EXERCISES 2.5
2.5.1. Solve Laplace's equation inside a rectangle 0 < x < L, 0 < y < H, with the
following boundary conditions:
*(a) u_x=0 at x=0 and x=L, u=0 at y=0, u=f(x) at y=H
(b) u_x=0 at x=0 and x=L, u=0 at y=0 and y=H
*(c) u_x=0 at x=0, u=g(y) at x=L, u=0 at y=0 and y=H
(d) u=g(y) at x=0, u=0 at x=L and y=H, u_x=0 at y=0
*(e) u=0 at x=0 and x=L, u-u_x=0 at y=0, u=f(x) at y=H
(f) u=f(y) at x=0, u=0 at x=L, u_y=0 at y=0 and y=H
(g) u_x=0 at x=0 and x=L, u=f(x) at y=0, u_y=0 at y=H,
   where f(x)=0 for x<L/2 and f(x)=1 for x>L/2

2.5.2.
Consider u(x, y) satisfying Laplace's equation inside a rectangle (0 < x <
L, 0 < y < H) subject to the boundary conditions
u_x=0 at x=0 and x=L, u_y=0 at y=0, u_y = f (x) at y=H.

*(a) Without solving this problem, briefly explain the physical condition
under which there is a solution to this problem.

(b) olve this problem by the method of separation of variables. Show that
the method works only under the condition of part (a).

(c) The solution [part (b)] has an arbitrary constant. Determine it by
consideration of the time-dependent heat equation u_t=k(u_{xx} + u_{yy})  subject to
the initial condition
u(x,y,0) = g(x,y)

*2.5.3. Solve Laplace's equation outside a circular disk (r > a) subject to the
boundary condition
(a) u(a,theta) = ln( 2 ) + 4 cos(3 theta)
(b) u(a,theta) = f(theta)

You may assume that u(r, theta) remains finite at r=infinity.

*2.5.4. For Laplace's equation inside a circular disk (r < a), using (2.5.45) and
(2.5.47), show that
00
u(r,9)= f(6) 2+E(a)ncosn(9-8)1 dB.
a L n_0
Using cos z = Re [ei=], sum the resulting geometric series to obtain Poisson's
integral formula.
2.5.5. Solve Laplace's equation inside the quarter-circle of radius 1 (0 < 0 <-
7r/2, 0 < r < 1) subject to the boundary conditions
* (a) (r, 0) = 0, u (r, 2) = 0, u(1,0) = f (O)
(b) Ou (r, 0) = 0, 6u (r, z) = 0, u(1, 0) = f (0)
* (c) u(r, 0) = 0, u (r, z) = 0, Ou (1, 9) = f (O)
(d) (r, o) = o, (r, 2) = o, (1, e) = g(e)
Show that the solution [part (d)] exists only if fo 2 g(9) d9 = 0. Explain
this condition physically.
2.5.6. Solve Laplace's equation inside a semicircle of radius a(0 < r < a, 0 < 9 <
a) subject to the boundary conditions
*(a) u = 0 on the diameter and u(a, 9) = g(9)
(b) the diameter is insulated and u(a, 0) = g(9)
2.5.7. Solve Laplace's equation inside a 60° wedge of radius a subject to the boundary
conditions
(a) u(r, 0) = 0, u (r, a) = 0, u(a, 9) = f (0)
* (b) (r, 0) = 0, (r, 3 ) = 0, u(a, 9) = f (0)
2.5. Laplace's Equation 87
2.5.8. Solve Laplace's equation inside a circular annulus (a < r < b) subject to
the boundary conditions
* (a) u(a, 9) = f (O), u(b, 9) = g(9)
(b) 67 (a,0) = 0, u(b,0) = g(9)
(c) Wr- (a,0) = f(0), 3T (b,0) = g(0)
If there is a solvability condition, state it and explain it physically.
*2.5.9. Solve Laplace's equation inside a 90° sector of a circular annulus (a < r <
b, 0 < 0 < ir/2) subject to the boundary conditions
(a) u(r, 0) = 0, u(r, it/2) = 0, u(a, 9) = 0, u(b, 0) = f (0)
(b) u(r,0) = 0, u(r,ir/2) = f(r), u(a,0) = 0, u(b,9) = 0
2.5.10. Using the maximum principles for Laplace's equation, prove that the solution
of Poisson's equation, V2u = g(x), subject to u = f (x) on the
boundary, is unique.
2.5.11. Do Exercise 1.5.8.
2.5.12. (a) Using the divergence theorem, determine an alternative expression for
ffu02udxdydz.
(b) Using part (a), prove that the solution of Laplace's equation V2u = 0
(with u given on the boundary) is unique.
(c) Modify part (b) if 0 on the boundary.
(d) Modify part (b) if 0 on the boundary. Show that Newton's
law of cooling corresponds to h < 0.
2.5.13. Prove that the temperature satisfying Laplace's equation cannot attain its
minimum in the interior.
2.5.14. Show that the "backward" heat equation
au 02u
at = -k 8x2 ,
subject to u(0, t) = u(L, t) = 0 and u(x, 0) = f (x), is not well posed. (Hint:
Show that if the data are changed an arbitrarily small amount, for example,
1
f (x) -' f (x) + n srn _
for large n, then the solution u(x, t) changes by a large amount.]
2.5.15. Solve Laplace's equation inside a semi-infinite strip (0 < x < oo, 0 < y < H)
subject to the boundary conditions
88
(a) 8' (x, 0) = 0,
(b) u(x,0) = 0,
(c) u(x,0) = 0,
(d) (x, 0) = 0,
Chapter 2. Method of Separation of Variables
"' (x, H) = 0, u(0, y) = f (y)
u(x, H) = 0, u(0,y) = f(y)
u(x, H) = 0, (0,y) = f(y)
Ou (x, H) = 0, a: (0, y) = f (y)
Show that the solution [part (d)] exists only if fH f (y) dy = 0.
2.5.16. Consider Laplace's equation inside a rectangle 0 < x < L, 0 < y < H, with
the boundary conditions
8u
au
8u
&" 8x(0, y) = 0, 8x(L, y) = g(y),
8y(x, 0) = 0, 8y (x, H) = f (x)
(a) What is the solvability condition and its physical interpretation?
(b) Show that u(x, y) = A(x2 - y2) is a solution if f (x) and g(y) are
constants [under the conditions of part (a)].
(c) Under the conditions of part (a), solve the general case [nonconstant
f (x) and g(y)]. [Hints: Use part (b) and the fact that f (x) = f +
[f (x) - f.,.], where f.,. = L fL f (x) dx.]
2.5.17. Show that the mass density p(x, t) satisfies k + V (pu) = 0 due to conservation
of mass.
2.5.18. If the mass density is constant, using the result of Exercise 2.5.17, show
that
2.5.19. Show that the streamlines are parallel to the fluid velocity.
2.5.20. Show that anytime there is a stream function, V x u = 0.
2.5.21. From u and v=- ,derive u,-=rue=-
2.5.22. Show the drag force is zero for a uniform flow past a cylinder including
circulation.
2.5.23. Consider the velocity ug at the cylinder. Where do the maximum and
minimum occur?
2.5.24. Consider .the velocity ue at the cylinder. If the circulation is negative, show
that the velocity will be larger above the cylinder than below.
2.5.25. A stagnation point is a place where u = 0. For what values of the circulation
does a stagnation point exist on the cylinder?
2.5.26. For what values of 0 will u,. = 0 off the cylinder? For these 6, where (for
what values of r) will ue = 0 also?
2.5.27. Show that r/ = a 81T B satisfies Laplace's equation. Show that the streamlines
are circles. Graph the streamlines.