analytical solution of the hyperbolic heat conduction ... · this paper presents an analytical...

Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2009, Article ID 604695, 18 pagesdoi:10.1155/2009/604695

Research ArticleAnalytical Solution of the Hyperbolic HeatConduction Equation for Moving Semi-InfiniteMedium under the Effect of Time-DependentLaser Heat Source

R. T. Al-Khairy and Z. M. AL-Ofey

Department of Mathematics, King Faisal University, P.O. Box 1982, Dammam 31413, Saudi Arabia

Correspondence should be addressed to Z. M. AL-Ofey, [email protected]

Received 28 June 2008; Revised 2 March 2009; Accepted 12 May 2009

Recommended by George Jaiani

This paper presents an analytical solution of the hyperbolic heat conduction equation for movingsemi-infinite medium under the effect of time dependent laser heat source. Laser heating ismodeled as an internal heat source, whose capacity is given by g(x, t) = I(t)(1 − R)μe−μx whilethe semi-infinite body has insulated boundary. The solution is obtained by Laplace transformsmethod, and the discussion of solutions for different time characteristics of heat sources capacity(constant, instantaneous, and exponential) is presented. The effect of absorption coefficients on thetemperature profiles is examined in detail. It is found that the closed form solution derived fromthe present study reduces to the previously obtained analytical solution when the medium velocityis set to zero in the closed form solution.

Copyright q 2009 R. T. Al-Khairy and Z. M. AL-Ofey. This is an open access article distributedunder the Creative Commons Attribution License, which permits unrestricted use, distribution,and reproduction in any medium, provided the original work is properly cited.

1. Introduction

An increasing interest has arisen recently in the use of heat sources such as lasers andmicrowaves, which have found numerous applications related to material processing(e.g., surface annealing, welding and drilling of metals, and sintering of ceramics) andscientific research (e.g., measuring physical properties of thin films, exhibiting microscopicheat transport dynamics). Lasers are also routinely used in medicine. In literature, manyresearchers have investigated the heat transfer for moving medium under the effect of theclassical Fourier heat conduction model [1, 3–6].

In applications involving high heating rates induced by a short-pulse laser, the typicalresponse time is in the order of picoseconds [7–10]. In such application, the classical Fourierheat conduction model fails, and the use of Cattaneo-Vernotte constitution is essential [11,12].

2 Journal of Applied Mathematics

In this constitution, it is assumed that there is a phaselag between the heat flux vector(q) and the temperature gradient (∇T). As a result, this constitution is given as

q + τ∂q

∂t= −κ∇T, (1.1)

where κ is the thermal conductivity and τ is the relaxation time (phase lag in heat flux). Theenergy equation under this constitution is written as

ρCpτ∂2T

∂t2+ ρCp

∂T

∂t= κ∇2T +

(τ∂g

∂t+ g). (1.2)

In the literature, numerous works have been conducted using the microscopichyperbolic heat conduction model [10, 13–18]. To the authors’ knowledge, the thermalbehavior of moving semi-infinite medium subject to Time-Dependent laser heat source,under the effect of the hyperbolic heat conduction model, has not been investigated yet. Inthe present work, the thermal behavior of moving semi-infinite medium subject to Time-Dependent laser heat source, under the effect of the hyperbolic heat conduction model, isinvestigated.

2. Mathematical Model

In this paper heat distribution in a moving semi-infinite medium due to internal laser heatsource is considered. Our medium at t = 0 is occupying the region x ≥ 0 with insulatedsurface at x = 0. Moreover, at time t = 0, the temperature field within the medium is uniformwith a value T0 and stationary.

We consider first a semi-infinite medium moving with a constant velocity u in thedirection of the x-axis, if heat generation is present within the material, the balance law forthe internal energy can be expressed in terms of T as

ρCpDT

Dt+∂q

∂x= g(x, t), (2.1)

where

D

Dt≡ ∂

∂t+ u

∂

∂x, (2.2)

which denotes the material derivative.If the body is in motion, the Maxwell-Cattaneo law (1.1) leads to a paradoxical result so

that by replacing the partial time derivative in (1.1) with the material derivative operator, theparadox is removed, and the material form of the Maxwell-Cattaneo law is strictly Galileaninvariant. Therefore, (1.1) is replaced by [19]

q + τ(∂q

∂t+ u

∂q

∂x

)= −κ∂T

∂x. (2.3)

Journal of Applied Mathematics 3

Elimination of q between (2.1) and (2.3) yields the heat transport equation

τ∂2T

∂t2+∂T

∂t+ u

∂T

∂x+ 2τu

∂2T

∂x∂t+ τ(u2 − c2

)∂2T

∂x2=

1ρCp

[g + τ

∂g

∂t+ τu

∂g

∂x

], (2.4)

where the initial and boundary conditions are given by

T(x, 0) = T0,∂T

∂t

∣∣∣∣t=0

=g

ρCp, x ≥ 0, (2.5)

∂T

∂x(0, t)c = 0,

∂T

∂x(∞, t) = 0, t > 0. (2.6)

The relaxation time is related to the speed of propagation of thermal wave in the medium, c,by

τ =α

c2. (2.7)

The heat source term in (2.4) which describes the absorption of laser radiation is modeled as[20]

g(x, t) = I(t)(1 − R)μ exp(−μx), (2.8)

where I(t) is the laser incident intensity, R is the surface reflectance of the body, and μ is theabsorption coefficient.

We consider semi-infinite domains, which have initial temperature equal to theambient one. The following dimensionless variables are defined:

X =x

2cτ, η =

t

2τ, θ =

(T − T0)(Tm − T0)

, U =u

c, S =

τg

ρCp(Tm − T0). (2.9)

Equation (2.4) is expressed in terms of the dimensionless variables (2.9) as

2∂θ

∂η+ 2U

∂θ

∂X+∂2θ

∂η2+ 2U

∂2θ

∂η∂X−(

1 −U2) ∂2θ

∂X2=[

4S + 2∂S

∂η+ 2U

∂S

∂X

]. (2.10)

The dimensionless heat source capacity according to (2.8) is

S = ψ0φ(η)

exp(−βX), (2.11)

where

ψ0 =τIr(1 − R)μρCpT0

, φ(η)=I(2τη

)Ir

, β = 2cτμ. (2.12)


The dimensionless initial conditions for the present problem are

θ(X, 0) = 0, (2.13)

∂θ

∂η(X, 0) = 2ψ0φ(0) exp

(−βX). (2.14)

The results from the assumption are that there is no heat flow in the body at the initialmoment [21], that is,

q(X, 0) = 0. (2.15)

The dimensionless boundary conditions are

∂θ

∂X

(0, η

)= 0, (2.16)

∂θ

∂X

(∞, η)= 0, η > 0. (2.17)

We substitute (2.11) for S in (2.10) to obtain

2∂θ

∂η+ 2U

∂θ

∂X+∂2θ

∂η2+ 2U

∂2θ

∂X∂η−(

1 −U2) ∂2θ

∂X2= 2ψ0

[(2 −Uβ)φ(η) + ∂φ

∂η

]exp

(−βX).(2.18)

3. Analytical Solution

Taking the Laplace transform of (2.18), using the initial conditions given by (2.13) and (2.14),yields

(1 −U2

) ∂2θ

∂X2− 2U(1 + s)

∂θ

∂X− s(2 + s)θ = −2ψ0

(2 + s −Uβ)φ exp

(−βX), (3.1)

where

θ(X, s) = L[θ(X, η

)], (3.2)

φ(s) = L[φ(η)]. (3.3)

The transformed boundary conditions given by (2.16) and (2.17) are

dθ

dX(0, s) = 0, (3.4)

dθ

dX(∞, s) = 0. (3.5)


Equation (3.1) has homogeneous (θh) and particular (θp) solutions. Therefore, θ yields

θ = θh + θp. (3.6)

The mathematical arrangement of the solution of (3.1) is given in Appendix A.Consequently, (3.1) for X > 0 yields

θ(X, η

)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2ψ0 exp(−βX)f(η), for η ≤ X

1 +U,

2ψ0 exp(−βX)f(η) − 2βψ0

∫ηX/1+U

exp(−y)I0

⎛⎝√

a

√(y +

UX

a

)2

− X2

a2

⎞⎠

×h8(η − y)dy − 2βψ0(1 +U)

∫ηX/1+U

exp(−y)I0

⎛⎝√a

√(y+

UX

a

)2

− X2

a2

⎞⎠

×h7(η − y)dy, for η >

X

1 +U,

(3.7)

where

0 < U < 1, (3.8)

f(η)=

12γ

∫η0φ(r)

{(γp −Uβ

)exp

[γm(η − r)] + (γm +Uβ

)exp

[−γp(η − r)]}dr, (3.9)

h7(η)= f

(η)+U2

∫η0φ(r)

[D1 exp

(−2(η − r)) +D2

+D3 exp(γm(η − r)) +D4 exp

(−γp(η − r))]dr,(3.10)

h8(η)= U

√a

∫η0

exp(−v)I1(√

av)h7(η − v)dv, (3.11)

γ =√

1 + β2, (3.12)

γm = γ − (1 −Uβ), (3.13)

γp = γ +(1 −Uβ), (3.14)

D1 =−Uβ

2(2 + γm

)(−2 + γp) , D2 =

−2 +Uβ2γmγp

,

D3 =

(γ + 1

)2γγm

(2 + γm

) , D4 =

(γ − 1

)2γγp

(−2 + γp) .

(3.15)


4. Solutions for Special Cases of Heat Source Capacity

The temperature distributions resulting from any specified time characteristics of the heatsource φ(η) are available using the general hyperbolic solution (3.7)–(3.14). However, forsome particular φ(η) the general solution can be considerably simplified. Some of such casesare discussed below.

4.1. Source of Constant Strength: φ(η) = 1

This case may serve as a model of a continuously operated laser source. It may be also usedas a model of a long duration laser pulse when the short times (of the order of few or tens τ)are considered. For φ(η) = 1, (3.9) and (3.10) are reduced, respectively, to

f1(η)

=γp(γp +Uβ

)exp

(γmη

) − γm(γm −Uβ) exp(−γpη) − 2γ

(2 −Uβ)

2γγmγp, (4.1)

h7(1)(η)= f1

(η)+U2

{D1 exp

(−η) sinh(η)+D2η +

D3

γm

[exp

(γmη

) − 1]

+D4

γp

[1 − exp

(−γpη)]}.

(4.2)

4.2. Instantaneous Source: φ(η) = δ(η)

In this case, (3.9) and (3.10) take the form, respectively,

f2(η)=

12γ[(γp +Uβ

)exp

(γmη

)+(γm −Uβ) exp

(−γpη)], (4.3)

h7(2)(η)= f2

(η)+U2[D1 exp

(−2η)+D2 +D3 exp

(γmη

)+D4 exp

(−γPη)]. (4.4)

4.3. Exponential Source: φ(η) = exp(−νη)In this case (3.9) and (3.10) are as follows, respectively,

f3(η)=

12γ(ν − γp

)(ν + γm

) [(ν + γm)(γm −Uβ) exp

(−γpη) + (ν − γp)(γp +Uβ) exp(γmη

)

+2γ(2 − ν −Uβ) exp

(−νη)],(4.5)

h7(3)(η)= f3

(η)+U2

{D1

(−2 + ν)exp

(−2η)+D2

ν+

D3(ν + γm

) exp(γmη

)

+D4(

ν − γp) exp

(−γpη) +D5 exp(−νη)

},

(4.6)


0

0.5

1

1.5

2

2.5

3

3.5

θ U = 0.7

U = 0

HyperbolicParabolic

0 2 4 6 8

X

Figure 1: Dimensionless temperature distributions resulting from the hyperbolic and parabolic modelswith dimensionless velocity of the medium for the heat source of constant strength; φ(η) = 1, ψ0 =1, and β = 1, η = 3.

0

2

4

6

θ X = 0

X = 3

X = 5

HyperbolicParabolic

0 2 4 6 8 10

η

Figure 2: Variation of dimensionless temperature with dimensionless time at different points of the bodyfor the heat source of constant strength; φ(η) = 1, ψ0 = 1, and U = 0.1.

where

D5 =D1

(2 − ν) −D2

ν− D3(

ν + γm) − D4(

ν − γp) . (4.7)

5. Results and Discussion

Using the solutions for arbitrary φ(η) and the solutions for the special cases we calculated,with the aid of the program Mathematica 5.0, and we performed calculations for metalsputting ψ0 = 1 and β = 0.5 or 1, since we assumed that typical values of the model parameters


0

0.25

0.5

0.75

1

1.25

1.5

1.75

θ

β = 0.3

β = 1

β = 3

HyperbolicParabolic

0 1 2 3 4 5

X

Figure 3: Dimensionless temperature distributions at η = 1 for the heat source of constant strength andvarious values of β; φ(η) = 1, ψ0 = 1, and U = 0.1.

0

0.1

0.2

0.3

0.4

0.5

0.6

θU = 0

U = 0.5

HyperbolicParabolic

0 1 2 3 4

X

Figure 4: Dimensionless temperature distributions resulting from the hyperbolic and parabolic modelswith dimensionless velocity of the medium for the instantaneous heat source; φ(η) = δ(η), ψ0 = 1, β =3, and η = 2.

for metals are: μ of the order of 107–108 m−1,R of the order of 0.9, τ of the order of 10−13–10−11 s,and c of the order of 103–104 m/s [22–25]. Some solutions for other values of β are alsopresented to set off the specific features of our model. The results of calculations for varioustime characteristics of the heat source capacity are shown in Figures 1–9. Moreover, thevelocity of the medium was assumed not to exceed the speed of heat propagation.

The hyperbolic and parabolic solutions for the heat source of constant strength [φ(η) =1] are presented in Figures 1–3. Figure 1 shows the temperature distribution in the body forthe two values of dimensionless velocity of the medium, U = 0, 0.7. Figure 2 displays thetime variation of temperature at the three points of the body, X = 0, 3, 5. It is clearly seenthat for small X the temperatures predicted by the hyperbolic model are greater than thecorresponding values for the Fourier model, whereas in the region of intermediate valuesof X, the situation is just the opposite. For large X (X � η) the hyperbolic and parabolic


0

0.2

0.4

0.6

0.8

1

1.2

θ

U = 0

U = 0.6

HyperbolicParabolic

0 2 4 6 8

X

Figure 5: Dimensionless temperature distributions with dimensionless velocity of the medium for theinstantaneous heat source; φ(η) = δ(η), ψ0 = 1, β = 1, and η = 2.

0

0.1

0.2

0.3

0.4

θ

η = 1

η = 2

η = 4η = 3

0 1 2 3 4 5

X

Figure 6: Dimensionless temperature distributions resulting from the hyperbolic model for theinstantaneous heat source; φ(η) = δ(η), ψ0 = 1, β = 5, and U = 0.1.

solutions tend to overlap. This behaviour can be explained as follows. In both models, theheat production is concentrated at the edge of the body. The same amounts of energy aregenerated continuously in both models, but in the case of hyperbolic models, because of thefinite speed of heat conduction, more energy is concentrated at the origin of X axis. Thisresults in the higher “hyperbolic” temperature in this region and the lower in the regionof intermediate X values. In Figure 3, we compare the temperature distributions at η = 1resulting from the hyberbolic and parabolic for the three values of β (β = 0.3, 1, and 3).For large β, that is, when the slope of the space characteristics of the heat source capacityincreases, in the hyperbolic solution, a blunt wave front can be observed. Figures 4–7 depictthe results of calculations for the instantaneous heat source [φ(η) = δ(η)]. A striking featureof the hyperbolic solutions is that the instantaneous heat source gives rise to a thermal pulsewhich travels along the medium and decays exponentially with time while dissipating itsenergy. During a period η, the maximum of the pulse moves over a distance X = η(1 + U).These effects are shown pictorially in Figures 4 and 6. Figure 4 presents the temperature


0

0.2

0.4

0.6

0.8

1

θ

β = 1

β = 2

β = 5

0 1 2 3 4 5

X

HyperbolicParabolic

Figure 7: Dimensionless temperature distributions at η = 1 for the instantaneous heat source and variousvalues of β; φ(η) = δ(η), ψ0 = 1, and U = 0.1.

0

0.5

1

1.5

θ

U = 0.8

U = 0.4

U = 0.2

U = 0

0 2 4 6 8

X

Figure 8: Dimensionless temperature distributions with dimensionless velocity of the medium from thehyperbolic model for the exponential heat source ; φ(η) = exp(−0.4η), ψ0 = 1, β = 1, and η = 3.

distributions in the body for β = 5 and U = 0, 0.5, but Figure 6 presents the temperaturedistributions in the body for β = 5 and η = 1, 2, 3, 4. It is seen that the pulse is notsharp but blunt exponentially, which results from the fact that in our model the heat sourcecapacity decays exponentially along the x-axis. Figure 5 gives the hyperbolic and parabolictemperature distribution in the body at time η = 2 for the two values of dimensionlessvelocity of the medium, U = 0, 0.6. Figure 7 gives the hyperbolic and parabolic temperaturedistribution in the body at time η = 1 and velocity of the medium U = 0.1 for various valuesof β. As shown in Figure 7, the smaller β is, the more blunt the pulse and the shorter is thetime of its decay is . After the decay of the pulse, the differences between the hyperbolic andparabolic solutions become only quantitative, and they vanish in short time. Figures 8 and 9depict the results of calculations for the exponential heat source [φ(η) = exp(−νη)]. Figure 8gives the hyperbolic temperature distribution in the body at time η = 3 for the four valuesof dimensionless velocity of the medium, U = 0, 0.2, 0.4, 0.8. Figure 9 shows the temperaturedistribution in the body for the three values of dimensionless β (β = 0.3, 1, 3). The results are


0

0.2

0.4

0.6

0.8

1

1.2

1.4

θ

β = 0.3

β = 1

β = 3

HyperbolicParabolic

0 1 2 3 4 5

X

Figure 9: Dimensionless temperature distributions at η = 1 for the exponential heat source and variousvalues of β; φ(η) = exp(−0.4η), ψ0 = 1, and U = 0.1.

compared with those obtained from an analytical model by Lewendowska [21]. For U = 0,our results are the same as those reported by Lewendowska [21].

6. Conclusions

This paper presents an analytical solution of the hyperbolic heat conduction equation formoving semi-infinit medium under the effect of Time-Dependent laser heat source. Laserheating is modeled as an internal heat source, whose capacity is given by (2.8) while thesemi-infinit body was insulated boundary. The heat conduction equation together with itsboundary and initial conditions have been written in a dimensionless form. By employing theLaplace transform technique, an analytical solution has been found for an arbitrary velocityof the medium variation. The temperature of the semi-infinit body is found to increase at largevelocities of the medium. The results are compared with those obtained from an analyticalmodel by Lewendowska [21]. For U = 0, our results are the same as those reported byLewendowska [21]. A blunt heat wavefront can be observed when the slope of the spacecharacteristics of the heat source capacity (i.e., the value of β) is large.

Appendix

A. Solution of Heat Transfer Equation

The characteristic equation for the homogeneous solution can be written as

r2 − 2U(1 + s)1 −U2

r − s(2 + s)1 −U2

= 0, (A.1)

which yields the solution of

r1,2 =U(1 + s)1 −U2

± 11 −U2

√(1 + s)2 − (1 −U2), (A.2)

where 0 < U < 1.


Therefore, the homogeneous solution (θh) yields

θh = c1 exp(r1X) + c2 exp(r2X), (A.3)

or

θh =[c1 exp

(U(1 + s)

a− 1a

√(1 + s)2 − a

)X + c2 exp

(U(1 + s)

a+

1a

√(1 + s)2 − a

)X

],

(A.4)

where a = 1 −U2.For the particular solution, one can propose θp = A0 exp(−βX).Consequently, substitution of θp into (3.1) results in

(1 −U2

)β2A0 exp

(−βX) + 2U(1 + s)βA0 exp(−βX) − s(2 + s)A0 exp

(−βX)

= −2ψ0(2 + s −Uβ)φ exp

(−βX),(A.5)

where

A0 =−2ψ0

(2 + s −Uβ)φ[

(1 −U2)β2 + 2U(1 + s)β − s(2 + s)] , (A.6)

or

θ = c1 exp(U(1 + s)

a− 1a

√(1 + s)2 − a

)X

+ c2 exp(U(1 + s)

a+

1a

√(1 + s)2 − a

)X

+2ψ0


(−βX)(s − γm

)(s + γp

) .

(A.7)

Since Re(s) > 0, 0 < U < 1 and dθ/dX(∞, s) = 0, then c2 = 0.Therefore,

θ = c1 exp(U(1 + s)

a− 1a

√(1 + s)2 − a

)X +

2ψ0(2 + s −Uβ)φ exp

(−βX)(s − γm

)(s + γp

) . (A.8)


By applying the boundary condition (3.4), we can obtain c1, that is,

dθ

dX={c1

(U(1 + s)

a− 1a

√(1 + s)2 − a

)× exp

(U(1 + s)

a− 1a

√(1 + s)2 − a

)X

−2βψ0(2 + s −Uβ)φ exp

(−βX)(s − γm

)(s + γp

)}X=0

= 0,

(A.9)

or

c1 =2βψ0a

(2 + s −Uβ)φ(

U(1 + s) −√(1 + s)2 − a

)(s − γm

)(s + γp

) . (A.10)

Hence,

θ =2βψ0a


(U(1 + s)/a − (1/a)

√(1 + s)2 − a

)X

(U(1 + s) −

√(1 + s)2 − a

)(s − γm

)(s + γp

)

+2ψ0


(−βX)(s − γm

)(s + γp

) .

(A.11)

Let H1 and H2 be

H1 =a(2 + s −Uβ)φ exp

(U(1 + s)/a − (1/a)

√(1 + s)2 − a

)X

(U(1 + s) −

√(1 + s)2 − a

)(s − γm

)(s + γp

)

−

⎡⎢⎢⎣

exp(−(X(1 + s))/(1 +U)) exp(−(X/a)

(√(1 + s)2 − a − (1 + s)

))√(1 + s)2 − a

×U

[(1 + s) −

√(1 + s)2 − a

]√(1 + s)2 − a

×

((1 + s)2 − a

)(2 + s −Uβ)φ(

(1 + s)2 − 1)(s − γm

)(s + γp

)⎤⎥⎥⎦


− (1 +U)

⎡⎢⎢⎣

exp(−X(1 + s)/(1 +U)) exp(−(X/a)

(√(1 + s)2 − a − (1 + s)

))√(1 + s)2 − a

×

((1 + s)2 − a

)(2 + s −Uβ)φ(

(1 + s)2 − 1)(s − γm

)(s + γp

)⎤⎥⎦

= −H3 − (1 +U)H4,

(A.12)

H2 =

(2 + s −Uβ)φ(s − γm

)(s + γp

) , (A.13)

Consequently,

θ(X, η

)= £−1θ = 2βψ0£−1H1 + 2ψ0 exp

(−βX)£−1H2

= −2βψ0£−1H3 − 2βψ0(1 +U)£−1H4 + 2ψ0 exp(−βX)£−1H2.

(A.14)

To obtain the inverse Laplace transformation of functions H2,H3, and H4, we use theconvolution for Laplace transforms.

The Laplace inverse of H2 can be obtained as

£−1H2 =1

2γ

∫η0φ(r)

[(γp +Uβ

)exp

(γm(η − r)) + (γm −Uβ) exp

(−γp(η − r))]dr= f

(η).

(A.15)

To obtain the inverse Laplace transformation of function H3, we use the convolutionfor Laplace transforms:

£−1H3 = £−1[H5(s)H6(s)H7(s)] =∫η

0h5(y)∫η−y

0h6(v)h7

(η − y − v)dv dy, (A.16)

where

h5(η)= £−1H5 = £−1

⎧⎪⎪⎨⎪⎪⎩

exp(−(X(1 + s)/(1 +U))) exp(−(X/a)

(√(1 + s)2 − a − (1 + s)

))√(1 + s)2 − a

⎫⎪⎪⎬⎪⎪⎭

= exp(−η)£−1

⎧⎨⎩

exp(−(X/(1 +U))s) exp(−(X/a)

(√s2 − a − s

))√s2 − a

⎫⎬⎭.

(A.17)


It is noted from the Laplace inversion that [26]

£−1{H(s − b)} = exp(bη)h(η), (A.18)

£−1

⎧⎨⎩

exp(−k(√

s2 − c2 − s))

√s2 − c2

⎫⎬⎭ = I0

(c√η2 + 2kη

), k ≥ 0, (A.19)

£−1{exp(−bs)H(s)}=

⎧⎨⎩h(η − b) at η > b,

0 at η < b,b > 0. (A.20)

Therefore,

h5(η)= £−1H5 = exp

(−η)I0

⎛⎝√

a

√(η +

UX

a

)2

− X2

a2

⎞⎠, η >

X

1 +U. (A.21)

Similarly, £−1H6 can be obtained, that is,

h6(η)= £−1H6 = £−1

⎧⎪⎪⎨⎪⎪⎩U

[(1 + s) −

√(1 + s)2 − a

]√(1 + s)2 − a

⎫⎪⎪⎬⎪⎪⎭

= U exp(−η)£−1

⎧⎨⎩[s −

√s2 − a

]√s2 − a

⎫⎬⎭.

(A.22)

It is noted from the Laplace inversion that [26]

£−1

⎧⎪⎨⎪⎩

[s −

√s2 − c2

]ν√s2 − c2

⎫⎪⎬⎪⎭ = cνIν

(cη), ν > −1. (A.23)

Therefore,

h6(η)= £−1H6 =

√aU exp

(−η)I1(√

aη). (A.24)


To obtain the inverse transformation of function H7, we use the convolution forLaplace transforms:

h7(η)= £−1H7 = £−1

⎧⎨⎩

((1 + s)2 − a

)(2 + s −Uβ)φ(

(1 + s)2 − 1)(s − γm

)(s + γp

)⎫⎬⎭

= f(η)+U2

∫η0φ(r)

[D1 exp

(−2(η − r)) +D2 +D3 exp

(γm(η − r))

+D4 exp(−γp(η − r))]dr.

(A.25)

Substituting (A.21), (A.24), and (A.25) into (A.16), it yields

£−1H3 =∫ηX/(1+U)

exp(−y)I0

⎛⎝√

a

√(y +

UX

a

)2

− X2

a2

⎞⎠ × h8

(η − y)dy, (A.26)

where

h8(η)=∫η

0

√aU exp(−v)I1

(√av) × h7

(η − v)dv. (A.27)

Similarly, £−1H4 can be obtained, after using the convolution for Laplace transformsand (A.21) and (A.25):

£−1H4 = £−1[H5(s)H7(s)]

=∫ηX/(1+U)

exp(−y)I0

⎛⎝√

a

√(y +

UX

a

)2

− X2

a2

⎞⎠ × h7

(η − y)dy. (A.28)

Substituting (A.15), (A.26), and (A.28) into (A.14), it yields

θ(X, η

)=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

2ψ0 exp(−βX)f(η), for η ≤ X

1 +U,

2ψ0 exp(−βX)f(η) − 2βψ0

∫ηX/(1+U)

exp(−y)I0

⎛⎝√

a

√(y +

UX

a

)2

− X2

a2

⎞⎠

×h8(η − y)dy − 2βψ0(1 +U)

∫ηX/(1+U)

exp(−y)I0

⎛⎝√

a

√(y+

UX

a

)2

−X2

a2

⎞⎠

×h7(η − y)dy, for η >

X

1 +U.

(A.29)


Nomenclature

Cp: Specific heat at constant pressure, J/(kg K)g: Capacity of internal heat source, W/m3

I: Laser incident intensity, W/m2

Ir : Arbitrary reference laser intensityI0: Modified Bessel function, 0th orderI1: Modified Bessel function, 1th orderL: Laplace operatorR: Surfase reflectances: Laplace variableq: Heat flux vector, W/m2

t: Time, sT : Temperature, KTm, T0: Arbitrary reference temperatures, Kc: Speed of heat propagation = (α/τ)1/2, m/sx, y, z: Cartesian coordinates, mX,Y,Z: Dimensionless cartesian coordinatesS: Dimensionless capacity of internal heat sourceu: Velocity of the medium, m/sU: Dimensionless velocity of the medium, u/c.

Greek symbols

α: Thermal diffusivity = κ/(ρCp), m2/sκ: Thermal conductivity, W/(mK)τ : Relaxation time of heat flux, sβ: Dimensionless absorption coefficientγ, γm, γp: Auxiliary coefficients defined by (3.12), (3.13), (3.14), respectivelyφ: Dimensionless rate of energy absorbed in the mediumμ: Absorption coefficientθ: Dimensionless temperaturesρ: Densityη: dimensionless timeψ0: Internal heat sourceθ: Laplace transformation of dimensionless temperature.

References

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[12] P. Vernotte, “Some possible complications in the phenomenon of thermal conduction,” Compte Rendus,vol. 252, pp. 2190–2191, 1961.

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