2.3 Summary [Couette/Poiseuille top]

The Navier-Stokes equations can be solved analytically if certain simplifications were made. In particular, the convection term ⋅∇ is the most mathematically difficult term in the Navier-Stokes equations to handle, and this chapter examines flows for which the geometry was so simple that we can assume away any spatial dependence that leads to nonzero net convective flux. This approach leads to steady solutions for Couette flow, i.e., flow between two infinite parallel plates:

and Hagen-Poiseuille flow, i.e., pressure-driven flow through a circular channel:

The startup of these flows highlights differences between forces, velocities and accelerations. During startup, the acceleration is proportional to the force, which is a sum of the pressure and viscous forces. At equilibrium, the acceleration is by definition zero, and the concavity of the velocity distribution (which is proportional to the viscous force) is proportional to the local pressure gradient. Thus steady Couette flow, which has no pressure gradient, has no concavity in the velocity distribution, while steady Poiseuille flow, which has a uniform pressure gradient, has uniform concavity in the velocity profile. Development of these flows includes nonzero convective terms, which are nonlinear and preclude general analytical solution.

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