Page 327 - Intro to Tensor Calculus

321

Figure 2.5-8. Rayleigh impulsive flow

where erf and erfc are the error function and complimentary error function respectively. Pick a point on the
√
line y = y 0 =2 ν and plot the velocity as a function of time. How does the viscosity effect the velocity of
the fluid along the line y = y 0 ?

I 19. Simplify the Navier-Stokes-Duhem equations using the assumption that there is incompressible and
irrotational flow.
2
I 20. Let ζ = λ + µ and show the constitutive equations (2.5.21) for fluid motion can be written in the
∗
∗
3
form
2
σ ij = −pδ ij + µ ∗ v i,j + v j,i − δ ij v k,k + ζδ ij v k,k .
3
I 21. (a) Write out the Navier-Stokes-Duhem equation for two dimensional flow in the x-y direction under
the assumptions that
2
∗
∗
• λ + µ = 0 (This condition is referred to as Stoke’s flow.)
3
• The fluid is incompressible
~
• There is a gravitational force b = −g∇ h Hint: Express your answer as two scalar equations
involving the variables v 1 ,v 2 ,h,g,%,p,t,µ plus the continuity equation. (b) In part (a) eliminate
∗
the pressure and body force terms by cross differentiation and subtraction. (i.e. take the derivative
of one equation with respect to x and take the derivative of the other equation with respect to y

1 ∂v 2 ∂v 1
and then eliminate any common terms.) (c) Assume that ~ω = ω ˆ 3 where ω = − and
e
2 ∂x ∂y
derive the vorticity-transport equation
dω 2 dω ∂ω ∂ω ∂ω
= ν∇ ω where = + v 1 + v 2 .
dt dt ∂t ∂x ∂y
Hint: The continuity equation makes certain terms zero. (d) Define a stream function ψ = ψ(x, y)
∂ψ ∂ψ
satisfying v 1 = and v 2 = − and show the continuity equation is identically satisfied.
∂y ∂x
2
1
Show also that ω = − ∇ ψ and that
2
2 2 2
1 ∂∇ ψ ∂ψ ∂∇ ψ ∂ψ ∂∇ ψ
4
∇ ψ = + − .
ν ∂t ∂y ∂x ∂x ∂y
4
If ν is very large, show that ∇ ψ ≈ 0.