Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 285

so that
⎧
⎨

B mnk = − α mnk
β mnk

1
M mnk

ϕ ( r , θ , z ) J n ( k mn r ) Z k ( z ) r cos n θ d θ d r d z ,

Ω

(6.128)

D mnk = − α mnk
β mnk

1
M mnk

⎩

ϕ ( r , θ , z ) J n ( k mn r ) Z k ( z ) r sin n θ d θ d r d z .

Ω

On the other hand, by the structure of W ψ ( r , θ , z , t ) (Eq. (6.123)),

∂
∂

)= m , n , k ( α mnk sin β mnk t + β mnk cos β mnk t )

t W ϕ (

r , θ , z , t

B mnk cos n D mnk sin n e α mnk t J n ( · ( θ + θ ) k mn r ) Z k ( z ) , (6.129)

⎨

1
M mnk β mnk

B mnk =

ϕ ( r , θ , z ) J n ( k mn r ) Z k ( z ) r cos n θ d θ d r d z ,

Ω

(6.130)

1
M mnk β mnk

⎩

D mnk = ϕ ( r , θ , z ) J n ( k mn r ) Z k ( z ) r sin n θ d θ d r d z .

Ω

Substituting Eqs. (6.127), (6.128) and (6.130) into Eq. (6.129) yields

∂
∂

)= m , n , k { [( − B mnk e α mnk t sin β mnk t ) cos n θ

t W ϕ (

r , θ , z , t

D mnk e α mnk t sin A mnk e α mnk t cos +( − β mnk t ) sin n θ ]+[( β mnk t ) cos n θ

+ C mnk e α mnk t cos β mnk t sin n

θ ] } J n ( k mn r ) Z k ( z ) . (6.131)

Thus
m , n , k [( A mnk e α mnk t cos β mnk t ) cos n θ + C mnk e α mnk t cos β mnk t sin n θ ]

)= ∂
∂

)+ m , n , k [( B mnk e α mnk t sin β mnk t ) cos n θ · J n ( k mn r ) Z k ( z

t W ϕ (

r , θ , z , t

D mnk e α mnk t sin +( β mnk t ) sin n θ ] J n ( k mn r ) Z k ( z ) . (6.132)

Note that

1
τ 0 +(

k mn + λ

k

B 2 )

α mnk = − .

2