Page 15 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

PARTIAL DIFFERENTIAL EQUATIONS IN ENGINEERING 5

FIGURE 1.2: An element in cylindrical coordinates

variable, temperature the partial differential equation is linear. If q is zero, the equation is
2
homogeneous. It is easy to see that if a third dimension, z, were included, the term k∂ u/∂z 2
must be added to the right-hand side of the above equation.

1.3.2 Cylindrical Coordinates
A small element of volume r r z is shown in Fig. 1.2.
The method of developing the diffusion equation in cylindrical coordinates is much the
same as for rectangular coordinates except that the heat conducted into and out of the element
depends on the area as well as the heat flux as given by Fourier’s law, and this area varies in
r
the r-direction. Hence the heat conducted into the element at r is q r z, while the heat
r
r
conducted out of the element at r + r is q r z + ∂(q r z)/∂r( r) when terms
2
of order ( r) are neglected as r approaches zero. In the z-and θ-directions the area does
not change. Following the same procedure as in the discussion of rectangular coordinates,
expanding the heat values on the three faces in Tayor series’, and neglecting terms of order
2
2
( ) and ( z) and higher,
r
θ
ρcr θ r z∂u/∂t =−∂(q r θ z)/∂r r − ∂(q r z)/∂θ θ
z
− ∂(q r θ r)/∂z z + qr θ r z (1.10)