Dual-Phase-Lagging Heat-Conduction Equations - Heat Conduction: Mathematical Models and Analytical Solutions - page 320

2
m

Fig. 6.3 | Γ m ( t ) | at the critically-damped oscillation and ψ > − ω m φ m ψ m (after Xu and

Wang 2002)

however, as t increases from 0 to t m

and then decreases monotonically (Fig. 6.3). Therefore, although the temperature
field does not oscillate, its absolute value reaches the maximum value at t

| Γ m ( t ) | first increases from φ m to Max [ | Γ m ( t ) | ]

= >

t m 0

=

rather than at the initial time instant t 0.

Overdamped Oscillation

For this case ( ζ m > 1),

λ 1 , 2 = ω m 1

m

− ζ m ± ζ − . (6.234)

Thus the solution of Eq. (6.215) subject to Eq. (6.214) is

ψ

ω m + φ m

1 e ω m t √ ζ m

e − ζ m ω m t
2

m

Γ m ( t )= ζ m + ζ

m

−

m

ζ − 1

ω m + φ m

1 e − ω m t √ ζ

1

− ψ m

m −

m

+ − ζ m + ζ − . (6.235)