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.