Page 326 - Intro to Tensor Calculus

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Figure 2.5-7. Plane Poiseuille flow

with u = u(y)and v = w = 0 with boundary values u(0) = u(2`)= 0. The density is constant and ν = µ /%
∗
is the kinematic viscosity.
2
d u p 0
(a) Show the equation of fluid motion is ν + =0, u(0) = u(2`)=0
dy 2 %
(b) Find the velocity u = u(y) and find the maximum velocity in the x-direction. (c) Let M denote the
mass flow rate across the plane x = x 0 = constant, ,where 0 ≤ y ≤ 2`, and 0 ≤ z ≤ 1.
2 3
Show that M = %p 0 ` . Note that as µ increases, M decreases.
∗
3µ ∗
∂(δcu)
I 16. The heat equation (or diffusion equation) can be expressed div ( k grad u)+H = , where c is the
∂t
3
3
specific heat [cal/gm C], δ is the volume density [gm/cm ], H is the rate of heat generation [cal/sec cm ], u
is the temperature [C], k is the thermal conductivity [cal/sec cm C]. Assume constant thermal conductivity,
volume density and specific heat and express the boundary value problem
2
∂ u ∂u
k = δc , 0 <x <L
∂x 2 ∂t
u(0,t)=0, u(L, t)= u 1 , u(x, 0) = f(x)
in a form where all the variables are dimensionless. Assume u 1 is constant.

I 17. Simplify the Navier-Stokes-Duhem equations using the assumption that there is incompressible flow.

I 18. (Rayleigh impulsive flow) The figure 2.5-8 illustrates fluid motion in the plane where y> 0above a
plate located along the axis where y =0. The plate along y = 0 has zero velocity for all negative time and
at time t = 0 the plate is given an instantaneous velocity u 0 in the positive x-direction. Assume the physical
components of the velocity are ~v =(u, v, w) which satisfy u = u(y, t),v = w =0. Assume that the density
of the fluid is constant, the gradient of the pressure is zero, and the body forces are zero. (a) Show that the
velocity in the x-direction is governed by the differential equation
2
∂u ∂ u µ ∗
= ν , with ν = .
∂t ∂y 2 %
Assume u satisfies the initial condition u(0,t)= u 0 H(t)where H is the Heaviside step function. Also assume
there exist a condition at infinity lim y→∞ u(y, t). This latter condition requires a bounded velocity at infinity.
(b) Use any method to show the velocity is

y y
u(y, t)= u 0 − u 0 erf √ = u 0 erfc √
2 νt 2 νt