Rajneesh Kumar, 1 S. C. Rajvanshi, 2,3 and Mandeep Kaur 2,3. 1. Introduction

Transcription

1 International Scholarly Research Notices Volume 4, Article ID 384, 6 pages Research Article Effect of Stiffness on Reflection and Transmission of Waves at an Interface between Heat Conducting Elastic Solid and Micropolar Fluid Media Rajneesh Kumar, S. C. Rajvanshi,,3 and Mandeep Kaur,3 Department of Mathematics, Kurukshetra University, Kurukshetra 369, India Research Centre, Punjab Technical University, Kapurthala, India 3 Department of Applied Sciences, Gurukul Vidyapeeth Institute of Engineering and Technology, Banur, Sector 7, Patiala District, Punjab 46, India Correspondence should be addressed to Mandeep Kaur; mandeep5@yahoo.com Received 8 March 4; Accepted 8 July 4; Published 9 October 4 Academic Editor: Giuseppe Maruccio Copyright 4 Rajneesh Kumar et al. 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. 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.. 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 []. 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 [] 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 withrigidmolecules,magneticfluids,cloudwithdust,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]studiedthetheory of thermo-microstretch fluids and bubbly liquid. Aydemir and Venart [] investigated flow of a thermomicropolar fluid with stretch. Ciarletta [] 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

2 International Scholarly Research Notices and angular velocity profile are prescribed together with the temperaturefieldatthefiniteendofthepipe.hsiaand Cheng [] 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. [3]. Different researchers studied the problems of reflection and transmission of plane waves at an interface of micropolar elastic half-spaces (Tomar and Gogna [4 6]and Kumar et al. [7, 8]). Singh and Tomar [9]discussedthelongitudinal waves at an interface of micropolar fluid/micropolar solid half-spaces. Sun et al. [] studied propagation characteristics of longitudinal displacement wave in micropolar fluid with micropolar elastic plate. Xu et al. [] discussed reflection and transmission characteristics of coupled wave through micropolar elastic solid interlayer in micropolar fluid. Dang et al. [] investigated propagation characteristics of coupled wave through micropolar fluid interlayer in micropolar elastic solid. Fu and Wei [3] investigated wave propagation through the imperfect interface between two micropolar solids. Sharma and Marin [4] studied reflection and transmission of waves from imperfect boundary between two heat conducting micropolar thermoelastic solids. Vishwakarma et al. [5] 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 timesontheamplituderatiosforincidenceofvariousplane 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.. 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 [6] and Green and Lindsay [7]as (λ + μ) ( u) μ( ( u)) u ] ( + τ t ) T=ρ t, K ΔT = ρc ( T t +τ T t )+]T ( t +η τ t ) ( u), () and the constitutive relations are t ij =λu r,r δ ij +μ(u i,j +u j,i ) ] (T + τ T t )δ ij, i, j, r =,, 3, where ] = (3λ+μ)α T and the meaning of symbols is defined in the list at the end of the paper. The thermal relaxation times τ and τ satisfy the inequalities τ τ for G-L theory () only and Δ= / x + / x 3 is the Laplacian operator. For Lord Shulman (L-S) theory η =, τ =and for Green- Lindsay (G-L) theory η =, τ >. Following Ciarletta [], the field equations and the constitutive relations for heat conducting micropolar fluids without body forces, body couples, and heat sources are given by where D V +(λ f +μ f ) ( V) +K f ( Ψ) b T f D c φ f =, Ψ+(α f +β f ) ( Ψ) + K f ( V) =, (3) K ΔTf bt f ( V) =ρf at f T f t, (4) ρ f φ f t = V, (5) D =(μ f +K f )Δ ρ f t, D =γ f Δ I t Kf, (6) such that superscript f denotes physical quantities and material constants related to the fluid and the constitutive relations are t f ij = pδ ij +σ f ij, p=bt f +c φ f, σ f ij =λf γ rr δ ij +(μ f +K f )γ ij +μ f γ ji, m f ij =αf υ rr δ ij +β f υ ji +γ f υ ij, where γ ij = V j,i +ε jir Ψ r, υ ij =Ψ j,i, b = (3λ f +μ f +K f )α T f andsymbolsaredefinedinthelistattheendofthepaper. 3. Formulation of the Problem An imperfect interface of a homogeneous, isotropic generalized thermoelastic half-space (medium M ) in contact with heat conducting micropolar fluid half-space (medium M ) is considered. The rectangular Cartesian coordinate system Ox x x 3 having origin on the surface x 3 =seperating the two media is taken. Let us take the x -axis along the interface between two half-spaces, namely, M ( < x 3 < )and M ( < x 3 <), in such a way that x 3 -axis is pointing vertically downward into the medium M.Thegeometryof the problem is shown in Figure. For two dimensional problem in x x 3 -plane,wetakethe displacement vector u, velocityvectorv, andmicrorotation velocity vector Ψ as u=(u (x,x 3 ),,u 3 (x,x 3 )), V =(V (x,x 3 ),,V 3 (x,x 3 )), (7) Ψ=(,Ψ (x,x 3 ),). (8)

3 International Scholarly Research Notices 3 The following nondimensional quantities are defined: where x i = ω x i c, u i = ρω c ]T u i, V i = ρc ]T V i, ψ = ρc ω ]T ψ, (t,τ,τ )=(ω t, ω τ,ω τ ), a = ρc μ, ε = ] T K ω ρ, b bρc = (λ f +μ f +K f )ω ], b = c c (λ f +μ f +K f )ω ]T, (T,T f )=( T T, Tf T ), (t ij,tf ij )= ]T (t ij,t f ij ), b 3 = ρ f c (λ f +μ f +K f )ω, b K f 4 = μ f +K, f m f ij = ω c ]T m f ij, φ f =ρφ f, K n = c ]T K n, K t = c ]T K t, K θ = ]c K θ, (9) b 5 = ρ f c (μ f +K f )ω, b 6 = Kf c γ f ω, b 7 =b 6, b 8 = Ic γ f ω, b 9 = b]tf K ρω, where ω =ρc c /K, c = (λ + μ)/ρ. The displacement components u, u 3 and velocity components V, V 3 arerelatedtothepotentialfunctionsφ, φ f and ψ, ψ f in dimensionless form as (u, V )=( (φ, φ f ) (ψ, ψ f )), x x 3 (u 3, V 3 )=( (φ, φ f )+ (ψ, ψ f )). x 3 x () Using () in() and(3) (5) and with the aid of (8) and(9) (after suppressing the primes), we obtain φ (+τ t )T φ t =, b = ρf at f c K, b ω = ]T ρ f c. 4. Boundary Conditions () The boundary conditions at the interface x 3 =are defined as K t f 33 =K n ( u 3 t V 3), t f 3 =K t ( u t V ), T f x 3 =K θ (T T f ), t 33 =t f 33, t 3 =t f 3, m f 3 =, T K =K T f, x 3 x 3 (3) ψ a ψ t =, T=(+τ t ) T t +ε ( t +η τ t ) φ, φ f b T f b φ f φ f b 3 =, t ψ f ψ f +b 4 Ψ b 5 =, t T f b 9 φ f T f b =, t Ψ b 6 ψ f b 7 Ψ b 8 Ψ t b φ f t φ f =, =, () where K n, K t,andk θ arethenormalforcestiffness,transverse force stiffness, and thermal contact conductance coefficients of unit layer thickness having dimensions N sec/m 3, Nsec/m 3,andN/msecK. 5. Reflection and Transmission The longitudinal wave (P-wave) or thermal wave (T-wave) or transverse wave (SV-wave) propagating through the medium M which is designated as the region x 3 >is considered. The plane wave is taken to be incident at the plane x 3 = with its direction of propagation with angle θ normal to the surface. Each incident wave corresponds to reflected P-wave, T-wave, and SV-wave in medium M 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 M as shown in Figure.

4 4 International Scholarly Research Notices C-II ( S 4 ) C-I (S 3 ) ω E 3 =, (b 8 + (ι/ω) b 7 )b 5 A x 3 -axis θ θ θ θ3 θ 4 θ 3 θ θ SV(S 3 ) Figure : Geometry of the problem. T(S ) T(S ) M P(S ) L( S ) M x -axis V = ω k, τ =( ι ω +τ ). (7) Here V, V are the velocities of P-wave and T-wave in medium M and these are roots of (5) andv 3 = / a is the velocity of SV-wave in medium M. V, V, V 3,and V 4 are the velocities of transmitted longitudinal wave (Lwave), thermal wave (T-wave), and transverse longitudinal wave coupled with microrotational wave (C-I and C-II) in medium M. These are roots of (6). In view of (4), the appropriate solutions of () for medium M and medium M are taken as follows. Medium M is as follows: φ, T = i=, f i [S i e ιk i(x sin θ i x 3 cos θ i ) ω i t +P i ], In order to solve (), we assume the solutions of the system of the form φ,t,ψ,φ f,φ f,t f,ψ f,ψ =φ, T, ψ, φ f, φ f, T f, ψ f, Ψ e ιk(x sin θ x 3 cos θ) ωt, (4) where k isthewavenumber,ω is the angular frequency, and φ, T, ψ, φ f, φ f, T f, ψ f, Ψ are arbitrary constants. Making use of (4)in(), we obtain where V 4 +D V +E =, (5) V 4 +D V +E =, V 4 +D 3 V +E 3 =, D = [ + ( ιτ τ ω) ε ( ι ω +η τ )], E =, τ D = ιω b 3 ( ιb b ω ιω )+ ( + ιb b 9 ), b ωb 3 E = ω b 3 b ( ιb b ω ), D 3 = ιω b 5 +ιω[ ( ιb 4b 6 /ωb 5 ) (b 8 + (ι/ω) b 7 ) ], (6) ψ=s 3 e ιk 3(x sin θ 3 x 3 cos θ 3 ) ω 3 t +S 3 e ιk 3(x sin θ 3 +x 3 cos θ 3 ) ω 3 t, (8) where ε f i = ω i (ι/ω i +η τ ) /Vi +(ι/ω i +τ ) ιε ω i (ι/ω i +η τ )(ι/ω i +τ ), where P i =S i e ιk i(x sin θ i +x 3 cos θ i ) ω i t. Medium M is as follows: φ f,t f,φ f = i= ψ f,ψ = 4 j=3, f i, g i S i e ιk i(x sin θ i x 3 cos θ i ) ω i t,, f j S j e ιk j(x sin θ j x 3 cos θ j ) ω j t, b f i = 3 b 9, b b 9 (ι/ω i )+ιω i ( ιb b /ω i )(/V i b (ι/ω i )) (b f j = 6 b 5 (ι/ω i )) /V j b7 /ωj +b 8 (ι/ω i )+b 4 b 6 /ωj (9) () (), g i = ιb ω i V i, () and S i, S 3 are the amplitudes of incident (P-wave, T-wave) and SV-wave, respectively. S i and S 3 are the amplitudes of reflected (P-wave, T-wave) and SV-wave and S i, S j are the amplitudes of transmitted longitudinal wave, thermal wave,

5 International Scholarly Research Notices 5 and transverse longitudinal wave coupled with transverse microrotational wave, respectively. Equation () represents the relation between (φ f and T f )and(φ f and φ f ). Snell s law is given by sin θ V = sin θ V = sin θ V = sin θ 3 V 3 = sin θ V = sin θ V = sin θ 3 V 3 = sin θ 4 V 4, (3) wave (L-wave), thermal wave (T-wave), and transverse longitudinal wave coupled with transverse microrotational wave (C-I and C-II waves) in medium M. 6. Particular Cases 6.. Case I: Normal Force Stiffness. K n =, K t,and K θ in (5) yield the resulting quantities for normal force stiffness and lead a system of seven nonhomogeneous equations given by (A.)withthechangedvaluesofa ij as where k V =k V =k 3 V 3 = k V = k V = k 3 V 3 = k 4 V 4 =ω, a i = ω V sin θ, a 3 = ω V 3 V 3 sin θ, at x 3 =. (4) Making use of (8) ()in theboundaryconditions(3)and with the help of (), (7), (9), (), (3), and (4), we obtain a system of seven nonhomogeneous equations which can be written as 7 j= a ij Z j =Y i ; (i =,, 3, 4, 5, 6, 7), (5) where the values of a ij are given in the Appendix. () For incident P-wave, A =S, S =S 3 =, Y =a, Y = a, Y 3 = a 3, Y 4 = a 4, Y 5 =a 5, Y 6 =, Y 7 =a 7. () For incident T-wave, A =S, S =S 3 =, Y =a, (6) a 4 =ι ω V sin θ, a 5 =ι ω V sin θ, a 6 =ι ω V 3 V 3 sin θ, a 7 =ι ω V 4 V 4 sin θ, a 3i =f i, a 33 =, a 34 = f, a 35 = f, a 36 =a 37 =. (9) 6.. Case II: Transverse Force Stiffness. K t =, K n,and K θ in (5) provide the case of transverse force stiffness giving a system of seven nonhomogeneous equations given by (A.)withthechangedvaluesofa ij as a i = ω V i sin θ V, i V Y = a, Y 3 = a 3, Y 4 = a 4, Y 5 =a 5, Y 6 =, Y 7 =a 7. (3) For incident SV-wave, A =S 3, S =S =, Y = a 3, Y =a 3, Y 3 =a 33 =, Y 4 = a 43, Y 5 = a 53, Y 6 =, Y 7 =a 73 =, Z = S A, Z = S A, Z 3 = S 3 A, Z 4 = S A, Z 5 = S A, Z 6 = S 3 A, Z 7 = S 4 A, (7) (8) a 3 = ω sin θ V, a 4 = ι ω ( V V sin θ ), a 5 = ι ω ( V V sin θ ), a 6 =ι ω sin θ V, a 7 =ι ω V sin θ, a 3i =f i, a 33 =, a 34 = f, a 35 = f, a 36 =a 37 =. (3) where Z, Z,andZ 3 aretheamplituderatiosofreflectedpwave, T-wave, and SV-wave in medium M and Z 4, Z 5, Z 6, and Z 7 are the amplitude ratios of transmitted longitudinal 6.3. Case III: Thermal Contact Conductance. Taking K θ =, K t,andk n in (5) corresponds to the case of thermal contact conductance and yields a system of

6 6 International Scholarly Research Notices seven nonhomogeneous equations as given by (A.)with the changed values of a ij as a 5 =ι ω sin θ V, a 6 =ι ω V 3 V 3 sin θ, a i = ω V i sin θ V, i V a 3 = ω V sin θ, a 7 =ι ω V 4 V 4 sin θ, a 3i =f i, a 4 = ι ω ( V V sin θ ), a 5 = ι ω ( V V sin θ ), a 6 =a 7 =ι ω V sin θ, a i = ω V sin θ, a 3 = ω V 3 V 3 sin θ, a 4 =ι ω sin θ V, a 5 =ι ω sin θ V, a 6 =ι ω V 3 V 3 sin θ, a 7 =ι ω V 4 V 4 sin θ. (3) a 33 =, a 34 = f, a 35 = f, a 36 =a 37 =. (3) 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 [8] for one relaxation time (L- S theory) and Sinha and Elsibai [9] for two relaxation times (G-L theory). (ii) In the absence of upper medium M,weobtainthe amplitude ratios at the free surface of thermoelastic solid for CT-theory (η =τ =τ =). These results are similar to those obtained by Deresiewicz [3] for CT-theory. 7. Special Cases 6.4. Case IV: Perfect Bonding. If we take K n, K t, and K θ in (5), we obtain the case of perfect bonding and yield a system of seven nonhomogeneous equations as given by (A.)withthechangedvaluesofa ij as a i = ω V i sin θ V, i V a 3 = ω V sin θ, a 4 = ι ω ( V V sin θ ), a 5 = ι ω ( V V sin θ ), a 6 =a 7 =ι ω V sin θ, a i = ω V sin θ, (i) If η =, τ = in (5), (A.), (9), (3), and (3) 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 η =, τ > in (5), (A.), (9), (3), and (3) 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 halfspaces for numerical computations are taken. Following Singh and Tomar [9], the values of elastic constants for medium M are taken as λ =.973 Nm, μ =.98 9 Nm, a 3 = ω V 3 V 3 sin θ, a 4 =ι ω sin θ V, ρ =.34 3 Kg m 3, (33)

7 International Scholarly Research Notices 7 and thermal parameters are taken from Dhaliwal and Singh [3]: ] =.68 7 Nm K, c =.4 3 Nm Kg K, K =.7 Nsec K, T =.98 K, τ =.63 sec, ω=. τ =.83 sec, (34) Following Singh and Tomar [9], the values of micropolar constants for medium M are taken as λ f =.5 8 Nsecm, μ f =.3 8 Nsecm, Amplitude ratio Z K f =.3 8 Nsecm, γ f =. Nsec, ρ f =.8 3 Kg m 3, I =.4 6 Nsec m. (35) Thermal parameters for the medium M are taken as of comparable magnitude: T f =.96 K, K =.89 Nsec K, c =.5 Nsec m 6, a =.5 5 m sec K, b =.6 5 Nm K. (36) The values of amplitude ratios have been computed at different angles of incidence. In Figures, for L-S theory, we represent the solid line for stiffness (), small dashes line for normal force stiffness (), medium dashes line for transverse force stiffness (), and dash dot dash line for thermal contact conductance (). For G-L theory, we represent the dash double dot dash line for stiffness (), solid line with center symbol plus for normal force stiffness (), solid line with center symbol diamond for transverse force stiffness (), and solid line with center symbol cross for thermal contact conductance (). 8.. P-Wave Incident. Variations of amplitude ratios Z i, i 7, with the angle of incidence θ, for incident P-wave are shown in Figures 8. Figure shows that the values of Z for and increaseinthewholerange.thevaluesfor,, 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 remain more than the values for,,,, and in the whole range. From Figure 3 it is evident that the values of Z for all the stiffnesses, except and, decrease in the whole range. The values of Z for and are magnified by multiplying by Angle of incidence θ Figure : Variation of Z with angle of incidence (P-wave). Figure 4 shows that the values of Z 3 for all the stiffnesses for L-S theory and G-L theory first increase up to intermediaterangeandthendecreasewiththeincreaseinθ and the values for are greater than the values for,,,,,, and in the whole range. The values of Z for and are magnified by multiplying by. From Figure 5 it is noticed that the values of Z 4 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 and are more than thevaluesforand,respectively,inthewholerange. There is slight difference in the values of and in the whole range. The values of Z 4 for,,,andare magnified by multiplying by and the values for,,, and are magnified by multiplying by. Figure 6 shows that the values of Z 5 for all the boundary stiffnesses decrease with increase in θ and the values of amplitude ratio for 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 Z 5 for all the stiffnesses are magnified by multiplying by. From Figure 7 it is evident that the amplitude of Z 6 for normal force stiffness increases in the range < θ < 48 and then decreases in the further range. The values for transverse force stiffness increase in the range <θ <59 and for thermal contact conductance increase in the range < θ < 56 and then decrease. The values of Z 6 for all the stiffnesses, except and, are magnified by

8 8 International Scholarly Research Notices.6.8 Amplitude ratio Z.4. Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure 3: Variation of Z with angle of incidence (P-wave). Figure 5: Variation of Z 4 with angle of incidence (P-wave) Amplitude ratio Z 3.4 Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure 4: Variation of Z 3 with angle of incidence (P-wave). Figure 6: Variation of Z 5 with angle of incidence (P-wave).

9 International Scholarly Research Notices Amplitude ratio Z Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure 7: Variation of Z 6 with angle of incidence (P-wave). Figure 8: Variation of Z 7 with angle of incidence (P-wave). multiplying by 3, while the values for and are magnified by multiplying by. Figure 8 depicts that the values of Z 7 for,, and increase in the range <θ <56 and decrease in the remaining range. The values for and are greater than the values for and 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 Z 7 for, are magnified by multiplying by 8,the values for, are magnified by multiplying by 6 and the values for,, and are magnified by multiplying by T-Wave Incident. The values of amplitude ratio Z for transverse force stiffness and thermal contact conductance decreaseinthewholerangewithslightincreaseintheinitial range. The values for and increase in the range < θ <48 and then decrease. These variations have been shown in Figure 9. The values for,, and are reduced by dividing by. The values of amplitude ratio Z for and increase in the whole range, while the values for,,,,, first decrease and then increase to attain maximum value at grazing incidence. These variations are shown in Figure. From Figure, it is noticed that the values of Z 3 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 are greater than the values for in the range <θ <38 and, in the further range, the values for are more. Figure depicts that the values of amplitude ratios Z 4 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,,, and decrease in the whole range. The values of Z 4 for and are magnified by multiplying by a factor of and,,,,, and are magnified by a factor of. Figure 3 shows that the behavior of variation of Z 5 for all the boundary stiffnesses is similar to that of Z 4,butthe magnitude of variation is different. The values of Z 5 for and are magnified by multiplying by a factor of and,,,,, and are magnified by a factor of. From Figure 4 it is evident that the values of Z 6 for all the boundary stiffnesses increase to attain maximum value and then decrease up to grazing incidence. The values for and are greater than the values for and, respectively, that reveals the thermal relaxation time effect. The values of Z 5 for and are magnified by multiplying by a factor of 3 and,,,,, and are magnified by a factor of. Figure 5 depicts that the values of Z 7 for and increase in the interval <θ <39 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.

10 International Scholarly Research Notices.6.8 Amplitude ratio Z..8 Amplitude ratio Z Angle of incidence θ Figure 9: Variation of Z with angle of incidence (P-wave) Angle of incidence θ Figure : Variation of Z 3 with angle of incidence (T-wave)..8 Amplitude ratio Z.8.6 Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure : Variation of Z with angle of incidence (P-wave). Figure : Variation of Z 4 with angle of incidence (T-wave).

11 International Scholarly Research Notices.6.8 Amplitude ratio Z Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure 3: Variation of Z 5 with angle of incidence (T-wave). Figure 4: Variation of Z 6 with angle of incidence (T-wave). The values of Z 7 for and are magnified by multiplyingbyafactorof 8 and,,,,, and are magnified by a factor of SV Wave Incident. Variations of amplitude ratios Z i, i 7, with the angle of incidence θ, for incident SV-wave are shown in Figures 6. Figure 6 depicts that the values of Z for transverse force stiffness and thermal contact conductance increase to attain peak values in the range 5 < θ < 5 and then decrease in the further range. The values for and increase in the range < θ < 35 and 45 < θ < 66, respectively, and decrease in the remaining range. The values for normal force stiffness increase from normal incidence to attain maximum value in the range 45 <θ <55 and then decrease up to grazing incidence. The values for,,, and are reduced by dividing by. Figure 7 shows that values of amplitude ratio Z for all the boundary stiffnesses follow oscillatory pattern in the whole range. The maximum value is attained by in the range 5 <θ <5. It is seen that the values for L-S theory are greater than the values for G-L theory in the whole range. Figure 8 shows that the values of Z 3 for transverse force stiffness increase from normal incidence to grazing incidence and attain peak value in the interval 5 < θ < 5.The values for normal stiffness and thermal contact conductance decrease in the intervals <θ <56 and <θ <7, respectively, and then increase in the further range. Figure 9 shows that the values of Z 4 for all the boundary stiffnesses oscillate in the whole range. The values for are greaterthanthevaluesforinthewholerange,except some intermediate range. The values of Z 4 for and aremagnifiedbymultiplyingbyafactorof 3 and,,, and by a factor of and and are magnified by a factor of. Figure depicts that the behavior of variation of Z 5 for all the boundary stiffnesses is similar to that of Z 4,butthe magnitude of variation is different. The values of Z 5 for, are magnified by multiplying by a factor of 3 and,,,,, and are magnified by a factor of. From Figure, it is noted that the values of Z 6 for and decrease in the range < θ < 45 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 6 <θ <4 and 7 <θ <3,respectively.The amplitude ratio for normal force stiffness attains maximum valueatnormalincidence.thevaluesof Z 6 for, aremagnifiedbymultiplyingbyafactorof 3 and,,,,,andaremagnifiedbyafactorof. Figure depicts that the values of Z 7 for all the boundary stiffnesses attain maximum value at the normal incidence andthendecreasewithoscillationtoattainminimumvalue at the grazing incidence. The values of Z 7 for and aremagnifiedbymultiplyingbyafactorof 8 and,,,,, and are magnified by a factor of 6.

12 International Scholarly Research Notices Amplitude ratio Z 7.8 Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure 5: Variation of Z 7 with angle of incidence for T-wave. Figure 7: Variation of Z with angle of incidence (SV-wave) Amplitude ratio Z.8 Amplitude ratio Z Angle of incidence θ Figure 6: Variation of Z with angle of incidence (SV-wave) Angle of incidence θ Figure 8: Variation of Z 3 with angle of incidence (SV-wave).

13 International Scholarly Research Notices Amplitude ratio Z 4.8 Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure 9: Variation of Z 4 with angle of incidence (SV-wave). Figure : Variation of Z 6 with angle of incidence (SV-wave)...8 Amplitude ratio Z Amplitude ratio Z Angle of incidence θ Angle of incidence θ Figure : Variation of Z 5 with angle of incidence (SV-wave). Figure : Variation of Z 7 with angle of incidence (SV-wave).

14 4 International Scholarly Research Notices 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 LDwave 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: ω a i = c K n V i sin ω θ V, a 3 = c K n sin θ i V, V a 4 = [d f +df 3 ( V V sin θ )] ω V +d ff f + g d f ω +ιc K n ( V V sin θ ), a 5 = [d f +df 3 ( V V sin θ )] ω V a 6 =d f 3 a 7 =d f 3 +d ff f + g d f ω +ιc K n ( V V sin θ ), ω sin θ V 3 V 3 V sin ω θ +ιc K n sin θ V, ω sin θ V 4 V 4 V sin ω θ +ιc K n sin θ V, a i = c K t ω V sin θ, a 3 =c K t ω V 3 V 3 V sin θ, a 4 = (d f 4 +df 5 ) ω sin θ V V V sin ω θ +ιc K t sin θ V, a 5 = (d f 4 +df 5 ) ω sin θ V V V sin ω θ +ιc K t sin θ V, a 6 =d f 4 a 7 =d f 4 ω V3 ( V 3 sin θ )+d f 5 ω +ιc K t V 3 V 3 sin θ, ω V4 ( V 4 sin θ )+ d f 5 ω +ιc K t V 4 V 4 sin θ, a 3i =c 4 K θ f i, a 33 =, [ [ [ [ V 3 V 4 a 34 = [ ι ω V V V sin ] θ c 4 K θ f, [ ] sin θ f 3 ] ] ] sin θ f 4 ] ] ] a 35 = [ ι ω V V V sin ] θ c 4 K θ f, a 36 =a 37 =, [ ] a 44 a 4i =(d +d ( V i a 43 =d ω sin θ )) ω Vi +( τ ιω) f i, V 3 V sin θ V 3 V sin θ, = [(d f +df 3 ( V sin θ )) ω +(d ff V f +d f g )], a 45 = [(d f +df 3 ( V sin θ )) ω +(d ff V f +d f g )], a 46 =d f 3 ω sin θ V 3 V 3 V sin θ,

15 International Scholarly Research Notices 5 a 47 =d f 3 ω sin θ V 4 V 4 V sin θ, a 5 = (d 4 ) ω sin θ V V V sin θ, a 5 = (d 4 ) ω sin θ V V V sin θ, a 53 =d 4 ω V 3 ( V 3 sin θ ), a 54 = (d f 4 +df 5 ) ω sin θ V V V sin θ, a 55 = (d f 4 +df 5 ) ω sin θ V V V sin θ, a 56 = [ [ a 57 = [ [ d f ω 4 V3 +d f 5 ( ω d f ω 4 V4 ( V 3 sin θ ) V 3 ( V 3 V ( V 4 sin θ ) +d f 5 ( ω V 4 ( V 4 V a 6i =a 63 =a 64 =a 65 =, a 66 =ι ω f 3 V 3 V 3 sin θ, a 67 =ι ω f 4 V 4 V 4 sin θ, sin θ ) f 3 ) ], ] sin θ ) f 4 ) ], ] a 7i =ι ω V i V i sin θ f i, a 73 =, ω a 74 =ιp V V 4 sin θ f, ω a 75 =ιp V V 5 sin θ f, a 76 =a 77 =, (i =,), Symbols d = λ ρc, d = μ ρc, d 3 = d, d ff = b ], df = c ρ]t, d f = λf ω ρc, d f 3 = (μf +K f )ω, d f 4 = μf ω, ρc d f 5 = Kf ω ρc, p = K K, ρc c =c = ]T ρc, c 4 = ]c ω K. λ, μ: Lame s constants t ij : Components of the stress tensor u: Displacement vector ρ: Density K : Thermalconductivity c : Specific heat at constant strain T : Uniformtemperature T: Temperaturechange α T : Coefficient of linear thermal expansion δ ij : Kronecker delta ε ijr : Alternatingsymbol λ f, μ f, K f, α f, β f, γ f, c : Material constants of the fluid σ f ij : Components of stress tensor in the fluid m f ij : Components of couple stress tensor in the fluid V: Velocityvector Ψ: Microrotation velocity vector ρ f : Density I: Scalarconstantwiththedimensionof moment of inertia of unit mass p: Pressure K : Thermalconductivity at f : Specific heat at constant strain T f : Absolutetemperature T f : Temperaturechange φ f : Variation in specific volume α T f: Coefficient of linear thermal expansion. Conflict of Interests (A.) 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

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