Journal of Applied Mathematics

Journal of Applied Mathematics / 2009 / Article

Research Article | Open Access

Volume 2009 |Article ID 604695 | 18 pages | https://doi.org/10.1155/2009/604695

Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium under the Effect of Time-Dependent Laser Heat Source

Academic Editor: George Jaiani
Received28 Jun 2008
Revised02 Mar 2009
Accepted12 May 2009
Published06 Aug 2009

Abstract

This paper presents an analytical solution of the hyperbolic heat conduction equation for moving semi-infinite medium under the effect of time dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by 𝑔(𝑥,𝑡)=𝐼(𝑡)(1−𝑅)𝜇𝑒−𝜇𝑥 while the semi-infinite body has insulated boundary. The solution is obtained by Laplace transforms method, 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 the temperature profiles is examined in detail. It is found that the closed form solution derived from the present study reduces to the previously obtained analytical solution when the medium velocity is set to zero in the closed form solution.

1. Introduction

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

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

In this constitution, it is assumed that there is a phaselag between the heat flux vector (ğ‘ž) and the temperature gradient (∇𝑇). As a result, this constitution is given as ğ‘ž+ğœğœ•ğ‘žğœ•ğ‘¡=−𝜅∇𝑇,(1.1) where 𝜅 is the thermal conductivity and 𝜏 is the relaxation time (phase lag in heat flux). The energy equation under this constitution is written as 𝜌𝐶𝑝𝜏𝜕2𝑇𝜕𝑡2+𝜌𝐶𝑝𝜕𝑇𝜕𝑡=𝜅∇2𝜏𝑇+𝜕𝑔𝜕𝑡+𝑔.(1.2)

In the literature, numerous works have been conducted using the microscopic hyperbolic heat conduction model [10, 13–18]. To the authors' knowledge, the thermal behavior 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. In the 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, is investigated.

2. Mathematical Model

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

We consider first a semi-infinite medium moving with a constant velocity 𝑢 in the direction of the 𝑥-axis, if heat generation is present within the material, the balance law for the internal energy can be expressed in terms of 𝑇 as 𝜌𝐶𝑝𝐷𝑇+ğ·ğ‘¡ğœ•ğ‘žğœ•ğ‘¥=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, the paradox is removed, and the material form of the Maxwell-Cattaneo law is strictly Galilean invariant. Therefore, (1.1) is replaced by [19] î‚µğ‘ž+ğœğœ•ğ‘žğœ•ğ‘¡+ğ‘¢ğœ•ğ‘žî‚¶ğœ•ğ‘¥=−𝜅𝜕𝑇𝜕𝑥.(2.3) Elimination of ğ‘ž between (2.1) and (2.3) yields the heat transport equation 𝜏𝜕2𝑇𝜕𝑡2+𝜕𝑇𝜕𝑡+𝑢𝜕𝑇𝜕𝜕𝑥+2𝜏𝑢2𝑇𝑢𝜕𝑥𝜕𝑡+𝜏2−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|||𝜕𝑡𝑡=0=𝑔𝜌𝐶𝑝,𝑥≥0,(2.5)𝜕𝑇𝜕𝑥(0,𝑡)=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 𝛼𝜏=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)𝜇exp(−𝜇𝑥),(2.8) where 𝐼(𝑡) is the laser incident intensity, 𝑅 is the surface reflectance of the body, and 𝜇 is the absorption coefficient.

We consider semi-infinite domains, which have initial temperature equal to the ambient one. The following dimensionless variables are defined: 𝑥𝑋=𝑡2𝑐𝜏,𝜂=2𝜏,𝜃=𝑇−𝑇0𝑇𝑚−𝑇0𝑢,𝑈=𝑐,𝑆=𝜏𝑔𝜌𝐶𝑝𝑇𝑚−𝑇0.(2.9) Equation (2.4) is expressed in terms of the dimensionless variables (2.9) as 2𝜕𝜃𝜕𝜂+2𝑈𝜕𝜃+𝜕𝜕𝑋2𝜃𝜕𝜂2𝜕+2𝑈2𝜃−𝜕𝜂𝜕𝑋1−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)𝜇𝜌𝐶𝑝𝑇0𝐼,𝜙(𝜂)=(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 initial moment [21], that is, ğ‘ž(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+𝜕𝜕𝑋2𝜃𝜕𝜂2𝜕+2𝑈2𝜃−𝜕𝑋𝜕𝜂1−𝑈2𝜕2𝜃𝜕𝑋2=2𝜓0(2−𝑈𝛽)𝜙(𝜂)+𝜕𝜙𝜕𝜂exp(−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)𝜙exp(−𝛽𝑋),(3.1) where [],𝜃(𝑋,𝑠)=𝐿𝜃(𝑋,𝜂)(3.2)[]𝜙(𝑠)=𝐿𝜙(𝜂).(3.3) The transformed boundary conditions given by (2.16) and (2.17) are 𝑑𝜃𝑑𝑑𝑋(0,𝑠)=0,(3.4)𝜃𝑑𝑋(∞,𝑠)=0.(3.5)

Equation (3.1) has homogeneous (ğœƒâ„Ž) and particular (𝜃𝑝) solutions. Therefore, 𝜃 yields 𝜃=ğœƒâ„Ž+𝜃𝑝.(3.6)

The mathematical arrangement of the solution of (3.1) is given in Appendix A. Consequently, (3.1) for 𝑋>0 yields âŽ§âŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽ©ğœƒ(𝑋,𝜂)=2𝜓0𝑋exp(−𝛽𝑋)𝑓(𝜂),for𝜂≤,1+𝑈2𝜓0exp(−𝛽𝑋)𝑓(𝜂)−2𝛽𝜓0𝜂𝑋/1+𝑈exp(−𝑦)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğ‘¦+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2×ℎ8(𝜂−𝑦)𝑑𝑦−2𝛽𝜓0(1+𝑈)𝜂𝑋/1+𝑈exp(−𝑦)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğ‘¦+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2×ℎ7𝑋(𝜂−𝑦)𝑑𝑦,for𝜂>,1+𝑈(3.7) where 10<𝑈<1,(3.8)𝑓(𝜂)=2𝛾𝜂0𝛾𝜙(𝑟)𝑝𝛾−𝑈𝛽exp𝑚+𝛾(𝜂−𝑟)𝑚+𝑈𝛽expâˆ’ğ›¾ğ‘â„Ž(𝜂−𝑟)𝑑𝑟,(3.9)7(𝜂)=𝑓(𝜂)+𝑈2∫𝜂0𝜙𝐷(𝑟)1exp(−2(𝜂−𝑟))+𝐷2+𝐷3𝛾exp𝑚(𝜂−𝑟)+𝐷4expâˆ’ğ›¾ğ‘â„Ž(𝜂−𝑟)𝑑𝑟,(3.10)8√(𝜂)=ğ‘ˆğ‘Žî€œğœ‚0exp(−𝑣)𝐼1î‚€âˆšî‚â„Žğ‘Žğ‘£7√(𝜂−𝑣)𝑑𝑣,(3.11)𝛾=1+𝛽2,𝛾(3.12)𝑚𝛾=𝛾−(1−𝑈𝛽),(3.13)𝑝𝐷=𝛾+(1−𝑈𝛽),(3.14)1=−𝑈𝛽2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 heat source 𝜙(𝜂) are available using the general hyperbolic solution (3.7)–(3.14). However, for some particular 𝜙(𝜂) the general solution can be considerably simplified. Some of such cases are 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 used as 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 𝑓1𝛾(𝜂)=𝑝𝛾𝑝𝛾+𝑈𝛽exp𝑚𝜂−𝛾𝑚𝛾𝑚−𝑈𝛽exp−𝛾𝑝𝜂−2𝛾(2−𝑈𝛽)2𝛾𝛾𝑚𝛾𝑝,ℎ(4.1)7(1)(𝜂)=𝑓1(𝜂)+𝑈2𝐷1exp(−𝜂)sinh(𝜂)+𝐷2𝐷𝜂+3𝛾𝑚𝛾exp𝑚𝜂+𝐷−14𝛾𝑝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, 𝑓21(𝜂)=𝛾2𝛾𝑝𝛾+𝑈𝛽exp𝑚𝜂+𝛾𝑚−𝑈𝛽exp−𝛾𝑝𝜂,ℎ(4.3)7(2)(𝜂)=𝑓2(𝜂)+𝑈2𝐷1exp(−2𝜂)+𝐷2+𝐷3𝛾exp𝑚𝜂+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, 𝑓31(𝜂)=2𝛾𝜈−𝛾𝑝𝜈+𝛾𝑚𝜈+𝛾𝑚𝛾𝑚−𝑈𝛽exp−𝛾𝑝𝜂+𝜈−𝛾𝑝𝛾𝑝𝛾+𝑈𝛽exp𝑚𝜂,ℎ+2𝛾(2−𝜈−𝑈𝛽)exp(−𝜈𝜂)(4.5)7(3)(𝜂)=𝑓3(𝜂)+𝑈2𝐷1𝐷(−2+𝜈)exp(−2𝜂)+2𝜈+𝐷3𝜈+𝛾𝑚𝛾exp𝑚𝜂+𝐷4𝜈−𝛾𝑝exp−𝛾𝑝𝜂+𝐷5,exp(−𝜈𝜂)(4.6) where 𝐷5=𝐷1−𝐷(2−𝜈)2𝜈−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 metals putting 𝜓0=1 and 𝛽=0.5 or 1, since we assumed that typical values of the model parameters for metals are: 𝜇 of the order of 107-108m−1, 𝑅 of the order of 0.9, 𝜏 of the order of 10−13-10−11s, and 𝑐 of the order of 103-104m/s [22–25]. Some solutions for other values of 𝛽 are also presented to set off the specific features of our model. The results of calculations for various time characteristics of the heat source capacity are shown in Figures 1–9. Moreover, the velocity 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 for the two values of dimensionless velocity of the medium, 𝑈=0,0.7. Figure 2 displays the time variation of temperature at the three points of the body, 𝑋=0,3,5. It is clearly seen that for small 𝑋 the temperatures predicted by the hyperbolic model are greater than the corresponding values for the Fourier model, whereas in the region of intermediate values of 𝑋, the situation is just the opposite. For large 𝑋   (𝑋≫𝜂) the hyperbolic and parabolic solutions tend to overlap. This behaviour can be explained as follows. In both models, the heat production is concentrated at the edge of the body. The same amounts of energy are generated continuously in both models, but in the case of hyperbolic models, because of the finite speed of heat conduction, more energy is concentrated at the origin of 𝑋 axis. This results in the higher “hyperbolic” temperature in this region and the lower in the region of intermediate 𝑋 values. In Figure 3, we compare the temperature distributions at 𝜂=1 resulting from the hyberbolic and parabolic for the three values of 𝛽(𝛽=0.3,1,and3). For large 𝛽, that is, when the slope of the space characteristics of the heat source capacity increases, in the hyperbolic solution, a blunt wave front can be observed. Figures 4–7 depict the results of calculations for the instantaneous heat source [𝜙(𝜂)=𝛿(𝜂)]. A striking feature of the hyperbolic solutions is that the instantaneous heat source gives rise to a thermal pulse which travels along the medium and decays exponentially with time while dissipating its energy. 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 distributions in the body for 𝛽=5 and 𝑈=0,0.5, but Figure 6 presents the temperature distributions in the body for 𝛽=5 and 𝜂=1,2,3,4. It is seen that the pulse is not sharp but blunt exponentially, which results from the fact that in our model the heat source capacity decays exponentially along the 𝑥-axis. Figure 5 gives the hyperbolic and parabolic temperature distribution in the body at time 𝜂=2 for the two values of dimensionless velocity of the medium, 𝑈=0,0.6. Figure 7 gives the hyperbolic and parabolic temperature distribution in the body at time 𝜂=1 and velocity of the medium 𝑈=0.1 for various values of 𝛽. As shown in Figure 7, the smaller 𝛽 is, the more blunt the pulse and the shorter is the time of its decay is . After the decay of the pulse, the differences between the hyperbolic and parabolic solutions become only quantitative, and they vanish in short time. Figures 8 and 9 depict the results of calculations for the exponential heat source [𝜙(𝜂)=exp(−𝜈𝜂)]. Figure 8 gives the hyperbolic temperature distribution in the body at time 𝜂=3 for the four values of dimensionless velocity of the medium, 𝑈=0,0.2,0.4,0.8. Figure 9 shows the temperature distribution in the body for the three values of dimensionless 𝛽(𝛽=0.3,1,3). The results are 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 for moving semi-infinit medium under the effect of Time-Dependent laser heat source. Laser heating is modeled as an internal heat source, whose capacity is given by (2.8)while the semi-infinit body was insulated boundary. The heat conduction equation together with its boundary and initial conditions have been written in a dimensionless form. By employing the Laplace transform technique, an analytical solution has been found for an arbitrary velocity of the medium variation. The temperature of the semi-infinit body is found to increase at large velocities of the medium. The results are 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]. A blunt heat wavefront can be observed when the slope of the space characteristics 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−𝑈2𝑟−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 (ğœƒâ„Ž) yields ğœƒâ„Ž=𝑐1𝑟exp1𝑋+𝑐2𝑟exp2𝑋,(A.3) or ğœƒâ„Ž=𝑐1exp𝑈(1+𝑠)ğ‘Žâˆ’1ğ‘Žâˆš(1+𝑠)2î‚¶âˆ’ğ‘Žğ‘‹+𝑐2exp𝑈(1+𝑠)ğ‘Ž+1ğ‘Žâˆš(1+𝑠)2î‚¶ğ‘‹î‚¹âˆ’ğ‘Ž,(A.4) where ğ‘Ž=1−𝑈2.

For the particular solution, one can propose 𝜃𝑝=𝐴0exp(−𝛽𝑋).

Consequently, substitution of 𝜃𝑝 into (3.1) results in 1−𝑈2𝛽2𝐴0exp(−𝛽𝑋)+2𝑈(1+𝑠)𝛽𝐴0exp(−𝛽𝑋)−𝑠(2+𝑠)𝐴0exp(−𝛽𝑋)=−2𝜓0(2+𝑠−𝑈𝛽)𝜙exp(−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exp𝑈(1+𝑠)ğ‘Žâˆ’1ğ‘Žî”(1+𝑠)2î‚¶ğ‘‹âˆ’ğ‘Ž+𝑐2exp𝑈(1+𝑠)ğ‘Ž+1ğ‘Žî”(1+𝑠)2𝑋+âˆ’ğ‘Ž2𝜓0(2+𝑠−𝑈𝛽)𝜙exp(−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exp𝑈(1+𝑠)ğ‘Žâˆ’1ğ‘Žî”(1+𝑠)2î‚¶âˆ’ğ‘Žğ‘‹+2𝜓0(2+𝑠−𝑈𝛽)𝜙exp(−𝛽𝑋)𝑠−𝛾𝑚𝑠+𝛾𝑝.(A.8)

By applying the boundary condition (3.4), we can obtain 𝑐1, that is, 𝑑𝜃=𝑐𝑑𝑋1𝑈(1+𝑠)ğ‘Žâˆ’1ğ‘Žâˆš(1+𝑠)2î‚¶î‚µâˆ’ğ‘ŽÃ—exp𝑈(1+𝑠)ğ‘Žâˆ’1ğ‘Žâˆš(1+𝑠)2î‚¶ğ‘‹âˆ’âˆ’ğ‘Ž2𝛽𝜓0(2+𝑠−𝑈𝛽)𝜙exp(−𝛽𝑋)𝑠−𝛾𝑚𝑠+𝛾𝑝𝑋=0=0,(A.9) or 𝑐1=2𝛽𝜓0ğ‘Ž(2+𝑠−𝑈𝛽)𝜙√𝑈(1+𝑠)−(1+𝑠)2î‚î€·âˆ’ğ‘Žğ‘ âˆ’ğ›¾ğ‘šî€¸î€·ğ‘ +𝛾𝑝.(A.10) Hence, 𝜃=2𝛽𝜓0ğ‘Ž(2+𝑠−𝑈𝛽)√𝜙exp𝑈(1+𝑠)/ğ‘Žâˆ’(1/ğ‘Ž)(1+𝑠)2î‚ğ‘‹âˆ’ğ‘Žî‚€âˆšğ‘ˆ(1+𝑠)−(1+𝑠)2î‚î€·âˆ’ğ‘Žğ‘ âˆ’ğ›¾ğ‘šî€¸î€·ğ‘ +𝛾𝑝+2𝜓0(2+𝑠−𝑈𝛽)𝜙exp(−𝛽𝑋)𝑠−𝛾𝑚𝑠+𝛾𝑝.(A.11) Let 𝐻1 and 𝐻2 be 𝐻1=ğ‘Ž(2+𝑠−𝑈𝛽)√𝜙exp𝑈(1+𝑠)/ğ‘Žâˆ’(1/ğ‘Ž)(1+𝑠)2î‚ğ‘‹âˆ’ğ‘Žî‚€âˆšğ‘ˆ(1+𝑠)−(1+𝑠)2î‚î€·âˆ’ğ‘Žğ‘ âˆ’ğ›¾ğ‘šî€¸î€·ğ‘ +ğ›¾ğ‘î€¸âˆ’âŽ¡âŽ¢âŽ¢âŽ£î‚€î‚€âˆšexp(−(𝑋(1+𝑠))/(1+𝑈))exp−(𝑋/ğ‘Ž)(1+𝑠)2âˆ’ğ‘Žâˆ’(1+𝑠)√(1+𝑠)2î‚ƒâˆšâˆ’ğ‘ŽÃ—ğ‘ˆ(1+𝑠)−(1+𝑠)2î‚„âˆ’ğ‘Žâˆš(1+𝑠)2Ã—î€·âˆ’ğ‘Ž(1+𝑠)2î€¸âˆ’ğ‘Ž(2+𝑠−𝑈𝛽)𝜙(1+𝑠)2−1𝑠−𝛾𝑚𝑠+ğ›¾ğ‘î€¸âŽ¤âŽ¥âŽ¥âŽ¦âŽ¡âŽ¢âŽ¢âŽ£î‚€âˆ’î‚€âˆšâˆ’(1+𝑈)exp(−𝑋(1+𝑠)/(1+𝑈))exp(𝑋/ğ‘Ž)(1+𝑠)2âˆ’ğ‘Žâˆ’(1+𝑠)√(1+𝑠)2Ã—î€·âˆ’ğ‘Ž(1+𝑠)2î€¸âˆ’ğ‘Ž(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+2𝜓0exp(−𝛽𝑋)£−1𝐻2=−2𝛽𝜓0£−1𝐻3−2𝛽𝜓0(1+𝑈)£−1𝐻4+2𝜓0exp(−𝛽𝑋)£−1𝐻2.(A.14)

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

The Laplace inverse of 𝐻2 can be obtained as £−1𝐻2=12𝛾𝜂0𝛾𝜙(𝑟)𝑝𝛾+𝑈𝛽exp𝑚+𝛾(𝜂−𝑟)𝑚−𝑈𝛽exp−𝛾𝑝(𝜂−𝑟)𝑑𝑟=𝑓(𝜂).(A.15)

To obtain the inverse Laplace transformation of function 𝐻3, we use the convolution for Laplace transforms: £−1𝐻3=£−1𝐻5(𝑠)𝐻6(𝑠)𝐻7=(𝑠)𝜂0ℎ5(𝑦)0ğœ‚âˆ’ğ‘¦â„Ž6(𝑣)ℎ7(𝜂−𝑦−𝑣)𝑑𝑣𝑑𝑦,(A.16) where ℎ5(𝜂)=£−1𝐻5=£−1⎧⎪⎨⎪⎩√exp(−(𝑋(1+𝑠)/(1+𝑈)))exp−(𝑋/ğ‘Ž)(1+𝑠)2âˆ’ğ‘Žâˆ’(1+𝑠)√(1+𝑠)2âŽ«âŽªâŽ¬âŽªâŽ­âˆ’ğ‘Ž=exp(−𝜂)£−1⎧⎪⎨⎪⎩√exp(−(𝑋/(1+𝑈))𝑠)exp−(𝑋/ğ‘Ž)𝑠2âˆ’ğ‘Žâˆ’ğ‘ î‚î‚âˆšğ‘ 2⎫⎪⎬⎪⎭.âˆ’ğ‘Ž(A.17) It is noted from the Laplace inversion that [26] £−1{𝐻(𝑠−𝑏)}=exp(𝑏𝜂)ℎ(𝜂),(A.18)£−1⎧⎪⎨⎪⎩√exp−𝑘𝑠2−𝑐2−𝑠√𝑠2−𝑐2⎫⎪⎬⎪⎭=𝐼0𝑐√𝜂2+2𝑘𝜂,𝑘≥0,(A.19)£−1{exp(−𝑏𝑠)𝐻(𝑠)}=ℎ(𝜂−𝑏)at𝜂>𝑏,0at𝜂<𝑏,𝑏>0.(A.20) Therefore, ℎ5(𝜂)=£−1𝐻5=exp(−𝜂)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğœ‚+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2𝑋,𝜂>1+𝑈.(A.21) Similarly, £−1𝐻6 can be obtained, that is, ℎ6(𝜂)=£−1𝐻6=£−1âŽ§âŽªâŽ¨âŽªâŽ©ğ‘ˆî‚ƒâˆš(1+𝑠)−(1+𝑠)2î‚„âˆ’ğ‘Žâˆš(1+𝑠)2âŽ«âŽªâŽ¬âŽªâŽ­âˆ’ğ‘Ž=𝑈exp(−𝜂)£−1âŽ§âŽªâŽ¨âŽªâŽ©î‚ƒâˆšğ‘ âˆ’ğ‘ 2î‚„âˆ’ğ‘Žâˆšğ‘ 2⎫⎪⎬⎪⎭.âˆ’ğ‘Ž(A.22) It is noted from the Laplace inversion that [26] £−1âŽ§âŽªâŽ¨âŽªâŽ©î‚ƒâˆšğ‘ âˆ’ğ‘ 2−𝑐2𝜈√𝑠2−𝑐2⎫⎪⎬⎪⎭=𝑐𝜈𝐼𝜈(𝑐𝜂),𝜈>−1.(A.23) Therefore, ℎ6(𝜂)=£−1𝐻6=âˆšğ‘Žğ‘ˆexp(−𝜂)𝐼1î‚€âˆšî‚ğ‘Žğœ‚.(A.24)

To obtain the inverse transformation of function 𝐻7, we use the convolution for Laplace transforms: ℎ7(𝜂)=£−1𝐻7=£−1(1+𝑠)2î€¸âˆ’ğ‘Ž(2+𝑠−𝑈𝛽)𝜙(1+𝑠)2−1𝑠−𝛾𝑚𝑠+𝛾𝑝=𝑓(𝜂)+𝑈2𝜂0𝐷𝜙(𝑟)1exp(−2(𝜂−𝑟))+𝐷2+𝐷3𝛾exp𝑚(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)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğ‘¦+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2×ℎ8(𝜂−𝑦)𝑑𝑦,(A.26) where ℎ8(𝜂)=𝜂0âˆšğ‘Žğ‘ˆexp(−𝑣)𝐼1î‚€âˆšî‚ğ‘Žğ‘£Ã—â„Ž7(𝜂−𝑣)𝑑𝑣.(A.27)

Similarly, £−1𝐻4 can be obtained, after using the convolution for Laplace transforms and (A.21) and (A.25): £−1𝐻4=£−1𝐻5(𝑠)𝐻7=(𝑠)𝜂𝑋/(1+𝑈)exp(−𝑦)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğ‘¦+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2×ℎ7(𝜂−𝑦)𝑑𝑦.(A.28)

Substituting (A.15), (A.26), and (A.28) into (A.14), it yields âŽ§âŽªâŽªâŽªâŽªâŽ¨âŽªâŽªâŽªâŽªâŽ©ğœƒ(𝑋,𝜂)=2𝜓0𝑋exp(−𝛽𝑋)𝑓(𝜂),for𝜂≤,1+𝑈2𝜓0exp(−𝛽𝑋)𝑓(𝜂)−2𝛽𝜓0𝜂𝑋/(1+𝑈)exp(−𝑦)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğ‘¦+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2×ℎ8(𝜂−𝑦)𝑑𝑦−2𝛽𝜓0(1+𝑈)𝜂𝑋/(1+𝑈)exp(−𝑦)𝐼0îƒ©âˆšğ‘Žî‚™î‚€ğ‘¦+ğ‘ˆğ‘‹ğ‘Žî‚2−𝑋2ğ‘Ž2×ℎ7𝑋(𝜂−𝑦)𝑑𝑦,for𝜂>.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 intensity
𝐼0:Modified Bessel function, 0th order
𝐼1:Modified Bessel function, 1th order
𝐿:Laplace operator
𝑅:Surfase reflectance
𝑠:Laplace variable
ğ‘ž:Heat flux vector, W/m2
𝑡:Time, s
𝑇:Temperature, K
𝑇𝑚,𝑇0:Arbitrary reference temperatures, K
𝑐:Speed of heat propagation = (𝛼/𝜏)1/2, m/s
𝑥,𝑦,𝑧:Cartesian coordinates, m
𝑋,𝑌,𝑍:Dimensionless cartesian coordinates
𝑆:Dimensionless capacity of internal heat source
𝑢:Velocity of the medium, m/s
𝑈:Dimensionless velocity of the medium, 𝑢/𝑐.

Greek symbols

𝛼:Thermal diffusivity = 𝜅/(𝜌𝐶𝑝), m2/s
𝜅:Thermal conductivity, W/(mK)
𝜏:Relaxation time of heat flux, s
𝛽:Dimensionless absorption coefficient
𝛾,𝛾𝑚,𝛾𝑝∶Auxiliary coefficients defined by (3.12), (3.13), (3.14), respectively
𝜙:Dimensionless rate of energy absorbed in the medium
𝜇:Absorption coefficient
𝜃:Dimensionless temperatures
𝜌:Density
𝜂:dimensionless time
𝜓0:Internal heat source
𝜃:Laplace transformation of dimensionless temperature.

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


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