Abstract
The present investigation is concerned with the propagation of waves at an imperfect boundary of heat conducting elastic solid and micropolar fluid media. The amplitude ratios of various reflected and transmitted waves are obtained in a closed form due to incidence of longitudinal wave (P-wave), thermal wave (T-wave), and transverse wave (SV-wave). The variation of various amplitude ratios with angle of incidence is obtained for normal force stiffness, transverse force stiffness, thermal contact conductance, and perfect bonding. Numerical results are shown graphically to depict the effect of stiffness and thermal relaxation times on resulting quantities. Some particular cases are also deduced in the present investigation.
1. Introduction
The fluids in which coupling between the spin of each fluid particle and the microscopic velocity field is taken into account are termed as micropolar fluids. They represent fluids consisting of rigid, randomly oriented, or spherical particles suspended in a viscous medium, where the deformation of fluid particles is ignored. The theory of microfluids was introduced by Eringen [1]. A microfluid possesses three gyration vector fields in addition to its classical translatory degrees of freedom. As a subclass of these fluids, Eringen introduced the micropolar fluids [2] to describe the physical systems, which do not fall in the realm of viscous fluids. In micropolar fluids the local fluid elements were allowed to undergo only rigid rotations without stretch. These fluids support couple stress, the body couples, and asymmetric stress tensor and possess a rotational field, which is independent of the velocity of fluid. A large class of fluids such as anisotropic fluids, liquid crystals with rigid molecules, magnetic fluids, cloud with dust, muddy fluids, biological fluids, and dirty fluids (dusty air, snow) can be modeled more realistically as micropolar fluids. The importance of micropolar fluids in industrial applications has motivated many researchers to extend the study in numerous ways to explain various physical effects.
Ariman et al. [3, 4] studied microcontinuum fluid mechanics and its applications and Říha [5] investigated the theory of heat-conducting micropolar fluid with microtemperature. Eringen and Kafadar [6] discussed polar field theories, Brulin [7] studied linear micropolar media, Gorla [8] investigated combined forced and free convection in the boundary layer flow of a micropolar fluid on a continuous moving vertical cylinder, and Eringen [9] studied the theory of thermo-microstretch fluids and bubbly liquid. Aydemir and Venart [10] investigated flow of a thermomicropolar fluid with stretch. Ciarletta [11] established two sorts of spatial decay estimates for describing the spatial behaviour of the solutions for the flow of a heat-conducting micropolar fluid in a semi-infinite cylindrical pipe. The lateral surface is taken to be thermal insulated and the adherence of the fluid to the lateral boundary is assumed. A time-dependent velocity and angular velocity profile are prescribed together with the temperature field at the finite end of the pipe. Hsia and Cheng [12] studied longitudinal plane waves propagation in elastic micropolar porous media. Propagation of transverse waves in elastic micropolar porous semispaces is discussed by Hsia et al. [13].
Different researchers studied the problems of reflection and transmission of plane waves at an interface of micropolar elastic half-spaces (Tomar and Gogna [14–16] and Kumar et al. [17, 18]). Singh and Tomar [19] discussed the longitudinal waves at an interface of micropolar fluid/micropolar solid half-spaces. Sun et al. [20] studied propagation characteristics of longitudinal displacement wave in micropolar fluid with micropolar elastic plate. Xu et al. [21] discussed reflection and transmission characteristics of coupled wave through micropolar elastic solid interlayer in micropolar fluid. Dang et al. [22] investigated propagation characteristics of coupled wave through micropolar fluid interlayer in micropolar elastic solid. Fu and Wei [23] investigated wave propagation through the imperfect interface between two micropolar solids. Sharma and Marin [24] studied reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids. Vishwakarma et al. [25] studied torsional wave propagation in an earth’s crustal layer under the influence of imperfect interface.
The purpose of the present study is to study the problem of reflection and transmission of plane waves at an imperfect interface between heat conducting elastic solid and micropolar fluid media. Effects of stiffness and thermal relaxation times on the amplitude ratios for incidence of various plane waves, for example, longitudinal wave (P-wave), thermal wave (T-wave), and transverse wave (SV-wave), are depicted numerically and shown graphically with angle of incidence.
2. Basic Equations
The field equations in a homogeneous and isotropic elastic medium in the context of generalized theories of thermoelasticity, without body forces and heat sources, are given by Lord and Shulman [26] and Green and Lindsay [27] as and the constitutive relations are where and the meaning of symbols is defined in the list at the end of the paper. The thermal relaxation times and satisfy the inequalities for G-L theory only and is the Laplacian operator. For Lord Shulman (L-S) theory , and for Green-Lindsay (G-L) theory , .
Following Ciarletta [11], the field equations and the constitutive relations for heat conducting micropolar fluids without body forces, body couples, and heat sources are given by where such that superscript denotes physical quantities and material constants related to the fluid and the constitutive relations are where , , and symbols are defined in the list at the end of the paper.
3. Formulation of the Problem
An imperfect interface of a homogeneous, isotropic generalized thermoelastic half-space (medium ) in contact with heat conducting micropolar fluid half-space (medium ) is considered. The rectangular Cartesian coordinate system O having origin on the surface seperating the two media is taken. Let us take the -axis along the interface between two half-spaces, namely, () and (), in such a way that -axis is pointing vertically downward into the medium . The geometry of the problem is shown in Figure 1.
For two dimensional problem in -plane, we take the displacement vector , velocity vector , and microrotation velocity vector as The following nondimensional quantities are defined: where , .
The displacement components , and velocity components , are related to the potential functions , and , in dimensionless form as Using (10) in (1) and (3)–(5) and with the aid of (8) and (9) (after suppressing the primes), we obtain where
4. Boundary Conditions
The boundary conditions at the interface are defined as where , , and are the normal force stiffness, transverse force stiffness, and thermal contact conductance coefficients of unit layer thickness having dimensions , , and .
5. Reflection and Transmission
The longitudinal wave (P-wave) or thermal wave (T-wave) or transverse wave (SV-wave) propagating through the medium which is designated as the region is considered. The plane wave is taken to be incident at the plane with its direction of propagation with angle normal to the surface. Each incident wave corresponds to reflected P-wave, T-wave, and SV-wave in medium and transmitted longitudinal wave (L-wave), thermal wave (T-wave), and transverse longitudinal wave coupled with transverse microrotational wave (C-I and C-II waves) in medium as shown in Figure 1.
In order to solve (11), we assume the solutions of the system of the form where is the wave number, is the angular frequency, and , , , , , , , are arbitrary constants.
Making use of (14) in (11), we obtain where Here , are the velocities of P-wave and T-wave in medium and these are roots of (15) and is the velocity of SV-wave in medium . , , , and are the velocities of transmitted longitudinal wave (L-wave), thermal wave (T-wave), and transverse longitudinal wave coupled with microrotational wave (C-I and C-II) in medium . These are roots of (16).
In view of (14), the appropriate solutions of (11) for medium and medium are taken as follows.
Medium is as follows: where
Medium is as follows: where and , are the amplitudes of incident (P-wave, T-wave) and SV-wave, respectively. and are the amplitudes of reflected (P-wave, T-wave) and SV-wave and , are the amplitudes of transmitted longitudinal wave, thermal wave, and transverse longitudinal wave coupled with transverse microrotational wave, respectively.
Equation (20) represents the relation between ( and ) and ( and ).
Snell’s law is given by where Making use of (18)–(21) in the boundary conditions (13) and with the help of (2), (7), (9), (10), (23), and (24), we obtain a system of seven nonhomogeneous equations which can be written as where the values of are given in the Appendix.(1)For incident P-wave, (2)For incident T-wave, (3)For incident SV-wave, where , , and are the amplitude ratios of reflected P-wave, T-wave, and SV-wave in medium and , , , and are the amplitude ratios of transmitted longitudinal wave (L-wave), thermal wave (T-wave), and transverse longitudinal wave coupled with transverse microrotational wave (C-I and C-II waves) in medium .
6. Particular Cases
6.1. Case I: Normal Force Stiffness
, , and in (25) yield the resulting quantities for normal force stiffness and lead a system of seven nonhomogeneous equations given by (A.1) with the changed values of as
6.2. Case II: Transverse Force Stiffness
, , and in (25) provide the case of transverse force stiffness giving a system of seven nonhomogeneous equations given by (A.1) with the changed values of as
6.3. Case III: Thermal Contact Conductance
Taking , , and in (25) corresponds to the case of thermal contact conductance and yields a system of seven nonhomogeneous equations as given by (A.1) with the changed values of as
6.4. Case IV: Perfect Bonding
If we take , , and in (25), we obtain the case of perfect bonding and yield a system of seven nonhomogeneous equations as given by (A.1) with the changed values of as
6.5. Subcases
(i)If micropolar heat conducting fluid medium is absent, we obtain the amplitude ratios at the free surface of thermoelastic solid with one relaxation time and two relaxation times. These results are similar to those obtained by A. N. Sinha and S. B. Sinha [28] for one relaxation time (L-S theory) and Sinha and Elsibai [29] for two relaxation times (G-L theory).(ii)In the absence of upper medium , we obtain the amplitude ratios at the free surface of thermoelastic solid for CT-theory ().
These results are similar to those obtained by Deresiewicz [30] for CT-theory.
7. Special Cases
(i)If , in (25), (A.1), (29), (30), and (31) then we obtain the corresponding amplitude ratios at an interface of thermoelastic solid with one relaxation time and heat conducting micropolar fluid half-space for normal force stiffness, transverse force stiffness, and thermal contact conductance.(ii)If , in (25), (A.1), (29), (30), and (31) then we obtain the corresponding amplitude ratios at an interface of thermoelastic solid with two relaxation times and heat conducting micropolar fluid half-space for normal force stiffness, transverse force stiffness, and thermal contact conductance.
8. Numerical Results and Discussion
The following values of relevant parameters for both the half-spaces for numerical computations are taken.
Following Singh and Tomar [19], the values of elastic constants for medium are taken as and thermal parameters are taken from Dhaliwal and Singh [31]:
Following Singh and Tomar [19], the values of micropolar constants for medium are taken as
Thermal parameters for the medium are taken as of comparable magnitude:
The values of amplitude ratios have been computed at different angles of incidence.
In Figures 2–22, for L-S theory, we represent the solid line for stiffness (ST1), small dashes line for normal force stiffness (NS1), medium dashes line for transverse force stiffness (TS1), and dash dot dash line for thermal contact conductance (TCS1). For G-L theory, we represent the dash double dot dash line for stiffness (ST2), solid line with center symbol “plus” for normal force stiffness (NS2), solid line with center symbol “diamond” for transverse force stiffness (TS2), and solid line with center symbol “cross” for thermal contact conductance (TCS2).
8.1. P-Wave Incident
Variations of amplitude ratios , , with the angle of incidence , for incident P-wave are shown in Figures 2–8.
Figure 2 shows that the values of for NS1 and NS2 increase in the whole range. The values for ST1, ST2, transverse force stiffness, and thermal contact conductance decrease in the whole range, except near the grazing incidence, where the values get increased. The values for ST1 remain more than the values for TS1, TS2, ST2, TCS1, and TCS2 in the whole range.
From Figure 3 it is evident that the values of for all the stiffnesses, except ST1 and ST2, decrease in the whole range. The values of for NS1 and NS2 are magnified by multiplying by 10.
Figure 4 shows that the values of for all the stiffnesses for L-S theory and G-L theory first increase up to intermediate range and then decrease with the increase in and the values for ST2 are greater than the values for ST1, NS1, NS2, TS1, TS2, TCS1, and TCS2 in the whole range. The values of for NS1 and NS2 are magnified by multiplying by 10.
From Figure 5 it is noticed that the values of for all the stiffnesses start with maximum value at normal incidence and then decrease to attain minimum value at grazing incidence. The values of amplitude ratios for ST1 and NS1 are more than the values for ST2 and NS2, respectively, in the whole range. There is slight difference in the values of TCS1 and TCS2 in the whole range. The values of for ST1, ST2, NS1, and NS2 are magnified by multiplying by 102 and the values for TS1, TS2, TCS1, and TCS2 are magnified by multiplying by 10.
Figure 6 shows that the values of for all the boundary stiffnesses decrease with increase in and the values of amplitude ratio for TS1 are greater than the values for all other boundary stiffnesses in the whole range that shows the effect of transverse force stiffness. The values of for all the stiffnesses are magnified by multiplying by 102.
From Figure 7 it is evident that the amplitude of for normal force stiffness increases in the range and then decreases in the further range. The values for transverse force stiffness increase in the range and for thermal contact conductance increase in the range and then decrease. The values of for all the stiffnesses, except TCS1 and TCS2, are magnified by multiplying by 103, while the values for TCS1 and TCS2 are magnified by multiplying by 102.
Figure 8 depicts that the values of for ST1, ST2, TS1 and TS2 increase in the range and decrease in the remaining range. The values for ST2 and TS2 are greater than the values for ST1 and TS2 respectively in the whole range. The values for normal force stiffness for G-L theory remain more than the values for L-S theory. The values of for ST1, ST2 are magnified by multiplying by 108, the values for TCS1, TCS2 are magnified by multiplying by 106 and the values for NS1, NS2, TS1 and TS2 are magnified by multiplying by 107.
8.2. T-Wave Incident
The values of amplitude ratio for transverse force stiffness and thermal contact conductance decrease in the whole range with slight increase in the initial range. The values for ST1 and ST2 increase in the range and then decrease. These variations have been shown in Figure 9. The values for TS1, TS2, TCS1 and TCS2 are reduced by dividing by 10.
The values of amplitude ratio for NS1 and NS2 increase in the whole range, while the values for TS1, TS2, ST1, ST2, TCS1, TCS2 first decrease and then increase to attain maximum value at grazing incidence. These variations are shown in Figure 10.
From Figure 11, it is noticed that the values of for normal force stiffness, transverse force stiffness, and thermal contact conductance for G-L theory are greater than the corresponding values for L-S theory. It is seen that the values for ST1 are greater than the values for ST2 in the range and, in the further range, the values for ST2 are more.
Figure 12 depicts that the values of amplitude ratios for transverse force stiffness and thermal contact conductance oscillate up to intermediate range and then decrease in the further range to attain minimum value at grazing incidence. The values for ST1, ST2, NS1, and NS2 decrease in the whole range. The values of for ST1 and ST2 are magnified by multiplying by a factor of 102 and NS1, NS2, TS1, TS2, TCS1, and TCS2 are magnified by a factor of 10.
Figure 13 shows that the behavior of variation of for all the boundary stiffnesses is similar to that of , but the magnitude of variation is different. The values of for ST1 and ST2 are magnified by multiplying by a factor of 102 and NS1, NS2, TS1, TS2, TCS1, and TCS2 are magnified by a factor of 10.
From Figure 14 it is evident that the values of for all the boundary stiffnesses increase to attain maximum value and then decrease up to grazing incidence. The values for NS1 and TS1 are greater than the values for NS2 and TS2, respectively, that reveals the thermal relaxation time effect. The values of for ST1 and ST2 are magnified by multiplying by a factor of 103 and NS1, NS2, TS1, TS2, TCS1, and TCS2 are magnified by a factor of 102.
Figure 15 depicts that the values of for ST1 and ST2 increase in the interval and then decrease in the further range. It is noticed that there is slight difference in the values of amplitude ratio for L-S and G-L theory. The values of for ST1 and ST2 are magnified by multiplying by a factor of 108 and NS1, NS2, TS1, TS2, TCS1, and TCS2 are magnified by a factor of 106.
8.3. SV Wave Incident
Variations of amplitude ratios , , with the angle of incidence , for incident SV-wave are shown in Figures 16–22.
Figure 16 depicts that the values of for transverse force stiffness and thermal contact conductance increase to attain peak values in the range and then decrease in the further range. The values for ST1 and ST2 increase in the range and , respectively, and decrease in the remaining range. The values for normal force stiffness increase from normal incidence to attain maximum value in the range and then decrease up to grazing incidence. The values for TS1, TS2, TCS1, and TCS2 are reduced by dividing by 10.
Figure 17 shows that values of amplitude ratio for all the boundary stiffnesses follow oscillatory pattern in the whole range. The maximum value is attained by TCS1 in the range . It is seen that the values for L-S theory are greater than the values for G-L theory in the whole range.
Figure 18 shows that the values of for transverse force stiffness increase from normal incidence to grazing incidence and attain peak value in the interval . The values for normal stiffness and thermal contact conductance decrease in the intervals and , respectively, and then increase in the further range.
Figure 19 shows that the values of for all the boundary stiffnesses oscillate in the whole range. The values for ST1 are greater than the values for ST2 in the whole range, except some intermediate range. The values of for ST1 and ST2 are magnified by multiplying by a factor of 103 and NS1, NS2, TS1, and TS2 by a factor of 102 and TCS1 and TCS2 are magnified by a factor of 10.
Figure 20 depicts that the behavior of variation of for all the boundary stiffnesses is similar to that of , but the magnitude of variation is different. The values of for ST1, ST2 are magnified by multiplying by a factor of 103 and NS1, NS2, TCS1, TCS2, TS1, and TS2 are magnified by a factor of 102.
From Figure 21, it is noted that the values of for ST1 and ST2 decrease in the range and then increase with increase in angle of incidence. The values of amplitude ratio for transverse force stiffness and thermal contact conductance decrease in the whole range, except the ranges and , respectively. The amplitude ratio for normal force stiffness attains maximum value at normal incidence. The values of for ST1, ST2 are magnified by multiplying by a factor of 103 and NS1, NS2, TCS1, TCS2, TS1, and TS2 are magnified by a factor of 102.
Figure 22 depicts that the values of for all the boundary stiffnesses attain maximum value at the normal incidence and then decrease with oscillation to attain minimum value at the grazing incidence. The values of for ST1 and ST2 are magnified by multiplying by a factor of 108 and NS1, NS2, TS1, TS2, TCS1, and TCS2 are magnified by a factor of 106.
9. Conclusion
Reflection and transmission at an interface between heat conducting elastic solid and micropolar fluid media are discussed in the present paper. Effect of normal force stiffness, transverse force stiffness, thermal contact conductance, and thermal relaxation times is observed on the amplitude ratios for incidence of various plane waves (P-wave, T-wave, and SV-wave). When P-wave is incident, it is noticed that the values of amplitude ratio for transverse force stiffness for transmitted T-wave are greater than all the other boundary stiffnesses. When plane wave (SV-wave) is incident, the trend of variation of amplitude ratio for transmitted transverse wave coupled with transverse microrotational wave, that is, C-I and C-II waves, is similar, but magnitude of oscillation is different. The values of amplitude ratio of transmitted LD-wave and T-wave for L-S theory are greater than the value for G-L theory (when T-wave is incident). The model considered is one of the more realistic forms of earth models and it may be of interest for experimental seismologists in exploration of valuable materials such as minerals and crystal metals.
Appendix
Consider the following:
Symbols
| , : | Lame’s constants |
| : | Components of the stress tensor |
| : | Displacement vector |
| : | Density |
| : | Thermal conductivity |
| : | Specific heat at constant strain |
| : | Uniform temperature |
| : | Temperature change |
| : | Coefficient of linear thermal expansion |
| : | Kronecker delta |
| : | Alternating symbol |
| , , , | |
| , , , : | Material constants of the fluid |
| : | Components of stress tensor in the fluid |
| : | Components of couple stress tensor in the fluid |
| : | Velocity vector |
| : | Microrotation velocity vector |
| : | Density |
| : | Scalar constant with the dimension of moment of inertia of unit mass |
| : | Pressure |
| : | Thermal conductivity |
| : | Specific heat at constant strain |
| : | Absolute temperature |
| : | Temperature change |
| : | Variation in specific volume |
| : | Coefficient of linear thermal expansion. |
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
Thanks are due to Punjab Technical University, Kapurthala, for providing research facilities and enrolling one of the authors (Mandeep Kaur) as a research scholar to Ph.D. Programme.