力学作业（英文版）- (2).pdf

Challenge Problem 2 (OPTIONAL) Name: ____________________

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


Consider steady flow in a circular tube whose radius ax ( ) varies slowly with x along the
centerline. A constant pressure difference is maintained between the two ends of the tube,
dp
and the axial pressure gradient = −Gx()also varies slowly with x. The flow is
K dx
axisymmetric so that Vvv= (xr , ) . Because ax ( ) and Gx ( ) vary slowly, the local flow
near some point x in the tube is approximately described by Poiseuille flow and the axial
ponent is given by

Gx()
vxr(,)= (() ax22− r )
x 4µ

where r is the radial distance from the tube centerline.

1. What terms must be neglected in the full Navier-Stokes equation for vx in order to
obtain this approximate solution?

2. What is the constant volume flux Q along the tube in terms of Gx ( ) and ax ( ) ?

3. Using your answer for Q, show tha