By Wipf A.
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1) Lastly we need some boundary conditions. The simplest ones are to have one constant temperature at infinity and another on the cylinder,2 so T → T∞ as r → ∞, T = T0 on r = a. 1. 1 The usage is changing in the loose direction; convection is often used for both processes, subdivided where necessary into forced convection for advection, and natural convection for buoyancy-driven heat transport. A lot of the heat lost by a hot person in still air is by (natural) convection. 2 The conditions on the cylinder are not especially realistic; a Newton condition of the form −k∂T /∂n + h(T − T0 ) = 0 would be better; see the exercises on page 55.
The Kirchhoff transformation. Suppose that the thermal conductivity of a material depends on the temperature. Show that the steady heat equation ∇ · (k(T )∇T )) = 0 can be transformed into Laplace’s equation for the new variable u = T k(s) ds. 5. Newton’s law of cooling and Biot numbers. The process of cooling a hot object is a complicated one. In addition to conduction to the surroundings, it may involve both forced and natural convection if the body is immersed in a liquid or gas; there may be boiling, or thermal 13 A shortcut: because φ + iψ is an analytic (holomorphic) function w(z) of z = x + iy, the Cauchy–Riemann equations let us simplify the Laplacian operator to ∂2 ∂2 dw + = ∂x2 ∂y 2 dz 2 ∂2 ∂2 + ∂φ2 ∂ψ 2 .
Hectares per megasecond?? CHAPTER 4. DIMENSIONAL ANALYSIS 48 [ν] = [L]2 [T]−1 . Suppose we have flow past a body of typical size L, with a free-stream velocity U e1 . As in the advection-diffusion problem, we scale all distances with L, time with L/U and velocities with U , writing x = Lx , t = (L/U∞ )t , u = Uu . Only p has not yet been scaled, and in the absence of any obvious exogenous scale we let the equations tell us what the possibilities are. For now, let’s write p = P0 p and substitute all these into the momentum equation (clearly the mass conservation just becomes ∇ · u = 0).