Chapter 2 455
Chapter 2
Section 2.1
- One possibility:u(x,t)is the temperature in a rod of lengthawhose lat-
eral surface is insulated. The temperature at the left end is held constant
atT 0. The right end is exposed to a medium at temperatureT 1 .Initially
the temperature isf(x).
3.A xg=hC x(U−u(x,t)),wherehis a constant of proportionality and
Cis the circumference. Eq. (4) becomes
∂^2 u
∂x^2 +hC
κA(U−u)=1
k∂u
∂t.- If∂∂ux( 0 ,t)is positive, then heat is flowing to the left, sou( 0 ,t)is greater
thanT(t). - The second factor is approximately constant ifTis much larger thanuor
ifTanduare approximately equal.
Section 2.2
1.v′′−γ^2 (ν−U)=0, 0<x<a,
v( 0 )=T 0 ,v(a)=T 1 ,
v(x)=U+Acosh(γx)+Bsinh(γx),A=T 0 −U,B=(T 1 −U)−(T 0 −U)cosh(γa)
sinh(γa).
One interpretation:uis the temperature in a rod, with convective heat
transfer from the cylindrical surface to a medium at temperatureU.
3.v(x)=T. Heat is being generated at a rate proportional tou−T.Ifγ=
π/a, the steady-state problem does not have a unique solution.
5.v(x)=Aln(κ 0 +βx)+B,A=(T 1 −T 0 )/ln( 1 +aβ/κ 0 ),
B=T 0 −Aln(κ 0 ).
7.v(x)=T 0 +r( 2 a−x)x/2.
9.Du′′−Su′=0, 0<x<a;u( 0 )=U,u(a)=0,
u(x)=U(eSx/D−eSa/D)/( 1 −eSa/D).