Heat and Mass Transfer Equations with Complicating Factors

6.3.2. Heat and Mass Transfer Equations with Complicating Factors

– f (w). This equation describes steady-state mass transfer with a volume chemical reaction in a three-

dimensional translational-shear fluid flow.

b 3 = 0,

and the constants

A, B, and C solve the degenerate system of linear algebraic equations

One of the equations follows from the other two and, hence, can be omitted. Solution:

w = w(ζ),

ζ = Ax + By + Cz,

where the function w(ζ) is determined by the ordinary differential equation ( kζ + Ad 1 + Bd 2 + Cd 3 ) w ′ ζ =( 2 A + 2 B 2 + C ) w ′′ ζζ − f (w). Remark. In the case of an incompressible fluid, some of the equation coefficients must satisfy

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

= f (w).

∂z Solutions are sought in the form

∂z

w = w(ζ),

ζ = Ax + By + Cz + D,

where the constants

A, B, C, and D are determined by solving the algebraic system of equations

The first three equations are first solved for

A, B, and C. The resulting expressions are then substituted into the last equation to evaluate

D. The desired function w(ζ) is determined by the ordinary differential equation

ζw ζζ ′′ +( a 1 A+b 2 B+c 3 C)w ′ ζ = f (w).

This equation describes steady-state heat/mass transfer or combustion processes in inhomogeneous anisotropic media. Here, f (w), g(w), and h(w) are the principal thermal diffusivities (diffusion coefficients) dependent on the temperature w.

1 ◦ . Suppose w(x, y, z) is a solution of the equation in question. Then the functions

w 1 = w( ✢ C 1 x+C 2 , ✢ C 1 y+C 3 , ✢ C 1 z+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation (the plus or minus signs in front of C 1 are chosen arbitrarily).

1 f (w) + k 2 2 g(w) + k 2 3 h(w)

1 ( k 1 x+k 2 y+k 3 z) + C 2 ,

where C 1 , C 2 , k 1 , k 2 , k 3 , and λ are arbitrary constants.

3 ◦ . Solution:

C 1 y+C 2 z+C 3

w = w(θ), θ=

x+C 4

where C 1 , ...,C 4 are arbitrary constants, and the function w(θ) is determined by the ordinary differential equation

θ ′ ] ′ θ + C 2 [ h(w)w θ ′ ] ′ θ = 0, which admits the first integral

[ θ 2 f (w)w θ ′ ] ′ θ + C 1 2 [ g(w)w

[ 2 2 θ 2 f (w) + C 1 g(w) + C 2 h(w)]w ′ θ = C 5 .

For C 5 ≠ 0, treating w as the independent variable, one obtains a Riccati equation for θ = θ(w):

(2) For exact solutions of this equation, which can be reduced to a second-order linear equation, see

C 5 θ w ′ = θ 2 f (w) + C 2 1 g(w) + C 2 2 h(w).

Polyanin and Zaitsev (2003). Relations (1) and equation (2) can be used to obtain two other “one-dimensional” solutions by means of the following cyclic permutations of variables and determining functions:

4 ◦ . “Two-dimensional” solution ( a and b are arbitrary constants):

w(x, y, z) = U (x, ζ), ζ = ay + bz,

where the function U = U (x, ζ) is determined by a differential equation of the form 5.4.4.8:

which can be reduced to a linear equation. Relations (4) and equation (5) can be used to obtain two other “two-dimensional” solutions by means of the cyclic permutations of variables and determining functions; see (3).

5 ◦ . There are “two-dimensional” solutions of the following forms: w(x, y, z) = V (z 1 , z 2 ), z 1 = a 1 x+a 2 y+a 3 z, z 2 = b 1 x+b 2 y+b 3 z;

w(x, y, z) = W (ξ, η), ξ = y/x, η = z/x, where the a n and b n are arbitrary constants (the first solution generalizes the one of Item 3 ◦ ).

6 ◦ . Let g(w) = af (w). Then, there is a “two-dimensional” solution of the form

2 w(x, y, z) = u(r, z), 2 r = ax + y .

7 ◦ . Let g(w) = af (w) and h(w) = bf (w). Then, the transformation

leads to the Laplace equation

2 ∂y + ∂z 2 = 0.

∂x 2

For solutions of this linear equation, see Tikhonov and Samarskii (1990) and Polyanin (2002).

∂z This equation describes steady-state anisotropic heat/mass transfer with a volume chemical reaction

∂x

∂y

in a three-dimensional translational-shear fluid flow. Let k be a root of the cubic equation

b 3 = 0,

and the constants

A, B, and C solve the degenerate system of linear algebraic equations

One of the equations follows from the other two and, hence, can be omitted. Solution:

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

w = w(ζ),

ζ = Ax + By + Cz,

[ ϕ(w)w ′ ζ ] ′ ζ =( kζ + Ad 1 + Bd 2 + Cd 3 ) w ′ ζ , ϕ(w) = A 2 f 1 ( w) + B 2 f 2 ( w) + C 2 f 3 ( w).

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 a solution of the form (1). Remark 2. In the case of an incompressible fluid, some of the equation coefficients must satisfy

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