Page 323 - Intro to Tensor Calculus

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EXERCISE 2.5

2
I 1. Let p = p(x, y, z), [dyne/cm ] denote the pressure at a point (x, y, z) in a fluid medium at rest
(hydrostatics), and let ∆V denote an element of fluid volume situated at this point as illustrated in the
figure 2.5-5.

Figure 2.5-5. Pressure acting on a volume element.

(a) Show that the force acting on the face ABCD is p(x, y, z)∆y∆z ˆ 1 .
e
(b) Show that the force acting on the face EFGH is

2 2
∂p ∂ p (∆x)
e
−p(x +∆x, y, z)∆y∆z ˆ 1 = − p(x, y, z)+ ∆x + 2 + ··· ∆y∆z ˆ 1 .
e
∂x ∂x 2!
(c) In part (b) neglect terms with powers of ∆x greater than or equal to 2 and show that the resultant force
∂p
e
in the x-direction is − ∆x∆y∆z ˆ 1 .
∂x
(d) What has been done in the x-direction can also be done in the y and z-directions. Show that the
∂p ∂p
e
e
resultant forces in these directions are − ∆x∆y∆z ˆ 2 and − ∆x∆y∆z ˆ 3. (e) Show that −∇p =
∂y ∂z

∂p ∂p ∂p
− ˆ e 1 + ˆ e 2 + ˆ e 3 is the force per unit volume acting at the point (x, y, z) of the fluid medium.
∂x ∂y ∂z
I 2. Follow the example of exercise 1 above but use cylindrical coordinates and find the force per unit volume
at a point (r, θ, z). Hint: An element of volume in cylindrical coordinates is given by ∆V = r∆r∆θ∆z.
I 3. Follow the example of exercise 1 above but use spherical coordinates and find the force per unit volume
2
at a point (ρ, θ, φ). Hint: An element of volume in spherical coordinates is ∆V = ρ sin θ∆ρ∆θ∆φ.
I 4. Show that if the density % = %(x, y, z, t) is a constant, then v r ,r =0.
I 5. Assume that λ and µ are zero. Such a fluid is called a nonviscous or perfect fluid. (a) Show the
∗
∗
Cartesian equations describing conservation of linear momentum are
∂u ∂u ∂u ∂u 1 ∂p
+ u + v + w = b x −
∂t ∂x ∂y ∂z % ∂x
∂v ∂v ∂v ∂v 1 ∂p
+ u + v + w = b y −
∂t ∂x ∂y ∂z % ∂y
∂w ∂w ∂w ∂w 1 ∂p
+ u + v + w = b z −
∂t ∂x ∂y ∂z % ∂z

where (u, v, w) are the physical components of the fluid velocity. (b) Show that the continuity equation can
be written
∂% ∂ ∂ ∂
+ (%u)+ (%v)+ (%w)= 0
∂t ∂x ∂y ∂z