Page 16 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

6 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS

FIGURE 1.3: An element in spherical coordinates

Dividing through by the volume, we find after using Fourier’s law for the heat fluxes

2
2
2
2
2
ρc ∂u/∂t = (1/r)∂(r∂u/∂r)/∂r + (1/r )∂ u/∂θ + ∂ u/∂z + q (1.11)
1.3.3 Spherical Coordinates
An element in a spherical coordinate system is shown in Fig. 1.3. The volume of the element is
2
r sin θ rr θ = r sin θ r θ . The net heat flows out of the element in the r, θ,and
directions are respectfully

r 2
q r sin θ θ (1.12)
θ
q r sin θ r (1.13)

q r θ r (1.14)

It is left as an exercise for the student to show that

2
2
2
2
2
ρc ∂u/∂t = k[(1/r )∂/∂r(r ∂u/∂r) + (1/r sin θ)∂ u/∂ 2
2
+ (1/r sin θ)∂(sin θ∂u/∂θ)/∂θ + q (1.15)
The Laplacian Operator
The linear operator on the right-hand side of the heat equation is often referred to as the
2
Laplacian operator and is written as ∇ .