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.