EXERCISES 1.5

1.5.1. Let c(x,y,z,t) denote the concentration of a pollutant (the amount per unit
volume).

(a) What is an expression for the total amount of pollutant in the region
R?
(b) Suppose that the flow J of the pollutant is proportional to the gradient
of the concentration. (Is this reasonable?) Express conservation of the
pollutant.

(c) Derive the partial differential equation governing the diffusion of the
pollutant.

1.5.2. For conduction of thermal energy, the heat flux vector is phi = - K0 grad(u). If
in addition the molecules move at an average velocity V, a process called
convection, then briefly explain why phi = - K0 grad(u) + c rho u V. Derive the
corresponding equation for heat flow, including both conduction and convection
of thermal energy (assuming constant thermal properties with no
sources).

1.5.3. Consider the polar coordinates
x = r cos(theta)
y = r sin(theta)

(a) Since r^2 = x^2 + y^2, show that dr/dx = cos(theta), dr/dy = sin(theta), 
(d/dy) theta = (1/r) cos(theta) and (d/dx) theta = (-1/r) sin(theta).

(b) Show that vec(r) = cos(theta) vec(i) + sin(theta) vec(j) and vec(theta) = - sin(theta) vec(i) + cos(theta) vec(j) are orthogonal unit vectors.

(c) Using the chain rule, show that grad(u) = vec(r) (du/dr) + \vec(theta) (1/r) (du/d theta).

(d) If A = C vec(r) + D vec(theta), then show that grad.A = (1/r)(d/dr)(r C) + (1/r)(d/d theta)(D), since (d/d theta) vec(r) = vec(theta) and (d/d theta) vec(theta)= - vec(r) follow from part (b).

(e) Show that grad^2(u) = (1/r)(d/dr)(r(du/dr)) + (1/r^2)(d/d theta)^2(u).

1.5.4. Using Exercise 1.5.3(a) and the chain rule for partial derivatives, derive the
special case of Exercise 1.5.3(e) when u(r,theta) depends only on r, i.e., it is independent of theta.

1.5.5. Assume that the temperature is circularly symmetric: u = u(r,t), where
r^2 = x^2 + y^2. We will derive the heat equation for this problem. Consider
any circular annulus a <= r <= b.
(a) Show that the total heat energy is 2 Pi int(c rho u r, r=a..b).

(b) Show that the flow of heat energy per unit time out of the annulus at
r = b is - 2 Pi b K0 (du/dr)(0,t). A similar result holds at r = a.

(c) Use parts (a) and (b) to derive the circularly symmetric heat equation
without sources:

du/dt = (k/r)(d/dr)(r du/dr)

1.5.6. Modify Exercise 1.5.5 if the thermal properties depend on r.

1.5.7. Derive the heat equation in two dimensions by using Green's theorem in the plane,
(1.5.16), the two-dimensional form of the divergence theorem.

1.5.8. If Laplace's equation is satisfied in three dimensions, then show using
the divergence theorem that
for any closed surface S, the surface integral over S of the normal component of grad(u) is zero. 

 Give a physical interpretation of this result in the context of heat flow.

1.5.9. Determine the equilibrium temperature distribution inside a circular annulus,
r1 <= r <= r2, 
(a) if the outer radius is at temperature T2 and the inner at T1

(b) if the outer radius is insulated and the inner radius is at temperature
T1

1.5.10. Determine the equilibrium temperature distribution inside a circle, r <= ro,
if the boundary is fixed at temperature T0.

1.5.11. Consider du/dt = (k/r)(d/dr)(r du/dr) on a < r < b subject to
u(r,0)=f(r), (du/dr)(a,t)=beta, (du/dr)(b,t)=1.
Using physical reasoning, for what value(s) of beta does an equilibrium temperature
distribution exist?

1.5.12. Assume that the temperature is spherically symmetric, u = u(r,t), where r
is the distance from a fixed point (r^2 = x^2 + y^2 + z^2). Consider the heat
flow (without sources) between any two concentric spheres of radii a and b.

(a) Show that the total heat energy is 4 Pi int(c rho u r^2, r=a..b).

(b) Show that the flow of heat energy per unit time out of the spherical
shell at r = b is - 4 Pi b^2 K0 (du/dr)(b,t). A similar result holds at r = a.

(c) Use parts (a) and (b) to derive the spherically symmetric heat equation
du/dt = (k/r^2)(d/dr)(r^2 (du/dr)).

1.5.13. Determine the steady-state temperature distribution between two concentric
spheres with radii 1 and 4, respectively, if the temperature of the outer
sphere is maintained at 80 C and the inner sphere at 0 C (see Exercise 1.5.12).

1.5.14. Isobars are lines of constant temperature. Show that isobars are perpendicular
to any part of the boundary that is insulated.

1.5.15. Derive the heat equation in three dimensions assuming constant thermal
properties and no sources.

1.5.16. Express the integral conservation law for any three-dimensional object. Assume
there are no sources. Also assume the heat flow is specified,
g(x,y,z), on the entire boundary and does not depend on time. By integrating
with respect to time, and using the initial condition, determine the total thermal energy.

1.5.17. Derive the integral conservation law for any three dimensional object (with
constant thermal properties) by integrating the heat equation (1.5.11) (assuming
no sources). Show that the result is equivalent to (1.5.1).

Orthogonal curvilinear coordinates. A coordinate system (u,
v, w) may be introduced and defined by x = x(u,v,w), y = y(u,v,w) and
z = z(u,v,w). The radial vector is vec(r) = x vec(i) + y vec(j) + z vec(k). Partial derivatives of
vec(r) with respect to a coordinate are in the direction of the coordinate. Thus,
for example, a vector in the u-direction (d/du) vec(r) can be made a unit vector vec(e)
in the u-direction by dividing by its length h(u) = |(d/du) vec(r)| called the scale
factor: vec(e)=(1/h(u))(d/du) vec(r).

1.5.18. Determine the scale factors for cylindrical coordinates.

1.5.19. Determine the scale factors for spherical coordinates.

1.5.20. The gradient of a scalar can be expressed in terms of the new coordinate
system grad(g) = a (d/du) vec(r) + b (d/dv) vec(r) + c (d/dw) vec(r), where you will determine the
scalars a, b, c. Using dg = grad(g),d vec(r), derive that the gradient in an orthogonal
curvilinear coordinate system is given by

grad(g) = (1/h(u))^2(dg/du)(d/du) vec(r)+(1/h(v))^2(dg/dv)(d/dv) vec(r)
          +(1/h(w))^2(dg/dw)(d/dw) vec(r)                               (1.5.23)

An expression for the divergence is more difficult to derive, and we will just
state that if a vector is expressed in terms of this new coordinate system, 
then the divergence satisfies (1.5.24) [equations omitted due to complexity].

1.5.21. Using (1.5.23) and (1.5.24), derive the Laplacian in an orthogonal curvilinear
coordinate system:

[equation omitted due to complexity]   (1.5.25)

1.5.22. Using (1.5.25), derive the Laplacian for cylindrical coordinates.

1.5.23. Using (1.5.25), derive the Laplacian for spherical coordinates.