力学作业（英文版）- (12).pdf

Problem Set 3A Name: ____________________

Assigned: Tuesday, September 18
Due: Tuesday, September 25



1. Recall from Lecture 3 that the Kinematic Transport Theorem (KTT) is:

d ∂f v
(1) f (xv,t)dV = dV + fU (xv,t) ⋅ nˆdS
∫∫∫∫∫∫∂∫∫
dt V (t ) V (t) t S (t)

where V(t) is an arbitrary time-varying control volume, S(t) is its bounding surface,
v
Uxt(,)v is the absolute velocity of the surface S with respect to a fixed frame, and nˆ is
the normal to the surface S pointing out of the volume V.

(a) In many applications, the control volume for a flow system can be chosen as fixed in
space. Choose the property f to be the mass per unit volume, or density ρ(,)xv t . Denote
the fixed control volume as VC and the corresponding control surface as SC . Write the
simplified KTT for this case, leaving it in terms of general integrals as above:




(b) Now assume the flow is also steady. This further simplifies the KTT to the following:




(c) Now, start again with equation (1) but let the control volume be a material volume
Vtm ()moving with the fluid and having a material surface Stm ( ) whose points move with
v
the fluid velocity Vxt (v , ) . Write equation (1) for unsteady flow, again leaving it in terms
of general integrals:




(d) Why must the right and left-hand sides of the expression you deriv