# Thermophoresis and MHD mixed convection three-dimensional flow of viscoelastic fluid with Soret and Dufour effects

- 106 Downloads
- 1 Citations

## Abstract

Heat and mass transfer effects in three-dimensional mixed convection flow of viscoelastic fluid over a stretching surface with convective boundary conditions are investigated. The fluid is electrically conducting in the presence of constant applied magnetic field. Conservation laws of energy and concentration are based upon the Soret and Dufour effects. First order chemical reaction effects are also taken into account. By using the similarity transformations, the governing boundary layer equations are reduced into the ordinary differential equations. The transformed boundary layer equations are computed for the series solutions. Dimensionless velocity, temperature, and concentration distributions are shown graphically for different values of involved parameters. Numerical values of local Nusselt and Sherwood numbers are computed and analyzed. It is found that the behaviors of viscoelastic, mixed convection, and concentration buoyancy parameters on the Nusselt and Sherwood numbers are similar. However, the Nusselt and Sherwood numbers have qualitative opposite effects for Biot number, thermophoretic parameter, and Soret-Dufour parameters.

## Keywords

Soret-Dufour effects Thermophoretic effect Mixed convection Chemical reaction Convective condition Three-dimensional flow## Notes

### Compliance with ethical standards

### Conflict of interest

The authors declare that there is no conflict of interest regarding publication of this research paper.

## References

- 1.Rashidi MM, Chamkha AJ, Keimanesh M (2011) Application of multi-step differential transform method on flow of a second-grade fluid over a stretching or shrinking sheet. Am J Comput Math 6:119–128CrossRefGoogle Scholar
- 2.Ahmad A, Asghar S (2011) Flow of a second grade fluid over a sheet stretching with arbitrary velocities subject to a transverse magnetic field. Appl Math Lett 24:1905–1909MathSciNetCrossRefzbMATHGoogle Scholar
- 3.Hayat T, Shehzad SA, Qasim M, Obaidat S (2011) Flow of a second grade fluid with convective boundary conditions. Therm Sci 15:S253–S261CrossRefGoogle Scholar
- 4.Jamil M, Rauf A, Fetecau C, Khan NA (2011) Helical flows of second grade fluid due to constantly accelerated shear stresses. Commun Nonlinear Sci Numer Simul 16:1959–1969MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Nazar, M., Fetecau, C., Vieru, D., and Fetecau, C. New exact solutions corresponding to the second problem of Stokes for second grade fluids. Nonlinear Analysis: Real World Applications, 11
**,**584 591 (2010).Google Scholar - 6.Tan WC, Masuoka T (2005) Stokes’ first problem for a second grade fluid in a porous half-space with heated boundary. Int J Non-Linear Mech 40:515–522CrossRefzbMATHGoogle Scholar
- 7.Nazar R, Latip NA (2009) Numerical investigation of three-dimensional boundary layer flow due to a stretching surface in a viscoelastic fluid. Eur J Sci Res 29:509–517Google Scholar
- 8.Sakidis BC (1961) Boundary layer behavior on continuous solid surfaces. AICHE J 7:26–28CrossRefGoogle Scholar
- 9.Crane LJ (1970) Flow past a stretching plate. Z Angew Math Phys 21:645–647CrossRefGoogle Scholar
- 10.Hayat T, Shehzad SA, Alsaedi A (2012) Study on three-dimensional flow of Maxwell fluid over a stretching surface with convective boundary conditions. Int J Phys Sci 7:761–768Google Scholar
- 11.Sahoo B (2010) Flow and heat transfer of a non-Newtonian fluid past a stretching sheet with partial slip. Commun Nonlinear Sci Numer Simul 15:602–615MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Bhattacharyya K, Uddin MS, Layek GC, Malek MA (2010) Effect of chemically reactive solute diffusion on boundary layer flow past a stretching surface with suction or blowing. J Math Math Sci 25:41–48Google Scholar
- 13.Turkyilmazoglu M (2011) Thermal radiation effects on the time-dependent MHD permeable flow having variable viscosity. Int J Therm Sci 50:88–96CrossRefGoogle Scholar
- 14.Makinde OD, Aziz A (2011) Boundary layer flow of a nanofluid past a stretching sheet with a convective boundary condition. Int J Therm Sci 50:1326–1332CrossRefGoogle Scholar
- 15.Motsa SS, Hayat T, Aldossary OM (2012) MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method. Appl Math Mech Engl Ed 33:975–990MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Makinde OD, Chinyoka T (2010) MHD transient flows and heat transfer of dusty fluid in a channel with variable physical properties and Navier slip condition. Comput Math Appl 60:660–669MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Rashidi MM, Erfani E (2011) A new analytical study of MHD stagnation-point flow in porous media with heat transfer. Comput Fluids 40:172–178CrossRefzbMATHGoogle Scholar
- 18.Turkyilmazoglu M (2015) An analytical treatment for the exact solutions of MHD flow and heat over two-three dimensional deforming bodies. Int J Heat Mass Transf 90:781–789CrossRefGoogle Scholar
- 19.Hayat T, Shehzad SA, Qasim M, Obadiat S (2011) Thermal radiation effects on the mixed convection stagnation-point flow in a Jeffery fluid. Zeitschrift fur Naturforschung 66a:606–614Google Scholar
- 20.Abbas Z, Wang Y, Hayat T, Oberlack M (2010) Mixed convection in the stagnation-point flow of a Maxwell fluid towards a vertical stretching surface. Nonlinear Anal Real World Appl 121:3218–3228MathSciNetCrossRefzbMATHGoogle Scholar
- 21.Makinde OD (2011) MHD mixed-convection interaction with thermal radiation and nth order chemical reaction past a vertical porous plate embedded in a porous medium. Chem Eng Commun 198:590–608CrossRefGoogle Scholar
- 22.Hayat T, Qasim M (2010) Influence of thermal radiation and Joule heating on MHD flow of a Maxwell fluid in presence of thermophoresis. Int J Heat Mass Transf 53:4780–4788CrossRefzbMATHGoogle Scholar
- 23.Kandasamy R, Muhaimin I, Saim H (2010) Lie group analysis for the effect of temperature-dependent fluid viscosity with thermophoresis and chemical reaction on MHD free convective heat and mass transfer over a porous stretching surface in the presence of heat source/sink. Commun Nonlinear Sci Numer Simul 15:2109–2123MathSciNetCrossRefzbMATHGoogle Scholar
- 24.Hayat T, Alsaedi A (2011) On thermal radiation and joule heating effects in MHD flow of an Oldroyd-B fluid with thermophoresis. Arab J Sci Eng 36:1113–1124CrossRefGoogle Scholar
- 25.Eckert ERG, Drake RM (1972) Analysis of heat and mass transfer. McGraw-Hill, New YorkzbMATHGoogle Scholar
- 26.Tsai R, Haung JS (2009) Heat and mass transfer for Soret and Dufour effects on Hiemenz flow through porous medium onto a stretching surface. Int J Heat Mass Transf 52:2399–2406CrossRefzbMATHGoogle Scholar
- 27.Hayat T, Shehzad SA, Alsaedi A (2012) Soret and Dufour effects in the magnetohydrodynamic (MHD) flow of Casson fluid. Appl Math Mech Engl Ed 33:1301–1312CrossRefzbMATHGoogle Scholar
- 28.Rahman GMA (2010) Thermal-diffusion and MHD for Soret and Dufour’s effects on Hiemenz flow and mass transfer of fluid flow through porous medium onto a stretching surface. Physica B 405:2560–2569CrossRefGoogle Scholar
- 29.Aziz A (2009) A similarity solution for thermal boundary layer over a flat plate with a convective surface boundary condition. Commun Nonlinear Sci Numer Simul 14:1064–1068CrossRefGoogle Scholar
- 30.Makinde OD, Aziz A (2010) MHD mixed convection from a vertical plate embedded in a porous medium with a convective boundary condition. Int J Therm Sci 49:1813–1820CrossRefGoogle Scholar
- 31.Hayat T, Shehzad SA, Alseadi A, Alhothuali MS (2012) Mixed convection stagnation point flow of Casson fluid with convective boundary conditions. Chin Phys Lett 29:114704CrossRefGoogle Scholar
- 32.Turkyilmazoglu M (2011) An optimal analytic approximate solution for the limit cycle of Duffing-van der Pol equation. J Appl Mech Trans ASME 78:021005CrossRefGoogle Scholar
- 33.Liao SJ (2012) Homotopy analysis method in nonlinear differential equations. Higher Edu. Press, BeijingCrossRefzbMATHGoogle Scholar
- 34.Zheng L, Niu J, Zhang X, Gao Y (2012) MHD flow and heat transfer over a porous shrinking surface with velocity slip and temperature jump. Math Comput Model 56:133–144MathSciNetCrossRefzbMATHGoogle Scholar
- 35.Rashidi MM, Pour SAM, Hayat T, Obaidat S (2012) Analytic approximate solutions for steady flow over a rotating disk in porous medium with heat transfer by homotopy analysis method. Comput Fluids 54:1–9MathSciNetCrossRefzbMATHGoogle Scholar
- 36.Zhang X, Tang B, Hea Y (2011) Homotopy analysis method for higher-order fractional integro-differential equations. Comput Math Appl 62:3194–3203MathSciNetCrossRefzbMATHGoogle Scholar
- 37.Hayat T, Kiran A, Imtiaz M, Alsaedi A Effect of homogeneous–heterogeneous reactions in stagnation point flow of third grade fluid past a variable thickness stretching sheet. Neural Comput Appl. doi: 10.1007/s00521-017-2913
- 38.Turkyilmazoglu M (2012) Solution of Thomas-Fermi equation with a convergent approach. Commun Nonlinear Sci Numer Simul 17:4097–4103MathSciNetCrossRefzbMATHGoogle Scholar
- 39.Hayat T, Ashraf MB, Alsulami HH, Alhuthali MS (2014) Three dimensional mixed convection flow of viscoelastic fluid with thermal radiation and convective conditions. PLoS One 9:e90038CrossRefGoogle Scholar
- 40.Ashraf MB, Hayat T, Alsulami H (2016) Mixed convection falkner-skan wedge flow of an oldroyd-b fluid in presence of thermal radiation. J Appl Fluid Mech 9(4):1753–1762Google Scholar
- 41.Turkyilmazoglu M (2016) Determination of the correct range of physical parameters in the approximate analytical solutions of nonlinear equations using the Adomian decomposition method. Mediterr J Math 13:4019–4037MathSciNetCrossRefzbMATHGoogle Scholar
- 42.Turkyilmazoglu M (2016) Equivalences and correspondences between the deforming body induced flow and heat in two-three dimensions. Phys Fluids 28:043102CrossRefGoogle Scholar