## THERMAL SCIENCE

International Scientific Journal

### MODELLING LAMINAR TRANSPORT PHENOMENA IN A CASSON RHEOLOGICAL FLUID FROM AN ISOTHERMAL SPHERE WITH PARTIAL SLIP

**ABSTRACT**

The laminar boundary layer flow and heat transfer of Casson non-Newtonian fluid from a permeable isothermal sphere in the presence of thermal and hydrodynamic slip conditions is analyzed. The surface of the sphere is maintained at a constant temperature. The boundary layer conservation equations, which are parabolic in nature, are normalized into non-similar form and then solved numerically with the well-tested, efficient, implicit, stable Keller-box finite-difference scheme. Increasing velocity slip induces acceleration in the flow near the surface of the sphere and the reverse effect further from the surface. Increasing velocity slip consistently enhances temperatures throughout the boundary layer regime. An increase in thermal slip parameter strongly decelerates the flow and also reduces temperatures in the boundary layer regime. An increase in Casson rheological parameter acts to elevate considerably the skin friction (non-dimensional wall shear stress) and this effect is pronounced at higher values of tangential coordinate. Temperatures are however very slightly decreased with increasing values of Casson rheological parameter. Increasing mass flow injection (blowing) at the sphere surface causes a strong acceleration, whereas increasing suction is found to induce the opposite effect. The study finds applications in rheological chocolate food processing.

**KEYWORDS**

PAPER SUBMITTED: 2012-08-28

PAPER REVISED: 2013-05-17

PAPER ACCEPTED: 2013-07-16

PUBLISHED ONLINE: 2013-08-04

**THERMAL SCIENCE** YEAR

**2015**, VOLUME

**19**, ISSUE

**Issue 5**, PAGES [1507 - 1519]

- Anwar Bég. O., Abdel Malleque. K and Islam. M.N, Modelling of Ostwald-deWaele non-Newtonian flow over a rotating disk in a non-Darcian porous medium, International Journal Applied Mathematics and Mechanics, 8 (2012), pp. 46-67.
- Anwar Bég. O. and Makinde. O.D, Viscoelastic flow and species transfer in a Darcian high-permeability channel, Journal of Petroleum Science and Engineering,76(2011), pp.93-99
- Gouse Mohiddin. S, Prasad. V. R., Anwar Bég. O, Numerical study of unsteady free convective heat and mass transfer in a Walters-B viscoelastic flow along a vertical cone, International Journal of Applied Mathematics and Mechanics, 6(2010), pp.88-114.
- Prasad. V.R, Vasu . B, Anwar Bég. O. and Prashad. R., Unsteady free convection heat and mass transfer in a Walters-B viscoelastic flow past a semi-infinite vertical plate: a numerical study, Thermal Science-International Scientific Journal., 15, (2011) 2, S291-S305
- Tripathi. D., Anwar Bég. O.and Curiel-Sosa. J., Homotopy semi-numerical simulation of peristaltic flow of generalised Oldroyd-B fluids with slip effects, Computer Methods in Biomechanics Biomedical Engineering (2012). DOI:10.1080/10255842.2012.688109
- Anwar Bég. O, Takhar. O, Bharagava. R., Rawat, S. and Prasad, V.R., Numerical study of heat transfer of a third grade viscoelastic fluid in non-Darcian porous media with thermophysical effects, Physica Scripta, 77(2008), pp.1-11
- Rashidi. M.M., Anwar Bég. O. and Rastegari. M.T, A study of non-Newtonian flow and heat transfer over a non-isothermal wedge using the Homotopy Analysis Method, Chemical Engineering Communications, 199(2012), pp.231-256
- Huilgol. R.R, You. Z., Application of the augmented Lagrangian method to steady pipe flows of Bingham, Casson and Herschel-Bulkley fluids, Journal of Non-Newtonian Fluid Mechanics, 128(2005), pp.126-143.
- Wilson. L.L, Speers. A. and Tung. M.A., Yield stresses in molten chocolates, Journal of Texture Studies, 24(1993), pp.269-286.
- Steffe. J.F, Rheological methods in Food Process Engineering, 2nd edn, Freeman Press, Michigan, USA (2001).
- Casson. N., A Flow equation for pigment oil-suspensions of the printing ink type, Rheolgy of Disperse Systems (C.C. Mill, ed.), pp. 84. Pergamon Press, London (1959).
- Bird. R. B, Dai. G. C and Yarusso. B. J, The rheology and flow of viscoplastic materials, Reviews in Chemical Engineering, 1(1983), pp.1-83.
- Chaturani. P, Ponnalagarsamy. R. Pulsatile flow of Casson's fluid through stenosed arteries with applications to blood flow. Biorheology, 23(1986), pp.499-511.
- Das, B. and Batra, R.L., Secondary flow of a Casson fluid in a slightly curved tube, International Journal of Non-Linear Mechanics, 28(1993), pp.567-580.
- Dash. R. K, Jayaraman. G.and Mehta. K. N, Shear-augmented dispersion of a solute in a Casson fluid flowing in a conduit, Annals of Biomedical Engineering. 28(2000), pp.373-385.
- Batra. R.L, Das B, Flow of Casson fluid between two rotating cylinders, Fluid Dynamics Research 9(1992), pp.133-141.
- Neofytou. P, Transition to asymmetry of generalised Newtonian fluid flows through a symmetric sudden expansion, Journal of Non-Newtonian Fluid Mechanics, 133(2006), pp.132- 140.
- Kandasamy. A, Karthik. K., Phanidhar P. V., Entrance region flow heat transfer in concentric annuli for a Casson fluid, International Conference of Thermal Issues in Emerging Technologies, Theory and Application (ThETA), Cairo, Egypt, January 3rd- 6th (2007).
- Nagarani. P, Sarojamma. G, Jayaraman G, On the dispersion of a solute in a Casson fluid flow in an annulus with boundary absorption, American Conference of Applied Mathematics (MATH'08), Harvard University, Harvard, Massachusetts, USA, March 24-26,(2008), pp. 265-273.
- Shaw. S, Gorla. R. S. R, Murthy. P. V. S. N and Ng. C. O, Effect of stenosis on the Casson fluid flow through a bifurcated artery, International Journal of Fluid Mechanics Research, 36(2009), pp.43-63.
- Attia. H and Sayed-Ahmed. M. E, Transient MHD Couette flow of a Casson fluid between parallel plates with heat transfer, Italian Journal of Pure Applied Mathematics, 27(2010), pp.19-38.
- Hayat. T, Pop. I and Hendi. A. A, Stagnation-point flow and heat transfer of a Casson fluid towards a stretching sheet, Zeitschrift für Naturforschung.,67(2012), pp.70-76.
- Mustafa. M, Hayat. T, Pop. I and Aziz .A, Unsteady boundary layer flow of a Casson fluid due to an impulsively started moving flat plate, Heat Transfer-Asian Research, 40(2011), pp.563-576.
- Sparrow. E.M, Lin. S.H, Laminar heat transfer in tubes under slip-flow conditions, ASME Journal of Heat Transfer, 84(1962), pp.363-639.
- Larrode. F.E, Housiadas. C, Drossinos. Y, Slip-flow heat transfer in circular tubes, International Journal of Heat Mass Transfer, 43(2000), pp.2669-2680.
- Spillane. S, A study of boundary layer flow with no-slip and slip boundary conditions, PhD Thesis, Dublin Institute of Technology, Ireland (2007).
- Crane. L. J and McVeigh. A. G, Slip flow on a microcylinder, Z. Angew. Math. Phys (Zeitschrift für angewandte Mathematik und Physik) 61(2010), 3, pp.579-582.
- Crane. L. J and McVeigh. A. G, Uniform slip flow on a cylinder, PAMM: Proceedings of Applied Mathematics and mechanics., 10(2010), pp.477-478.
- Wang. C.Y and Ng. C-O, Slip flow due to a stretching cylinder, International Journal of Non-Linear Mechanics , 46(2011), pp.1191-1194.
- Wang. C.Y, Stagnation flow on a cylinder with partial slip-an exact solution of the Navier-Stokes equations, IMA Journal of Applied Mathematics, 72(2007), pp.271- 277.
- Yih. K.A, Viscous and Joule Heating effects on non-Darcy MHD natural convection flow over a permeable sphere in porous media with internal heat generation, International Communications in Heat and mass Transfer, 27(4), (2000), pp.591-600.
- Cebeci T., Bradshaw P., Physical and Computational Aspects of Convective Heat Transfer, Springer, New York, 1984.
- J.H.Merkin, Free convection boundary layer on an isothermal Horizontal cylinder. ASME/AICHE Heat Transfer Conference. August 9-11 St. Louis USA (1976).
- Prasad. V.R, Vasu. B, Prashad. R and Anwar Bég. O, Thermal radiation effects on magneto-hydrodynamic heat and mass transfer from a horizontal cylinder in a variable porosity regime, Journal of Porous Media, 15(2012), pp.261-281.
- Aldoss, T.K.,Ali, Y.D. and A-Nimr, M.A. MHD Mixed convection from a horizontal circular cylinder. Numerical heat transfer, part A. 30: 379-396 (1996).
- Ajadi. S. O, Adegoke. A and Aziz. A, Slip boundary layer flow of non-Newtonian fluid over a flat plate with convective thermal boundary condition, International Journal of Nonlinear Science, 8(2009), pp.300-306.
- Aziz. A, Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition, Communications in Nonlinear Science and Numerical Simulation, 15(2010), pp.573-580.