Influence of initial stress and gravity on refraction and reflection of SV wave at interface between two viscoelastic liquid under three thermoelastic theories

Technical Paper
  • 18 Downloads

Abstract

This approach is to examine the impact of magnetic field, viscosity, gravity and initial stress on SV wave while traveling through the interface of two visco-thermoelastic liquid layers. The basic equations in context of three theories have been discussed to drive results for refracted thermal and P waves and reflected thermal, SV and P waves. After using the boundary conditions the amplitude ratios have been computed in matrix form.

Keywords

Initial stress Viscosity Magnetic field Reflection Refraction 

List of symbols

P

Initial stress

\(\overrightarrow {\acute\omega }\)

Local rotation

\({\acute{s}}_{11}\)

Principal or incremental stress component along x axis

\({\acute{s}}_{22}\)

Principal or incremental stress components along y axis

\({\acute{s}}_{21}\)

Shear stress component

S11

Normal initial stress component along x axis

S22

Normal initial stress component along y axis

ρ

Density of medium

u

Component of displacement along x axis

v

Component of displacement along y axis

F1

Lorent’z force along x axis

F2

Lorent’z force along y axis

 \(\gimel\), μ

Lame’s constant

dxy

Shear strain component

dxx

Principal strain component

αt

Coefficient of linear thermal expansion

K

Bulk modulus

T

Absolute temperature

τ1

Relaxation time

ce

Specific heat per unit mass

ϑ

Thermal conductivity

\({\mathbf{\mathcal{B}}}\)

Electric intensity vector

°

Perturbed magnetic field vector

\(\varvec{H}\)

Magnetic field vector

\(\varvec{J}\)

Electric current density vector

\(\in_{0}\)

Electric permeability

μe

Magnetic permeability

RH

Magnetic pressure number

c1

Elastic wave velocity

c2

Rotational wave velocity

ζ

Initial stress parameter

ω

Circular frequency

k

Wave number

ɛT

Thermoelastic coupling constant

τθ

Phase lag of gradient of temperature

d

Strain component

References

  1. 1.
    Dhaliwal RS, Singh A (1980) Dynamic coupled thermoelasticity Hindustan Publ. Corp, New DelhiGoogle Scholar
  2. 2.
    Lord HW, Shulman Y (1967) A generalized dynamical theory of thermoelasticity. J Mech Phys Solids 15(5):299–309CrossRefMATHGoogle Scholar
  3. 3.
    Green AE, Lindsay KA (1972) Thermoelasticity. J Elast 2(1):1–7CrossRefMATHGoogle Scholar
  4. 4.
    RoyChoudhuri SK (2007) One-dimensional thermoelastic waves in elastic half-space with dual phase-lag effects. J Mech Mater Struct 2(3):489–503CrossRefGoogle Scholar
  5. 5.
    Abouelregal Ahmed E, Abo-Dahab SM (2012) Dual phase lag model on magneto-thermoelasticity infinite non-homogeneous solid having a spherical cavity. J Therm Stresses 35(9):820–841CrossRefGoogle Scholar
  6. 6.
    Sheikholeslami Mohsen (2017) Magnetic field influence on nanofluid thermal radiation in a cavity with tilted elliptic inner cylinder. J Mol Liq 229:137–147CrossRefGoogle Scholar
  7. 7.
    Sheikholeslami Mohsen (2017) Numerical simulation of magnetic nanofluid natural convection in porous media. Phys Lett A 381(5):494–503CrossRefGoogle Scholar
  8. 8.
    Sheikholeslami M, Rokni HB (2017) Nanofluid two phase model analysis in existence of induced magnetic field. Int J Heat Mass Transf 107:288–299CrossRefGoogle Scholar
  9. 9.
    Sheikholeslami M (2017) Influence of Coulomb forces on Fe 3 O 4–H 2 O nanofluid thermal improvement. Int J Hydrogen Energy 42(2):821–829CrossRefGoogle Scholar
  10. 10.
    Kandelousi Mohsen Sheikholeslami (2014) Effect of spatially variable magnetic field on ferrofluid flow and heat transfer considering constant heat flux boundary condition. Eur Phys J Plus 11(129):1–12Google Scholar
  11. 11.
    Sheikholeslami M, Hayat T, Alsaedi A (2017) Numerical simulation of nanofluid forced convection heat transfer improvement in existence of magnetic field using lattice Boltzmann method. Int J Heat Mass Transf 108:1870–1883CrossRefGoogle Scholar
  12. 12.
    Sheikholeslami M, Bhatti MM (2017) Forced convection of nanofluid in presence of constant magnetic field considering shape effects of nanoparticles. Int J Heat Mass Transf 111:1039–1049CrossRefGoogle Scholar
  13. 13.
    Sheikholeslami M, Hayat T, Alsaedi A (2017) Numerical study for external magnetic source influence on water based nanofluid convective heat transfer. Int J Heat Mass Transf 106:745–755CrossRefGoogle Scholar
  14. 14.
    Sheikholeslami M, Houman B (2017) Rokni. “Numerical modeling of nanofluid natural convection in a semi annulus in existence of Lorentz force.”. Comput Methods Appl Mech Eng 317:419–430CrossRefGoogle Scholar
  15. 15.
    Abd-Alla AM, Abo-Dahab SM, Kilany AA (2016) SV-waves incidence at interface between solid-liquid media under electromagnetic field and initial stress in the context of three thermoelastic theories. J Therm Stresses 39(8):960–976CrossRefGoogle Scholar
  16. 16.
    Sharma JN, Chauhan RS (1999) On the problems of body forces and heat sources in thermoelasticity without energy dissipation. Indian J Pure Appl Math 30:595–610MATHGoogle Scholar
  17. 17.
    Chakraborty N, Singh MC (2011) Reflection and refraction of a plane thermoelastic wave at a solid–solid interface under perfect boundary condition, in presence of normal initial stress. Appl Math Model 35(11):5286–5301MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Deswal S, Tomar SK, Kumar R (2000) Effect of fluid viscosity on wave propagation in a cylindrical bore in micropolar elastic medium. Sadhana 25(5):439–452CrossRefGoogle Scholar
  19. 19.
    Singh MC, Chakraborty N (2016) Reflection and refraction of plane waves at the interface of two visco-thermoelastic liquid layers in presence of magnetic field and compressional stress, with the outer core inside the earth as a model. Int J Appl Comput Math 3(3):2107–2124MathSciNetCrossRefGoogle Scholar
  20. 20.
    De SN, Sen-Gupta PR (1974) Influence of gravity on wave propagation in an elastic layer. J Acoust Soc Am 55(5):919–921CrossRefGoogle Scholar
  21. 21.
    Acharya DP, Roy I, Sengupta S (2009) Effect of magnetic field and initial stress on the propagation of interface waves in transversely isotropic perfectly conducting media. Acta Mech 202(1):35–45CrossRefMATHGoogle Scholar
  22. 22.
    Biot MA, Drucker DC (1965) Mechanics of incremental deformation. J Appl Mech 32:957CrossRefGoogle Scholar
  23. 23.
    Ewing WM, Jardetzky WS, Press F, Beiser A (1957) Elastic waves in layered media. Phys Today 10:27CrossRefMATHGoogle Scholar
  24. 24.
    Dziewonski AM, Anderson DL (1981) Preliminary reference Earth model. Phys Earth Planet Inter 25(4):297–356CrossRefGoogle Scholar

Copyright information

© The Brazilian Society of Mechanical Sciences and Engineering 2018

Authors and Affiliations

  1. 1.Department of Mathematics and StatisticsInternational Islamic UniversityIslamabadPakistan

Personalised recommendations