Unsteady hydromagnetic flow of Oldroyd-B fluid over an oscillatory stretching surface: a mathematical model
Sami Ullah Khan
Nasir Ali
Abstract
In the present work, we have studied an unsteady, two-dimensional boundary layer flow of a magnetohydrodynamics (MHD) Oldroyd-B fluid over an oscillatory stretching surface. The problem is modeled by using constitutive equations. The number of independent variables in the governing equations are reduced by using appropriate dimensionless variables. The analytical solution is computed by using homotopy analysis method. The influences of various physical parameters such as Deborah numbers, ratio of angular frequency to stretching rate parameter and Hartmann number on time-series of velocity and transverse velocity profiles at different time instants are investigated and discussed quantitatively with the help of various graphs. It is observed that amplitude of velocity increases by increasing ratio of oscillating frequency to stretching rate parameter while decreases by increasing Hartmann number. It is further observed that the magnitude of velocity decreases by increasing Hartmann number and Deborah numbers in the terms of relaxation time parameter.
Keywords:
Oldroyd-B fluid, oscillatory stretching sheet, homotopy analysis methodReferences
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ELSHEHAWEY E.F., ELSAYED M.E., ELBARBARY N.E. 2003. Effect of inclined magnetic field on magneto fluid flow through a porous medium between two inclined wavy porous plates (numerical study). Applied Mathematics and Computation, 135(1): 85-103.
HAYAT T., MUHAMMAD T., SHEHZAD S.A., ALSAEDI A. 2015a. Temperature and Concentration Stratification Effects in Mixed Convection Flow of an Oldroyd-B Fluid with Thermal Radiation and Chemical Reaction. PLoS ONE 10(6): e0127646, doi: 10.1371/journal.pone.0127646.
HAYAT T., MUSTAFA M., POP I. 2010. Heat and mass transfer for Soret and Dufour;s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Commun. Nonlinear Sci. Numer. Simul., 15: 1183-1196.
HAYAT T., SHAFIQ A., ALSAEDI A., ASGHAR S. 2015b. Effect of inclined magnetic field in flow of third grade fluid with variable thermal conductivity. AIP Advances, 5: 087108, doi: http://dx.doi.org/10.1063/1.4928321.
HAYAT T., SHEHZAD S.A., MEZEL S.A., ALSAEDI A. 2014. Three-dimensional flow of an Oldroyd-B fluid over a bidirectional stretching surface with prescribed surface temperature and prescribed surface heat flux. J. Hydrol. Hydromech., 62: 117-125.
LIAO S.J. 2004. On the homotopy analysis method for nonlinear problems. App. Mathematics and Computation, 147: 499-513.
MUKHOPADHYAY S., RANJAN DE P., LAYEK G.C. 2013. Heat transfer, characteristics for the Maxwell fluid flow past an unsteady stretching permeable surface embedded in a porous medium with thermal radiation. Journal of Applied Mechanics and Technical Physics, 54(3): 385-396.
POP I., NA T.Y. 1996. Unsteady flow past a stretching sheet. Mechanics Research Communications, 23: 413-422.
RAJAGOPAL K.R., BHATNAGAR R.K. 1995. Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mechanica,113: 233-239.
RAJU C.S.K., SANDEEP N., SULOCHANA C., SUGUNAMMA V., JAYACHANDRA BABU M. 2015. Radiation, inclined magnetic field and cross-diffusion effects on flow over a stretching surface. Journal of the Nigerian Mathematical Society, 34: 169-180.
SAJID M., ABBAS Z., JAVED T., ALI N. 2010. Boundary layer flow of an Oldroyd-B fluid in the region of stagnation point over a stretching sheet. Canadian Journal of Physics, 88: 635-640.
SHEIKH M., ABBAS Z. 2015. Effects of thermophoresis and heat generation/absorption on MHD flow due to an oscillatory stretching sheet with chemically reactive species. Journal of Magnetism and Magnetic Materials, S0304-8853(15): 3043.
TURKYILMAZOGLU M. 2009. Purely Analytic Solutions of the Compressible Boundary Layer Flow due to a Porous Rotating Disk with Heat Transfer. Phys. Fluids, 21: 106104.
TURKYILMAZOGLU M. 2012. Solution of the Thomas-Fermi equation with a convergent approach. Commun. Nonlinear Sci. Numer. Simul., 17: 4097-4103.
TURKYILMAZOGLU M. 2013. The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface. Int. J. Mech. Sci., 77: 263-268.
VARSHNEY C.L. 1979. Fluctuating flow of viscous fluid through a porous medium bounded by a porous plate. Indian J. Pure Appl. Math., 10(12): 1558-1564.
WANG C.Y. 1988. Nonlinear streaming due to the oscillatory stretching of a sheet in a Viscous fluid. Acta Mech. 72: 261-268.
ZHENG L., LIU Y., ZHANG X. 2011. Exact solutions for MHD flow of generalized Oldroyd-B fluid due to an infinite accelerating plate. Mathematical and Computer Modelling, 54: 780-788.
ZHENG L.C., JIN X., ZHANG X.X., ZHANG J. H. 2013. Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects. Acta Mechanica Sinica, 29(5): 667-675.
ABBAS Z., WANG Y., HAYAT T., OBERLACK M. 2009. Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface. Int. J. Numer. Meth. Fluids, 59: 443-458.
ABBASBANDY S. 2007. Homotopy Analysis Method for Heat Radiation Equations. Int. Commun. Heat Mass Transfer, 34: 380-387.
ALI N., KHAN S.U., ABBAS Z. 2015. Hydromagnetic Flow and Heat Transfer of a Jeffrey Fluid over an Oscillatory Stretching Surface. Z. Naturforsch., A, 70(7): 567-576.
ALI N., KHAN S.U., ABBAS Z., SAJID M. 2016. Soret and Dufour effects on hydromagnetic flow of viscoelastic fluid over porous oscillatory stretching sheet with thermal radiation. J. Braz. Soc. Mech. Sci. Eng., 38: 2533-2546.
CRANE L.J. 1970. Flow past a stretching plate. Z. Angew. Math. Phys. (ZAMP), 21: 645-647.
ELSHEHAWEY E.F., ELSAYED M.E., ELBARBARY N.E. 2003. Effect of inclined magnetic field on magneto fluid flow through a porous medium between two inclined wavy porous plates (numerical study). Applied Mathematics and Computation, 135(1): 85-103.
HAYAT T., MUHAMMAD T., SHEHZAD S.A., ALSAEDI A. 2015a. Temperature and Concentration Stratification Effects in Mixed Convection Flow of an Oldroyd-B Fluid with Thermal Radiation and Chemical Reaction. PLoS ONE 10(6): e0127646, doi: 10.1371/journal.pone.0127646.
HAYAT T., MUSTAFA M., POP I. 2010. Heat and mass transfer for Soret and Dufour;s effect on mixed convection boundary layer flow over a stretching vertical surface in a porous medium filled with a viscoelastic fluid. Commun. Nonlinear Sci. Numer. Simul., 15: 1183-1196.
HAYAT T., SHAFIQ A., ALSAEDI A., ASGHAR S. 2015b. Effect of inclined magnetic field in flow of third grade fluid with variable thermal conductivity. AIP Advances, 5: 087108, doi: http://dx.doi.org/10.1063/1.4928321.
HAYAT T., SHEHZAD S.A., MEZEL S.A., ALSAEDI A. 2014. Three-dimensional flow of an Oldroyd-B fluid over a bidirectional stretching surface with prescribed surface temperature and prescribed surface heat flux. J. Hydrol. Hydromech., 62: 117-125.
LIAO S.J. 2004. On the homotopy analysis method for nonlinear problems. App. Mathematics and Computation, 147: 499-513.
MUKHOPADHYAY S., RANJAN DE P., LAYEK G.C. 2013. Heat transfer, characteristics for the Maxwell fluid flow past an unsteady stretching permeable surface embedded in a porous medium with thermal radiation. Journal of Applied Mechanics and Technical Physics, 54(3): 385-396.
POP I., NA T.Y. 1996. Unsteady flow past a stretching sheet. Mechanics Research Communications, 23: 413-422.
RAJAGOPAL K.R., BHATNAGAR R.K. 1995. Exact solutions for some simple flows of an Oldroyd-B fluid. Acta Mechanica,113: 233-239.
RAJU C.S.K., SANDEEP N., SULOCHANA C., SUGUNAMMA V., JAYACHANDRA BABU M. 2015. Radiation, inclined magnetic field and cross-diffusion effects on flow over a stretching surface. Journal of the Nigerian Mathematical Society, 34: 169-180.
SAJID M., ABBAS Z., JAVED T., ALI N. 2010. Boundary layer flow of an Oldroyd-B fluid in the region of stagnation point over a stretching sheet. Canadian Journal of Physics, 88: 635-640.
SHEIKH M., ABBAS Z. 2015. Effects of thermophoresis and heat generation/absorption on MHD flow due to an oscillatory stretching sheet with chemically reactive species. Journal of Magnetism and Magnetic Materials, S0304-8853(15): 3043.
TURKYILMAZOGLU M. 2009. Purely Analytic Solutions of the Compressible Boundary Layer Flow due to a Porous Rotating Disk with Heat Transfer. Phys. Fluids, 21: 106104.
TURKYILMAZOGLU M. 2012. Solution of the Thomas-Fermi equation with a convergent approach. Commun. Nonlinear Sci. Numer. Simul., 17: 4097-4103.
TURKYILMAZOGLU M. 2013. The analytical solution of mixed convection heat transfer and fluid flow of a MHD viscoelastic fluid over a permeable stretching surface. Int. J. Mech. Sci., 77: 263-268.
VARSHNEY C.L. 1979. Fluctuating flow of viscous fluid through a porous medium bounded by a porous plate. Indian J. Pure Appl. Math., 10(12): 1558-1564.
WANG C.Y. 1988. Nonlinear streaming due to the oscillatory stretching of a sheet in a Viscous fluid. Acta Mech. 72: 261-268.
ZHENG L., LIU Y., ZHANG X. 2011. Exact solutions for MHD flow of generalized Oldroyd-B fluid due to an infinite accelerating plate. Mathematical and Computer Modelling, 54: 780-788.
ZHENG L.C., JIN X., ZHANG X.X., ZHANG J. H. 2013. Unsteady heat and mass transfer in MHD flow over an oscillatory stretching surface with Soret and Dufour effects. Acta Mechanica Sinica, 29(5): 667-675.
Khan, S. U., & Ali, N. (2017). Unsteady hydromagnetic flow of Oldroyd-B fluid over an oscillatory stretching surface: a mathematical model. Technical Sciences, 20(1), 87–100. https://doi.org/10.31648/ts.2913
Sami Ullah Khan
Nasir Ali
