## THERMAL SCIENCE

International Scientific Journal

### TIME-SPACE DEPENDENT FRACTIONAL VISCOELASTIC MHD FLUID FLOW AND HEAT TRANSFER OVER ACCELERATING PLATE WITH SLIP BOUNDARY

**ABSTRACT**

The MHD flow and heat transfer of viscoelastic fluid over an accelerating plate with slip boundary are investigated. Different from most classical works, a modified time-space dependent fractional Maxwell fluid model is proposed in depicting the constitutive relationship of the fluid. Numerical solutions are obtained by explicit finite difference approximation and exact solutions are also presented for the limiting cases in integral and series forms. Furthermore, the effects of parameters on the flow and heat transfer behavior are analyzed and discussed in detail.

**KEYWORDS**

PAPER SUBMITTED: 2015-06-14

PAPER REVISED: 2015-09-13

PAPER ACCEPTED: 2015-09-15

PUBLISHED ONLINE: 2015-09-26

**THERMAL SCIENCE** YEAR

**2017**, VOLUME

**21**, ISSUE

**6**, PAGES [2337 - 2345]

- Renardy, M., Stability of Creeping Flows of Maxwell Fluids, Archive for Rational Mechenics and Analysis, 198 (2010), 2, pp. 723-733
- Savelev, E., Renardy, M., Control of Homogeneous Shear Flow of Multimode Maxwell Fluids, Journal of Non-Newtonian Fluid Mechanics, 165 (2010), 3-4, pp. 136-142
- Karra, S., et al., On Maxwell Fluids with Relaxation Time and Viscosity Depending on the Pressure, International Journal of Non-Linear Mechanics, 46 (2011), 6, pp. 819-827
- Hayat, T., et al., Radiation Effects on MHD Flow of Maxwell Fluid in a Channel with Porous Medium, International Journal of Heat and Mass Transfer, 54 (2011), 4, pp. 854-862
- Salah, F., et al., New Exact Solution for Rayleigh-Stokes Problem of Maxwell Fluid in a Porous Medium and Rotating Frame, Results in Physics, 1 (2011), 1, pp. 9-12
- Podlubny, I., Fractional Differential Equations, Academic Press, New York, USA, 1999
- Nazar, M., et al., Flow Through an Oscillating Rectangular Duct for Generalized Maxwell Fluid with Fractional Derivatives, Commun Nonlinear Sci Numer Simulat., 17 (2012), 8, pp. 3219-3234
- Jamil, M., et al., New Exact Analytical Solutions for Stokes' First Problem of Maxwell Fluid with Fractional Derivative Approach, Computers and Mathematics with Applications, 62 (2011), 3, pp. 1013-1023
- Fetecau, C., et al., Flow of Fractional Maxwell Fluid between Coaxial Cylinders, Archive of Applied Mechanics, 81 (2011), 8, pp. 1153-1163
- Yang, D., Zhu, K. Q., Start-up Flow of a Viscoelastic Fluid in a Pipe with a Fractional Maxwell's Model, Computers and Mathematics with Applications, 60 (2010), 8, pp. 2231-2238
- Cao, L. L., et al., Time Domain Analysis of the Fractional Order Weighted Distributed Parameter Maxwell Model, Computers and Mathematics with Applications, 66 (2013), 5, pp. 813-823
- Hayat, T., et al., Soret and Dufour Effects in Three-dimensional Flow of Maxwell Fluid with Chemical Reaction and Convective Condition, International Journal of Numerical Method for Heat & Fluid Flow, 25 (2015), 1, pp. 98-120.
- Vieru, D., et al., Time-fractional Free Convection Flow near a Vertical Plate with Newtonian Heating and Mass Diffusion, Thermal Science, 19 (2015), Suppl. 1, pp. S85-S98
- Mustafa, M., et al., Sakiadis Flow of Maxwell Fluid Considering Magnetic Field and Convective Boundary Conditions, AIP Advances, 5 (2015), pp. 027106
- Fetecau, C., et al., Unsteady Flow of a Generalized Maxwell Fluid with Fractional Derivative due to a Constantly Accelerating Plate, Computers and Mathematics with Applications, 57 (2009), 4, pp. 596-603
- Jamil, M., Fetecau, C., Helical Flows of Maxwell Fluid between Coaxial Cylinders with Given Shear Stresses on the Boundary, Nonlinear Analysis: Real World Applications, 11 (2010), 5, pp. 4302-4311
- Laadj, T., Renardy, M., Initial Value Problems for Creeping Flow of Maxwell Fluids. Nonlinear Analysis, 74 (2011), 11, pp. 3614-3632
- Liu, Q. S., et al., Time Periodic Electroosmotic Flow of the Generalized Maxwell Fluids between Two Micro-parallel Plates, Journal of Non-Newtonian Fluid Mechanics, 166 (2011), 9-10, pp. 478-486
- Yuste, S. B., Weighted Average Finite Difference Methods for Fractional Diffusion Equations, Journal of Computational Physics, 216 (2006), 1, pp. 264-274
- Jia, J. H., Wang, H., Fast Finite Difference Methods for Space-fractional Diffusion Equations with Fractional Derivative Boundary, Journal of Computational Physics, 293 (2015), pp. 359-369
- Xu, Y. F., et al., Numerical and Analytical Solutions of New Generalized Fractional Diffusion Equation, Computers and Mathematics with Applications, 66 (2013), 10, pp. 2019-2029
- Zheng, L. C., et al., Exact Solutions for Generalized Maxwell Fluid Flow due to Oscillatory and Constantly Accelerating Plate. Nonlinear Analysis: Real World Applications, 11 (2010), 5, pp. 3744-3751
- Gómez-Aguilar, J.F., et al., A Physical Interpretation of Fractional Calculus in Observables Terms: Analysis of the Fractional Time Constant and the Transitory Response, Revista Mexicana de Física, 60 (2014), 1, pp. 32-38
- Fetecau, C., et al., A Note on the Flow Induced by a Constantly Accelerating Plate in an Oldroyd-B Fluid, Applied Mathematical Modelling, 31 (2007), 4, pp. 647-654
- Fetecau, C., et al., General Solutions for Magnetohydrodynamic Natural Convection Flow with Radiative Heat Transfer and Slip Condition over a Moving Plate, Zeitschrift für Naturforschung A, 68a (2013), 10-11, pp. 659-667