-Stokes equation simplifies significantly to

(5) $\begin{equation*} 0 = - \nabla p + \mu \nabla^2 \vec{V} + \vec{f_b} \end{equation*}$

where $-\nabla p$ represents the pressure gradient, $\mu \nabla^2 \vec{V}$ accounts for viscous forces, and $\vec{f_b}$ denotes body forces such as gravity. If body forces are negligible ( $\vec{f_b} = 0$ ), the equation further simplifies to

(6) $\begin{equation*} \nabla p = \mu \nabla^2 \vec{V} \end{equation*}$

This reduced form highlights a direct balance between pressure gradients and viscous forces, which is the essence of viscous-dominated or creeping flows. The absence of inertial terms ensures that the flow is dominated entirely by viscosity, resulting in a laminar, predictable motion. The incompressible form of the continuity equation, given by $\nabla \bigcdot \vec{V} = 0$ , remains valid for Stokes’ flow.

Linearity of Stokes Flow

In Stokes flow, the governing equation is linear because the nonlinear inertial term $(\vec{V} \bigcdot \nabla) \vec{V}$ is absent. This linearity arises because only the viscous and pressure terms remain in the governing equation. The linear nature of these equations permits the use of the superposition principle, allowing complex flow solutions to be constructed by combining simpler ones, akin to the process used in potential flows. For example, if $\vec{V}_1$ and $\vec{V}_2$ are solutions to the Stokes equations, their sum $\vec{V} = \vec{V}_1 + \vec{V}_2$ is also a valid solution.

A classic formulation for two-dimensional incompressible Stokes flow uses the stream function $\psi(x, y)$ . The velocity components $u$ and $v$ are expressed as

(7) $\begin{equation*} u = \frac{\partial \psi}{\partial y} \quad \text{and} \quad v = -\frac{\partial \psi}{\partial x} \end{equation*}$

By definition, this stream function automatically satisfies the continuity equation $\dfrac{\partial u}{\partial x} + \dfrac{\partial v}{\partial y} = 0$ . The vorticity, $\omega$ , is defined as

(8) $\begin{equation*} \omega = \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \end{equation*}$

Substituting $u = \dfrac{\partial \psi}{\partial y}$ and $v = -\dfrac{\partial \psi}{\partial x}$ into the vorticity definition gives

(9) $\begin{equation*} \omega = -\nabla^2 \psi \end{equation*}$

where $\nabla^2$ is the Laplacian operator, defined as

(10) $\begin{equation*} \nabla^2 = \frac{\partial^2}{\partial x^2} + \frac{\partial^2}{\partial y^2} \end{equation*}$

In the Stokes flow regime, where viscous forces dominate, the governing equations are simplified to

(11) $\begin{equation*} \nabla^2 \omega = 0 \end{equation*}$

Substituting $\omega = -\nabla^2 \psi$ into this equation leads to

(12) $\begin{equation*} \nabla^2 (\nabla^2 \psi) = 0 \implies \nabla^4 \psi = 0 \end{equation*}$

where $\nabla^4$ is the biharmonic operator, defined as

(13) $\begin{equation*} \nabla^4 = \frac{\partial^4}{\partial x^4} + 2 \frac{\partial^4}{\partial x^2 \partial y^2} + \frac{\partial^4}{\partial y^4} \end{equation*}$

The preceding result underscores the tractability of Stokes flow, where the stream function formulation provides a systematic way to analyze complex flow configurations. The linearity of the governing equations simplifies the modeling of flows involving multiple particles, boundaries, or sources by enabling decomposition into more straightforward, analytically solvable problems.

Stokes’ Drag Law for Spheres

George Stokes established a foundational result for the drag force $D_S$ on a spherical particle moving through a viscous fluid at a constant velocity $V_{\infty}$ in the Stokes flow regime. For very low Reynolds numbers, where inertial effects are negligible, the drag force is given by

(14) $\begin{equation*} D_S = 6 \pi \, \mu \, R \, V_{\infty} = 3 \pi \, \mu \, D \, V_{\infty} \end{equation*}$

where $\mu$ is the fluid’s dynamic viscosity, $R$ is the sphere’s radius ( $R = D/2$ ), and $V$ is the velocity of the particle relative to the fluid. This relationship, known as Stokes’ drag law, is derived based on the balance of viscous forces acting on the sphere and has been experimentally validated.

Using the Buckingham Pi method, the drag force $D_S$ can be expressed as

(15) $\begin{equation*} D_S = \phi(\mu, R, V_{\infty}) \quad \implies \quad \phi_1(D_S, \mu, D, V_{\infty}) = 0 \end{equation*}$

In this case, $D_S$ represents the drag force with dimensions of $\rm M \, L \, T^{-2}$ , $\mu$ is the dynamic viscosity of the fluid with dimension $\rm M \, L^{-1} \, T^{-1}$ , $D$ is the diameter of the sphere with dimensions $\rm L$ , and $V_{\infty}$ is the velocity of the sphere with dimensions $\rm L \, T^{-1}$ . The number of variables is $N = 4$ , and the number of fundamental dimensions ( $\rm M$ , $\rm L$ , and $\rm T$ ) is $K = 3$ . Therefore, using the Buckingham $\Pi$ method, the number of dimensionless groups is $N - K = 1$ .

The Stokes drag on a particle can be obtained using dimensional analysis.

This dimensionless group $\Pi$ is formed as

(16) $\begin{equation*} \Pi = \phi_2(D_S, \mu, D, V_{\infty}) \quad \implies \quad \Pi = D_S \, \mu^\alpha \, D^\beta \, V_{\infty}^\gamma \end{equation*}$

To make $\Pi$ dimensionless, the dimensions of each variable are substituted, giving

(17) $\begin{equation*} [\Pi] = 1 = \rm M^0 L^0 T^0 = (\rm M L T^{-2}) (\rm M L^{-1} T^{-1})^\alpha (\rm L)^\beta (\rm L T^{-1})^\gamma \end{equation*}$

Equating the exponents of $\rm M$ , $\rm L$ , and $\rm T$ on both sides yields $1 + \alpha = 0$ , $1 - \alpha + \beta + \gamma = 0$ , and $-2 - \alpha - \gamma = 0$ . Solving these simultaneous algebraic equations gives $\alpha = -1$ , $\gamma = -1$ , and $\beta = -1$ . Substituting these numerical values into the dimensionless grouping results in

(18) $\begin{equation*} \Pi = \frac{D_S}{\mu D V_{\infty}} \end{equation*}$

Because $\Pi$ is dimensionless, the drag force can be expressed as $D_S = k \mu D V_{\infty}$ , where $k$ is a dimensionless constant. Experiments have determined that $k = 3 \pi$ for a sphere, leading to Stokes’ drag law, i.e.,

(19) $\begin{equation*} D_S = 3 \pi \, \mu \, D \, V_{\infty} = 6 \pi \, \mu \, R \, V_{\infty} \end{equation*}$

This latter result is valid for very low Reynolds numbers ( $Re \ll 1$ ), where viscous forces dominate over inertial forces. In general, for any body shape of characteristic length $L$ in Stokes flow, then the drag can be written as

(20) $\begin{equation*} D_S = k \, \mu \, L \, V_{\infty} \end{equation*}$

where $k$ depends on the body shape.

Drag Coefficient

The drag coefficient $C_D$ for a sphere in Stokes flow can be derived from the conventional drag force expression by normalizing the drag force by free-stream dynamic pressure and cross-sectional area, i.e., $A = \pi R^2$ . This drag coefficient is defined as

(21) $\begin{equation*} C_D = \frac{D_S}{\dfrac{1}{2} \varrho V_{\infty}^2 \, A} = \frac{D_S}{\dfrac{1}{2} \varrho V_{\infty}^2 \pi R^2} \end{equation*}$

Substituting Stokes’ drag law into this definition yields

(22) $\begin{equation*} C_D = \frac{6 \pi \mu R V_{\infty}}{\dfrac{1}{2} \varrho V_{\infty}^2 \pi R^2} \end{equation*}$

Simplifying this latter expression gives

(23) $\begin{equation*} C_D = \dfrac{12 \mu}{\varrho V_{\infty} R} = \dfrac{24}{Re} \end{equation*}$

where $Re = \dfrac{\varrho V_{\infty} R}{\mu}$ is the Reynolds number for the sphere. This latter relationship illustrates that the drag coefficient increases as the Reynolds number decreases, emphasizing the dominance of viscous forces at low Reynolds numbers. Alternatively, another form of drag coefficient for Stokes flow can be written as

(24) $\begin{equation*} C_{D_{S}} = \frac{D_S}{ \, \mu \, R \, V_{\infty}} \end{equation*}$

and so using Eq. 19 then $C_{D_{S}} = 3 \pi$ for a sphere.

Derivation of Stokes’ Drag Law

It has been shown that for low Reynolds numbers (creeping flow), the Navier-Stokes equation simplifies to

(25) $\begin{equation*} \nabla p = \mu \nabla^2 \vec{V} \end{equation*}$

which represents a balance between pressure forces and viscous forces. In spherical coordinates $(r, \theta, \phi)$ centered on the spherical particle, the velocity field is expressed as

(26) $\begin{equation*} \vec{V}(r) = u_r(r, \theta) \, \vec{e_r} + u_\theta(r, \theta) \, \vec{e_{\theta}} \end{equation*}$

where $u_r(r, \theta)$ and $u_\theta(r, \theta)$ are the radial and tangential velocity components, respectively. This problem has azimuthal symmetry, so the dependency on $\phi$ is removed.

The boundary conditions for the flow are:

At the surface of the sphere where $r = R$ , then $\vec{V}(r = R) = V_{\infty}$ , where $V_{\infty}$ is the velocity of the particle through the flow.
Far from the surface where $r \to \infty$ , then $\vec{V}(r \to \infty) = 0$ .

By solving the Stokes flow equations under these boundary conditions, the velocity field components are found to be

(27) $\begin{equation*} u_r(r, \theta) = V_{\infty} \left(1 - \frac{3R}{2r} + \frac{R^3}{2r^3}\right) \cos \theta \end{equation*}$

(28) $\begin{equation*} u_\theta(r, \theta) = -V_{\infty} \left(1 - \frac{3R}{4r} - \frac{R^3}{4r^3}\right) \sin \theta \end{equation*}$

These results show that the velocity field does not depend on the viscosity, $\mu$ , even though the drag depends on $\mu$ because the shear stresses depend on the velocity gradients.

Notice that by substituting $r = R$ into Eq. 27, then

(29) $\begin{equation*} u_r(R, \theta) = V_{\infty} \left(1 - \frac{3R}{2R} + \frac{R^3}{2R^3}\right) \cos \theta = V_{\infty} \left(1 - \frac{3}{2} + \dfrac{1}{2}\right) \cos\theta = 0 \end{equation*}$

i.e., the radial velocity is zero, satisfying the flow tangency (no-penetration) condition on the surface. Also, $r = R$ is substituted for the tangential flow component in Eq. 28 gives

(30) $\begin{equation*} u_\theta(R, \theta) = -V_{\infty} \left(1 - \frac{3R}{4R} - \frac{R^3}{4R^3}\right) \sin\theta = -V_{\infty} \left(1 - \frac{3}{4} - \frac{1}{4}\right) \sin\theta = 0 \end{equation*}$

So, the tangential velocity is also zero at $r = R$ , i.e., the no-slip condition is satisfied.

The stress tensor components are required to compute the drag force $D_S$ on the sphere. The pressure, $p$ , acts inward and normal to the surface so that the total radial stress at the sphere’s surface is given by

(31) $\begin{equation*} \sigma_{rr} = -p + 2 \mu \frac{\partial u_r}{\partial r} \end{equation*}$

where $\dfrac{\partial u_r}{\partial r}$ is the radial velocitiy gradient. If the velocity field at $r = R$ is substituted into the radial derivative, then

(32) $\begin{equation*} \sigma_{rr} = -p + 3 \mu V \cos \theta \end{equation*}$

The total (pressure plus viscous) drag force is obtained by integrating the radial stress over the sphere’s surface. Therefore, the drag force is expressed as

(33) $\begin{equation*} D_S= \oiint_S \sigma_{rr} \cos \theta \, dA \end{equation*}$

where $dA = R^2 \sin \theta \, d\theta \, d\phi$ is the elemental surface area on the sphere. Substituting $\sigma_{rr}$ and integrating over $\theta$ (from 0 to $\pi$ ) and $\phi$ (from 0 to $2\pi$ ) yields

(34) $\begin{equation*} D_S = \int_0^{2\pi} \int_0^\pi \left(-p + 3 \mu V_{\infty} \cos \theta\right) \, \cos \theta \, R^2 \sin \theta \, d\theta \, d\phi \end{equation*}$

The surface integral can be separated into two parts:

The azimuthal integral over $\phi$ from $0$ to $2\pi$ , i.e.,
(35) $\begin{equation*}\int_0^{2\pi} d\phi = 2\pi \end{equation*}$
The polar integral over $\theta$ from $0$ to $\pi$ , i.e.,
(36) $\begin{equation*}\int_0^\pi \left(-p + 3 \mu V_\infty \cos \theta\right) \cos \theta \sin \theta \, d\theta \end{equation*}$

Using the trigonometric identity $\sin \theta \cos \theta = \dfrac{1}{2} \sin(2\theta)$ , and the symmetry of the pressure term, the polar integral simplifies to

(37) $\begin{equation*} \int_0^\pi -p \cos \theta \sin \theta \, d\theta = 0 \end{equation*}$

because the pressure distribution is symmetric for and aft and side to side about the equators. For the viscous stress term, then

(38) $\begin{equation*} \int_0^\pi 3 \mu V_\infty \cos^2 \theta \sin \theta \, d\theta, \end{equation*}$

and using the substitution $u = \cos \theta$ and so $du = -\sin \theta \, d\theta$ , gives

(39) $\begin{equation*} \int_0^\pi \cos^2 \theta \sin \theta \, d\theta = \int_{-1}^1 u^2 \, du = \frac{2}{3} \end{equation*}$

Combining the results gives

(40) $\begin{equation*} D_S = (2\pi R^2) (3 \mu V_{\infty}) \left( \frac{2}{3} \right) = 6 \pi \, \mu \, R \, V_{\infty} \end{equation*}$

or in coefficient form

(41) $\begin{equation*} C_{D_{S}} = \frac{D_S}{\mu \, R \, V_{\infty}} = 6 \pi \end{equation*}$

This latter expression is called Stokes’ drag law. Once again, it applies only to a sphere moving through a viscous fluid at very low Reynolds numbers ( $Re \ll 1$ ), where inertial forces are negligible and viscous forces dominate.

Oseen Equations

The Oseen equations are an extension of the Stokes equations for creeping flows, which include minor inertial effects while remaining linear. They are particularly useful for analyzing flows around objects such as spheres or cylinders at low but finite Reynolds numbers.

The starting point is again the incompressible Navier-Stokes equation, i.e.,

(42) $\begin{equation*} \varrho \left( \frac{\partial \vec{V}}{\partial t} + (\vec{V} \bigcdot \nabla) \vec{V} \right) = -\nabla p + \mu \nabla^2 \vec{V} \end{equation*}$

For Oseen flow, it is assumed that the flow is steady just as Stokes flow, so $\dfrac{\partial \vec{V}}{\partial t} = 0$ . At low Reynolds numbers, inertial effects are small compared to viscous effects. Additionally, a uniform far-field velocity $\vec{U}$ is assumed, where the flow has a constant velocity far from the object.

To simplify the convective term $(\vec{V} \bigcdot \nabla) \vec{V}$ , the velocity can be re-expressed as

(43) $\begin{equation*} \vec{V} = \vec{V} + \vec{v'} \end{equation*}$

where $\vec{v'}$ is the disturbance velocity from the object. Substituting this expression into the Navier-Stokes equation and linearizing the convective term results in the inertial term $\varrho (\vec{V} \bigcdot \nabla) \vec{v'}$ . The final Oseen equations then take the form

(44) $\begin{equation*} -\nabla p + \mu \nabla^2 \vec{v'} - \varrho (\vec{V} \bigcdot \nabla) \vec{v'} = 0 \quad \text{and} \quad \nabla \bigcdot \vec{v'} = 0 \end{equation*}$

Unlike the Stokes equations, which neglect all inertial terms and assume $Re \to 0$ , the Oseen equations include the linearized inertial term $\varrho (\vec{U} \bigcdot \nabla) \vec{v'}$ , extending their validity to slightly higher Reynolds numbers.

For example, in the flow around a sphere, the Oseen solution for the velocity field becomes

(45) $\begin{equation*} u_r = U \left( 1 - \frac{3a}{2r} + \frac{3a^2}{2r^2} \right) \cos\theta \end{equation*}$

which can be compared with Eq. 27 for pure Stokes flow. For the radial component, then

(46) $\begin{equation*} u_\theta = -U \left( 1 - \frac{3a}{4r} - \frac{3a^2}{4r^2} \right) \sin\theta \end{equation*}$

which can be compared with Eq. 28 for Stokes flow. At larger distances, i.e., $r \to \infty$ , the velocity approaches the uniform far-field flow of $V_{\infty}$ , decaying as $1/r$ , as opposed to the $1/r^2$ decay of the Stokes solution. The Oseen correction accounts for weak inertial effects, providing a more physically representative solution for low Reynolds numbers.

The resulting correction to Stokes’ drag law is given by

(47) $\begin{equation*} D_S = 6\pi \mu R V_{\infty} \left(1 + \frac{3}{8} Re\right) \end{equation*}$

where $R$ is the sphere’s radius, and the $\dfrac{3}{8} Re$ term is the Reynolds number correction for weak inertial effects. In this case, the Stokes drag coefficient, $C_{D_{S}}$ for a sphere in creeping flow is

(48) $\begin{equation*} C_{D_{S}} = \dfrac{D_S}{\mu (2R) V_{\infty} } = \dfrac{ 3\pi \, \mu \, R \, V_{\infty} \left(1 + \dfrac{3}{8} Re\right)}{\mu (2R) V_{\infty}} = 3\pi \left(1 + \frac{3}{8} Re\right) \end{equation*}$

The conventional drag coefficient, including the Oseen correction, is

(49) $\begin{equation*} C_D = \frac{24}{Re} \bigg( 1 + \frac{3}{16} Re \bigg) \end{equation*}$

This latter result aligns with the creeping flow behavior at low values of $Re$ and includes the inertial correction. The results for both the Stokes drag coefficient and the Oseen drag coefficient are shown in the plot below.

The validity of the Stokes drag model becomes increasingly invalid for $Re > 1$ , which can then be replaced by the Oseen model.

The Oseen equations provide improved predictions for the flow and drag behavior at low but finite Reynolds numbers, typically below 100. They are especially useful for studying flows around objects, wake formation, and flow through porous media. For example, Oseen’s analysis shows how inertial effects result in streamlines that deviate slightly from purely viscous Stokes flow. However, the Oseen equations are limited to low Reynolds numbers where linear inertial effects can be linearized. At higher Reynolds numbers, the solutions to the Navier-Stokes equation are more difficult because of the need to solve for both nonlinear inertial effects and viscous effects.

Comparison of Stokes Flow & Potential Flow

Stokes (creeping) and potential flows may look superficially similar, but they are distinctly different flow conditions that describe fluid behavior around bodies under different physical assumptions and conditions. Indeed, their characteristics, mathematical descriptions, and physical implications differ significantly. Creeping flow occurs at very low Reynolds numbers, i.e., $Re \ll 1$ , where viscous forces dominate and inertial effects are negligible. In contrast, potential flow represents an idealized, inviscid, and irrotational flow regime, where inertial forces dominate and viscous effects are ignored. Potential flow solutions apply to high Reynolds number flows outside boundary layers.

Coordinate definitions and sign conventions for the Stokes flow about a sphere.

On the one hand, as previously shown, creeping flow is derived from the Stokes equations, which simplify the Navier-Stokes equation by neglecting inertial terms. The velocity components for creeping flow are axisymmetric, so they depend only on the $r$ and $\theta$ coordinates, i.e., referring to the figure below then

(50) $\begin{equation*} u_r(r, \theta) = V_{\infty} \left(1 - \frac{3R}{2r} + \frac{R^3}{2r^3}\right) \cos \theta \end{equation*}$

(51) $\begin{equation*} u_\theta(r, \theta) = -V_{\infty} \left(1 - \frac{3R}{4r} - \frac{R^3}{4r^3}\right) \sin \theta \end{equation*}$

Notice that these latter equations include terms proportional to $\dfrac{1}{r}$ and $\dfrac{1}{r^3~}$ , reflecting the rapid reduction in viscous effects away from the surface.

On the other hand, potential flow is derived from the Laplace equation, which is an inviscid, irrotational flow. In this case, the velocity components for potential flow are

(52) $\begin{equation*} u_r(r, \theta) = V_{\infty} \left(1 - \frac{R^3}{r^3}\right) \cos \theta \end{equation*}$

(53) $\begin{equation*} u_\theta(r, \theta) = -V_{\infty} \left(1 + \frac{R^3}{2r^3}\right) \sin \theta \end{equation*}$

In this case, the velocity components decay faster overall at a rate proportional to $\dfrac{1}{r^3~}$ compared to creeping flow because of the absence of viscosity.

The plot below shows the predicted streamline patterns for Stokes flow (left) and potential flow (right) around a sphere. Comparing the results highlights several differences. The streamlines in potential flow are closer together than those in Stokes flow. While the streamline patterns are qualitatively similar, the far-field effects are markedly different. Stokes flow exhibits a much more significant far-field effect. Notice that in creeping flow, the inclusion of terms proportional to $\dfrac{1}{r}$ reflects the slower decay of viscous effects, which dominate at intermediate distances. Viscous drag leads to a flow disturbance that extends further from the sphere.

The Stokes flow (blue streamlines) exhibits a slower decay with distance. The streamlines bend around the sphere more prominently because of viscous effects, whereas the potential flow (red streamlines) shows that flow velocity decays faster with distance.

In potential flow, all velocity terms decay as $\dfrac{1}{r^3~}$ , showing a faster reduction of the flow disturbance with distance. The absence of viscosity limits the spatial extent of the disturbance. The velocity near the sphere is slower than the free stream velocity everywhere, unlike potential flow, where velocities can exceed the freestream velocity in some locations. The velocity profile in potential flow is also distinct. Substituting $r = R$ into Eq. 52 gives

(54) $\begin{equation*} u_r(R, \theta) = V_{\infty} \left(1 - \frac{R^3}{R^3}\right) \cos \theta = V_{\infty} \left(1 - 1\right) \cos \theta = 0 \end{equation*}$

Therefore, the radial velocity satisfied the flow tangency (no-penetration) condition. For the tangential velocity, then substituting $r = R$ into Eq. 53 gives

(55) $\begin{equation*} u_\theta(R, \theta) = -V_{\infty} \left(1 + \frac{R^3}{2R^3}\right) \sin \theta = -V_{\infty} \left(1 + \dfrac{1}{2}\right) \sin \theta = -\frac{3}{2} V_{\infty} \sin \theta \end{equation*}$

Therefore, in potential flow at the sphere’s surface, the local velocity is $\dfrac{3}{2} V_{\infty}$ at its equator. In contrast, the velocity is zero everywhere on the surface in Stokes flow because of the no-slip condition.

Drag is also where the two flows differ. Potential flow predicts zero drag (d’Alembert’s paradox), i.e., no pressure drag and no viscous drag, so $C_D = 0$ , while creeping flow exhibits significant viscous drag, as given by

(56) $\begin{equation*} C_D = \dfrac{24}{Re} \end{equation*}$

As previously discussed, an alternative drag coefficient can be defined for creeping flow. The Stokes drag coefficient, $C_{D_{S}}$ for a sphere in creeping flow is

(57) $\begin{equation*} C_{D_{S}} = \dfrac{D_S}{\mu \, (2R) \, V_{\infty} } = \dfrac{6 \pi \mu \, (2R) \, V_{\infty}}{\mu \, (2R) \, V_{\infty} } = 3\pi \end{equation*}$

Stokes Flow Around a 2-D Ellipse

Other than the sphere, solutions to flow about other bodies must be performed numerically. Consider Stokes flow about a unit ellipse, given by

(58) $\begin{equation*} \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \end{equation*}$

where $a$ and $b$ are the fractions of the semi-major and semi-minor axes, respectively. The stream function $\psi$ for Stokes flow satisfies the biharmonic equation

(59) $\begin{equation*} \nabla^4 \psi = 0 \end{equation*}$

The velocity components are

(60) $\begin{equation*} u_x = \frac{\partial \psi}{\partial y} \quad \text{and} \quad u_y = -\frac{\partial \psi}{\partial x} \end{equation*}$

The boundary conditions are:

The flow tangency (no-penetration) boundary condition on the ellipse: $\psi = 0$ on the surface of the ellipse.
No-slip boundary condition on the ellipse, i..e., $\dfrac{\partial \psi}{\partial n} = 0$ .
Uniform flow is the far-field boundary condition, i.e., $\psi = V_\infty y$ , where $V_\infty$ is the freestream velocity.

The computational domain can be discretized using a uniform Cartesian grid for the numerical solution. The values of $\psi$ are first initialized with the far-field condition $\psi = V_\infty y$ . A finite-difference relaxation method can then be used to solve the biharmonic equation iteratively, i.e., using a stencil of the form

(61) $\begin{equation*} \psi^{(n+1)}_{i,j} = \frac{1}{4} \bigg( \psi^{(n)}_{i-1,j} + \psi^{(n)}_{i+1,j} + \psi^{(n)}_{i,j-1} + \psi^{(n)}_{i,j+1} \bigg) \end{equation*}$

where the index $i$ runs in the $x$ direction and $j$ runs in the $y$ direction. The boundary condition that $\psi = 0$ is enforced on the ellipse at each iteration number $n$ , and the process continues iterating until the solution converges to some reasonable numerical tolerance. The results shown in the figure below seem reasonable enough. Again, notice how the effects of viscosity diminish rapidly away from the body, causing the streamlines to become more parallel to each other.

The numerical solution for Stokes flow about an ellipse.

Stokes Particle Behavior in a Carrier Flow

Suspensions of particles in a carrier flow, while often modeled as continuous media for practical purposes, consist of discrete particles immersed in a continuous fluid, where interactions like hydrodynamic forces and Brownian motion play a role. Depending on the scale and application, this dual-phase nature influences their behavior, requiring a balance between continuum approximations and discrete particle modeling. A classic application of Stokes flow is the steady settling of small, dispersed spherical particles in a carrier fluid, as shown in the schematic below. This phenomenon occurs under three primary forces acting on the particles. The first is the particle’s weight (gravitational force), $F_g$ . The second is the buoyancy force, $F_b$ , which arises from the weight of the fluid displaced by the particle. The third is the viscous drag force, $D_S$ . The balance of these three forces collectively determines the behavior of the settling particles.

The principles of Stokes flow govern the settling out of suspended particles in a carrier media.

When any one particle reaches its terminal velocity $V_t$ , the motion becomes steady, and the forces acting on the particle balance. This force equilibrium can be expressed as

(62) $\begin{equation*} F_g - F_b - D_S = 0 \end{equation*}$

Substituting the expressions for the individual forces, the gravitational force is given by the particle’s weight, $F_g = \varrho_p \frac{4}{3} \pi R^3 g$ , where $\varrho_p$ is the particle’s density, $R$ is its radius, and $g$ is the acceleration under gravity. The buoyancy force exerted by the displaced fluid is $F_b = \varrho_f \dfrac{4}{3} \pi R^3 g$ , where $\varrho_f$ is the density of the fluid. Stokes’ drag law determines the viscous drag force, $D_S = 6 \pi \mu R V_t$ , where $\mu$ is the fluid’s dynamic viscosity. Combining these expressions, the force balance equation becomes

(63) $\begin{equation*} \varrho_p \frac{4}{3} \pi R^3 g - \varrho_f \frac{4}{3} \pi R^3 g - 6 \pi \mu R V_t = 0 \end{equation*}$

Simplifying, the gravitational and buoyancy terms can be grouped, leading to

(64) $\begin{equation*} \frac{4}{3} \pi R^3 (\varrho_p - \varrho_f) g = 6 \pi \mu R V_t \end{equation*}$

Solving for the terminal velocity $V_t$ , the result is

(65) $\begin{equation*} V_t = \frac{2 R^2 (\varrho_p - \varrho_f) g}{9 \mu} \end{equation*}$

This latter expression shows that the terminal velocity, $V_t$ , depends on the particle’s radius, the density difference between the particle and the fluid, the acceleration under gravity, and viscosity. Larger particles or more significant density differences lead to higher terminal velocities, while increased fluid viscosity reduces the settling rate.

The theory of particle settling under Stokes flow provides critical insights into sedimentation processes, particle dynamics in suspensions, and various industrial and environmental applications. It is particularly relevant in systems where Reynolds numbers are sufficiently low for viscous forces to dominate over inertial effects, ensuring that the assumptions of Stokes flow remain valid. Tiny particles in the air have extremely low terminal velocities, meaning they can stay suspended almost indefinitely.

One relevant aerospace application is the problem of rotorcraft “brownout,” where a rotorcraft operating over loose sediment, such as sand, stirs up a large dust cloud, as shown in the figure below. This dust cloud causes many problems, such as a loss of visibility and spatial-optical illusion for the pilot and erosion of the rotor blades and turbine blades in the engines.

The brownout problem for rotorcraft is a suspended cloud of small, dust-like particles, to which the principles of Stokes flow apply.

Hindered Settling in Suspensions

The settling behavior of particles in suspensions depends significantly on the concentration of particles within the fluid. In dilute suspensions, where particle interactions are minimal, the settling time $t_s$ for a particle to descend from an initial height $h$ can be described by the terminal velocity $V_t$ derived from Stokes’ flow. The settling time is given by

(66) $\begin{equation*} t_s = \frac{h}{V_t} = \frac{9 \mu H}{2 R^2 (\varrho_p - \varrho_f) g} \end{equation*}$

where $h$ is the height, $\mu$ is the dynamic viscosity of the fluid, $R$ is the particle radius, $\varrho_p$ and $\varrho_f$ are the densities of the particle and fluid, respectively, and $g$ is the gravitational acceleration. Each particle settles independently in this scenario, and the individual particle properties and fluid characteristics govern the overall dynamics.

In concentrated suspensions, however, the settling behavior is significantly altered due to interactions between particles. These interactions include hydrodynamic effects, collisions, and disturbances in the local flow field, collectively hindering the settling process. As a result, the settling velocity decreases compared to the terminal velocity of a single particle in isolation. This phenomenon is known as hindered settling.

The hindered settling velocity $V_h$ in concentrated suspensions can be described empirically as

(67) $\begin{equation*} V_h = V_t (1 - \phi)^n \end{equation*}$

where $\phi$ is the particle volume fraction, representing the fraction of the suspension volume occupied by the particles, and $n$ is an empirical exponent that depends on the specific suspension system and the nature of particle interactions. The term $(1 - \phi)^n$ captures the reduction in velocity due to the increased particle concentration. As $\phi$ approaches 1, the velocity $V_h$ approaches zero, reflecting the complete suppression of settling in a densely packed system.

The relationship between hindered settling and particle concentration is critical in sedimentation, industrial slurry transport, and wastewater treatment applications. Understanding and modeling hindered settling provide insights into the behavior of particle-laden flows in both natural and engineered systems, helping to optimize processes involving suspensions of varying concentrations.

Stokes Number

The Stokes number, $St$ , is a dimensionless parameter that characterizes a solid particle’s behavior in a carrier flow. It is defined as the ratio of the particle relaxation time to a characteristic flow time scale $T_f$ , such that

(68) $\begin{equation*} St = \frac{\tau_p}{T_f} \end{equation*}$

Here, $T_f$ represents the time scale over which significant changes in the fluid velocity occur, often associated with the most significant flow structures in the system, e.g., vortices. The Stokes number determines how well the particles can follow the fluid flow, which is important in several applications.

For example, Particle Image Velocimetry (PIV) is a widely used experimental technique for analyzing fluid flow by tracking the motion of small particles suspended in the fluid. These particles are assumed to follow the flow, representing the velocity field accurately, but errors can occur. The ability of particles to respond to changes in the fluid motion is governed by the particle relaxation time, $\tau_p$ , which quantifies the time lag for a particle to catch up to the surrounding fluid’s velocity. The relaxation time is given by

(69) $\begin{equation*} \tau_p = \frac{2 \varrho_p R^2}{9 \mu} \end{equation*}$

where $\varrho_p$ is the particle’s density, $R$ is the particle’s radius, and $\mu$ is the dynamic viscosity of the fluid. Smaller particles with lower densities or those in highly viscous fluids exhibit shorter relaxation times. If $St < 1$ , the particle relaxation time is much smaller than the fluid time scale, allowing particles to follow the fluid motion closely. This is ideal for PIV applications, as the particle trajectories faithfully represent the fluid’s velocity field. Conversely, when $St > 1$ , the particle relaxation time is larger than the fluid time scale, causing the particles to lag behind rapid changes in the fluid velocity. In this case, the particles do not accurately track the fluid motion, leading to errors in velocity measurements.

Selecting appropriate particles for PIV experiments is critical to ensuring accurate flow analysis. Particles must be small enough to maintain $St < 1$ but large enough to produce Mie scattering of the laser light illumination. Understanding the interplay between the particle relaxation time, fluid time scales, and the Stokes number is essential for optimizing PIV performance and interpreting the results in complex flow environments.

Taylor–Couette Flow

Taylor–Couette flow refers to the flow of a viscous fluid between two concentric cylinders, where one or both cylinders may rotate. This configuration is a classical problem in fluid mechanics, often studied to understand flow stability, transitions between flow regimes, and turbulence. The behavior of Taylor–Couette flow is determined by the geometry of the cylinders, their relative rotational speeds, and the fluid properties.

The radii of the inner and outer cylinders are denoted by $R_{\text{in}}$ and $R_{\text{out}}$ , respectively, while their rotational speeds are denoted by $\Omega_{\text{in}}$ and $\Omega_{\text{out}}$ . Two dimensionless Reynolds numbers are defined to characterize the contributions of each cylinder to the flow, i.e.,

(70) $\begin{equation*} Re_{\text{in}} = \frac{\Omega_{\text{in}} R_{\text{in}} (R_{\text{out}} - R_{\text{in}})}{\nu} \quad \text{and} \quad Re_{\text{out}} = \frac{\Omega_{\text{out}} R_{\text{out}} (R_{\text{out}} - R_{\text{in}})}{\nu} \end{equation*}$

where $\nu = \mu/\varrho$ is the kinematic viscosity of the fluid. These Reynolds numbers help quantify the relative importance of inertial and viscous forces in the system for each cylinder.

The Taylor number, $Ta$ , is a dimensionless parameter that characterizes the onset of instability in the Taylor–Couette flow. It is defined as

(71) $\begin{equation*} Ta = \frac{4\Omega^2 R_{\text{in}}^2 (R_{\text{out}} - R_{\text{in}})^3}{\nu^2 R_{\text{out}}^2} \end{equation*}$

where $\Omega$ is an effective angular velocity that combines the rotational speeds of the two cylinders. The Taylor number represents the balance between the centrifugal forces that drive instability and the viscous forces that stabilize the flow.

At low Taylor numbers ( $Ta \le 1$ ), the flow is stable and laminar, with the velocity profile purely azimuthal. The flow adheres to a steady, circular motion dictated by the boundary conditions at the cylinder surfaces. As the Taylor number increases and exceeds a critical value $Ta_c$ , the flow becomes unstable due to centrifugal forces. Axisymmetric vortices, known as Taylor vortices, emerge in the annular gap between the cylinders. These vortices appear as toroidal structures stacked along the axis of the cylinders. Further increases in $Ta$ result in more complex flow patterns, including wavy vortices, modulated wavy vortices, and eventually turbulent flow. These transitions depend on the relative rotational speeds of the cylinders and the gap width, i.e., $R_{\rm out} - R_{\rm in}$ .

Calculations of Taylor-Couette flow between two concentric rotating cylinders. The attainment of a critical condition changes the flow from radial streamlines to a series of vortices.

Taylor–Couette flow has numerous engineering, geophysics, and astrophysics applications. It is relevant to mixing processes in chemical reactors, where controlled shear and turbulence enhance reaction rates. In turbulence and nonlinear dynamics research, Taylor–Couette flow provides a well-defined system for studying transitions between laminar flow, vortex structures, and chaotic behavior. In astrophysics, it models accretion disk dynamics, planetary formation, and other phenomena influenced by rotational flows. The study of Taylor–Couette flow remains a cornerstone in fluid mechanics due to its well-defined geometry and the wide variety of flow phenomena it exhibits, from stable laminar motion to fully developed turbulence.

Who were Taylor and Couette?

Maurice Couette (1858–1943) and Geoffrey Taylor (1886–1975) were pioneers in the field of fluid mechanics. Couette, a French physicist, is renowned for describing the motion of viscous fluid between two concentric cylinders or parallel plates and inventing the Couette apparatus to measure fluid viscosity. Taylor, a British physicist, extended the study to analyze the stability and transitions in fluid motion between rotating cylinders. Taylor’s discovery of Taylor vortices and his broader contributions to turbulence, instabilities, and wave propagation solidified his role as one of the most influential figures in the field of fluid dynamics. Specifically, their combined work laid the groundwork for a deeper understanding of complex shear-driven flows.

Summary & Closure

Stokes flow, characterized by the dominance of viscous forces and negligible inertia, plays a crucial role in aerospace, mechanical engineering, and other scientific applications. In aerospace, it is used to model the behavior of fine particles in low-speed airflows, such as dust dispersion caused by helicopter rotor downwash or particulate mobilization in planetary environments, contributing to improved safety and operational efficiency. In mechanical engineering, Stokes flow principles aid in designing lubrication systems, where precise control of viscous forces ensures optimal performance under minimal inertia. The simplified governing equations of Stokes flow also underpin microfluidic technologies, enabling precise fluid manipulation for diagnostics and research and sedimentation processes, where particle settling dynamics are modeled for industrial and natural systems. These diverse applications highlight the critical role in understanding Stokes’s flow for advancing understanding and innovation in mechanical and aerospace engineering alongside other scientific discipline.

5-Question Self-Assessment Quickquiz

For Further Thought or Discussion

Explain the significance of the Reynolds number in distinguishing flows about spheres.
How do the characteristics of viscous-dominated flows differ from inertial-dominated flows?
What are some real-world applications where viscous-dominated flows are found?
How do the Stokes equations simplify the analysis flows at low Reynolds numbers?
Why is time reversibility a characteristic of viscous-dominated flows?
How might non-Newtonian fluids behave in viscous-dominated flow regimes?
How does energy dissipation differ in viscous-dominated versus inertial-dominated flows?

Other Useful Online Resources

To learn more about viscous-dominated flows, take a look at some of these online resources:

Graduate Fluid Mechanics Lesson Series – Stokes Flow Over a Sphere.
Graduate Fluid Mechanics Lesson Series – Observations: Stokes Flow Over a Sphere
A demonstration of the reversibility of Taylor-Couette flow.
Low Reynolds number flows: Lecture on the derivation of the Stokes equation.

Nothing in fluid mechanics is ever easy. ↵
See: Happel, J. and Brenner, H. (1983), "Low Reynolds Number Hydrodynamics: With Special Applications to Particulate Media," Martinus Nijhoff Publishers, Leiden. ↵

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Introduction to Aerospace Flight Vehicles Copyright © 2022–2025 by J. Gordon Leishman is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License, except where otherwise noted.