DIRICHLET SERIES AND APPROXIMATE ANALYTICAL METHOD FOR THE SOLUTION OF MHD BOUNDARY LAYER FLOW OF CASSON FLUID OVER A STRETCHING/SHRINKING SHEET

Year 2017, Volume: 7 Issue: 2, 343 - 353, 01.12.2017

Viswanath B. Avati

Abstract

The paper presents analytical and semi-numerical solution for magnetohy- drodynamic MHD boundary layer ow of Casson uid over a exponentially permeable shrinking sheet. The governing partial dierential equations of momentum equations are reduced to ordinary dierential equations by using a classical similarity transformation along with appropriate boundary conditions. Both nonlinearity and innite interval de- mand novel mathematical tools for their analysis. We use fast converging Dirichlet series and approximate analytical solution by the Method of stretching of variables for the solution of the nonlinear dierential equation. These methods have the advantages over pure numerical methods for obtaining the derived quantities accurately for various values of the parameters involved at a stretch and also they are valid in much larger parameter domain as compared with HAM, HPM, ADM and the classical numerical schemes.

Keywords

magnetohydrodynamics MHD, boundary layer ow, Casson uid, shrink- ing /stretching sheet, wall mass transfer, Dirichlet series, Powells method, method of stretching of variables.

References

• Sajid,M., and Hayat,T., (2009),The application of homotopy analysis method for MHD viscous flow due to shrinking sheet, Chaos Solitons and Fractals, 39, pp.1317-1323.
• Hayat,T., Abbas,Z., Javed,T., and Sajid,M., (2009),Three-dimensional rotating flow induced by a shrinking sheet for suction, Chaos Solitons and Fractals, 39(4), pp.1615-1626.
• Fang,T.,and Zhang,J., (2009), Closed-form exact solutions of MHD viscous flow over a shrinking sheet,Commun Nonlinear Sci. Neumer Simulat, 14(7), pp.2853-2857.
• Fang,T., (2008),Boundary layer flow over a shrinking sheet with power-law velocity, Int. J. Heat Mass Tran, 51(25-26), pp.5838-5843.
• Nadeem,S., and Awais,M., (2008), Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity, Phys. Lett. A, 372(30), pp.4965-4972.
• Hayat,T., Javed,T., and Sajid,M., (2008), Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface, Phys. Lett. A., 372(18), pp.3264-3273.
• Wang C.Y., (2008), Stagnation flow towards a shrinking sheet, Int. J. Non-Linear Mech, 43(5), pp.377
• Hayat,T., Abbas,Z., and Ali,N., (2008), MHD flow and mass transfer of an upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Phys. Lett.A., 732(26), pp.4698
• Crane,L.J., (1970), Flow past a stretching plate, Z Angew Math. Mech., 21, pp.645-647.
• Carragher,P., and Carane,L.J., (1982), Heat transfer on a continuous stretching sheet, Z Angew Math. Mech., 62, pp.564-565.
• Vajravelu,K., and Rollins,D.,(1992), Heat transfer in an electrically conducting fluid over a stretching surface, Int. J. Nonlinear Mech., 27, pp.265-277.
• Salem,A.M., and Fathy,R., (2012), Effects of variable properties on MHD heat and mass transfer near a stagnation point towards a stretching sheet in a porous medium with thermal radiation, Chin Phys. B, 21, 054701.
• Battacharyya,K., and Layek,G.C.,(2011), Slip effects on diffusion of chemically reactive species in boundary layer flow over a vertical stretching sheet with suction and blowing, Chem. Eng. Commun., , pp.1354-1365.
• Hayat,T., Qasim,M., and Mesloub,S., (2011),MHD flow and heat transfer over permeable stretching sheet with slip conditions, Int. J. Numer. Methods Fluids, 66, pp.963-975.
• Wang,C.Y., (1990), Liquid film on an unsteady stretching sheet, Quart Appl Math , 48, pp.601-610.
• Miklavcic,M., and Wang,C. Y.,(2006), Viscous flow due to a shrinking sheet, Quart. Appl. Math. 64, pp.283-290.
• Fang,T., Zhang,J.,and Yao,S., (2010), Slip magnetohydrodynami viscous flow over a permeable shrink- ing sheet, Chin. Phys. Lett. 27, 124702.
• Ali,F., Nazar,R., Arifin,N., (2010), MHD viscous flow and heat transfer induced by permeable shrink- ing sheet with prescribed surface heat flux, WSAS Trans Math, 5(9), pp.365-375.
• Noor,N.F.M., Kechil,S.A.,and Hashim,I., (2010), Simple non-perturbative solution for MHD viscous flow due to a shrinking sheet, Commun Nonlinar Sci. Numer Simulat , 15, pp.144-148.
• Raftari,B., Yildirim,A.,(2011), A series solution of nonlinear ODE arising in MHD by HPM-Pade technique, Comp. Math. Appl., 61, pp.1676-1681.
• Bhattacharyya, K., (2011),Effects of heat source/ sink on MHD flow and heat transfer over a shrinking sheet with mass suction, Chem. Eng. Res. Bull 15, pp.12-17.
• Fredrickson,A.G., (1964), Principles and applications of rheology, (1964) (Englewood Cliffs, N.J; Prentice-Hall).
• Mustafa,M., Hayat,T., Pop,I. and Hendi,A., (2012), Stagnation point flow and heat transfer of a Casson fluid towards a stretching, Z. Naturforsch, 67a, pp.70-76.
• Thiagarajan,M., and Senthilkumar,K., (2013), DTM-Pade approximants of MHD boundary layer flow of a Casson fluid over a shrinking sheet, United States of America Research Journal (USARJ), Vol.1, No.1, pp.1-7.
• Nadeem, S., Haq, R. U., and Lee, C.,(2013), MHD flow of a Casson fluid over an exponentially shrinking sheet, Scientia Iranica B, 19(6), pp.1550-1553.
• Abel,M.S., Sujit Kumar Khan, and Prasad,K.V.,(2002), Study of visco-elastic fluid flow and heat transfer over a stretching sheet with variable viscosity, International journal of non-linear mechanics, (1), pp.81-88.
• Prasad,K.V., Vajravelu,K., and Pop,I., (2013) ,Flow and heat transfer at a nonlinearly shrinking porous sheet: the case of asymptotically large power law shrinking rates, IJAME, 18(3), pp.779-791.
• Prasad,K.V., Vajravelu,K., and Vaidya,H., (2016), MHD Casson Nanofluid Flow and Heat Transfer at a Stretching Sheet with Variable Thickness, Journal of Nanofluids, 5(3), pp.423-435.
• Kravchenko,T.K., and Yablonskii,A.I., (1965), Solution of an infinite boundary value problem for third order equation, Differentialnye Uraneniya. 327, 1.
• Kravchenko,T.K., and Yablonskii,A.I., (1972), A boundary value problem on a semi-infinite interval,DifferentialnyeUraneniya, 8(12), pp.2180-2186.
• Riesz,S., (1957), Introduction to Dirichlet series, Camb. Univ. Press.
• Sachdev,P.L., Bujurke,N.M., and Awati,V.B.,(2005), Boundary value problems for third order non- linear ordinary differential equations, Stud. Appl. Math, 115, pp.303-318.
• Awati,V.B., Bujurke,N.M., and Kudenatti,R.B., (2011), An exponential series method for the solution of the free convection boundary layer flow in a saturated porous medium. AJCM , 1, pp.104-110.
• Awati,V.B., Bujurke,N.M., and Kudenatti,R.B., (2011), Dirichlet series method for the solution ofMHD flow over a nonlinear stretching sheet. IJAMES, 5(1), pp.07-12.
• Kudenatti,R.B., Awati,V.B., and Bujurke,N.M.,(2011), Exact analytical solutions of class of boundary layer equations for a stretching surface, Appl Math Comp, 218, pp.2952-2959.
• Press,W.H.H., Flannery,B.P., Teulosky,S.A., and Vetterling,W.T., (1987), Numerical Recipes in C, Camb. Univ. Press, UK.
• Ariel,P.D., (1994), Stagnation point flow with suction; an approximate solution, Journal of Applied Mechanics, 61(4), pp.976-978.