We derive the Stokes flow equations by neglecting the unsteady and convective terms when the Reynolds number is
low. The Navier-Stokes equations are:
and, rewritten in nondimensional form where the pressure is normalized by ηU∕ℓ, are given by
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
The Stokes flow approximation is valid for Re≪ 1, in which case we neglect the unsteady and convective terms,
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
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
VIDEO: Different forms of the Stokes 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
The divergence of the Laplacian of is equal to the Laplacian of the divergence of (Equation C.85),
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
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
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
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.