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Newtonian Viscous Fluid 377

the gradient of (p+pgy) where y is the vertical height measured from some reference datum,
and the piezometric head (p/pg+y) is a constant along any direction perpendicular to the
flow, [see Example 6.7.2].

6.14 Plane Couette Flow of Two Layers of Incompressible Fluids
Let the viscosity and the density of the top layer be^i andpj and those of the bottom layer
=
be/^ andp2- Let JCj be the direction of flow and let JC2 0 be the interface. We look for steady
unidirectional flows of the two layers between the infinite plates x.^ = +&iand*2 — ~&2- Th e
plateX2 = —bi is fixed and the plate^ = +b\ is moving on its own plane with velocity v 0. The
pressure gradient in the flow direction is assumed to be zero. (Fig. 6.9).
Let the velocity distribution in the top layer be

and that in the bottom layer be

the equation of continuity is clearly satisfied for each layer. The Navier-Stokes equations give:

For layer 1,

For layer 2,