力学作业（英文版）- (3).pdf

Challenge Problem 3 (OPTIONAL) Name: ____________________

Assigned: Sunday, December 2
Due: Tuesday, December 11


1. Consider a real fluid (ν≠ 0 ) of constant density. Use the Navier-Stokes Equations to
show that
ddΓ JK KK KK K
= Vdl⋅= −∇××()ων V⋅ ndSˆ+ ∇⋅2 Vdl (1)
dt dt ∫∫∫∫vvCA C

where C is a closed curveKK fixed in space K and A KKis any surface bounded by C.
K 1
Hint: Use the identity ()VV⋅∇=×+ω V∇⋅(2 VV ).

2. A real, constant density fluid flows over a thin flat plate placed in a free stream of
K
velocity VUi= ˆ, creating a boundary layer. Assume that the flow is steady. Consider a
rectangular curve fixed with respect to the plate and boundary layer as shown:
curve C
y

boundary
layer
p0 h


p1 p2
U x
xpss, x1 x2


The point xs at the leading edge of the plate is a stagnation point where the stagnation
pressure is ps . The pressure at point x1 is p1 . Point x2 is assumed to be far enough
downstream in the boundary layer that p2 equals the free stream pressure p0 .

(a) Use equation (1) in Problem 1 to show that

hh
p2
= ()udyudyζζxx==−() xx (2)
∫∫0021
p1

K K
where Vuivj=+ˆˆ and ωζ= kˆ for two-dimensional flow.

Hint: Evaluate the Navier-Stokes equations at y = 0 on the plate to obtain a relationship
between the pressure gradient and velocity gradient.