Miscellaneous Exercises 289
24.Find a polynomial of second degree inxandy,
v(x,y)=A+Bx+Cy+Dx^2 +Exy+Fy^2 ,
that satisfies the potential equation and these boundary conditions:
v( 0 ,y)= 0 , 0 <y<b,
v(x, 0 )= 0 ,v(x,b)=x, 0 <x<a.25.Find the problem (partial differential equation and boundary condi-
tions) satisfied byw(x,y)=v(x,y)−u(x,y),whereuandvare the so-
lutions of the problems in Exercises 23 and 24. Solve the problem. Is this
problem easier to solve than the one in Exercise 23?
26.Solve the potential equation in the quarter-plane 0<x,0<y,subjectto
the boundary conditions
u(x, 0 )=f(x), 0 <x,
u( 0 ,y)=f(y), 0 <y.
The functionfthat appears in both boundary conditions is given by the
equation
f(x)=
{ 1 , 0 <x<a,
0 , a<x.27.(Flow past a plate) A fluid occupies the half-planey>0 and flows past
(left to right, approximately) a plate located near thex-axis. If thexandy
components of velocity areU 0 +u(x,y)andv(x,y),respectively(U 0 =
constant free-stream velocity), under certain assumptions, the equations
of motion, continuity, and state can be reduced to
∂u
∂y=
∂v
∂x,(
1 −M^2
)∂u
∂x+∂v
∂y=^0 ,
valid for allxandy>0.Mis the free-stream Mach number. Define the
velocity potentialφby the equationsu=∂φ/∂xandv=∂φ/∂y.Show
that the first equation is automatically satisfied and the second is a partial
differential equation that is elliptic ifM<1 or hyperbolic ifM>1.28.If the plate is wavy — say, its equation isy=cos(αx)— then the
boundary condition, that the vector velocity be parallel to the wall, is
v
(
x,cos(αx))
=−αsin(αx)(
U 0 +u(
x,cos(αx)))
.
This equation is impossible to use, so it is replaced by
v(x, 0 )=−αU 0 sin(αx)