Equations of Heat and Mass Transfer in Anisotropic Media

2.5.3. Equations of Heat and Mass Transfer in Anisotropic Media

+ f (w).

1 ◦ . Solution for m ≠ 2 and n ≠ 2:

, where the function w(ξ, t) is determined by the one-dimensional nonstationary equation

w = w(ξ, t),

+ f (w),

A=

(2 − m)(2 − n) For solutions of this equation with

A = 0 and various f (w), see Subsections 1.1.1 to 1.1.3 and equations 1.2.1.1 to 1.2.1.3, 1.4.1.2, 1.4.1.3, 1.4.1.7, and 1.4.1.8.

2− m

w = w(x, ξ, t),

b(2 − m)

c(2 − n)

where the function w(x, ξ) is determined by the two-dimensional nonstationary equation

+ f (w),

A=

(2 − m)(2 − n) ∂w

l 2. ∂w = ax + by + cz + f (w). ∂t

Solution for n ≠ 2, m ≠ 2, and l ≠ 2:

2 + b(2 − m) 2 a(2 − n) + c(2 − l) 2 , where the function w(ξ, t) is determined by the one-dimensional nonstationary equation

w = w(ξ, t),

+ f (w),

A=2

2− m 2− l For solutions of this equation with

A = 0 and various f (w), see Subsections 1.1.1 to 1.1.3 and equations 1.2.1.1 to 1.2.1.3, 1.4.1.2, 1.4.1.3, 1.4.1.7, and 1.4.1.8.

+ f (w). ∂t

Solution for λ ≠ 0, µ ≠ 0, and ν ≠ 0:

w = w(ξ, t),

2 aλ + bµ cν 2 ,

where the function w(ξ, t) is determined by the one-dimensional nonstationary equation

∂w 2 ∂ w

1 ∂w

+ f (w).

νz 4. ∂w = ax + by + ce + f (w). ∂t

Solution for n ≠ 2, m ≠ 2, and ν ≠ 0:

w = w(ξ, t),

, a(2 − n) 2 b(2 − m) 2 cν 2

where the function w(ξ, t) is determined by the one-dimensional nonstationary equation

+ f (w),

A=

(2 − n)(2 − m) For solutions of this equation with

A = 0 and various f (w), see Subsections 1.1.1 to 1.1.3 and equations 1.2.1.1 to 1.2.1.3, 1.4.1.2, 1.4.1.3, 1.4.1.7, and 1.4.1.8.

+ f (w). ∂t

Solution for n ≠ 2, µ ≠ 0, and ν ≠ 0:

2 a(2 − n) , bµ cν where the function w(ξ, t) is determined by the one-dimensional nonstationary equation

w = w(ξ, t),

+ f (w).

∂t

2− n ξ ∂ξ

+ g(w). ∂t

For group classification and exact solutions of this equation for some f n ( w) and g(w), see Dorod- nitsyn, Knyazeva, and Svirshchevskii (1983).

∂y ∂z ∂z This equation describes unsteady anisotropic heat or mass transfer in a three-dimensional steady

translational-shear fluid flow.

1 ◦ . Let λ be a root of the cubic equation

b 3 = 0,

and let the constants A 1 , A 2 , and A 3 solve the degenerate system of linear algebraic equations

One of these equations is redundant and can be omitted. Suppose w(x, y, z, t) is a solution of the equation in question. Then the function

w 1 = wx+A 1 Ce λt , y+A 2 Ce λt , z+A 3 Ce λt , t

where

C is an arbitrary constant, λ is a root of the cubic equation (1), and A 1 , A 2 , and A 3 are the corresponding solution of the algebraic system (2), is also a solution of the equation.

2 ◦ . Solution:

(3) where

w = w(ξ), ξ=A 1 x+A 2 y+A 3 z + Ce λt ,

C is an arbitrary constant, λ is a root of the cubic equation (1), and A 1 , A 2 , and A 3 are the corresponding solution of the algebraic system (2), and the function w(ξ) is determined by the ordinary differential equation

2 2 ( 2 λξ + A

1 d 1 + A 2 d 2 + A 3 d 3 ) w ξ =[ ϕ(w)w ξ ] ξ ,

ϕ(w) = A 1 f 1 ( w) + A 2 f 2 ( w) + A 3 f 3 ( w).

3 ◦ . Let λ be a root of the cubic equation (1) and let A 1 , A 2 , and A 3 be the corresponding solution of the algebraic system (2). “Two-dimensional” solutions:

(4) where the function U (ζ, t) is determined by the differential equation ∂U

ϕ(U ) = A 1 f 1 ( U)+A 2 f 2 ( U)+A 3 f 3 ( U ). ∂t

Remark 1. A more general equation, with an additional term g(w) on the right-hand side, where

g is an arbitrary function, also has solutions of the forms (3) and (4). Remark 2. In the case of an incompressible fluid, the equation coefficients must satisfy the

condition a 1 + b 2 + c 3 = 0.