Couette flow
The relative motion of the surfaces imposes a shear stress on the fluid and induces flow.
Depending on the definition of the term, there may also be an applied pressure gradient in the flow direction.
The Couette configuration models certain practical problems, like the Earth's mantle and atmosphere,[1] and flow in lightly loaded journal bearings.
[2][3] It is named after Maurice Couette, a Professor of Physics at the French University of Angers in the late 19th century.
Couette flow is frequently used in undergraduate physics and engineering courses to illustrate shear-driven fluid motion.
A simple configuration corresponds to two infinite, parallel plates separated by a distance
This equation reflects the assumption that the flow is unidirectional — that is, only one of the three velocity components
The exact solution can be found by integrating twice and solving for the constants using the boundary conditions.
A notable aspect of the flow is that shear stress is constant throughout the domain.
Then, applying separation of variables leads to the solution:[4] The timescale describing relaxation to steady state is
The time required to reach the steady state depends only on the spacing between the plates
A more general Couette flow includes a constant pressure gradient
[5] In incompressible flow, the velocity profile is linear because the fluid temperature is constant.
When the upper and lower walls are maintained at different temperatures, the velocity profile is more complicated.
Denote fluid properties at the lower wall with subscript
The properties and the pressure at the upper wall are prescribed and taken as reference quantities.
Introduce the non-dimensional variables In terms of these quantities, the solutions are where
[clarification needed] Then the solution is If the specific heat is constant, then
are constant everywhere, thus recovering the incompressible Couette flow solution.
that is both accurate and general, there are several approximations for certain materials — see, e.g., temperature dependence of viscosity.
is not constant) have also been studied; in that case the recovery temperature is reduced by the dissociation of molecules.
However, the infinite length in the streamwise direction must be retained in order to ensure the unidirectional nature of the flow.
As an example, consider an infinitely long rectangular channel with transverse height
, subject to the condition that the top wall moves with a constant velocity
Without an imposed pressure gradient, the Navier–Stokes equations reduce to with boundary conditions Using separation of variables, the solution is given by When
, the planar Couette flow is recovered, as shown in the figure.
[8] The original problem was solved by Stokes in 1845,[9] but Geoffrey Ingram Taylor's name was attached to the flow because he studied its stability in a famous 1923 paper.
Assuming the cylinders rotate at constant angular velocities
-direction is[11] This equation shows that the effects of curvature no longer allow for constant shear in the flow domain.
, the finite-length problem can be solved using separation of variables or integral transforms, giving:[12] where
