9-104 Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between two infi- nitely long and wide, parallel flat plates (Fig. P9-104). The flow is steady, incompressible, parallel, and laminar. The top plate moves at velocity V to the right, and the bottom plate is stationary. Gravity acts in the -z-direction (downward in the figure). There is no forced pressure gradient pushing the flu- ids through the channel-the flow is set up solely by viscous effects created by the moving upper plate. You may ignore surface tension effects and assume that the interface is hori- zontal. The pressure at the bottom of the flow (z = 0) is equal to P- (a) List all the appropriate boundary conditions on both velocity and pressure. (Hint: There are six required boundary conditions.) (b) Solve for the velocity field. (Hint: Split up the solution into two portions, one for each fluid. Generate expressions for u, as a function of z and u, as a function of z.) (c) Solve for the pressure field. (Hint: Again split up the solution. Solve for P, and P) (d) Let fluid 1 be water and let fluid 2 be unused engine oil, both at 80°C.

9-104 Consider a modified form of Couette flow in which there are two immiscible fluids sandwiched between two infi- nitely long and wide, parallel flat plates (Fig. P9-104). The flow is steady, incompressible, parallel, and laminar. The top plate moves at velocity V to the right, and the bottom plate is stationary. Gravity acts in the -z-direction (downward in the figure). There is no forced pressure gradient pushing the flu- ids through the channel-the flow is set up solely by viscous effects created by the moving upper plate. You may ignore surface tension effects and assume that the interface is hori- zontal. The pressure at the bottom of the flow (z = 0) is equal to P- (a) List all the appropriate boundary conditions on both velocity and pressure. (Hint: There are six required boundary conditions.) (b) Solve for the velocity field. (Hint: Split up the solution into two portions, one for each fluid. Generate expressions for u, as a function of z and u, as a function of z.) (c) Solve for the pressure field. (Hint: Again split up the solution. Solve for P, and P) (d) Let fluid 1 be water and let fluid 2 be unused engine oil, both at 80°C.

Structural Analysis

6th Edition

ISBN:9781337630931

Author:KASSIMALI, Aslam.

Publisher:KASSIMALI, Aslam.

Chapter2: Loads On Structures

Section: Chapter Questions

Problem 1P

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Question

9-104 Consider a modified form of Couette flow in which
there are two immiscible fluids sandwiched between two infi-
nitely long and wide, parallel flat plates (Fig. P9–104). The
flow is steady, incompressible, parallel, and laminar. The top
plate moves at velocity V to the right, and the bottom plate is
stationary. Gravity acts in the -z-direction (downward in the
figure). There is no forced pressure gradient pushing the flu-
ids through the channel–the flow is set up solely by viscous
effects created by the moving upper plate. You may ignore
surface tension effects and assume that the interface is hori-
zontal. The pressure at the bottom of the flow (z = 0) is
equal to Pg. (a) List all the appropriate boundary conditions
on both velocity and pressure. (Hint: There are six required
boundary conditions.) (b) Solve for the velocity field. (Hint:
Split up the solution into two portions, one for each fluid.
Generate expressions for u, as a function of z and uz as a
function of z.) (c) Solve for the pressure field. (Hint: Again
split up the solution. Solve for P, and P2.) (d) Let fluid 1
be water and let fluid 2 be unused engine oil, both at 80°C.
Also let h, = 5.0 mm, h, = 8.0 mm, and V = 10.0 m/s. Plot u
as a function of z across the entire channel. Discuss the
results.
Moving wall
Interface
Fluid 2
P2. 42
Fluid 1
FIGURE P9-104

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