Page 248 - Intro to Tensor Calculus

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I 22. Show that for an orthogonal coordinate system the ith component of the convective operator can be
written
3 3
X V (m) ∂A(i) X A(m) ∂h i ∂h m
~
~
[(V ·∇) A] i = m + V (i) m − V (m) i
h m ∂x h mh i ∂x ∂x
m=1 m=1
m6=i
I 23. Consider a parallelepiped with dimensions `, w, h which has a uniform pressure P applied to each
face. Show that the volume strain can be expressed as
∆V ∆` ∆w ∆h −3P(1 − 2ν)
= + + = .
V ` w h E

The quantity k = E/3(1 − 2ν) is called the bulk modulus of elasticity.
I 24. Show in Cartesian coordinates the continuity equation is

∂% ∂(%u) ∂(%v) ∂(%w)
+ + + =0,
∂t ∂x ∂y ∂z
where (u, v, w) are the velocity components.

I 25. Show in cylindrical coordinates the continuity equation is

∂% 1 ∂(r%V r ) 1 ∂(%V θ ) ∂(%V z )
+ + + =0
∂t r ∂r r ∂θ ∂z
where V r ,V θ ,V z are the velocity components.

I 26. Show in spherical coordinates the continuity equation is
2
∂% 1 ∂(ρ %V ρ ) 1 ∂(%V θ sin θ) 1 ∂(%V φ )
+ + + =0
∂t ρ 2 ∂ρ ρ sin θ ∂θ ρ sin θ ∂φ
where V ρ ,V θ ,V φ are the velocity components.

I 27. (a) Apply a stress σ yy to both ends of a square element in a x, y continuum. Illustrate and label
all changes that occur due to this stress. (b) Apply a stress σ xx to both ends of a square element in a
x, y continuum. Illustrate and label all changes that occur due to this stress. (c) Use superposition of your
results in parts (a) and (b) and explain each term in the relations

σ xx σ yy σ yy σ xx
e xx = − ν and e yy = − ν .
E E E E

x, y, z dJ
~
I 28. Show that the time derivative of the Jacobian J = J satisfies = J div V where
X, Y, Z dt
~
div V = ∂V 1 + ∂V 2 + ∂V 3 and V 1 = dx , V 2 = dy , V 3 = dz .
∂x ∂y ∂z dt dt dt
Hint: Let (x, y, z)= (x 1 ,x 2 ,x 3 )and (X, Y, Z)=(X 1 ,X 2 ,X 3), then note that

∂V 1 ∂x 2 ∂x 3 ∂V 1 ∂x m ∂x 2 ∂x 3 ∂x 1 ∂x 2 ∂x 3 ∂V 1
e ijk = e ijk = e ijk , etc.
∂X i ∂X j ∂X k ∂x m ∂X i ∂X j ∂X k ∂X i ∂X j ∂X k ∂x 1