Couette Flow - Couette Flow With Pressure Gradient

Couette Flow With Pressure Gradient

A more general Couette flow situation arises when a pressure gradient is imposed in a direction parallel to the plates. The Navier–Stokes equations, in this case, simplify to

\frac{d^2 u}{d y^2} = \frac{1}{\mu} \frac{dp}{dx},

where is the pressure gradient parallel to the plates and is fluid viscosity. Integrating the above equation twice and applying the boundary conditions (same as in the case of Couette flow without pressure gradient) to yield the following exact solution

u (y) = u_0\frac{y}{h} + \frac{1}{2\mu} \left(\frac{dp}{dx}\right) \left(y^2 - hy\right).

The shape of the above velocity profile depends on the dimensionless parameter

P = - \frac{h^2}{2\mu u_0} \left(\frac{dp}{dx}\right).

The pressure gradient can be positive (adverse pressure gradient) or negative (favorable pressure gradient).

It may be noted that in the limiting case of stationary plates, the flow is referred to as plane Poiseuille flow with a symmetric (with reference to the horizontal mid-plane) parabolic velocity profile.

Read more about this topic:  Couette Flow

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