1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

36 Chapter 0 Ordinary Differential Equations

Figure 9 Poiseuille flow.

10.Verify that the solution of the problem given in Eqs. (17) and (18) can also

be written as follows, withμ^2 =Ch/Aκ:

u(x)=T 0

cosh(μ(x−a/ 2 ))

cosh(μa/ 2 ).

11.(Poiseuille flow) A viscous fluid flows steadily between two large paral-

lel plates so that its velocity is parallel to thex-axis. (See Fig. 9.) The

x-component of velocity of the fluid at any point(x,y)is a function of

yonly. It can be shown that this componentu(x)satisfies the differential

equation

d^2 u

dy^2

=−g

μ

, 0 <y<L,

whereμis the viscosity and−gis a constant, negative pressure gradi-

ent. Findu(y), subject to the “no-slip” boundary conditions,u( 0 )=0,

u(L)=0.

12.If the beam mentioned in Exercise 6 is subjected to axial compression in-

stead of tension, the boundary value problem foru(x)becomes the one

here. Solve foru(x).

d^2 u

dx^2 +

P

EIu=−

w

EI

Lx−x^2

2 ,^0 <x<L,

u( 0 )= 0 , u(L)= 0.

13.For what value(s) of the compressive loadPin Exercise 12 does the prob-

lem have no solution or infinitely many solutions?

14.The pressurep(x)in the lubricant under a plane pad bearing satisfies the

problem

d

dx

(

x^3 dpdx

)

=−K, a<x<b,

p(a)= 0 , p(b)= 0.