Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions

2.4.1. Heat and Mass Transfer Equations in Quiescent or Moving Media with Chemical Reactions

∂w

1. = a ∂t

+ f (w).

∂x 2

∂y 2

This is a two-dimensional equation of unsteady heat/mass transfer or combustion in a quiescent medium.

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

w 1 = w( ✩ x+C 1 , ✩ y+C 2 , t+C 3 ), w 2 = w(x cos β − y sin β, x sin β + y cos β, t),

where C 1 , C 2 , C 3 , and β are arbitrary constants, are also solutions of the equation (the plus or minus signs in w 1 can be chosen independently of each other).

2 ◦ . Traveling-wave solution:

w = w(ξ),

ξ = Ax + By + λt,

where

A, B, and λ are arbitrary constants, and the function w(ξ) is determined by the autonomous ordinary differential equation

2 a(A 2 + B ) w

ξξ ′′ − λw ξ ′ + f (w) = 0.

For solutions of this equation, see Polyanin and Zaitsev (1995, 2003).

3 ◦ . There are “two-dimensional” solutions of the following forms:

w(x, y, t) = F (z, t),

z=k 1 x+k 2 y;

w(x, y, t) = G(r, t), p r= x 2 + y 2 ;

w(x, y, t) = H(ξ 1 , ξ 2 ), ξ 1 = k 1 x+λ 1 t, ξ 2 = k 2 y+λ 2 t,

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

2. + a + b =

– f (w).

This is a two-dimensional equation of unsteady heat/mass transfer with a volume chemical reaction in a steady translational fluid flow.

The transformation

leads to a simpler equation of the form 2.4.1.1:

– f (w). This is a two-dimensional equation of unsteady heat/mass transfer with a volume chemical reaction

in a steady translational-shear fluid flow.

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

w 1 = w x + Cb 1 e λt , y + C(λ − a 1 ) e λt , t

where

C is an arbitrary constant and λ = λ 1,2 are roots of the quadratic equation

(1) are also solutions of the equation.

λ 2 −( a 1 + b 2 ) λ+a 1 b 2 − a 2 b 1 = 0,

2 ◦ . Solutions:

w = w(z),

z=a 2 x + (λ − a 1 ) y + Ce λt ,

where λ=λ 1,2 are roots of the quadratic equation (1), and the function w(z) is determined by the ordinary differential equation

2 c 1 2 +( 2 λ−a 1 ) c 2 z ′ = 2 +( λ−a 1 ) ′′ zz − f (w).

3 ◦ . “Two-dimensional” solutions:

w = U (ζ, t),

ζ=a 2 x + (λ − a 1 ) y,

where λ=λ 1,2 are roots of the quadratic equation (1), and the function U (ζ, t) is determined by the differential equation

∂U 2

2 − ∂t f (U ). ∂ζ ∂ζ Remark. In the case of an incompressible fluid, the equation coefficients must satisfy the

2 c 1 +( λ−a 1 ) c 2 = 2 +( λ−a 1 )

condition a 1 + b 2 = 0. ∂w

– g(w).

This equation describes mass transfer with volume chemical reaction in an unsteady translational fluid flow.

The transformation

f 2 ( t) dt, leads to a simpler equation of the form 2.4.1.1:

w = U (ξ, η, t),

ξ=x−

f 1 ( t) dt,

η=y−

2 ∂U 2 ∂ U ∂ U

− g(U ).

∂t

2.4.2. Equations of the Form ∂w =∂

f (x) ∂w +∂ g(y) ∂w +h(w)

∂t ∂x

∂x

∂y ∂y

+ f (w).

This is a two-dimensional equation of unsteady heat (mass) transfer or combustion in an anisotropic case with power-law coordinate-dependent principal thermal diffusivities (diffusion coefficients).

Solution for n ≠ 2 and m ≠ 2:

w = w(ξ, t), ξ =

a(2 − n)

b(2 − m)

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).

This is a two-dimensional equation of unsteady heat (mass) transfer or combustion in an anisotropic case with exponential coordinate-dependent principal thermal diffusivities (diffusion coefficients).

Solution for β ≠ 0 and λ ≠ 0:

2 4 − βx

w = w(ξ, t), ξ =

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

∂w 2 ∂ w

1 ∂w

f (w).

+ f (w).

Solution for n ≠ 2 and λ ≠ 0:

w = w(ξ, t),

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

∂w 2 ∂ w

n 1 ∂w

f (w).

This is a two-dimensional equation of unsteady heat (mass) transfer or combustion in an anisotropic case with arbitrary coordinate-dependent principal thermal diffusivities (diffusion coefficients) and

a logarithmic source.

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

w 1 = exp( C 1 e at ) w(x, y, t + C 2 ), where C 1 and C 2 are arbitrary constants, is also a solution of the equation.

w(x, y, t) = exp(C 1 e at ) U (x, y),

where the function U (x, y) is determined by the stationary equation

3 ◦ . “Two-dimensional” solution with incomplete separation of variables (the solution is separable in the space variables x and y, but is not separable in time t):

w(x, y, t) = ϕ(x, t)ψ(y, t),

where the functions ϕ(x, t) and ψ(y, t) are determined from the two independent one-dimensional nonlinear parabolic differential equations

+ aϕ ln ϕ + C(t)ϕ,

+ aψ ln ψ + bψ − C(t)ψ,

and ✪☎✫ C(t) is an arbitrary function.

Reference : A. D. Polyanin (2000).

2.4.3. Equations of the Form ∂w =∂

f (w)∂w +∂ g(w)∂w +h(t,w)

∂t ∂x

∂x ∂y ∂y

+ f (t)w.

1 ◦ . Suppose w(x, y, t) is a solution of this equation. Then the functions

where C 1 , C 2 , C 3 , and β are arbitrary constants, are also solutions of the equation (the plus or minus signs in w 1 can be chosen independently of each other).

2 ◦ . Multiplicative separable solution:

(1) where the function Θ( x, y) is a solution of the Laplace equation

w(x, y, t) = exp

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

3 ◦ . Multiplicative separable solution:

(2) where the function ϕ(t) is a solution of the Bernoulli equation

w(x, y, t) = ϕ(t)

n+1 ,

ϕ ′ t − f (t)ϕ + Aaϕ n+1 = 0,

A is an arbitrary constant, and the function Θ(x, y) is determined by the stationary equation

The general solution of equation (3) is given by

Z ϕ(t) = exp

−1 /n

f (t) dt, where

Aan exp

F (t) =

B is an arbitrary constant.

f (t) dt leads to a simpler equation of the form 2.1.2.4:

w(x, y, t) = F (t)U (x, y, τ ), τ=

µw e ∂w + f (t).

1 ◦ . Suppose w(x, y, t) is a solution of this equation. Then the functions

where C 1 , C 2 , C 3 , and β are arbitrary constants, are also solutions of the equation.

2 ◦ . Additive separable solution:

1 w(x, y, t) = ϕ(t) + ln Θ( x, y),

where the function ϕ = ϕ(t) is determined by the ordinary differential equation

(1) and the function Θ( x, y) is a solution of the two-dimensional Poisson equation

ϕ ′ t + A(a/µ) exp(µϕ) − f (t) = 0,

The general solution of equation (1) is given by

Z ϕ(t) = F (t) −

For solutions of the linear stationary equation (2), see the books by Tikhonov and Samarskii (1990) and Polyanin (2002).

Note that equations (1), (2) and relation (3) involve arbitrary constants

A and B.

3 ◦ . The transformation

Z w(x, y, t) = U (x, y, τ ) + F (t), τ=

f (t) dt, leads to a simpler equation of the form 2.2.2.1:

exp[ µF (t)] dt,

This is a two-dimensional nonlinear heat and mass transfer equation for an anisotropic medium.

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

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

w 2 = w(x cos β − y sin β, x sin β + y cos β, t), where C 1 , ...,C 4 and β are arbitrary constants, are also solutions of the equation.

2 Z 2 f (w) dw

= k 1 x+k 2 y + λt + C 2 ,

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

3 ◦ . Solution:

w(x, y, t) = U (ξ),

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

[ 1 ξf (U )U

ξ ] ξ + 4 ξU ξ ′ = 0.

4 ◦ . “Two-dimensional” solutions (for the axisymmetric problems):

w(x, y, t) = V (r, t), p r= x 2 + y 2 ,

where the function

V = V (r, t) is determined by the differential equation

✮☎✯ 5 ◦ . For other “two-dimensional” solutions, see equation 2.4.3.4 with g(w) = f (w).

Reference : V. A. Dorodnitsyn, I. V. Knyazeva, and S. R. Svirshchevskii (1983).

This is a two-dimensional unsteady heat and mass transfer equation in an anisotropic case with arbitrary coordinate-dependent principal thermal diffusivities (diffusion coefficients).

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

1 = w(C 1 x+C 2 , ✰ C 1 y+C 3 , C 1 t+C 4 ),

where C 1 , ...,C 4 are arbitrary constants, are also solutions of the equation.

2 ◦ . Traveling-wave solution in implicit form:

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

dw = k 1 x+k 2 y + λt + C 2 , λw + C 1

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

3 ◦ . “Two-dimensional” solution:

w(x, y, t) = U (z, t),

z=k 1 x+k 2 y,

where the function U = U (z, t) is determined by a differential equation of the form 1.6.15.1:

4 ◦ . There are more complicated “two-dimensional” solutions of the form

w(x, y, t) = V (ζ 1 , ζ 2 ),

ζ 1 = a 1 x+a 2 y+a 3 t,

ζ 2 = b 1 x+b 2 y+b 3 t.

5 ◦ . “Two-dimensional” solution: