30120130405018
Upcoming SlideShare
Loading in...5
×
 

30120130405018

on

  • 415 views

 

Statistics

Views

Total Views
415
Views on SlideShare
415
Embed Views
0

Actions

Likes
0
Downloads
1
Comments
0

0 Embeds 0

No embeds

Accessibility

Categories

Upload Details

Uploaded via as Adobe PDF

Usage Rights

© All Rights Reserved

Report content

Flagged as inappropriate Flag as inappropriate
Flag as inappropriate

Select your reason for flagging this presentation as inappropriate.

Cancel
  • Full Name Full Name Comment goes here.
    Are you sure you want to
    Your message goes here
    Processing…
Post Comment
Edit your comment

    30120130405018 30120130405018 Document Transcript

    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 156 MHD BOUNDARY LAYER FLOW OF A MAXWELL FLUID PAST A POROUS STRETCHING SHEET IN PRESENCE OF VISCOUS DISSIPATION Anand H. Agadi1* , M. Subhas Abel2 , Jagadish V. Tawade3 and Ishwar Maharudrappa4 1* Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102,INDIA 2 Department of Mathematics, Gulbarga University, Gulbarga- 585 106, INDIA 3 Department of Mathematics, Bheemanna Khandre Institute of Technology,Bhalki-585328 4 Department of Mathematics, Basaveshwar Engineering College, Bagalkot-587102. ABSTRACT Present study deals flow of MHD boundary layer of a Maxwell fluid over stretching sheet with non-uniform heat source in porous medium. The effects of various values of the emerging dimensionless parameters are discussed in two different cases namely, (i) a surface with prescribed wall temperature (PST) (ii) a surface with prescribed wall heat flux (PHF). The partial differential equations governing the momentum and heat transfer are converted in to ordinary differential equations by suitable similarity transformations. Numerical solutions for boundary value problems carried out by shooting technique with fourth order Runge-Kutta scheme. The results so obtained are presented in the form of graphs for different non-dimensional parameters for both PST and PHF cases and discussed. Key words: Boundary layer, MHD, Maxwell fluid, porous stretching sheet, viscous dissipation. 1. INTRODUCTION The study of flow induced by a stretching surface has scientific and engineering applications such as aerodynamic extrusion of plastic sheets and fibers, drawing-annealing- tinning of copper wire, paper production, crystal growing and glass blowing. Such applications involve cooling of a molten liquid by drawing it into a cooling system. In drawing the liquid into the cooling system it is sometimes stretched as in the case of polymer extrusion process. The fluid mechanical properties desired for the penultimate outcome of such a process depend mainly on two things one being the rate of cooling and other being the rate of stretching. The choice of an appropriate cooling liquid is crucial as it has a direct impact on rate of cooling and care must be taken to exercise optimum INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING AND TECHNOLOGY (IJMET) ISSN 0976 – 6340 (Print) ISSN 0976 – 6359 (Online) Volume 4, Issue 5, September - October (2013), pp. 156-163 © IAEME: www.iaeme.com/ijmet.asp Journal Impact Factor (2013): 5.7731 (Calculated by GISI) www.jifactor.com IJMET © I A E M E
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 157 stretching rate otherwise sudden stretching may spoil the properties desired for the final outcome. These two aspects demand for a thorough understanding of flow and heat transfer characteristics which is the main theme of the present investigation. With the stand point of many applications akin to polymer extrusion process Crane [1] initiated the analytical study of boundary layer flow due to a stretching sheet. He assumed the velocity of the sheet to be a linear function of the distance from the slit. The solution so obtained by Crane for the flow driven by a stretching sheet belongs to an important class of exact solutions of Navier-Stokes equations and the uniqueness of the solution is well-established. The existence and uniqueness of solution for the flow caused by a stretching sheet is addressed by many authors (see McLeod and Rajagopal [2] and Troy et al [3]). The analytical study of McLeod and Rajagopal [2] throws light on the specification of infinity for solving non-linear differential equations in case of unbounded domains. Of late the work of Crane was extended by many authors to both Newtonian and non-Newtonian boundary layer flow subjected to various physical situations. Gupta and Gupta[4] investigated heat transfer from an isothermal stretching sheet with suction/blowing effects. Chan and Char [5] extended the works of Gupta and Gupta to that of a non-isothermal stretching sheet. Grubka and Bobba [6] carried out heat transfer studies by considering the power law variation of surface temperature. Chiam [7] investigated the MHD heat transfer from a non-isothermal stretching sheet. Heat source/sink effect is an important factor that requires attention as it exerts strong influence on the heat transfer characteristics in such an exothermic process. Many of the authors have studied heat transfer by considering a uniform heat source/sink or a temperature dependent heat source/sink (see Vajravelu and Rollins [8], Vajravelu and Hadjinicolaou [9]). Eldahab and El-Aziz [10] included the effect of non-uniform heat source/sink (space and temperature dependent heat source/sink) on the heat transfer. A non-Newtonian second grade fluid does not give meaningful results for highly elastic fluids (polymer melts) which occur at high Deborah numbers (Refs. Hayat et.al [11], [12]).Therefore, the significance of the results reported in the above works are limited, at least as far as polymer industry is concerned. Obviously, for the theoretical results to become of any industrial significance, more realistic viscoelastic fluid models such as Upper-Convected Maxwell model or Oldroyd-B model should be invoked in the analysis. Indeed, these two fluid models have recently been used to study the flow of viscoelastic fluids above stretching and non-stretching sheets but with no heat transfer effects involved Hayat et al. [11] and Sadeghy et. al [13]. Sadeghy et.al ([13], [14]), Alizadeh-Pahlavan et. al [15], Renardy [16], Rao and Rajagopal[17], Aliakbar et.al[18] have done the work related to UCM fluid by using HAM- method and by using numerical methods with no heat transfer. But the effect of thermal conductivity and non-uniform heat source/sink is very important and cannot be ignored. This fact motivates us to propose the effect of thermal conductivity and non-uniform heat source/sink on the heat transfer characteristics of a boundary layer flow of a Maxwell fluid over the stretching sheet in the present paper. The various effects of different parameters such as Elastic parameter, MHD parameter, porous parameter, Prandtl number and non-uniform heat source/sink parameter are discussed and shown with the aid of graphs. 2. MATHEMATICAL FORMULATION The boundary layer equations can be derived for any viscoelastic fluid starting from Cauchy equations of motion. For steady two-dimensional flows, these equations governing transport of heat and momentum can be written as (Refs. Sadeghy et.al[7], Alizadeh-Pahlavan and Sadeghy[11]).
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 158 0, u v x y ∂ ∂ + = ∂ ∂ (1) 2 2 2 2 2 2 2 2 2 2 , ' u u u u u u u v u v uv u x y x y x y y k ν λ υ  ∂ ∂ ∂ ∂ ∂ ∂ + + + + = − ∂ ∂ ∂ ∂ ∂ ∂ ∂  (2) 22 2 . p p T T k T u u v x y C y C y µ ρ ρ  ∂ ∂ ∂ ∂ + = +   ∂ ∂ ∂ ∂  (3) where u and v are the velocity components along x and y directions respectively, t is the temperature of the fluid, σ is the density, υ is the kinematic viscosity, k′ is the porosity parameter, Cp is the specific heat at constant pressure, k is the thermal conductivity of the liquid far away from the sheet, 0B , is the strength of the magnetic field, υ is the kinematic viscosity of the fluid andλ is the relaxation time Parameter of the fluid. The boundary conditions applicable to the flow problem are 2 2 , 0 0 0, 0, , w w y x u bx v T T T A PST Case l T x K Q D PHF Case at y y l u u T T as y ∞ ∞   = = = = +     ∂   − = = =  ∂   → → → → ∞ (4) Where A and D are constants, b is the constant known as the stretching rate, l the characteristic length, Tw is the wall temperature and T∞ constant temperature far away from the sheet. In order to obtain dimensionless form of the solution we define following variables y b Wherefbvfxbu γ ηηγηη =−== ),(),( 2 2 ( ) , w w T T x where T T A PST Case T T l x D PHF Case l θ η ∞ ∞ ∞ −   = − =   −     =     (5) where subscript η denotes the derivative with respect to η. Clearly u & v satisfy the equation (1) identically. Substituting these new variables in equation (2) and (3), we have, ( ) ( ) 2 22 0,f f f ff f ff k fβ′′′ ′ ′′ ′ ′′ ′′′ ′− + + − + = (6) [ ] 2 Pr 2 Pr ,f f Ec fθ θ θ′ ′ ′′ ′′− = + (7) [ ] 2 Pr 2 Pr ,f g g f g Ec f′ ′ ′′ ′′− = + (8)
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 159 Boundary conditions of equation (4) transform to PST CASE ( ) 1, ( ) 1 ( ) 0 0 ( ) 0, ( ) 0 ( ) 0 f f at f f as η η ηη η θ η η η η θ η η η = = = = → → → →∞ (9) PHF CASE ( ) 1, ( ) 1 ( ) 0 0 ( ) 0, ( ) 0 ( ) 0 f f at f f as η η η ηη η θ η η η η θ η η η = = − = = → → → →∞ (10) Where subscript η denotes the differentiation with respect to η . β denotes elastic parameter, 2k is the porosity parameter, Pr and Ec denotes the Prandtl number and Eckert number respectively, The physical quantities are defined as, kb k ′ = γ 2 , ∞ = K Cpµ Pr , 2 2 p b l Ec Ac = PHYSICAL QUANTITIES Our interest lies in investigation of the flow behavior and heat transfer characteristics by analyzing the non-dimensional local shear stress )( wτ and Nusselt number (Nu). These non- dimensional parameters are defined as : 0 ),0( = ∗ ∗       ∂ ∂ −=== y w y u Wheref bxb µτ γµ τ τ ηη (11)       = − − = ∞ CasePHF CasePST T TT h Nu y w )0(1 )0( θ θη (12) 3. NUMERICAL SOLUTION OF THE PROBLEM We adopt the most effective shooting method (see Refs. Conte and De Boor[20], Cebeci and Bradshaw[21]) with fourth order Runge-Kutta integration scheme to solve boundary value problems in PST and PHF cases mentioned in the previous section. The non-linear equations (1) and (3) in the PST case are transformed into a system of five first order differential equations as follows: ( ) [ ] 0 1 1 2 2 1 0 2 0 1 2 22 2 0 0 1 21 1 0 1 0 2 , , 2 , 1 , Pr 2 EcPr . df f d df f d f f f f f f k fdf d f d d d f f f d η η β η β θ θ η θ θ θ η = = ′− − − = − = = − − (13)
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 160 Subsequently the boundary conditions (8) take the form, 0 1 1 2 0 0 (0) 0, (0) 1, ( ) 0, (0) 0, (0) 0, ( ) 0. f f f f θ θ = = ∞ = = = ∞ = (14) Here 0 0( ) and ( ).f f η θ θ η= = Aforementioned boundary value problem is first converted into an initial value problem by appropriately guessing the missing slopes 2 1(0) and (0)f θ . The resulting IVP is solved by shooting method for a set of parameters appearing in the governing equations with a known value of 2 1(0) and (0)f θ . The convergence criterion largely depends on fairly good guesses of the initial conditions in the shooting technique. The iterative process is terminated until the relative difference between the current iterative values of 2 (0)f matches with the previous iterative value of 2 (0)f up to a tolerance of 6 10− . Once the convergence is achieved we integrate the resultant ordinary differential equations using standard fourth order Runge–Kutta method with the given set of parameters to obtain the required solution. 4. RESULTS AND DISCUSSION Numerical computation has been carried out for different physical parameters like Elastic parameter ( )β , Porosity parameter (k2), Prandtl number (Pr), Eckert number (Ec), which are presented graphically in figures (1 – 7). The non-linear ordinary differential equations (6), (7) and (8) subject to the boundary conditions (4), (8) and (9) were solved numerically using the most effective numerical fourth-order Runge-Kutta method with efficient shooting technique. Appropriate similarity transformation is adopted to transform the governing partial differential equations of flow and heat transfer into a system of non-linear ordinary differential equations. The effect of several parameters controlling the velocity and temperature profiles are shown graphically and discussed briefly. Figs.1 and 2 show the effect of Elastic parameter β , on the velocity profile above the sheet. An increase in the Elastic parameter is noticed to decrease both u- and v- velocity components at any given point above the sheet. Figs.3 and 4 revels that, the effect of porosity γ in presence of magnetic number and Elastics parameter (at 1)β = on the velocity profile above the sheet. An increase in the porous parameter leads to increase both u- and v- velocity components above the sheet. Fig.5(a) and 5(b) demonstrate the effect of Prandtl number Pr on the temperature profiles for two different PST and PHF. These plots reveals the fact that for a particular value of Pr the temperature increases monotonically from the free surface temperature sT to wall velocity the 0T . The thermal boundary layer thickness decreases drastically for high values of Pr i.e., low thermal diffusivity. Fig.6(a) and 6(b) project the effect of Eckert number Ec on the temperature profiles for both PST and PHF cases. The effect of viscous dissipation is to enhance the temperature of the fluid. i.e., increasing values of Ec contributes in thickening of thermal boundary layer. For effective cooling of the sheet a fluid of low viscosity is preferable. Figs.7(a) and 7(b) revels that, the effect of porosity γ in presence of magnetic number and Elastics parameter (at 1)β = on the temperature profile above the sheet. An increase in the porous parameter leads to decrease the temperature in PST case whereas opposite effect is seen in PHF case.
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 161 5. CONCLUSIONS The viscous dissipation effect is characterized by Eckert number (Ec) in the present analysis. Comparing to the results without viscous dissipation, one can see that the dimensionless temperature will increase when the fluid is being heated (Ec > 0) but decreases when the fluid is being cooled (Ec < 0). This reveals that effect of viscous dissipation is to enhance the temperature in the thermal boundary layer. 0 1 2 3 4 5 0.0 0.2 0.4 0.6 0.8 1.0 Fig.1. The effect of elastic parameter β on u-velocity component f' at β =1 f'(η) η β = 1 β = 2 β = 3 0 1 2 3 4 5 6 7 0.0 0.2 0.4 0.6 0.8 1.0 Fig.2. The effect of elastic parameter β on v-velocity component f at β =1f(η) η β = 1 β = 2 β = 3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 0.0 0.2 0.4 0.6 0.8 1.0 Fig.3. The effect of Porous parameter γ on u-velocity component f' at β=1 f'(η) η γ = 0.1 γ = 0.2 γ = 0.3 0 1 2 3 4 0.0 0.2 0.4 0.6 0.8 1.0 Fig.4. The effect of Porous parameter γ on v-velocity component f at M=β=1 f(η) η γ = 0.1 γ = 0.2 γ = 0.3 0 1 2 3 4 5 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Fig. 5b. The effect of Prandtl number Vs temperature profile g(η) η PHF-Case Pr=1 Pr=5 Pr=10 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Fig. 5a. The effect of Prandtl number Vs Temperature profile PST Case Pr = 5 Pr = 10 θ(η) η
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 162 0 1 2 3 4 5 6 7 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Fig. 6b. The effect of Eckert number Vs Temperature profile g(η) η PHF-Case Ec=0.2 Ec=1.0 Ec=2.0 0 1 2 3 4 5 6 0.0 0.2 0.4 0.6 0.8 1.0 Fig.7a. Effect of porous parameter γ on temperature profile in PST Case θ(η) η PST Case γ = 0.1 γ = 0.2 γ = 0.3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 -1.0 -0.8 -0.6 -0.4 -0.2 0.0 Fig.7b. Effect of Porous parameter γ on temperature profile in PHF Case g(η) η PHF Case γ=0.1 γ=0.2 γ=0.3 REFERENCES [1] L.J. Crane, flow past a stretching plate, Z. Angrew. Math. Phys. 21 (1970) 645-647. [2] J.B. McLeod, K.R. Rajagopal, On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary, Arch. Rational Mech. Anal. 98 (1987) 699-709. [3] W.C. Troy, W.A Overman II, G.B. Ermentrout, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math. 45 (1987) 753-755. [4] P.S. Gupta, A.S. Gupta, Heat and Mass transfer on a stretching sheet with suction or blowing, Can. J. Chem. Eng. 55 (1977) 744-746. [5] C.K. Chan, M.I. Char, Heat transfer of a Continuous stretching surface with suction or blowing, J. math. Anal. Appl. 135 (1988) 568-580. [6] L.G. Grubka, K.M, Bobba, Heat Transfer characteristics of a continuous stretching surface with variable temperature, J. Heat Transfer 107 (1985) 248-250. [7] T.C. Chiam, Magnetohydrodynamic heat transfer over a non-isothermal stretching sheet, Acta Mechanica 122 (1997) 169-179. [8] K. Vajravelu, D. Rollins, Heat transfer in electrically conducting fluid over a stretching surface, Int. J. Non-Linear Mech. 27 (2) (1992) 265-277. [9] K. Vajravelu, A. Hadjinicolaou, Heat transfer in a viscous fluid over a stretching sheet with viscous dissipation and internal heat generation, Int. Comm. Heat Mass Transfer. 20 (1993) 417-430. 0 1 2 3 4 5 6 7 8 9 0.0 0.2 0.4 0.6 0.8 1.0 1.2 Fig. 6a. The effect of Eckert number Vs temperature profile θ(η) η PST-Case Ec=1 Ec=2 Ec=5
    • International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print), ISSN 0976 – 6359(Online) Volume 4, Issue 5, September - October (2013) © IAEME 163 [10] Emad m. Abo-Eladahab, Mohamed A, El Aziz, Blowing/suction effect on hydromagnetic heat transfer by mixed convection from an inclined continuously stretching surface with internal heat generation /absorption, Int. J. Therm. Sci. 43 (2004) 709-719 [11] T. Hayat, Z. Abbas, M. Sajid. Series solution for the upper-convected Maxwell fluid over a porous stretching plate. Phys Lett A 358 (2006) 396-403. [12] T. Hayat, Z. Abbas, M. Sajid, and S. Asghar, “ The influence of thermal radiation on MHD flow of a second grade fluid,” International Journal of Heat and Mass Transfer, vol. 50, no. 5- 6, pp. 931–941, (2007). [13] K. Sadeghy, A.H.Najafi, M.Saffaripour. Sakiadis flow of an upper convected Maxwell fluid Int J Non-Linear Mech 40 (2005) 1220. [14] K. Sadeghy, Hadi Hajibeygi, Seyed-Mahammad Taghavi. Stagnation point flow of upper- convected Maxwell fluids. I.J. Non-Linear Mechanics. 41 (2006) 1242-1247. [15] Alizadeh-Pahlavan A, Sadeghy K (2009) on the use of homotopy analysis Method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheet. Commun Nonlinear Sci Numer Simul 14(4):1355-1365. [16] M. Renardy. High Weissenberg number boundary layers for the Upper Convected Maxwell fluid. J. Non-Newtonian Fluid Mech. 68 (1997) 125. [17] I.J. Rao and K. R. Rajgopal. On a new interpretation of the classical Maxwell model. Mechanics Research Communications. 34 (2007) 509-514. [18] Aliakbar V, Alizadeh-Pahlavan A, Sadeghy K (2009) The influence of Thermal radian on MHD flow of Maxwellian fluids above stretching sheets. Commun Nonlinear Sci Numer Simul 14(3):779-794. [19] A. Alizadeh-Pahlavan, K. Sadeghy. On the use of homotopy analysis method for solving unsteady MHD flow of Maxwellian fluids above impulsively stretching sheets. Commun Ninlinear Sci Numer Simulat. In press(2008). [20] S.D. Conte, C. de Boor, Elementary Numerical analysis, McGraw-Hill, New York, 1972. [21] T. Cebeci, P. Bradshaw, Physical and computational aspects of convective heat transfer, Springer-Verlag, New York, 1984. [22] Anand H. Agadi, M. Subhas Abel and Jagadish V. Tawade, “A Numerical Solution of MHD Heat Transfer in a Laminar Liquid Film on an Unsteady Flat Incompressible Stretching Surface with Viscous Dissipation and Internal Heating”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 5, 2013, pp. 49 - 62, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [23] Anand H. Agadi, M. Subhas Abel, Jagadish V. Tawade and Ishwar Maharudrappa, “Effect of Non-Uniform Heat Source for the UCM Fluid Over a Stretching Sheet With Magnetic Field”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 6, 2013, pp. 40 - 49, ISSN Print: 0976-6480, ISSN Online: 0976-6499. [24] Anand H. Agadi, M. Subhas Abel and Jagadish V. Tawade, “MHD and Heat Transfer in a Thin Film Over an Unsteady Stretching Surface with Combined Effect of Viscous Dissipation and Non-Uniform Heat Source”, International Journal of Mechanical Engineering & Technology (IJMET), Volume 4, Issue 4, 2013, pp. 387 - 400, ISSN Print: 0976 – 6340, ISSN Online: 0976 – 6359. [25] Anand H. Agadi, M. Subhas Abel, Jagadish V. Tawade and Ishwar Maharudrappa, “MHD Flow and Heat Transfer for the Upper Convected Maxwell Fluid Over a Stretching Sheet with Viscous Dissipation”, International Journal of Advanced Research in Engineering & Technology (IJARET), Volume 4, Issue 5, 2013, pp. 231 - 242, ISSN Print: 0976-6480, ISSN Online: 0976-6499.