Page 14 - Essentials of applied mathematics for scientists and engineers

book Mobk070 March 22, 2007 11:7

4 ESSENTIALS OF APPLIED MATHEMATICS FOR SCIENTISTS AND ENGINEERS

FIGURE 1.1: An element in three dimensional rectangular Cartesian coordinates

q x y z minus the rate at which heat is conducted out of the material. The flux of heat
y
x
conducted into the element at the x face is denoted by q while at the y face it is denoted by q .
x
At x + x the heat flux (i.e., per unit area) leaving the element in the x direction is q + q x
y
y
z
while at y + y the heat flux leaving in the y direction is q + q . Similarly for q . Expanding
x
x
x
x
x
the latter three terms in Taylor series, we find that q + q = q + q x + (1/2)q ( x) 2
x
xx
y
y
3
+ termsoforder ( x) or higher order. Similar expressions are obtained for q + q and
z
z
q + q Completing the heat balance
x
y
ρc x y z∂u/∂t = q x y z + q y z + q x z

x
2
x
x
− (q + q x + (1/2)q ( x) + ··· ) y z
x xx
y
2
y
y
− (q + q y + (1/2)q ( y) + ··· ) x z (1.8)
y
yy
z
2
z
z
− (q + q z + (1/2)q ( z) + ··· ) x y
zz
z
y
x
z
The terms q y z, q x z,and q x y cancel. Taking the limit as x, y,and z
2
2
2
approach zero, noting that the terms multiplied by ( x) ,( y) ,and ( z) may be neglected,
x
dividing through by x y z and noting that according to Fourier’s law q =−k∂u/∂x,
y
z
q =−k∂u/∂y,and q =−k(∂u/∂z) we obtain the time-dependent heat conduction equation
in three-dimensional rectangular Cartesian coordinates:
2
2
2
2
ρc ∂u/∂t = k(∂ u/∂x + ∂ u/∂y ) + q (1.9)
The equation is first order in t, and second order in both x and y. If the property values
ρ, c and k and the heat generation rate per unit volume q are independent of the dependent