1. Determine the material derivative of the flux of any vector property Q: through the spatial area S. Specifically, show that
dta
(o: + a Via - ava)nidS
in agreement with Eq 5.2-5.
2. Let the property P;.. in Eq 5.2-1 be the scalar 1 so that the integral in that equation represents the instantaneous volume V. Show that in this case
d
Pei = Tit = Iv udV
3. Verify the identity
. a . =2(141.+ w.v . . - w .v..)
qk k,j t,j
and, by using this identity as well as the result of Problem 5.1, prove that the material derivative of the vorticity flux equals one half the flux of the curl of the acceleration; that is, show that
-d w.n.dS = is E.. akj .nidS
dt s " 2 s
4. Making use of the divergence theorem of Gauss together with the identity
caw _1
at 2 Eakj - Eijk£kmq(Wmi) 9).i
5. Show that the material derivative of the vorticity of the material contained in a volume V is given by
dt SvividV = (2 soak + wivi)nidS
6. Given the velocity field
= ax1 - bx2, v2 = bx, + ax2, v3 = CVX2 + X2
7. Using the identity
E.. a (
k i =2 w.. . - w 1 1 .v. .)
,1
as well as the continuity equation, show that
d w, Eijkaki + 3/4 11)1,
dt p 2p
8. For a certain contiuum at rest, the stress is given by
aq = -Po(5ei
where po is a constant. Use the continuity equation to show that for this case the stress power may be expressed as
o-Y-.D.- = P°P
9. Consider the motion xi = (1 + t / k)X; where k is a constant. From the conservation of mass and the initial condition p = po at t = 0, determine p as a function of po, t, and k.
ok3
Answer: p = P
(k+
10. By combining Eqs 5.3-10b and 5.3-6, verify the result presented in Eq 4.11-6.