A viscous incompressible fluid is between two parallel plates (that are along the x-z plane), one...

A viscous incompressible fluid is between two parallel plates (that are along the x-z plane), one of them is at y = 0 and the other at y = d.

a) Considerthecasewheretheflowisstationaryinthezˆdirection, driven by a pressure gradient. Write down the Navier-Stokes equations and solve for the velocity field after imposing the relevant boundary conditions. [10 points]
b) The pressure gradient is now removed, but the plate at y = d is moved in an oscillatory fashion with velocity ue−iωt in the zˆ direction, while the other plate remains stationary. Write down the Navier-Stokes equations and solve for the velocity field after imposing the relevant boundary conditions. [15 points]

Solutions

Expert Solution

A). Consider the flow of a viscous Newtonian fluid between two parallel plates located at Z = 0 and Z = d. The upper plane is moving with velocity U. Calculate the flow field.

Assume the following: Steady flow:

Parallel, fully-developed flow:

Two-dimensional flow:

No pressure gradient:

The streamwise Navier-Stokes equation is

can be simplified using the above assumptions. We get

With the boundary conditions

u(0) = 0 ⇒ B = 0, u(d) = U ⇒ A = U/d

we finally obtain

u(y) = Uy/d

genius_generous answered 1 week ago

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