BE 110
An Introduction to Biomechanics
Fall 2005
Homework Assignment #8
Due THURSDAY 12/01/05
, 8AM before class
1. Practice Problems

Show that the material derivative of a volume integral
D
Dt
A
(
x
,
y
,
z
,
t
)
dV
V
(
t
)
∫
=
dA
dt
+
A
∂
v
i
∂
x
i
⎛
⎝
⎜
⎞
⎠
⎟
dV
V
(
t
)
∫
where A(x,y,z,t) is a continuously differentiable function.
Note that this relationship can be
applied to different physical principles, such as conservation of mass (in which case A =
ρ
the
mass density), or conservation of momentum (in which case A needs to be replaced by the vector
component A
i
=
ρ
v
i
).
See section 10.4. on page 215217.

The material derivative can be used to find the boundary condition for an object suspended in a
fluid.
This is a useful theorem for analysis of particle motions.
See problem 10.7. and 10.8.

Show that the permutation symbol is an isotropic tensor, i.e. the tensor components remain the
same under a permissible orthogonal transformation.
A permissible transformation refers to the
fact that the orientation of the axis system does not change during the transformation, i.e. a right
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 Spring '08
 SCHMIDSCHOENBEIN
 Fluid Dynamics, Derivative, Shear Stress, velocity profile vz, homogeneous biological fluid, homogeneous biological solids

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