Page 325 - Intro to Tensor Calculus

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Figure 2.5-6. Plane Couette flow

I 11. Find the physical components of the rate of deformation tensor D ij in Cartesian coordinates. (Hint:
See problem 10.)

I 12. Find the physical components of the rate of deformation tensor in cylindrical coordinates. (Hint: See
problem 10.)

I 13. (Plane Couette flow) Assume a viscous fluid with constant density is between two plates as illustrated
in the figure 2.5-6.
(a) Define ν = µ ∗ as the kinematic viscosity and show the equations of fluid motion can be written
%
∂v i i s 1 im jm i ij
+ v v = − g p ,m + νg v ,mj + g b j , i =1, 2, 3
,s
∂t %
(b) Let ~v =(u, v, w) denote the physical components of the fluid flow and make the following assumptions
• u = u(y),v = w =0
• Steady state flow exists
• The top plate, with area A, is a distance ` above the bottom plate. The bottom plate is fixed and
a constant force F is applied to the top plate to keep it moving with a velocity u 0 = u(`).
• p and % are constants
• The body force components are zero.
Find the velocity u = u(y)
F u 0
(c) Show the tangential stress exerted by the moving fluid is = σ 21 = σ xy = σ yx = µ ∗ . This
A `
example illustrates that the stress is proportional to u 0 and inversely proportional to `.

I 14. In the continuity equation make the change of variables
t % ~v x y z
t = , % = , ~ v = , x = , y = , z =
τ % 0 v 0 L L L
and write the continuity equation in terms of the barred variables and the Strouhal parameter.

I 15. (Plane Poiseuille flow) Consider two flat plates parallel to one another as illustrated in the figure
2.5-7. One plate is at y = 0 and the other plate is at y =2`. Let ~v =(u, v, w) denote the physical components
of the fluid velocity and make the following assumptions concerning the flow The body forces are zero. The
∂p ∂p ∂p
derivative = −p 0 is a constant and = =0. The velocity in the x-direction is a function of y only
∂x ∂y ∂z