Heat and Mass Transfer Equations of the Form h i h i h i
6.3.1. Heat and Mass Transfer Equations of the Form h i h i h i
∂ f 1 (x) ∂w +∂ f 2 (y) ∂w +∂ f 3 (z) ∂w = g(w)
∂x
∂x
∂y
∂y
∂z
∂z
◮ Equations of this type describe steady-state heat/mass transfer or combustion processes in inho- mogeneous anisotropic media. Here, f 1 ( x), f 2 ( y), and f 3 ( z) are the principal thermal diffusivities
(diffusion coefficients) dependent on coordinates, and g = g(w) is the kinetic function, which defines the law of heat (substance) release or absorption.
1. a 2 + b 2 + c
= f (w).
∂x ∂y
∂z 2
1 ◦ . Traveling-wave solution:
w = w(θ),
θ = Ax + By + Cz.
The function w(θ) is defined implicitly by Z
A, B, C, C 1 , and C 2 are arbitrary constants.
2 ◦ . Solution: ( x+C 1 ) 2 ( y+C 2 ) 2 ( z+C 3 ) 2
w = w(r),
where C 1 , C 2 , and C 3 are arbitrary constants, and the function w(r) is determined by the ordinary differential equation
2 rr + w ′ r r = f (w).
w ′′ w ′′
C is an arbitrary constant (C ≠ 0), and the function U = U (ξ, η) is determined by the equation
∂η Remark. The solution specified in Item 3 ◦ can be used to obtain other “two-dimensional”
solutions by means of the following cyclic permutations of variables and determining parameters:
( x, a)
ց ( z, c) ←− (y, b)
4 ◦ . “Two-dimensional” solution:
a b c a b b c c a where
A, B, and C are arbitrary constants and the function V = V (ζ, ρ) is determined by the equation
c ¯z brings the original equation to the form ∆ w = f (w).
5 ◦ . The transformation
= f (w).
1 ◦ . For n = m = 0, see equation 6.3.1.1.
2 ◦ . Functional separable solution for n ≠ 2 and m ≠ 2:
2 y 2−
w = w(r),
r =4
4 a b(2 − n) 2 c(2 − m) 2
where the function w(r) is determined by the ordinary differential equation
w ′ r = f (w),
A=
(2 − n)(2 − m)
3 ◦ . “Two-dimensional” solution for n ≠ 2 and m ≠ 2:
where the function U (x, ξ) is determined by the differential equation
(2 − n)(2 − m)
4 ◦ . There are “two-dimensional” solutions of the following forms:
Reference: A. D. Polyanin and A. I. Zhurov (1998).
ax n ∂w
= f (w). ∂x
1 ◦ . Functional separable solution for n ≠ 2, m ≠ 2, and k ≠ 2: y 2− 2 m x 2− n
z 2− k
w = w(r),
r =4
2 a(2 − n) + b(2 − m) 2 + c(2 − k) 2 ,
where the function w(r) is determined by the ordinary differential equation
w rr ′′
r = f (w),
2 ◦ . There are “two-dimensional” solutions of the following forms: y 2− m
Reference: A. D. Polyanin and A. I. Zhurov (1998).
ce λz = f (w). ∂x
1 ◦ . Functional separable solution for β ≠ 0, γ ≠ 0, and λ ≠ 0:
w = w(r),
where the function w(r) is determined by the ordinary differential equation
w ′′
rr − w r r = f (w).
2 ◦ . There are “two-dimensional” solutions of the following forms:
Reference: A. D. Polyanin and A. I. Zhurov (1998).
ce λz = f (w). ∂x
1 ◦ . Functional separable solution for n ≠ 2, m ≠ 2, and λ ≠ 0:
w = w(r),
where the function w(r) is determined by the ordinary differential equation
w rr ′′ +
w r ′ = f (w),
A=2
2− n 2− m 2− n 2− m
1 ◦ . Functional separable solution for n ≠ 2, β ≠ 0, and λ ≠ 0:
e − 2 βy x 2− n
e − λz
2 a(2 − n) + bβ 2 + cλ 2 , where the function w(r) is determined by the ordinary differential equation
w = w(r),
w r ′ = f (w).
2− n r
Example 1. For n = 0 and any f = f (w), equation (1) can be solved by quadrature to obtain
where C 1 and C 2 are arbitrary constants. Example 2. For n = 1 and f (w) = Ae βw , equation (1) has the one-parameter solution
1 8 w(r) = 2
where C is an arbitrary constant.
2 ◦ . There are “two-dimensional” solutions of the following forms:
Reference: A. D. Polyanin and A. I. Zhurov (1998).
This is a special case of equation 6.3.3.6 with g 1 ( x) = b and g 2 ( y) = g 3 ( z) = 0.