Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 315

6.10.1 Thermal Waves

Without loss of generality, Xu and Wang (2002) consider the one-dimensional
initial-boundary value problem of dual-phase-lagging heat conduction
⎧
⎨

∂

k F t

2 T
∂

2 T 3 T

1
α

T

t + τ 0 ∂

= ∂

x 2 + τ T ∂

1

+ τ 0 ∂ F
∂

x 2 +

, ( 0 , l ) × ( 0 , + ∞ ) ,

∂ t 2 ∂ ∂ t ∂

T ( 0 , t )= T ( l , t )= 0 ,

⎩

T ( x , 0 )= φ ( x ) , T t ( x , 0 )= ψ ( x ) ,

(6.209)

whose solution represents the temperature distribution in an infinitely-wide slab of
thickness l .Here t is the time, T is the temperature, α is the thermal diffusivity

of the medium, F is the volumetric heat source, φ and ψ are two given functions,

and τ 0 are the phase-lags of the temperature gradient and heat flux vector,

respectively.
For a free thermal oscillation, F

τ T and

= 0. By taking the boundary conditions into

account, let

∞
m = 1 Γ m ( t ) sin β m x ,

T ( x , t )= (6.210)

where

m l .

β m =

Using the Fourier sine series to express φ and ψ as

∞
m = 1 φ m sin β m x ,

φ ( x )= (6.211)

and

∞
m = 1 ψ m sin β m x ,

ψ ( x )= (6.212)

where

l
0 φ ( ξ )

2
l

φ m = sin β m ξ d ξ ,

and

l
0 ψ ( ξ )

2
l

=

ξ . ψ sin β ξ d m m

A substitution of Eqs. (6.210)-(6.212) into (6.209) yields, by making use of the