Flow Between Fixed Plates with Upper Plate Moving

David Naylor

Problem Definition

Consider steady laminar incompressible viscous flow between parallel plates a distance $h$ apart.

Flow driven by upper plate moving at velocity $V.$ No pressure gradient, $\frac{\text{d}\text{e}\text{l}\text{p}}{\text{d}\text{e}\text{l}\text{x}}=0.$

Plates are very wide and long. So$,$ the flow is purely axial: $v=w=0.$ Neglect gravity effects.

Continuity Equation: $($incompressible$)$

$\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}+\frac{\text{d}\text{e}\text{l}\text{v}}{\text{d}\text{e}\text{l}\text{y}}+\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{z}}=0\frac{\text{d}\text{e}\text{l}\text{u}}{\text{d}\text{e}\text{l}\text{x}}=0$

So$,$ the flow is fully developed.