Two-Phase Flow of Fluid-Particle Interaction over a Stretching Sheet in the Presence of Magnetic Field
-
J. Hasnain
,
Z. Abbas
Abstract
This article presents a theoretical study of magnetohydrodynamic boundary layer flow of a dusty viscoelastic fluid over a porous stretching sheet. The basic steady equations of the viscoelastic second grade fluid and dust phases are in the form of partial differential equations. A set of coupled nonlinear ordinary differential equations is obtained by using suitable similarity transformations. The approximate first order solutions of the resulting equations are obtained using the perturbation technique. The results are also verified with the well-known finite difference technique known as Keller box method. The physical insight of the involved parameters on the velocity of both fluid and dust phases and the skin-friction coefficient is shown through graphs and tables and discussed in detail. The study shows that an increased effective viscosity increases the velocity of both fluid and particle phase.
Nomenclature
| velocity field for fluid phase | dimensionless dust phase density | ||
| velocity field for dust phase | suction velocity | ||
| spatial coordinates | second grade parameter | ||
| velocity vector components for fluid in | magnetic parameter | ||
| velocity vector components for dust in | suction parameter | ||
| current density | Greek letters | ||
| magnetic field | material constants | ||
| number density of dust particle | Cauchy stress tensor | ||
| Stokes constant | dynamic viscosity | ||
| spherical radius of the dust particle | density of fluid | ||
| mass of dust particle | kinematic viscosity | ||
| pressure | electrical conductivity | ||
| identity tensor | fluid-particle interaction parameter | ||
| dimensionless velocity components for fluid phase | constant | ||
| dimensionless velocity components for dust phase | dimensionless quantity | ||
Acknowledgements
We are thankful to the anonymous reviewer for his/her suggestions to improve the version of paper.
References
[1] K. R. Rajagopal, On boundary conditions for fluids of the differential type, in: A. Sequeira (ed.), Navier Stokes equations and related non-linear problems, pp. 273–278, Plenum Press, New York, 1995.10.1007/978-1-4899-1415-6_22Search in Google Scholar
[2] K. R. Rajagopal and P. N. Kaloni, Some remarks on boundary conditions for fluids of the differential type, in: G. A. C. Graham and S. K. Malik (eds.), Continuum mechanics and its applications, pp. 935–942, Hemisphere, New York, 1989.Search in Google Scholar
[3] M. Sajid, I. Ahmad, T. Hayat and M. Ayub, Unsteady flow and heat transfer of a second grade fluid over a stretching sheet, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), 96–108.10.1016/j.cnsns.2007.07.014Search in Google Scholar
[4] M. Turkyilmazoglu, 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 (2013), 263–268.10.1016/j.ijmecsci.2013.10.011Search in Google Scholar
[5] R. Cortell, MHD (magneto-hydrodynamic) flow and radiative nonlinear heat transfer of a viscoelastic fluid over a stretching sheet with heat generation/absorption, Energy. 74 (2014), 896–905.10.1016/j.energy.2014.07.069Search in Google Scholar
[6] T. E. Akinbobola and S. S. Okoya, The flow of second grade fluid over a stretching sheet with variable thermal conductivity and viscosity in the presence of heat source/sink, J. Niger. Mathe. Soc. 34 (2015), 331–342.10.1016/j.jnnms.2015.10.002Search in Google Scholar
[7] N. S. Khan, S. Uslam, T. Gul, I. Khan and W. Khan, Thin film flow of a second grade fluid in a porous medium past a stretching sheet with heat transfer, Alexandria Eng. J. 57 (2018), 1019–1031.10.1016/j.aej.2017.01.036Search in Google Scholar
[8] M. Rafiq, M. Kamran, N. Ahmed, S. T. Mohyud-Din, Y. Bashir, S. A. Haider, S. Farwa and M. Tahir, Analytical solution for the flow of second grade fluid over a stretching sheet, AIP. Adv. 9 (2019), 055313.10.1063/1.5093158Search in Google Scholar
[9] S. S. Mosta, T. Hayat and O. M. Aldossary, MHD flow of upper-convected Maxwell fluid over porous stretching sheet using successive Taylor series linearization method, Appl. Math. Mech. Engl. Ed. 33 (2012), 975–990.10.1007/s10483-012-1599-xSearch in Google Scholar
[10] T. Hayat, M. Imtiaz, A. Alsaedi and R. Mansoor, MHD flow of nanofluids over an exponentially stretching sheet in a porous medium with convective boundary conditions, Chin. Phys. B. 23(5) (2014), 054701(1–8).10.1088/1674-1056/23/5/054701Search in Google Scholar
[11] A. S. Zaman, A. S. Abd Aziz and Z. Md Ali, Double slip effects of Magnetohydrodynamic (MHD) boundary layer flow over an exponentially stretching sheet with radiation, heat source and chemical reaction, IOP. Conf. Ser. J. Phys. Conf. Ser. 890 (2017), 012020. doi:10.1088/1742-6596/890/1/012020.Search in Google Scholar
[12] F. Mabood and S. Shateyl, Multiple slip effects on MHD unsteady flow heat and mass transfer impinging on permeable stretching sheet with radiation, Modell. Simul. Eng. 2019 (2019), 3052790 1–11. doi:10.1155/2019/3052790.Search in Google Scholar
[13] I. M. Alarifi, A. G. Abokhalil, M. Osman, L. A. Lund, M. B. Ayed, H. Hafedh Belmabrouk and I. Tlili, MHD flow and heat transfer over vertical stretching sheet with heat sink or source effect, Symmetry. 11 (2019), 297 1–14. doi:10.3390/sym11030297.Search in Google Scholar
[14] P. G. Saffman, On the stability of laminar flow of a dusty gas, J. Fluid. Mech. 13 (1962), 120–128.10.1017/S0022112062000555Search in Google Scholar
[15] K. Vajravelu and J. Nayfeh, Hydromagnetic flow of a dusty fluid over a stretching surface, Int. J. Non Linear Mech. 27 (1992), 937–945.10.1016/0020-7462(92)90046-ASearch in Google Scholar
[16] S. Manjunatha, B. J. Gireesha and C. S. Bagewadi, Effect of thermal radiation on boundary layer flow and heat transfer of dusty fluid over an unsteady stretching sheet, Int. J. Eng. Sci. Technol. 4 (2012), 36–48.10.4314/ijest.v4i4.5Search in Google Scholar
[17] B. J. Gireesha, B. Mahanthesh, R. S. R. Gorla and K. L. Krupalakshmi, Mixed convection two-phase flow of Maxwell fluid under the influence of non-linear thermal radiation, non-uniform heat source/sink and fluid-particle suspension, Ains Shams Eng. J. 9(4) (2018), 735–746.10.1016/j.asej.2016.04.020Search in Google Scholar
[18] M. Jalil, S. Asghar and S. Yasmeen, An exact solution of MHD boundary layer flow of dusty fluid over a stretching surface, Math. Prob. Eng. 2017 (2017), 2307469 1–5. doi:10.1155/2017/2307469.Search in Google Scholar
[19] M. Turkyilmazoglua, Magnetohydrodynamic two-phase dusty fluid flow and heat model over deforming isothermal surfaces, Phys. Fluids. 29 (2017), 013302.10.1063/1.4965926Search in Google Scholar
[20] Z. Abbas, J. Hasnain and M. Sajid, Effects of slip on MHD flow of a dusty fluid over a stretching sheet through porous space, J. Eng. Thermo. Phys. 28 (2019), 84–102.10.1134/S1810232819010077Search in Google Scholar
© 2019 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- Research article
- Improved Results on Adaptive Control Approach for Projective Synchronization of Neural Networks with Time-Varying Delay
- A Method to Solve the Reaction-Diffusion-Chemotaxis System
- Two-Phase Flow of Fluid-Particle Interaction over a Stretching Sheet in the Presence of Magnetic Field
- An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation
- Additional Extension of the Mathematical Model for BCG Immunotherapy of Bladder Cancer and Its Validation by Auxiliary Tool
- Extended Transformed Rational Function Method to Nonlinear Evolution Equations
- Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method
- Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System
- On the Numerical Computation of the Mittag–Leffler Function
Articles in the same Issue
- Frontmatter
- Research article
- Improved Results on Adaptive Control Approach for Projective Synchronization of Neural Networks with Time-Varying Delay
- A Method to Solve the Reaction-Diffusion-Chemotaxis System
- Two-Phase Flow of Fluid-Particle Interaction over a Stretching Sheet in the Presence of Magnetic Field
- An Algorithm for the Approximate Solution of the Fractional Riccati Differential Equation
- Additional Extension of the Mathematical Model for BCG Immunotherapy of Bladder Cancer and Its Validation by Auxiliary Tool
- Extended Transformed Rational Function Method to Nonlinear Evolution Equations
- Approximation of p-Biharmonic Problem using WEB-Spline based Mesh-Free Method
- Lie Symmetries, One-Dimensional Optimal System and Group Invariant Solutions for the Ripa System
- On the Numerical Computation of the Mittag–Leffler Function
