Heat and Mass Transfer Equations of the Form

5.4.3. Heat and Mass Transfer Equations of the Form

f (x) ∂w +∂ g(y) ∂w = h(w) ∂x

∂x

∂y

∂y

◮ Equations of this form describe steady-state heat/mass transfer or combustion processes in inhomogeneous anisotropic media. Here, f = f (x) and g = g(y) are the principal thermal diffusivities

(diffusion coefficients) dependent on coordinates; h = h(w) is the kinetic function (source function), which defines the law of heat (substance) release of absorption. The simple solutions dependent on

a single coordinate, w = w(x) or w = w(y), are not considered in this subsection. ∂

ax n ∂w

by m ∂w = f (w).

1 ◦ . Functional separable solution for n ≠ 2 and m ≠ 2:

2 2− m 1 w = w(ξ), /2 ξ= b(2 − m) x + a(2 − n) y . Here, the function w(ξ) is determined by the ordinary differential equation

2 2− n

w ′′ ξξ +

w ξ ′ = Bf (w),

(2 − n)(2 − m)

ab(2 − n) 2 (2 − m) 2

For m = 4/n, a family of exact solutions to the original equation with arbitrary f = f (w) follows from (1). It is given by

ab(2 − n) where C 1 and C 2 are arbitrary constants.

w ζζ ′′ =

2 ζ 1− A f (w).

(1 − A)

A large number of exact solutions to equation (2) for various f = f (w) can be found in Polyanin and Zaitsev (2003). ❍☎■

Reference : V. F. Zaitsev and A. D. Polyanin (1996).

= f (w).

The transformation ζ = x + c, η = y + s leads to an equation of the form 5.4.3.1:

= f (w).

= f (w).

The transformation ζ = |x| + c, η = |y| + s leads to an equation of the form 5.4.3.1:

= f (w).

= f (w).

∂x ∂y

∂y

Functional separable solution for µ ≠ 0:

1 w = w(ξ), /2 ξ= bµ ( x+C

1 ) +4 ae µy

where C 1 is an arbitrary constant and the function w(ξ) is determined by the autonomous ordinary differential equation

w ′′

ξξ = abµ 2 f (w).

The general solution of this equation with arbitrary kinetic function f = f (w) is defined implicitly by

where C 2 and C 3 are arbitrary constants. ∂ 2 w

∂w

5. a 2 +

be µ|y|

= f (w).

∂x ∂y

∂y

The substitution ζ = |y| leads to an equation of the form 5.4.3.4. ∂

= f (w).

Functional separable solution for βµ ≠ 0:

w = w(ξ), 1 ξ = bµ e βx /2 + 2 aβ − e µy ,

where the function w(ξ) is determined by the ordinary differential equation

w ′′

ξξ − w ′ ξ ξ = Af (w),

The substitution ζ=ξ 2 brings (1) to the generalized Emden–Fowler equation

′′ = 1 w −1 ζζ

4 Aζ

f (w),

whose solutions with f (w) = (kw + s) −1 and f (w) = (kw + s) −2 ( k, s = const) can be found in Polyanin and Zaitsev (2003). ❍☎■

Reference : V. F. Zaitsev and A. D. Polyanin (1996).

7. ae β|x| ∂w

be µ|y| ∂w = f (w).

The transformation ζ = |x|, η = |y| leads to an equation of the form 5.4.3.6. ∂

= f (w).

Functional separable solution for n ≠ 2 and µ ≠ 0:

2 a(2 − n) − e µy 1 /2 , where the function w(ξ) is determined by the ordinary differential equation

2 w = w(ξ), 2− ξ= bµ x n +

2− n ξ ξ = abµ 2 (2 − n) 2 f (w). ∂

Multiplicative separable solution:

w(x, y) = ϕ(x)ψ(y),

where the functions ϕ(x) and ψ(y) are determined by the ordinary differential equations [ f (x)ϕ x ′ ] ′ x = kϕ ln ϕ + Cϕ,

[ g(y)ψ ′ y ] ′ y = kψ ln ψ − Cψ, and ▲☎▼

C is an arbitrary constant.

Reference : A. D. Polyanin and V. F. Zaitsev (2002).

This is a special case of equation 5.4.4.6 with k = a, h 1 ( x) = b, and h 2 ( y) = 0.