8.1 Stokes flow equation [Stokes flow top]

We derive the Stokes flow equations by neglecting the unsteady and convective terms when the Reynolds number is low. The Navier-Stokes equations are:

(8.1)

and, rewritten in nondimensional form where the pressure is normalized by ηU∕ℓ, are given by

(8.2)

where Re is the Reynolds number, defined by Re = ρUℓ∕η, η and ρ are fluid properties in the governing equations, and U and ℓ come from the boundary conditions. This nondimensionalization is shown in more detail in the Appendix, Section E.2.1. If Re → 0, the unsteady and convective terms can be neglected.

The Stokes flow approximation is valid for Re ≪ 1, in which case we neglect the unsteady and convective terms, leaving

these are the Stokes flow equations. These equations are linear in both the velocity and the pressure, and they are much easier to solve than the Navier-Stokes equations. Because the unsteady term can be neglected, Stokes flows have the propertiesof instantaneity (no dependence on time except through time-dependent boundary conditions) andtime-reversibility (a time-reversed Stokes flow solves the Stokes equations). Time-reversibility also implies that Stokes flow around a symmetric body exhibits fore-aft symmetry. Because the nonlinear term can be neglected, Stokes flows also have the property of superposability both for the pressure and for the velocity.

All real flows have a finite Reynolds number, and so the Stokes flow equations are only an approximation of the real flows. While the Reynolds number cutoff depends on the flow, a good rule of thumb is that the Stokes flow solution is a good approximation when Re < 0.1.

8.1.1 Different forms of the Stokes flow equations

Equation 8.3 is a function of both the pressure and the velocity, which is sometimes inconvenient. Fortunately, we can convert the Stokes flow equations to two other forms, which are each a function of the pressure field or the velocity field exclusively. Taking the divergence of the Stokes flow equations results in

(8.4)

The divergence of the Laplacian of  is equal to the Laplacian of the divergence of  (Equation C.85), so

(8.5)

and, since the flow is incompressible, ∇⋅ = 0 and thus

For Stokes flow, the pressure satisfies Laplace’s equation. This form of the Stokes flow equations is useful if the boundary conditions are specified exclusively in terms of pressure. Alternately, we can take the curl of the Stokes flow equations, giving

(8.7)

which eliminates the left hand side because the curl of the gradient of a vector field is zero. The curl of the Laplacian is equal to the Laplacian of the curl (Equation C.86), so

This version of the Stokes flow equations is most useful if the boundary conditions are expressed exclusively in terms of velocities.

The Stokes flow equations can also be written in terms of the stream function. For a 2D flow with plane symmetry, the Stokes equations take the form

where ψ is defined by Equations ??–??. This is termedthe biharmonic equation and ∇⁴ is the biharmonic operator. For axisymmetric flow, the equation for the Stokes stream function (defined in Equations ??–??) can be written as

where E² here is a symbol for the second-order differential operator +. This operator plays a role similar to the Laplacian operator, but is slightly different owing to the use of curvilinear coordinates.

8.1.2 Analytical vs. numerical solutions of the Stokes flow equations

The Stokesflow equations are linear and simple to solve numerically. Thus, when studying low-Re flow for a complicated geometry, practicing engineers typically solve the Stokes equations numerically using any of a variety of numerical codes. We can also solve the Stokes equations analytically for several model problems in a way that is immediately useful for study of flows in microdevices. Sections 8.2 and 8.3 consider analytical solutions for flow in shallow channels and over spheres.

Some examples of our research where Stokes flow analysis is relevant include our circulating tumor cell capture microchips and our dielectrophoretic manipulation of microparticles.

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