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Axi-Symmetric Propagation in a Thermoelastic Diffusion with Phase Lags
Rajneesh Kumar1, Lajvinder Singh Reen2, S. K. Garg3
1Department of Mathematics, Kurukshetra University, Kurukshetra, India
2Department of Mathematics, Seth Jai Parkash Mukand Lal Institute of Engineering & Technology, Radaur (Yamunanagar), Haryana, India
3Department of Mathematics, Deen Bandhu Chhotu Ram University of Science and Technology, Sonipat, Haryana, India
Email address
(R. Kumar) 
(L. S. Reen) 
(S. K. Garg) Citation
Rajneesh Kumar, Lajvinder Singh Reen, S. K. Garg. Axi-Symmetric Propagation in a Thermoelastic Diffusion with Phase Lags. American Journal of Science and Technology. Vol. 3, No. 4, 2016, pp. 82-96.
Abstract
The purpose of this paper is to depict the effect of thermal and diffusion phase lags due to axisymmetric heat supply for a disc. The problem is discussed within the context of DPLT and DPLD models. The upper and lower surfaces of the disc are traction free and subjected to an axisymmetric heat supply. The solution is found by using Laplace and Hankel transform technique and a direct approach without the use of potential functions. The analytical expressions of displacements, stresses and chemical potential, temperature and mass concentration are computed in transformed domain. Numerical inversion technique has been applied to obtain the results in the physical domain. Numerically simulated results are depicted graphically. The effect of diffusion and thermal phase-lags are shown on the various components. Some particular cases of result are also deduced from the present investigation.
Keywords
Dual Phase Lag, Isotropic Thermoelastic, Laplace Transform, Hankel Transform, Plane Axisymmetric, Diffusion
1. Introduction
Classical Fourier heat conduction law implies an infinitely fast propagation of a thermal signal which is violated in ultra-fast heat conduction system due to its very small dimensions and short timescales. Catteno [1] and Vernotte [2] proposed a thermal wave with a single phase lag in which the temperature gradient after a certain elapsed time was given by 
, where 
denotes the relaxation time required for thermal physics to take account of hyperbolic effect within the medium. Here when 
, the thermal wave propagates through the medium with a finite speed of 
, where 
is thermal diffusivity. when approaches zero, the thermal wave has an infinite speed and thus the single phase lag model reduces to traditional Fourier model. The dual phase lag model of heat conduction was proposed by Tzou [3] 
, where the temperature gradient 
at a point P of the material at time 
corresponds to the heat flux vector 
at the same time at the time
. Here 
is thermal conductivity of the material. The delay time 
is interpreted as that caused by the microstructural interactions and is called the phase lag of temperature gradient. The other delay time interpreted as the relaxation time due to the fast transient effects of thermal inertia and is called the phase lag of heat flux. This universal model is claimed to be able to bridge the gap between microscopic and macroscopic approaches, covering a wide range of heat transfer models. If 
, Tzou [3] refers to the model as single phase model. Numerous efforts have been invested in the development of an explicit mathematical solution to the heat conduction equation under dual phase lag model. Quintanilla [4] compared two different mathematical hyperbolic models proposed by Tzou. Kumar and Mukhopadhyay [5] investigated the propagation of harmonic waves of assigned frequency by employing the thermoelasticity theory with three phase lags. Chou and Yang [6] discussed two dimensional dual phase lag thermal behaviour in single-/multi-layer structures using CESE method. Zhou et. al [7] proposed an axisymmetric dual-phase-lag bio heat model for laser heating of living tissues. Kumar et. al [8] discussed effect of viscosity on wave propagation in anisotropic thermoelastic medium with three phase lag model. Abbas et. al [9] investigated two-dimensional problem in a transversely isotropic thermoelastic medium without energy dissipation and with two temperatures due to a thermal source. Liu et. al [10] analysed thermal damage in a laser-Irradiated based on non-Fourier model. Ying and Yun [11] built a fractional dual-phase-lag model and the corresponding bio-heat transfer equation. Mondal et. al [12] investigated fractional order two-temperature dual-phase-lag thermoelasticity with variable thermal conductivity. Abdallah [13] used uncoupled thermoelastic model based on dual phase lag to investigate the thermoelastic properties of a semi-infinite medium. Bhattacharya and Kanoria [14] investigated the problem of elasto-thermo-diffusion inside a spherical shell. Kaushal et. al [15] analysed wave propagation in temperature rate dependent thermoelasticity with two temperatures. Kaushal et. al [16] discussed propagation of waves in generalized thermoelastic continua with two temperature. Rukolaine [17] investigated unphysical effects of the dual-phase-lag model of heat conduction. Tripathi et. al [18] analysed generalized thermoelastic diffusion problem in a thick circular plate with axisymmetric heat supply.
Diffusion is defined as the spontaneous movement of the particles from high concentration region to the low concentration region, and it occurs in response to a concentration gradient expressed as the change in concentration due to change in position. Thermal diffusion utilizes the transfer of heat across a thin liquid or gas to accomplish isotope separation. The thermodiffusion in elastic solids is due to coupling of fields of temperature, mass diffusion and that of strain in addition to heat and mass exchange with the environment. Dual phase lag diffusion model was considered by Kumar and Gupta [19, 20], Chiriţă, et. al [21] study the propagation of plane time harmonic waves in the infinite space filled by a time differential dual-phase-lag thermoelastic material. Sherief and Hamza [22] considered the two-dimensional problem of a thick plate whose lower and upper surfaces are traction free and subjected to a given axisymmetric temperature distribution is considered within the context of the theory of generalized thermoelasticity with one relaxation time. Kumar et. al [23] investigated thermomechanical interactions for dual –phase- lag in a homogeneous isotropic thick circular plate in the light of two-temperature thermoelasticity theory.
Here in this investigation, a generalized form of mass diffusion equation is introduced instead of classical Fick's diffusion theory by using two diffusion phase-lags in axisymmetric form. One phase-lag of diffusing mass flux vector, represents the delayed time required for the diffusion of the mass flux and the other phase-lag of chemical potential, represents the delayed time required for the establishment of the potential gradient. The basic equations for the isotropic thermoelastic diffusion medium in the context of dual-phase-lag heat transfer (DPLT) and dual-phase-lag diffusion (DPLD) models in axisymmetric form are presented. The components of displacements, stresses and chemical potential, Temperature and mass concentration are computed numerically. Numerically computed results are depicted graphically. The effect of diffusion and thermal phase-lags are shown on the various components.
2. Basic Equations
The basic equations of motion, heat conduction and mass diffusion in a homogeneous isotropic thermoelastic solid with DPLT and DPLD models in the absence of body forces, heat sources and mass diffusion sources are
(1)
(2)
(3)
and the constitutive relations are
(4)
) (5)
(6)
Where 
, 
are Lame's, 
is the density assumed to be independent of time, D is the diffusivity, P is the chemical potential per unit mass, C is the concentration, 
are components of displacement vector u, K is the coefficient of thermal conductivity, 
is the specific heat at constant strain, 
is small temperature increment, 
is the absolute temperature of the medium, 
is the reference temperature of the body such that 
, a and b are respectively, the coefficients describing the measure of thermodiffusion and mass diffusion effect respectively, 
and 
are the components of stress and strain respectively, 
is dilatation, S is the entropy per unit mass, 
=
=
, 
is the coefficient of linear diffusion expansion and 
is the coefficient of thermal linear expansion. , , 
,
are respectively phase lag of temperature gradient, the phase lag of heat flux, the phase lag of chemical potential, and phase lag of diffusing mass flux vector. In above equations, a comma followed by suffix denotes spatial derivative and a superposed dot denotes derivative with respect to time.
3. Formulation and Solution of the Problem
Consider a disc of diameter 2b occupying the space D defined by 
Let the ring be subjected to an axisymmetric heat supply depending on the radial and axial directions of the cylindrical co-ordinate system. The initial temperature of the disc is given by a constant temperature 
and the heat flux 
is prescribed on the upper and lower boundary surfaces. Under these conditions, the thermoelastic quantities for the disc are required to be determined. We take a cylindrical polar co-ordinate system 
with symmetry about 
–axis. As the problem considered is plane axisymmetric, the field component 
, and 
and C are independent of 
and restrict our analysis to the two dimensional problem with
(7)
Equations (1)-(6) with the aid of (7) take the form
(8)
(9)
(1+
K
(10)
(1+
(11)
and Constitutive relations
(12)
(13)
(14)
, 
(15)
(16)
where
, 
, 
, 
(17)
To facilitate the solution, the following dimensionless quantities are introduced
, 
, (
, 
, 
, 
, 
, 
(18)
Using (18) in equations (8)-(11) and after that suppressing the primes and then applying the Laplace transform defined by (19)
(19)
(20)
on the resulting quantities and simplifying we obtain
(21)
(
=0 (22)
(23)
where 
, 
, 
Eliminating 
from equations (21)-(23), we obtain
(24)
The solutions of the equation (24) can be written in the form
, 
(25)
where 
, 
, and 
are solutions of the following equation
, 
(26)
On taking Hankel transform defined by (20) on (26), we obtain
(27)
(28)
(29)
(30)
, 
, 
, 
, 
, 
, 
, 
Applying inversion of Hankel transform on (28), (29) and (30), we get
(31)
(32)
(33)
Using (8)-(11), (18) and (31)-(33), we obtain the displacement components in the transformed domain as
(34)
(35)
where
, 
, 
Substituting the values of (31)-(35) in (12)-(15) and with the aid of (18) yield
(36)
(37)
(38)
(39)
(40)
where 
, 
, 
4. Boundary Conditions
We consider a thermal source and chemical potential source (disc load). The disc load which emanates from origin of the coordinates and expands radically at constant rate ‘c’ over along with vanishing of stress components at the stress free surface at 
Mathematically, these can be written as
, (41)
(42)
(43)
=
= 
(44)
Applying Laplace transform and Hankel transform on both sides of the boundary conditions (41)-(44), we obtain
(45)
=0 (46)
0 (47)
(48)
Substitute the values of 
,
, 
, 
in (45)-(48), we obtain the values of unknown parameters as
, 
, 
, 
, 
, 
, 
, i=1, 2, 3 and 
is obtained from 
, by interchanging ith column with 
, where t denotes transpose.
5. Inversion of Double Transform
Due to the complexity of the solution in the Laplace transform domain, the inverse of the Laplace transform is obtained by using the Gaver-Stehfast algorithm. Graver [24] and Stehfast [25, 26] derived the formula given below. By this method, the inverse 
of Laplace transform 
is approximated by
with
Where K is an even integer, whose value depends on the word length of computer used. M=K/2, and m is an integer part of 
. The optimal value of K was chosen as described in Gaver-Stehfast algorithm, for the fast convergence of results with desired accuracy. The Romberg numerical integration technique [27] with variable step size was used to evaluate the results involved.
6. Particular Cases
(1) If we neglect the diffusion effect (i.e.,
, we obtain the expressions for components of displacement, stress and temperature distribution in thermoelastic isotropic half space.
(2) If 
, we obtain traditional Fourier model from dual phase lag model.
(3) If 
, then it reduces to DPLT model.
(4) If 
and 
, then DPLT and DPLD models reduce to single phase heat model (SPLT) and single phase diffusion model (SPLD)
7. Numerical Results and Discussion
The mathematical model is prepared with copper material for purposes of numerical computation. The material constants for the problem are taken from Dhaliwal and Singh [28]
, 
, 
, 
, 
, 
,
293K, 
An investigation has been conducted to compare the effect of phase lags of heat transfer and diffusion on normal displacement u3, Chemical potential function P, Temperature change T and mass concentration C by keeping one phase lag fixed and varying the values of other phase lag and vice versa in both the cases. The graphs have been plotted in the range 
.
We consider the following cases
a) 
, 
, 0.02 and 0.04
b) 
, = 
, 0.02 and 0.04
c) 
, 
0.02 and 0.04
d) =0.08, 
,0.02 and 0.04
In all figures solid line corresponds to the value of phase lag =0, small dashed line corresponds to the value of phase lag= 0.02, long dashed line with dots corresponds to value of phase lag= 0.04
Figures (1)-(4), exhibits variations of axial displacement uz with distance r corresponding to the cases (a), (b), (c), (d) respectively.
In Fig. 1. near the loading surface, there is a sharp decrease in range 0
r
corresponding to 
and variations are in wave form for the rest two curves.
In Fig. 2. Opposite oscillatory behaviour away from the loading surface is observed for 
whereas trends are similar for 
and .
In Fig. 3. for 
, variations are in wave form whereas for 
, there is a sharp increase near the loading surface and away from it, trend is oscillatory.
In Fig. 4. illustrates the variations of uzcorresponding to case (d). Here variations are similar as discussed in figure 3 with interchanging and .
Figures (5)-(8) represent the variations in chemical potential function P with distance r corresponding to the four cases. In these figures, we observe that ther is a sharp decrease in the range 0 r and variations are oscillatory in the rest with different amplitudes and trends.
Variations of temperature change T with distance r have been shown in figures (9)-(12) corresponding to the cases (a),(b),(c) and (d) respectively.
In Fig. 9. we observe that in the range 2 r
and 7 r
values corresponding to and =0.04 are smaller than for =0.02 whereas behaviour is opposite in the rest.
In Figures (10)-(12), there is a sharp increase in the range 0 r and afterwards behaviour is oscillatory with different magnitudes and patterns. Small variations are observed near zero in Fig. 12 in the range 4 r.
In Figures (13)-(16) show variations in mass concentration C with distance r corresponding to the four cases respectively. Here, we observe a descending behaviour in the cases. As r increases, values of mass concentration are decreasing.
In Fig. 13. we observe that, as increases, there is a deficiency in the wave.
In Fig. 14. For and trends are similar with difference in amplitude whereas for 
, trends are different. Small variations near zero are observed in Fig. 15 and Fig. 16. in the range 4 r
and trends are descending oscillatory in the rest.
Fig. 1. Variations of axial displacement 
with distance r (case a).
Fig. 2. Variations of axial displacement with distance r (case b).
Fig. 3. Variations of axial displacement with distance r (case c).
Fig. 4. Variations of axial displacement with distance r (case d).
Fig. 5. Variations of chemical potential function P with distance r (case a).
Fig. 6. Variations of chemical potential function P with distance r (case b).
Fig. 7. Variations of chemical potential function P with distance r(case c).
Fig. 8. Variations of chemical potential function P with distance r (case d).
Fig. 9. Variations of temperature change T with distance r(case a).
Fig. 10. Variations of temperature change T with distance r (case b).
Fig. 11. Variations of temperature change T with distance r (case c).
Fig. 12. Variations of temperature change T with distance r (case d).
Fig. 13. Variations of mass concentration C with distance r (case a).
Fig. 14. Variations of mass concentration C with distance r (case b).
Fig. 15. Variations of mass concentration C with distance r (case c).
Fig. 16. Variations of mass concentration C with distance r (case d).
8. Conclusion
From the graphs, effects of phase lags are computed and comparison of variations is made. It is observed that change in phase lags changes the behaviour of deformations of the various components of stresses, displacements, chemical potential function, temperature change and mass concentration. Small difference in phase lags results in big difference of thermal waves. A sound impact of diffusion and thermal phase-lags on the various quantities is observed. The use of diffusion phase-lags in the equation of mass diffusion gives a more realistic model of thermoelastic diffusion media as it allows a delayed response between the relative mass flux vector and the potential gradient. The result of the problem is useful in the two dimensional problem of dynamic response due to various sources of thermodiffusion which has various geophysical and industrial applications.
References
