Exact Solutions for Fractionalized Second Grade Fluid Flows with Boundary Slip Effects
More details
Hide details
Department of Mathematics and Statistics, Quaid-e-Awam University of Engineering, Science and Technology, 67480, Nawabshah, Pakistan
Department of Electrical Engineering, Quaid-e-Awam University of Engineering, Science and Technology, 67480, Nawabshah, Pakistan
Department of Business Administration, Shaheed Benazir Bhutto University, 67480, Nawabshah, Pakistan
Online publication date: 2021-01-29
Publication date: 2021-03-01
International Journal of Applied Mechanics and Engineering 2021;26(1):88-103
In this paper, an exact analytical solution for the motion of fractionalized second grade fluid flows moving over accelerating plate under the influence of slip has been obtained. A coupled system of partial differential equations representing the equation of motion has been re-written in terms of fractional derivatives form by using the Caputo fractional operator. The Discrete Laplace transform method has been employed for computing the expressions for the velocity field u(y, t) and the corresponding shear stress τ (y, t). The obtained solutions for the velocity field and the shear stress have been written in terms of Wright generalized hypergeometric function pψq and are expressed as a sum of the slip contribution and the corresponding no-slip contribution. In addition, the solutions for a fractionalized, ordinary second grade fluid and Newtonian fluid in the absence of slip effect have also been obtained as special case. Finally, the effect of different physical parameters has been demonstrated through graphical illustrations.
Rajagopal K.R. and Kaloni P.N. (1989): Continuum Mechanics and Its Applications.– Washington, DC, Hemisphere Press.
Fetecau Corina, Jamil M., Fetecau C. and Vieru D. (2009): The Rayleigh-Stokes problems for an edge in a generalized Oldroyd-B fluid.– Zeitschrift Frangewandte Mathematik und Physik (ZAMP), vol.60, pp.5921-933.
Jamil M. (2012): First problem of Stokes’ for generalized Burgers’ fluids.– ISRN Mathematical Physics, Article ID 831063.
Jamil M., Khan N. A. and Rauf A. (2012): Oscillating flows of fractionalized second grade fluid.– ISRN Mathematical Physics, Article ID 908386.
Khandelwal K. and Mathur V. (2015): Exact solutions for an unsteady flow of viscoelastic fluid in cylindrical domains using the fractional Maxwell model.– International Journal of Applied and Computational, vol.1, pp.143-156.
Liu Y., Zheng L. and Zhang X. (2011): Unsteady MHD Couette flow of a generalized Oldroyd-B fluid with fractional derivative.– Computers and Mathematics and Applications, vol.61, No.2, pp.443-450.
Tripathi D., Gupta P.K. and Das S. (2011): Influence of slip condition on peristaltic transport of a viscoelastic fluid with fractional Burgers’ model.– Thermal Science, vol.15 No.2, pp.501-515.
Kumar D., Singh J. and Kumar S. (2014): A fractional model of Navier–Stokes equation arising in unsteady flow of a viscous fluid.– Journal of the Association of Arab Universities for Basic and Applied Sciences, vol.17, pp.14-19.
Valério D., Machado J.T. and Kiryakova V. (2014): Some pioneers of the applications of fractional calculus.– Fractional Calculus and Applied Analysis, vol.17, pp.552-578.
Gupta S., Kumar D. and Singh J. (2014): Numerical study for systems of fractional differential equations via Laplace transform.– Journal of the Egyptian Mathematical Society, doi.org/10.1016/j.joems.2014.04.003.
Bagley R. L. (1983). A theoretical basis for the application of fractional calculus to viscoelasticity.– Journal of Rheology, vol.27, No.3, pp.201-210.
Friedrich C.H.R. (1991): Relaxation and retardation function of the Maxwell model with fractional derivatives.– Rheological Acta, vol.30, pp.151-158.
Junki H., Guangyu H. and Ciqun L. (1997): Analysis of general second order fluid flow in double cylinder rheometer.– Science in China Series A, vol.40, pp.183-190.
Guangyu H., Junki H. and Ciqun L. (1995): General second order fluid flow in a pipe.– Applied Mathematics and Mechanics, vol.16, pp.825-831.
Xu M. Y. and Tan W. C. (2001): Theoretical analysis of the velocity field, stress field and vortex sheet of generalized second order fluid with fractional anomalous diffusion.– Science and China Series A, vol.44, pp.1387-1399.
Xu M. Y. and Tan W. C. (2003): Representation of the constitutive equation of viscoelastic materials by the generalized fractional element networks and its generalized solution.– Science in China Series G Physics Mechanics and Astronomy, vol.46, No.2, pp.145-157.
Jamil M., Khan N.A. and Asjad M. I. (2013): New exact solutions for an Oldroyd-B fluid with fractional derivatives, Stokes’ first problem.– International Journal of Nonlinear Sciences & Numerical Simulation, vol.14, No.7-8, pp.443-451.
Jamil M., Khan N. A. and Shahid N. (2013): Fractional MHD Oldroyd-B fluid over an oscillating plate.– Thermal Science, vol.17, pp.997-1011.
Tan W.C., Pan W.X. and Xu M.Y. (2003): A note on unsteady flows of a viscoelastic fluid with fractional Maxwell model between two parallel plates.– International Journal of Non-linear Mechanics, vol.38, No.5, pp.645-650.
Tan W.C., Xian F. and Wei L. (2002): An exact solution of unsteady Couette flow of generalized second grade fluid.– Chinese Science Bulletin, vol.47, pp.1783-1785.
Tan W.C. and Xu M.Y. (2002): The impulsive motion of flat plate in a generalized second grade fluid.– Mechanics Research Communications, vol.29, No.1, pp.3-9.
Tan W.C. and Xu M.Y. (2004): Unsteady flows of a generalized second grade fluid with the fractional derivative model between two parallel plates.– Acta Mechinca Sinica, vol.20, pp.471-476.
Jamil M., Rauf A., Zafar A.A. and Khan N.A. (2011): New exact analytical solutions for Stokes’ first problem of Maxwell fluid with fractional derivative approach.– Computers and Mathematics with Applications, vol.62, No.3, pp.1013-1023.
Khan M. and Wang S. (2009): Flow of a generalized second-grade fluid between two side wall perpendicular to a plate with a fractional derivative model.– Nonlinear Analysis: Real World Applications, vol.10, No.1, pp.203-208.
Zheng L., Liu Y. and Zhang X. (2012): Slip effects on MHD flow of a generalized Oldroyd-B fluid with fractional derivative.– Nonlinear Analysis, Real World Applications, vol.13, No.2, pp.513-523.
Eldesoky I.M. (2012): Influence of slip condition on peristaltic transport of a compressible Maxwell fluid through porous medium in a tube.– International Journal of Applied Mathematics and Mechanics, vol.8, pp.99-117.
Ashmawy E.A. (2012): Unsteady Couette flow of a micropolar fluid with slip.– Meccanica, vol.47, No.1, pp.85-94.
Ellahi R. (2009): Effects of the slip boundary condition on non-Newtonian flows in a channel.– Communications in Nonlinear Science and Numerical Simulation, vol.14, No.4, pp.1377-1384.
Jamil M. and Khan N.A. (2011): Slip effects on fractional viscoelastic fluids.– International Journal of Differential Equations, Article ID 193813.
Hayat T., Ellahi R., Asghar S. and Siddiqui A.M. (2004): Flow induced by a non-coaxial rotation of a porous disk executing non-torsional oscillations and a second grade fluid at infinity.– Applied Mathematical Modelling, vol.28, No.6, pp.591-605.
Richardson S. (1973): On the no-slip boundary condition.– Journal of Fluid Mechanics, vol.59, No.4, pp.707-719.
Thompson P.A. and Troian S.M. (1997): A general boundary condition for liquid flow at solid surface.– Nature, vol.389, pp.360-362.
Basset A.B. (1961): A Treatise on Hydrodynamic with Numerous Tramples.– Dover, New York, NY, USA, vol.2. p.368.
O’Neill M.E., Ranger K.B. and Brenner H. (1986): Slip at the surface of a translating-rotating sphere bisected by a free surface bounding a semi-infinite viscous fluid, removal of the contact line singularity.– Physics of Fluids, vol.29, pp.913-924.
Devakar M., Sreenivasu D. and Shankar B. (2014): Analytical solutions of some fully developed flows of couple stress fluid between concentric cylinders with slip boundary conditions.– International Journal of Engineering Mathematics, Article ID 785396.
Fetecau C. and Fetecau Corina. (2005): Starting solutions for some unsteady unidirectional flows of a second grade fluid.– International Journal of Engineering Science, vol.43, No.10, pp.781-789.
Fetecau C. and Fetecau Corina. (2006): Starting solutions for the motion of second grade fluid due to longitudinal and torsional oscillations of a circular cylinder.– International Journal of Engineering Science, vol.44, No.11-12, pp.788-796.
Fetecau C., Fetecau Corina. and Zierep J. (2002): Decay of potential vortex and propagation of a heat wave in a second grade fluid.– International Journal of Non-linear Mechanics, vol.37, No.6, pp.1051-1056.
Hayat T., Khan M. and Ayub M. (2006): Some analytical solutions for second grade fluid flows for cylindrical geometries.– Mathematical and Computer Modelling, vol.43, No.1-2, pp.16-29.
Hayat T., Najam S., Sajid M., Ayub M. and Mesloub S. (2010): On exact solutions for oscillatory flows in a generalized burgers fluid with slip condition.– Z. Naturforschung, vol.65, No.5, pp.381-391.
Rajagopal K.R. (1984): On the creeping flow of the second order fluid.– Journal of Non Newtonian Fluid Mechanics, vol.15, No.2, pp.239-246.
Luchko Y. (2008): Algorithms for evaluation of the wright function for the real arguments values.– Fractional Calculus and Applied Analysis, vol.11, No.1, pp.57-75.
Mainardi F.(2010): Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models.– Imperial College Press, London, p.368.
Petras I. (2011): Fractional Derivatives, Fractional Integral and Fractional Differential Equations in Matlab.– Engineering Education and Research using MATLAB, p.28.
Fetecau C., Mahmood A. and Jamil M. (2010): Exact solutions for the flow of visco-elastic fluid induced by a circular cylinder subject to a time dependent shear stress.– Communication in Nonlinear Science and Numerical Simulation, vol.15, No.12, pp.3931-3938.
Jamil M., Khan N.A. and Zafar A.A. (2011): Translational flows of an Oldroyd-B fluid with fractional derivatives.– Computers and Mathematical Applications, vol.62, No.3, pp.1540-1553.
Journals System - logo
Scroll to top