Abstract
This paper presents a study of Walters’-B fluid with Caputo–Fabrizio fractional derivatives through an infinitely long oscillating vertical plate by Newtonian heating under the action of transverse magnetic field. The fractional calculus approach is employed to obtain a system of fractional partial differential equations. The governing equations of momentum and energy are converted first into dimensionless form and then solved by employing Laplace transformation. The Laplace inverse transform has been evaluated both analytically and numerically. The graphical illustrations represent the behavior of material parameters on the solutions. A comparison between exact and numerical solutions is presented in tabular and graphical form. The variation in Nusselt number with the change in fractional and physical parameters is also presented. The velocity and temperature of the fluid decreases with the enhancement in the fractional parameter for small values of time, and it has the opposite behavior for greater values of time.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Abbreviations
- \(u \) :
-
Fluid velocity, [\(\mbox{m}\,\mbox{s}^{ - 1}\)]
- \(T \) :
-
Fluid temperature, [K]
- \(g \) :
-
Gravitational acceleration, [\(\mbox{m}\,\mbox{s}^{ - 2}\)]
- \(c_{p} \) :
-
Specific heat at a constant pressure, [\(\mbox{J}\,\mbox{kg}^{ - 1}\,\mbox{K}^{ - 1}\)]
- \(T_{\infty } \) :
-
Temperature of the fluid away from the plate, [K]
- \(\mathit{Gr} \) :
-
Thermal Grashof number, [\(\beta T_{w}\)]
- \(k \) :
-
Fluid thermal conductivity, [\(\mbox{W}\,\mbox{m}^{ - 2}\,\mbox{K}^{ - 1}\)]
- \(\mathit{Pr} \) :
-
Prandtl number, [\(= \mu c_{p}/k\)]
- \(q \) :
-
Laplace transforms parameter, [–]
- \(h \) :
-
Heat transfer coefficient, [\(\mbox{W}\,\mbox{m}^{ - 2}\,\mbox{K}^{ - 1}\)]
- \(\mu \) :
-
Dynamic viscosity, [\(\mbox{kg}\,\mbox{m}^{ - 1}\,\mbox{s}^{ - 1}\)]
- \(\gamma _{T} \) :
-
Coefficient of volumetric thermal expansion, [\(\mbox{K}^{ - 1}\)]
- \(\rho \) :
-
Fluid density, [\(\mbox{kg}\,\mbox{m}\,\mbox{s}^{ - 3}\)]
- \(\nu \) :
-
Kinematics viscosity of fluid, [\(\mbox{m}^{2}\,\mbox{s}^{ - 1}\)]
- \(\beta \) :
-
Walter’s-B fluid parameter
References
Abdullah, M., Raza, N., Butt, A.R., Haque, E.U.: Semi-analytical technique for the solution of fractional Maxwell fluid. Can. J. Phys. 95, 472–478 (2017). https://doi.org/10.1139/cjp-2016-0817
Ahmad, B., Shah, S.I.A., Haq, S.U., Shah, N.A.: Analysis of unsteady natural convective radiating gas flow in a vertical channel by employing the Caputo time-fractional derivative. Eur. Phys. J. Plus 132, 380 (2017). http://doi.org/10.1140/epjp/i2017-11651-1
Ali, F., Saqib, M., Khan, I., Sheikh, N.A.: Application of Caputo–Fabrizio derivatives to MHD free convection flow of generalized Walters’-B fluid model. Eur. Phys. J. Plus 131, 377 (2016). https://doi.org/10.1140/epjp/i2016-16377-x
Ali, F., Saqib, M., Khan, I., Sheikh, N.A., Alam Jan, S.A.: Exact analysis of MHD flow of a Walters’-B fluid over an isothermal oscillating plate embedded in a porous medium. Eur. Phys. J. Plus 132, 95 (2017). https://doi.org/10.1140/epjp/i2017-11404-2
Andersson, H.I.: MHD flow of a viscoelastic fluid past a stretching sheet. Acta Mech. 95, 227–230 (1992). https://doi.org/10.1007/BF01170814
Ariel, P.D.: MHD flow of a viscoelastic fluid past a stretching sheet with suction. Acta Mech. 105, 49–56 (1994). https://doi.org/10.1007/BF01183941
Ariel, P.D.: Axisymmetric flow due to a stretching sheet with partial slip. Comput. Math. Appl. 54, 1169–1183 (2007). https://doi.org/10.1016/j.camwa.2006.12.063
Ariel, P.D., Hayat, T., Asghar, S.: The flow of an elastico-viscous fluid past a stretching sheet with partial slip. Acta Mech. 187, 29–35 (2006). https://doi.org/10.1007/s00707-006-0370-3
Bhattacharyya, K., Uddin, M.S., Layek, G.C., Ali, W.: Analysis of boundary layer flow and heat transfer for two classes of viscoelastic fluid over a stretching sheet with heat generation or absorption. Bangladesh J. Sci. Ind. Res. 64, 451–456 (2011). https://doi.org/10.3329/bjsir.v46i4.9590
Butt, A.R., Abdullah, M., Raza, N., Imran, M.: Influence of non-integer order parameter and Hartmann number on the heat and mass transfer flow of a Jeffery fluid over an oscillating vertical plate via Caputo–Fabrizio time fractional derivatives. Eur. Phys. J. Plus 132, 414 (2017). http://doi.org/10.1140/epjp/i2017-11713-4
Chang, T.B., Mehmood, A., Beg, O.A., Narahari, M., Islam, M.N., Ameen, F.: Numerical study of transient free convective mass transfer in a Walters-B viscoelastic flow with wall suction. Commun. Nonlinear Sci. Numer. Simul. 16, 216–225 (2011). https://doi.org/10.1016/j.cnsns.2010.02.018
Chaudhary, R.C., Jain, P.: Hall effect on MHD mixed convection flow of a viscoelastic fluid past an infinite vertical porous plate with mass transfer and radiation. Theor. Appl. Mech. 33, 281–309 (2006)
Friedrich, C.H.R.: Relaxation and retardation functions of the Maxwell model with fractional derivatives. Comput. Math. Appl. 1, 151–158 (1991). https://doi.org/10.1007/BF01134604
Gemant, A.: On fractional differentials. Comput. Math. Appl. 1, 540–549 (1938). http://doi.org/10.1155/2011/193813
Ghasemi, E., Bayat, M., Bayat, M.: Viscoelastic MHD flow of Walters’s liquid B fluid and heat transfer over a non-isothermal stretching sheet. Int. J. Phys. Sci. 6, 5022–5039 (2011)
Imran, M.A., Khan, I., Ahmad, M., Shah, N.A., Nazar, M.: Heat and mass transport of differential type fluid with non-integer order time-fractional Caputo derivatives. J. Mol. Liq. 229, 67–75 (2017). https://doi.org/10.1016/j.molliq.2016.11.095
Imran, M.A., Riaz, M.B., Shah, N.A., Zafar, A.A.: Boundary layer flow of MHD generalized Maxwell fluid over an exponentially accelerated infinite vertical surface with slip and Newtonian heating at the boundary. Results Phys. 8, 1061–1067 (2018). https://doi.org/10.1016/j.rinp.2018.01.036
Jiang, Y., Qi, H., Xu, H., Jiang, X.: Transient electro-osmotic slip flow of fractional Oldroyd-B fluids. Microfluid. Nanofluid. 21, 7 (2017). http://doi.org/10.1007/s10404-016-1843-x
Khan, I., Ali, F., Sharidan, S., Qasim, M.: Unsteady free convection flow in a Walters’-B fluid and heat transfer analysis. Bull. Malays. Math. Sci. Soc. 37, 437–445 (2014)
Khan, I., Ali, F., Shah, N.A.: Interaction of magnetic field with heat and mass transfer in free convection flow of a Walters’-B fluid. Eur. Phys. J. Plus 131, 77 (2016). https://doi.org/10.1140/epjp/i2016-16077-7
Khan, I., Shah, N.A., Mahsud, Y., Vieru, D.: Heat transfer analysis in a Maxwell fluid over an oscillating vertical plate using fractional Caputo–Fabrizio derivatives. Eur. Phys. J. Plus 132, 194–199 (2017). https://doi.org/10.1140/epjp/i2017-11456-2
Kumar, D., Srivastava, R.K.: Effects of chemical reaction on MHD flow of dusty viscoelastic (Walter’s liquid model-B) liquid with heat source/sink. In: Proceedings of the National Seminar on Mathematics and Computer Science, Meerut, India, pp. 105–112 (2005)
Mahapatra, T.R., Dholey, S., Gupta, A.S.: Momentum and heat transfer in the magnetohydrodynamic stagnation-point flow of a viscoelastic fluid toward a stretching surface. Meccanica 42, 263–272 (2007). https://doi.org/10.1007/s11012-006-9040-8
Nadeem, S., Akbar, N.S.: Peristaltic flow of Walters’-B fluid in a uniform inclined tube. J Biorheol. 24, 22–28 (2010). https://doi.org/10.1007/s.12573-010-0018-8
Nandeppanavar, M.M., Abel, M.S., Tawade, J.: Heat transfer in a Walter’s liquid B fluid over an impermeable stretching sheet with non-uniform heat source/sink and elastic deformation. Commun. Nonlinear Sci. Numer. Simul. 7, 1791–1802 (2010). http://doi.org/10.1016/j.cnsns.2009.07.009
Nanousis, N.: Unsteady magnetohydrodynamic flows in a rotating elasto-viscous fluid. Astrophys. Space Sci. 199, 317–321 (1993). http://doi.org/10.1007/BF00613205
Pal, D., Mondal, H.: Effects of Soret Dufour, chemical reaction and thermal radiation on MHD non-Darcy unsteady mixed convective heat and mass transfer over a stretching sheet. Commun. Nonlinear Sci. Numer. Simul. 16, 1942–1953 (2011). http://doi.org/10.1016/j.cnsns.2010.08.033
Prakash, O., Kumar, D., Dwivedi, Y.K.: Effects of thermal diffusion and chemical reaction on MHD flow of dusty viscoelastic (Walter’s Liquid Model-B) fluid. J. Electromagn. Anal. Appl. 2, 581–587 (2010). http://doi.org/10.4236/jemaa.2010.210075
Raptis, A.A., Takhar, H.S.: Heat transfer from flow of an elastico-viscous fluid. Int. Commun. Heat Mass Transf. 16, 193–197 (1989). http://doi.org/10.1016/j.cnsns.2010.02.018
Rath, R.S., Bastia, S.N.: Steady flow and heat transfer in a visco-elastic fluid between two coaxial rotating disks. Proc. Math. Sci. 87, 227–236 (1978). https://doi.org/10.1007/BF02837758
Raza, N., Abdullah, M., Butt, A.R., Awan, A.U., Haque, E.U.: Flow of a second-grade fluid with fractional derivatives due to a quadratic time dependent shear stress. Alex. Eng. J. (2017). http://doi.org/10.1016/j.aej.2017.04.004
Roy, J.S., Chaudhury, N.K.: Heat transfer by laminar flow of an elastico-viscous liquid along a plane wall with periodic suction. Czechoslov. J. Phys. 30, 1199–1209 (1980). https://doi.org/10.1007/BF01605620
Saqib, M., Ali, F., Khan, I., Sheikh, N.A., Alam Jan, S.A., Haq, S.U.: Exact solutions for free convection flow of generalized Jeffrey fluid: A Caputo–Fabrizio fractional model. Alex. Eng. J. (2017). https://doi.org/10.1016/j.aej.2017.03.017
Shah, N.A., Khan, I.: Heat transfer analysis in a second-grade fluid over and oscillating vertical plate using fractional Caputo-Fabrizio derivatives. Eur. Phys. J. C 76, 362 (2016). http://doi.org/10.1140/epjpc/s10052-016-4209-3
Shah, N.A., Mehsud, Y., Zafar, A.A.: Unsteady free convection flow of viscous fluids with analytical results by employing time-fractional Caputo–Fabrizio derivative (without singular kernel). Eur. Phys. J. Plus 132, 411 (2017). http://doi.org/10.1140/epjp/i2017-11711-6
Sharma, V., Rana, G.C.: Thermosolutal instability of Walters’ (Model B’) visco-elastic rotating fluid permeated with suspended particles and variable gravity field in porous medium. Int. J. Appl. Mech. Eng. 6, 843–860 (2001). https://doi.org/10.2478/ijame-2013-0007
Sharma, R.C., Kumar, P., Sharma, S.: Rayleigh–Taylor instability of Walters-B elastico-viscous fluid through porous medium. Int. J. Appl. Mech. Eng. 7, 433–444 (2002)
Sheikh, N.A., Ali, F., Saqib, M., Khan, I., Alam Jan, S.A.: A comparative study of Atangana–Baleanu and Caputo–Fabrizio fractional derivatives to the convective flow of a generalized Casson fluid. Eur. Phys. J. Plus 132, 54 (2017). https://doi.org/10.1140/epjp/i2017-11326-y
Sheng, H., Li, Y., Chen, Y.Q.: Application of numerical inverse Laplace transform algorithms in fractional calculus. J. Franklin Inst. 348, 317–330 (2011). https://doi.org/10.1016/j.jfranklin.2010.11.009
Soundalgekar, V.M., Puri, P.: On fluctuating flow of an elastico-viscous fluid past an infinite plate with variable suction. J. Fluid Mech. 35, 561–573 (1969). https://doi.org/10.1017/S0022112069001297
Stehfest, H.: Algorithm 368: numerical inversion of Laplace transforms. Commun. ACM 13, 47–49 (1970). http://doi.org/10.1145/361953.361969
Tong, D.K., Zhang, X.M., Zhang, X.H.: Unsteady helical flows of a generalized Oldroyd-B fluid. J. Non-Newton. Fluid Mech. 156, 75–83 (2009). https://doi.org/10.1016/j.jnnfm.2008.07.004
Tzou, D.Y.: Macro to Microscale Heat transfer: The Lagging Behavior. Taylor and Francis, Washington (1997)
Vieru, D., Fetecau, C., Fetecau, C.: Time fractional free convection flow near a vertical plate with Newtonian heating and mass discussion. Therm. Sci. 19, S85–S98 (2015). http://doi.org/10.2298/TSCI15S1S85V
Vieru, D., Fetecau, C., Fetecau, C., Mirza, I.A.: Effects of fractional order on convective flow of an Oldroyd-B fluid along a moving porous hot plate with thermal diffusion. Heat Transf. Res. (2017). http://doi.org/10.1615/HeatTransRes.2017016039
Walters, K.: Non-Newtonian effects in some elastico-viscous liquids whose behavior at small rates of shear is characterized by a general linear equation of state. J. Mech. Appl. Math. 15, 63–90 (1962)
Wang, C.Y.: Flow due to a stretching boundary with partial slip-an exact solution of the Navier–Stokes equation. Chem. Eng. Sci. 57, 37–45 (2002). https://doi.org/10.1016/S0009-2509(02)00267-1
Wang, C.Y.: Stagnation flows with slip: exact solutions of the Navier–Stokes equations. Z. Angew. Math. Phys. 54, 184–189 (2003). https://doi.org/10.1007/PL00012632
Wang, C.Y.: Analysis of viscous flow due to a stretching sheet with surface slip and suction. Nonlinear Anal., Real World Appl. 10, 375–380 (2009). https://doi.org/10.1016/j.nonrwa.2007.09.013
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Abdullah, M., Butt, A.R. & Raza, N. Heat transfer analysis of Walters’-B fluid with Newtonian heating through an oscillating vertical plate by using fractional Caputo–Fabrizio derivatives. Mech Time-Depend Mater 23, 133–151 (2019). https://doi.org/10.1007/s11043-018-9396-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11043-018-9396-x