Question

Consider the steady, laminar flow of two liquids, A and B, with viscosities HA-μ and μΒ 21, respectively, between infinite parallel plates at 2- a, as shown in the diagram below. The plate at 2 a is fixed, while the plate at 2a moves with constant velocity -Vi, where V0. The liquids do not mix, and each forms a layer of depth a. There is an applied pressure gradient acting on both liquids, given by ▽p--Ci (where C > 0 is constant), and the effects of gravitation can be assumed to be negligible. pressure gradient liquid A pressure gradient liquid B (a) Assuming that each fluid is of constant viscosity and is Newtonian, what other assumption must be made about the fluids in order to apply Equations (2.8) on page 179 of Unit 8? Given the problem statement, why is this additional assumption reasonable? (b) The following assumptions are to be made. Write down the mathematical consequences of each of them. 1. The flow is two-dimensional. 2. There is no variation in the direction into the page. 3. The flow is steady . There is no variation of velocity parallel to the plates. Hence write down the continuity equation for cither of the two liquids. (c) Show that the fluid velocities uin liquid A and u in liquid B are given by State the boundary conditions at the upper and lower plates, and at the interface z0. (Note that the shear stress must vary continuously between the plates.) (d) Write down the r-components of the Navier-Stokes equations for uA and uB. Solve them, using the boundary conditions from part (c), to show that UA (2)--(3:2-(1 + 47)az-2(1-27)a2) , 6AL uB(z)-- (3:2 _ (1 + 47)a-4(1-27)a*) , 120 where γ = μν/(Ca2)

Answer 1

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