Plane Waves and Fundamental Solutions in Heat Conducting Micropolar Fluid

In the present investigation, we study the propagation of plane waves in heat conducting micropolar fluid. The phase velocity, attenuation coefficient, specific loss, and penetration depth are computed numerically and depicted graphically. In addition, the fundamental solutions of the system of differential equations in case of steady oscillations are constructed. Some basic properties of the fundamental solution and special cases are also discussed.


Introduction
Eringen [1] developed the theory of microfluids, in which microfluids possess three gyration vector fields in addition to its classical translatory degrees of freedom represented by velocity field.Eringen introduced the micropolar fluids [2] which are subclass of these fluids, in which the local fluid elements possess rigid rotations without stretch.Micropolar fluids can support couple stress, the body couples, and asymmetric stress tensor and possess a rotational field, which is independent of the velocity of fluid.Anisotropic fluids, liquid crystals with rigid molecules, magnetic fluids, cloud with dust, muddy fluids, biologicaltropic fluids, and dirty fluids (dusty air and snow) over airfoil can be modeled more realistically as micropolar fluids.Ariman et al. [3,4] studied microcontinuum fluid mechanics.Říha [5] discussed the theory of heat conducting micropolar fluid with microtemperature.Eringen and Kafadar [6] developed polar field theories.Brulin [7] discussed linear micropolar media.Flow and heat transfer in a micropolar fluid past with suction and heat sources were discussed by Agarwal and Dhanapal [8].Payne and Straughan [9] investigated critical Rayleigh numbers for oscillatory and nonlinear convection in an isotropic thermomicropolar fluid.Gorla [10] studied combined forced and free convection in the boundary layer flow of a micropolar fluid on a continuous moving vertical cylinder.Eringen [11] investigated the theory of microstretch and bubbly liquids.Aydemir and Venart [12] investigated the flow of a thermomicropolar fluid with stretch.Yerofeyev and Soldatov [13] discussed a shear surface wave at the interface of an elastic body and a micropolar liquid.The theory of elastic and viscoelastic micropolar liquids was studied by Yeremeyev and Zubov [14].Hsia and Cheng [15] discussed longitudinal plane waves propagation in elastic micropolar porous media.Hsia et al. [16] studied propagation of transverse waves in elastic micropolar porous semispaces.
Construction of fundamental solution of systems of partial differential equations is necessary to investigate the boundary value problems of the theory of elasticity and thermoelasticity.The fundamental solutions in the classical theory of coupled thermoelasticity were firstly studied by Hetnarski [17,18].Hetnarski and Ignaczak [19] studied generalized thermoelasticity.Svanadze [20][21][22][23][24][25] constructed the fundamental solutions in the microcontinuum field theories.Kumar and Kansal [26] investigated the fundamental solution in the theory of thermomicrostretch elastic diffusive solids.Fundamental solution in the theory of micropolar thermoelastic diffusion with voids was studied by Kumar and Kansal [27].Recently, Kumar and Kansal [28] discussed plane waves and fundamental solution in the generalized theories of thermoelastic diffusion.Kumar and Kansal [29] studied propagation of plane waves and fundamental solution in the theories of thermoelastic diffusive materials with voids.The information related to fundamental solutions of differential equations is contained in the books of Hörmander [30,31].
The main objective of the present paper is to study the propagation of plane waves in heat conducting micropolar fluid.Several qualitative characterizations of the wave field, such as phase velocity, attenuation coefficient, specific loss, and penetration depth, are computed and depicted graphically for different values of frequency.The representation of fundamental solution of system of equations in the case of steady oscillations is obtained in terms of elementary functions.Some particular cases have also been deduced.
Following Ciarletta [32], the basic equations for homogeneous, isotropic heat conducting micropolar fluids without body forces, body couples, and heat sources are given by where where , , , , , , and  0 are material constants of the fluid.k and Ψ are the velocity vector and microrotation velocity vector,  0 is the density,  is a scalar constant with the dimension of moment of inertia of unit mass,  * is the thermal conductivity,  0 is the specific heat at constant strain,  0 is the absolute temperature,  is the temperature change,  * is the variation in specific volume,  = (3 + 2 + )  , where   is the coefficient of linear thermal expansion, and Δ is the Laplacian operator.For convenience, the following nondimensional quantities are introduced: where  * = /,  2 1 = ( + 2 + )/ 0 , and  * is the characteristic frequency of the medium.
Making use of ( 2) and (3) in basic equations (1) and after suppressing the primes, we obtain Making use of ( 7) in (4), we obtain where Journal of Fluids 3 For two-dimensional problem, we take The relation between dimensionless velocity components V 1 and V 3 and nondimensional velocity potential functions  and  is expressed as Making use of ( 10)-( 11) in ( 5), (6), and ( 8), we obtain where

Solution of Plane Waves
For plane harmonic waves, we assume the solution of the form where  is the circular frequency and  is the complex wave number., , Ψ 2 ,  are undetermined amplitude vectors that are independent of time  and coordinates   ( = 1, 3). 1 and  3 are the direction cosines of the wave normal onto  1  3 -plane with the property  2 1 +  2 3 = 1.Using ( 13) in (12), we obtain The system of ( 14) will have nontrivial solution, if the determinant of the coefficients , , Ψ 2 ,  vanishes which on expansion yield where Solving (15), we obtain eight roots of , in which four roots of , that is,  1 ,  2 ,  3 , and  4 , correspond to positive  3direction and other four roots of , that is, − 1 , − 2 , − 3 , and − 4 , correspond to negative  3 -direction.Now and after, we will restrict our work to positive  3 -direction.Corresponding to roots  1 ,  2 ,  3 , and  4 there exist four waves in descending order of their velocities, that is, two coupled longitudinal waves and two coupled transverse waves.The expressions for phase velocity, attenuation coefficient, specific loss, and penetration depth of above waves are derived as follows.
(i) Phase Velocity.The phase velocities are given by where  1 ,  2 ,  3 ,  4 are the phase velocities of two coupled longitudinal waves and two coupled transverse waves, respectively.
(ii) Attenuation Coefficient.The attenuation coefficients are defined as where  1 ,  2 ,  3 ,  4 are the attenuation coefficients of two coupled longitudinal waves and two coupled transverse waves, respectively.
(iii) Specific Loss.The specific loss is the ratio of energy () dissipated in taking a specimen through a rate of stress cycle, to the elastic energy () stored in the specimen when the rate of strain is maximum.The specific loss is the most direct method of defining internal friction for a material.For a sinusoidal plane wave of small amplitude, Kolsky [33] shows that the specific loss (/) equals 4 times the absolute value of the imaginary part of  to the real part of ; that is, where  1 ,  2 ,  3 ,  4 are the specific loss of two coupled longitudinal waves and two coupled transverse waves, respectively.
(iv) Penetration Depth.The penetration depths are defined by where  1 ,  2 ,  3 ,  4 are the penetration depths of two coupled longitudinal waves and two coupled transverse waves, respectively.
The matrix differential operator is taken as where The above system of ( 22) can be represented in the following form: where V = (k, Ψ,   ) is a seven component vector function on  3 .
Let us assume that F is an elliptic differential operator Hörmander [30], if condition ( 26) is fulfilled.
Definition 1.The fundamental solution of the system of ( 21)-( 22) (the fundamental matrix of operator F) is the matrix (x) = ‖ ℎ (x)‖ 7×7 , satisfying condition [30] Here  is the Dirac delta, I = ‖ ℎ ‖ 7×7 is the unit matrix, and x ∈  3 .Now further G(x) in terms of elementary functions is constructed.

Fundamental Solution of System of Equation of Steady Oscillations
We consider the following system of equations: where H  , H  are three-component vector function on  3 and  is scalar function on  3 .The system of ( 28)-( 30) can be written as where F tr is the transpose of matrix F, Q = (H  , H  , ), and x ∈  3 .
Lemma 2. The matrix Y defined above is the fundamental matrix of operator Θ(Δ); that is, Proof.To prove the lemma, it is proved that We find that Now consider In the similar way, (71b) and (71c) can be proved.
The following matrix is now introduced: From ( 66), (70), and (74), it is obtained that Hence, G(x) is a solution to (27).Hence, the following theorem has been proved.
Theorem 3. The matrix G(x) defined by ( 60) is the fundamental solution of system of ( 22).

Basic Properties of the Matrix G(x)
Property 1.Each column of the matrix G(x) is the solution of the system of ( 22) at every point x ∈  3 except the origin.(76)

Numerical Results and Discussion
The following values of relevant parameters for numerical computations are taken.Following Singh and Tomar [34], the values of micropolar constants are taken as Thermal parameters are taken as of comparable magnitude: The variations of phase velocities, attenuation coefficients, with respect to frequency have been shown in Figures 1-4 and 5-8, respectively.In Figures 1-8, solid line corresponds to heat conducting micropolar fluid for  = 0.2 (MT1) and dash line corresponds to heat conducting micropolar fluid for  = 0.4 (MT2).1-4 that the magnitudes of the phase velocities   ,  = 1, 2, 3, 4, for MT1 and MT2 increase with increase in frequency.The phase velocity  1 for MT1 is greater than the phase velocity for MT2, while for  2 ,  3 ,  4 the behavior is reversed.This shows that as the value of micropolar constant increases, the phase velocity  1 decreases, while other phase velocities increase.There is slight difference in the phase velocity  2 for MT1 and MT2.

Attenuation Coefficient.
Figures 5-8 depict that the magnitudes of attenuation coefficient   ,  = 1, 2, 3, 4, get increased with increase in wave number .It is noticed that, with increase in the value of , the magnitude of attenuation coefficient increases; that is, the values of attenuation coefficients for MT2 are greater than the values for MT1 that shows the effect of micropolarity.

Conclusion
In the present paper, we have studied the propagation of plane waves in heat conducting micropolar fluid.The magnitudes of phase velocities and attenuation coefficients are depicted  numerically and presented graphically with respect to frequency.Appreciable micropolarity effect is observed on these amplitudes.It is noticed that the magnitudes of phase velocity   ,  = 2, 3, 4, and attenuation coefficient   ,  = 1, 2, 3, 4, for  = 0.4 remain more than the magnitude for  = 0.2.This reveals that the more the value of micropolar constant, the more the magnitude of phase velocity and attenuation coefficient.The fundamental solution in heat conducting micropolar fluid in case of steady oscillations in terms of elementary functions is also constructed in the present study.

Figure 7 :
Figure 7: Variation of attenuation coefficient with frequency .

Figure 8 :
Figure 8: Variation of attenuation coefficient with frequency .