INFLUENCE OF CATTANEO-CHRISTOV HEAT FLUX MODEL ON MHD HYPERBOLIC TANGENT FLUID OVER A MOVING POROUS SURFACE

Transcription

1 Frontiers in Heat and Mass Transfer Available at INFLUENCE OF CATTANEO-CHRISTOV HEAT FLUX MODEL ON MHD HYPERBOLIC TANGENT FLUID OVER A MOVING POROUS SURFACE Z. Iqbal Ehtsham Azhar* E. N. Maraj and Bilal Ahmad Department of Mathematics HITEC Universit Taila Pakistan ABSTRACT Present investigation represent the stud of Cattaneo-Christov heat flu model on boundar laer flow of hperbolic tangent fluid which is generalized non-newtonian fluid model over a continuousl moving porous surface with a parallel free stream velocit. Mathematical formulation is completed in the presence of Magneto-hdrodnamics (MHD). Suitable relations transform the partial differential equations into the ordinar differential equations. Nonlinear flow analsis is computed and velocit and temperature profiles are obtained b shooting algorithm. Graphs are plotted to analze the behavior of various involved phsical parameters. Furthermore both tpe of flows Sakaidis ( λ = ) and Blasius flow (0 λ < ) are discussed significantl. Special emphasis has been given to flow patterns for both tpes of flows presented through stream functions contour and D plots. Ke finding includes: Boundar laer thickness is an increasing function of power law inde and Suction parameter for the case of Blasius flow opposite to Sakaidis flow and dwindle of thermal boundar laer is witness for rising values of γ Pr and S while augmented boundar laer is observed for increasing values of n M and fluid parameter. Kewords: Three Cattaneo-Christov heat flu Non-Newtonian fluid Moving surface Megnatohdrodnamics (MHD) Momentum transfer.. discussed b Ali (00) is among man other power law fluids which describe rheological behavior of pseudo plastic fluids. Harfoush et al. (989) contributed b numerical investigation on analzing electromagnetic wave scattering from moving surfaces in one and two dimensions. Grosan et al. (000) carried out similarit solutions for boundar laer flow on a moving surface in non-newtonian fluids. Elliott et al. (98) eplained breakdown of boundar laer in detail which turned to be a major contribution to the present dramaticall growing applications of flow on a moving surface. The consequence of transpiration on self alike border line laer flow on moving surfaces was eploded b Weidman et al. (00). Latterl Bachok et al. (00) inspected such flow of nanofluids in a flowing fluid. Keeping the important of heat transfer in engineering procedures and industries researchers and analst are ver keen in studing heat transfer attributes of different fluids in various flow problems. This phenomenon has been eamined through established Fourier law of heat conduction (8) till Cattaneo (98) floated an adjustment in it b incorporating thermal relaation effects. Christov (008) upgraded his contribution b adding Oldrod's upper convected derivatives. His modification is termed as Cattaneo-Christov heat flu model. Man researchers have eamined the heat transfer of various fluids for different flow problems few are Han et al. (0) Khan et al. (0) Tibullo et al. (0) Nadeem et al. (0) Muhammad et al. (07) and (07) Li et al. (0) Sui et al. (0) Liu et al. (07) and INTRODUCTION This article investigates pseudo plastic fluids flow over a continuousl moving surface taking into account Cattaneo-Christov heat flu. Applications of fluids flow on a moving surface are mostl encounter in polmer industries and man engineering processes such as cooling of polmer films or sheets metallic plates on conveers filament disentangle incessantl from a die long fiber travelling amid in feed roll and a wind up roll etc. Sakiadis (9) attempted to analze boundar laer flow for an incompressible Newtonian fluid on an invariantl moving surface. He discovered that these tpes of flow problems leads to substantiall different solutions compare to boundar laer flow on a stationar semi-infinite smooth shield eamined b Blasius in (908). Latterl Pop in (990) and (99) highlighted some particular aspects of this flow. It is well known that most of the fluids occurring in practical applications such as molten metal or plastics polmers pulps food etc. have rheological characteristics of non-newtonian fluids. Considerable effort has been directed towards understanding their friction and heat transfer characteristics because of the growing usage of such fluids. However in chemical engineering researchers and scientist encounter inelastic non-newtonian fluids known as power law fluids where shear varies to a power function of deformation rate. Tangent hperbolic fluid * Corresponding Author: * Corresponding author ehtsham@uaar.edu.pk address: ehtsham@uaar.edu.pk

2 Mukhopadhar (009). To the best of author's knowledge up till now no one has attempted to stud the heat transfer b taking into account Cattaneo-Christov heat flu model for pseudo plastic fluid on travelling surface with an equivalent free stream. Present analsis is ma be beneficial in academic research in the field of heat transfer and industr.. u u T T T v T v T + Λ u +u u T T T T T + uv + u + v = α associated boundar conditions are T = Tw ( ) at = 0; T T as. PROBLEM DEVELOPMENT We take boundar laer flow of an incompressible Hperbolic Tangent Fluid over a moving surface. We consider constant velocit u w in the () () Here α is thermal diffusivit Tw is temperature at the wall and T is the ambient fluid temperature. We introduce following dimensionless quantities same direction as that of the uniform free stream velocit u. It is assumed that the wall and free stream temperatures Tw and T are u = Uf (η ) v = constants with Tw > T. The geometr of present phsical flow phenomena is presented in figure. θ (η ) = U Uν f (η ) ηf (η ) η = ν T T Tw T (7) in which U = u w + u. Invoking above mathematical relations into Eq. () which is identicall satisfied and Eqs. () () () and () are reduced to ( n ) f + ff nwef f M f = 0 (8) ( ) Pr γ + f θ = 0 ff θ f (η ) = λ f (η ) = S θ (η ) = at η = 0 θ + Pr fθ (9) f (η ) = λ θ (η ) = 0 as η. where f (0) = S with S < 0 corresponds to suction case and S > 0 implies injection λ is a constant parameter Pr is Prandtl number We is Weissenberg number and γ is the Deborah number with respect to heat flu. Definitions of these phsical parameters are Fig. Engineering Flow Diagram Assume that a uniform magnetic field of strength B0 is applied in the positive direction normal to the plate and the induced magnetic field due to the magnetic Renolds number is taken to be small enough and assumed to be negligible in comparison to the applied magnetic field. The boundar laer equations governing the flow for hperbolic tangent fluid are u v + = 0 () u u u σb 0 u u u = v( n ) + vnγ u ρ subject to flow boundar conditions u = u w ( ) v = vw at = 0 u u ( ) u 0 as. (0) λ= uw U Pr = v α γ = ΛU We = U ν v. UΓ S = vu w It is worth mentioning here that λ = 0 corresponds to the flow over a stationar surface caused b the free stream velocit (Blasius flow). Whereas λ = corresponds to moving plate in fluid (Sakaidis flow). The case 0 < λ < is when plate and fluid are moving in same direction. If λ > free stream is directed toward negative direction whereas plate moves toward positive direction. Skin friction coefficient C f is defined as () Cf = τw ρu w () in which epressions of wall skin friction (τ w ) is defined b () τ w = ( n ) Where u and v are the velocit components along and directions respectivel. ρ is densit of fluid and is kinematic viscosit. Epression for Cattaneo-Christov heat flu model discussed b (009) is q q + Λ + V. q q. V (.V ) q = k T () t u nγ u + () =0 With the help of dimensionless variables (7) we have C f (Re ) / n = ( n ) f (η ) + We ( f (η )). η =0 where Re = U / v is the local Renolds number. in which q is heat flu Λ is relaation time of heat flu T is temperature k is thermal conductivit and V is the velocit vector. In view of above epression temperature profile governs following relation ()

3 . COMPUTATIONAL PROCEDURE parameter S also enhances velocit profile while boundar laer thickness decas in this case (see figure 0). Due to phsical aspect growth of boundar laer reduced b emploing the suckling particles in the porous wall. It is quite obvious that a suction cause reduces boundar laer thickness. Effects of various parameter on temperature and thermal boundar laer are eamined through figures - for Sakaidisflow (λ = ). The contribution of Prandtl number Pr and Deborah number with respect to the relaation time of heat flu γ on the temperature field θ (η ) can be seen in figures and. The effect of Prandtl number Pr on θ can be visualized in Fig.. It is obvious that an increase in the values of Pr greatl reduces the thermal diffusivit therefore temperature and the thermal boundar laer thickness are decreasing functions of Pr. It is also observed that deviation in the temperature profiles is more significant for small values of Pr when compared with its larger values. It is important to note that Pr(< ) corresponds to liquid metals which have higher thermal diffusivit. However while large values of Pr(> ) lead to high-viscosit oils. The impact of non-dimensional relaation time heat flu γ on the temperature field is analzed in figure. There is a decrease in temperature when enhances. Temperature profile θ (η ) in figure and are plotted against different values of power law inde n Weissenberg number We and Hartman number M respectivel. All these parameters contribute in rising in temperature profile with broadness in thermal boundar laer also observed. Viscous forces reduce for larger Hartman number M as a result thermal forces increases and hence temperature of the fluid enhances (see figure ). Figure depict the influence of Suction parameter S on temperature. Increasing suction effect tends to decrease the temperature and thermal boundar laer thickness. In accordance with the Mukhopadha (009) an increase in the suction parameter S corresponds to a decrease in the temperature and the thermal boundar laer thickness. Figures 7-0 are devoted to illustrate flow patterns for different values of We for Sakaidis and Blasius flows. It is depicted that flow pattern for Sakaidis flow is concave down (see Figs. 7 and 8) while it is concave up (see Fig. 9 and 0) in Blasius flow. The numerical values of skin friction for Sakaidis flow (λ = ) and Blasius flow (λ < ) visualized through table. In both cases (λ = ) and (λ < ) the rate of shear stress at wall decreases with increasing values of power law inde n where an increase can be observed in skin friction for higher values of We M and S. Solution of Eqs. (8) and (9) subject to boundar conditions (0) b using shooting method with fifth order RK procedure. Initiall using the similarit transform we get a sstem of nonlinear ordinar differential equation then we convert higher order nonlinear ordinar differential equations into sstem of first order ordinar differential equation b making suitable substitution in the form ( ) = ( f f f θ θ ) this ield following mathematical relations M = ( n + nwe ) Pr γ Pr ( Pr γ ) ( ( ) ) (0) S (0) λ (0 ) = f (0). (0) (0 ) θ (0) For shooting method we implemented Newton-Raphson method to find the targets and Runge-Kutta of order method is chosen for the time integration in MATLAB. A step size of 0.00 is selected satisfactor for a convergence criterion of 0- in nearl all cases.. ANALYSIS AND DISCUSSION This section presents the effects of embedding parameters on the velocit and temperature fields. Figure -0 shows the influence of various fluid parameters on velocit profile. Figure is plotted for velocit profiles for various values of λ. It is observed that velocit field decreases rapidl with an increase in λ. From the phsical aspect it is clear that an increase in plate velocit greatl reduces the velocit of fluid. The boundar laer thickness decreases b increasing in the range 0 λ < 0.. However when λ > 0. the boundar laer thickness grows with the increasing values of λ. Hartman number M is ratio of electromagnetic force to the viscous force. Increase in the values of M causes a decrease in viscous forces as a result velocit of fluid decrease and increase in boundar laer thickness. This trend can be depicted in figure for Sakaidis flow. For (λ = 0.) Blasius flow similar behavior can be observed through figure. It is also analzed that M is perpendicular to the moving surface so for rising values of Hartman number it reduces the velocit and enhances the boundar laer thickness. The effect of power law inde n Weissenberg number We and Suction parameter S on fluid velocit are demonstrated through figures 7 and 9 respectivel (Sakaidis flow λ < ) whereas figs. 8 and 0 demonstrate Blasius flow (λ < ) for different values of same phsical parameters. Weissenberg number is the ratio of relaation time of fluid and a specific process time. It increases thickness of the fluid so velocit profile decreases with an increase in We. The power law inde enhances the fluid velocit and boundar laer thickness while We shows an opposite trend for fluid velocit. The impact of Suction Fig. Effect of λ on f ' (η ).

4 Fig. Effect of M on f ' (η ) for Sakaidis flow Fig. 7 Effect of We on f ' (η ) for Sakaids flow Fig. Effect of M on f ' (η ) for Blasius flow Fig. 8 Effect of We on f ' (η ) for Blasius flow Fig. Effect of n on f ' (η ) for Sakaidis flow Fig. 9 Effect of S on f ' (η ) for Sakaidis flow Fig. Effect of n on f ' (η ) for Blasius flow Fig. 0 Effect of S on f ' (η ) for Blasius flow

5 Fig. Effect of Pr on θ (η ) Fig. Effect of Fig. Effect of γ on θ (η ) Fig. Effect of S on θ (η ) M on θ (η ) Fig. Effect of n on θ (η ) D Plot Fig. Effect of We on θ (η ) Fig. 7 Streamlines for Sakaidis flow when We = 0..

6 D Plot Fig. 8 Streamlines for Sakaidis flow when We = D Plot Fig. 0 Streamlines for Blasius flow when We = Table Tabulated values of shear stress at wall for different parameters n.08 We M S. Re C f (λ ) Re C f (λ ) Skaidis Flow Blasius Flow CLOSING REMARKS From the above stud we conclude the following findings: Boundar laer thickness trend was observed to be opposite to boundar each other for Blasius and Sakaidis flow. D Plot Fig. 9 Streamlines for Blasius flow when We = 0..

7 Khan J.A. Mustafa M. Haat T. and Alsaedi A. 0 Numerical Stud of Cattaneo-christov Heat Flu Model for Visco Elastic Flow Due To an Eponentiall Stretching Surface PLoS One Hartman number M and fluid parameter Weissenberg number We shows a decrease in velocit profile and boundar laer thickness for both tpe of flows. Boundar laer thickness enhances b increasing power law inde n and Suction parameter S for the case of Blasius flow opposite to Sakaidis flow. Dwindle of thermal boundar laer is witness for rising values of γ Pr and S while augmented boundar laer is observed for increasing values of n M and fluid parameter. Power law inde n lessen skin friction for Sakaidis and Blasius flow while for all other pertinent parameters an increased skin friction is observed. Li J. Zheng L. and Liu L. 0 MHD Viscoelastic Flow and Heat Transfer over a Vertical Stretching Sheet with Cattaneo-Christov Heat Flu Effects Journal of Mol. Liq Liu L. Zheng L. Liu F. and Zhang X. 07 Heat Conduction with Fractional Cattaneo Christov Upper-Convective Derivative Flu Model Int. Therm. Sci Muhammad N. Nadeem S. and Mustafa T. 07 Squeezed Flow of a Nanofluid with Cattaneo-Christov Heat and Mass Flues Res. Ph REFERENCES Ali L. and Vafai K. 00 An Investigation of Stokes Second Problem for Non-Newtonian Fluids Numerical Heat Transfer Part A: Applications Muhammad N. Nadeem S. and Haq R.U. 07 Heat Transport Phenomenon in the Ferromagnetic Fluid over a Stretching Sheet with Thermal Stratification Res. Ph Bachok N. Ishak A. and Pop I. 00 Boundar Laer Flow of Nanofluids over A Moving Surface in a Flowing Fluid International Journal of Thermal Science Mukhopadha S. 009 Effect of Thermal Radiation on Unstead Mied Convection Flow and Heat Transfer over a Stretching Surface in a Porous Medium International Journal of Heat and Mass Transfer -. Blasius H. 908 Grenzschichten in Flussigkeitenmitkleiner Reibung Zeitschriftfürangewandte Mathematik und Phsik -. Nadeem S. and Muhammad N. 0 Impact of Stratification and Cattaneo-Christov Heat Flu in the Flow Saturated with Porous Medium J. Mol. Liq Cattaneo C. 98 Sullaconduzion edelcalore in: Atti del Seminario Matematico e Fisico dell Universita di Modena e Reggio Emilia 80. Christov C.I. 008 On Frame in Different Formulation of the Mawell--Cattaneo Model of Finite-Speed Heat Conduction Mechanics Research Communication doi:0.0/j.mechrescom Pop I. and Gorla R.S.R. 990 Second Order Boundar Laer Solution for a Continuous Moving Surface in a Non-Newtonian Fluid International Journal of Engineering Science Elliott J.W. and SmithF.T. 98 Breakdown of Boundar Laers: (i) on Moving Surface; (ii) in Semi-Similar Unstead Flow; (iii) in Full Unstead Flow Geophsical & Astrophsical Fluid Dnamics Pop I.and Watanabe T.99 The Effects of Suction or Injection in Boundar Laer Flow and Heat Transfer on a Continuousl Moving Surface Technische Mechanik 9-. Sakiadis B.C. 9 Boundar Laer Cattaneo-Christov Doublediffusion Model of Heat and Mass Transfer in Upper-Convected Mawell Nanofluid Past a Stretching Sheet with Slip Velocit American Institute of Chemical Engineers Journal /aic Fourier J.B.J. 8 Theorie Analtique De La Chaleur Chez FirminDidot Paris. GrosanT.S. Pop I. and NaT.Y. 000 Similarit Solutions for Boundar Laer Flows on a Moving Surface in Non-Newtonian Power Law Fluids Technische Mechanik -0. Sui J. Zheng L. and Zhang X. 0 Boundar Laer CattaneoChristov Double-Diffusion Model of Heat And Mass Transfer in Upper-convected Mawell Nanofluid Past a Stretching Sheet with Slip Velocit Int. Therm. Sci Han S. Zheng I. Li C. and Zhang X. 0 Coupled Flow and Heat Transfer in Visco Elastic Fluid With Cattaneo-Chirstov Heat Flu Model App. Math. Lett Tibullo V. and Zampoli V. 0 A Uniqueness Result for the Cattaneo-Christov Heat Conduction Model Applied to Incompressible Fluids Mech. Res. Comm Harfoush F.Taflove A. and KroegsmannG.A.989 A Numerical Technique for Analzing Electromagnetic Wave Scattering from Moving Surface in One and Two Dimensions IEEE Transactions on Antennas and Propagation /8.9 Weidman P.D. Kubitschek D.G. and DavisA.M.J.00 The Effect of Transpiration on Self-Similar Boundar Laer over Moving Surfaces International Journal of Engineering Science