2.2 Steady-State Temperatures 143
Under what conditions might the second factor on the right be taken ap-
proximately constant? If the factor were constant, the boundary condition
would be linear.- Interpret this problem in terms of diffusion. Be sure to explain how the
boundary conditions could arise physically.
D
∂^2 u
∂x^2 +K=∂u
∂t,^0 <x<a,^0 <t,
u( 0 ,t)=C, ∂u
∂x(a,t)= 0 , 0 <t,
u(x, 0 )= 0 , 0 <x<a.2.2 Steady-State Temperatures
Before tackling a complete heat conduction problem, we shall solve a simpli-
fied version called the steady-state or equilibrium problem. We begin with this
problem:
∂^2 u
∂x^2=^1
k∂u
∂t, 0 <x<a, 0 <t, (1)
u( 0 ,t)=T 0 , 0 <t, (2)
u(a,t)=T 1 , 0 <t, (3)
u(x, 0 )=f(x), 0 <x<a. (4)We m a y t h i n k o fu(x,t)as the temperature in a cylindrical rod, with insulated
lateral surface, whose ends are held at constant temperaturesT 0 andT 1.
Experience indicates that after a long time under the same conditions, the
variation of temperature with time dies away. In terms of the functionu(x,t)
that represents temperature, we thus expect that the limit ofu(x,t),asttends
to infinity, exists and depends only onx,
tlim→∞u(x,t)=v(x),and also that
tlim→∞∂u
∂t=^0.
The functionv(x),calledthesteady-state temperature distribution,muststill
satisfy the boundary conditions and the heat equation, which are valid for all
t>0.