Temperature Dependent Viscosity of a thin film fluid on a Vertical Belt with slip boundary conditions

Transcription

1 J. Appl. Environ. Biol. Sci., 5(2) , , TextRoad Publication ISSN: Journal of Applied Environmental and Biological Sciences Temperature Dependent Viscosity of a thin film fluid on a Vertical Belt with slip boundary conditions Taza Gul 1, Noor Jamal 2, S.Islam 1, Muhammad Altaf Khan 1, Haroon Rashid 2 1 Department of mathematics, Abdul Wali K.U Mardan, KPK Pakistan 2 Department of mathematics, ISPaR Peshawar/Bacha Khan University, KPK Pakistan Received: November 24, 2014 Accepted: January 11, 2015 ABSTRACT In this research our goal as to study the effect of temperature on viscosity of a thin film flows with slip boundary condition. The problem is modeled in the form of coupled non-linear differential equations. The coupled non-linear differential equations are solved analytically by using Adomian decomposition method (ADM) for the approximate solutions of velocity and temperature fields. Both of these solutions are compared and are in good agreement as shown in graphs and tables. KEYWORDS: Thin Film Flows, Slip Boundary Conditions, Adomian Decomposition Method (ADM). I. INTRODUCTION The thin film flow of a fluid on vertical belt has many applications in industries processes, for examples petroleum industry, manufacturing and processing food and paper in chemical industries such as fiber coating, wire coating drawing of plastic sheet etc. Taza Gul et al [1] study the thin film layer flowing on a vertical oscillating belt. Ali and Awais [2] studied the solution of unsteady second grade fluid through porous medium. Hayat et al [3] investigated a thin film flow of fourth grade fluid vertical cylinder. Zheng et al [4] studied the effect of mass and heat transfer of fluid over moving and oscillating surface. The steady flow of a third grade fluid down an inclined plane was investigated by Aiyesimi et al [5, 6] and they observed the effect of Brinkmen number and magnetic parameter for temperature and velocity profile. A thin film flow on a vertical belt with slip boundary condition for lifting and drainage was investigated by Taza Gul et al [7]. Hashmi et al [8] observed the third grade fluid on inclined plane by using ordinary OHAM to solve the problem with slip boundary condition. Nadeem et al [9] examined thin-film unsteady flow of a third grade fluid under the state of constant and variable viscosity. Sahoo et al [10] discussed a thin film flow with slip condition. The effect of slip condition on thin film flow of third grade fluid through inclined plane was study by Khan and Mahmood [11]. The modified ADM method was used for the solution of nonlinear differential equation the work has been found in [12, 14]. G. Adomian [15] uses decomposition method to solve the differential equation. Wazwaz [16, 17] used ADM for reliable treatment of Bratu-type and Emden-Fowler equations. Noor et al [18] used HAM method to solve the problem. The related and recent work can also be seen [19-22]. II. Basic equations The governing equations div v = 0, (1) DΘ ρc p Dt κ 2 Θ + tr(t. L), (2) Where D = + (v. ) is the material time derivative, ρ is the fluid density,vis the velocity vector, g is the body force, Θ Dt t is temprature, kis the thermal conductivity, c p is specific heat,l = v and T is the caushy stress tensor and define for second grade fluid as T = pi + S, (3) Where pi denotes spherical stress and shear stress S is defined as S = μa 1 + α 1 A 2 + α 2 A 2 1. (4) Here α 1, α 2, are material coefficients for second grade fluid. A 1, A 2 and A 3 are the kinematical tensors given by A 0 = I, A 1 = ( v) + ( v) T A n = DA n 1 + A Dt n 1 ( v) + ( v) T A n 1, n 2. (5) III. Formulation of the lift problem Consider a thin layer on a lubricating vertical belt. The belt is moving upward with constant velocity V. Thickness δ of the layer is assumed to be uniform. The coordinate s axis is chosen in such a way that x-axis is parallel to the belt and y- axis is perpendicular to the belt. Flow is assumed to be incompressible, laminar and pressure is atmospheric. Velocity and temperature fields are v = (0, v(x),0) and Θ = Θ(x). (6) *Corresponding Author: Taza Gul, Department of Mathematics, Abdul Wali K.U Mardan, KPK Pakistan 157

2 Taza Gul et al., 2015 The slip boundary conditions are v(x) = V γt xy, Θ(x) = Θ 0,atx = 0, (7) dv(x) = 0, Θ(x) = Θ 1, at x = δ, (8) The continuity Eq. (1) satisfies identically and T xy component of stress tensor is T xy = μ dv, (9) So the momentum and energy equations reduce to the form 0= μ d2 v + dv dμ + ρg (10) 2 0 = κ d2 Θ + μ ( dv 2 )2 (11) Here we introduce the dimensionless parameters v = v V, x = x δ, Θ = Θ Θ 0, Θ 1 Θ 0 B r = μ = μ, μ 0 2 ρg γ = γ, γ 0 μ 0 V2, S k(θ 1 Θ 0 ) t = δ,, Λ = γμ μ 0 V 0. (12) The dimensionless viscosity for Reynold s model is μ = exp ( ηθ) (13) By making use of Taylor series expansion, one may represent viscosity and its derivative as dμ dθ μ 1 ηθ, η (14) The dimensionless form of eq (12, 13) dropping the bars are d 2 v η ( dv 2 dθ d + Θ(x) 2 v ) S 2 t = 0, (15) d 2 Θ + B 2 r (( dv )2 ηθ ( dv )2 ) = 0, (16) The boundary conditions for zero component solution are, v n = 1 Λ ( dv n ηθ dv n ), Θ n = 0,at x = 0, (17) dv n = 0, Θ n = 1, at x = 1, when n = 0, (18) Similarly the boundary conditions for first, second and third components solutions are v n = Λ ( dv n ηθ dv n ), Θ n = 0, at x = 0, (19) dv n = 0, Θ n = 0, at x = 1, when n = 1,2,3, (20) IV. The ADM Solution of lifting problem: Write Eq. (19, 20) in standard form of ADM and then apply the inverse operator L 1 = ) v(x) = φ + ηl 1 [ dv dθ + Θ d2 v ], (21) 2 Θ(x) = ψ B r L 1 ( dv )2 + B r ηl 1 Θ ( dv )2, (22) The integral constant φ, ψ is determined from the given boundary conditions. Series solution of v(x) = n=0 v n,, Θ(x) = n=0 Θ n are n=0 v n = φ + ηl 1 [ n=0 B n ] + ηl 1 [ n=0 C n ], (23) n=0 Θ n = ψ B r L 1 [ n=0 D n ] + B r ηl 1 [ n=0 E n ], (24) Where A n,b n,c n, D n,e n and F n, are called the Adomian polynomials and define as Components of Adomian polynomials are A 0 = ( dv 0 )2 d 2 v 0, B 2 0 = dv 0 dθ 0, C 0 = Θ 0 (x) d2 v 0, d x 2 A 1 = ( dv 0 )2 d 2 v dv 0 d x 2 D 1 = 2 dv 0 D 0 = ( dv 0 )2, E 0 = Θ 0 (x) ( dv 0 )2, F 0 = ( dv 0 )4, (25) dv 1 d 2 v 0, B 2 1 = dv 1 dθ 0 + dv 0 dθ 1 dv 1, E 1 = Θ 1 (x)( dv 0 )2 + 2Θ 0 (x) dv 1, C 1 = Θ 1 (x) d2 v 0 + Θ 2 0 (x) d2 v 1 dv 0, F 1 = 4 ( dv 0 )3 dv 1, (26) In series form Eq. (25, 26) becomes v 0 + v 1 + v 2... = φ + ηl 1 [B 0 + B 1 + B ] + ηl 1 [C 0 + C 1 + C ] (27) Θ 0 + Θ 1 + Θ = ψ B r L 1 [D 0 + D 1 + D ] + ηb r L 1 [E 0 + E 1 + E ]. (28) By comparing both sides of Eq. (27, 28), the zero, first, second component velocity and heat problems are v 0 (x) = φ = L 1 [S t ], (29) Θ 0 (x) = ψ, (30) v 1 (x) = ηl 1 [B 0 ] ηl 1 [C 0 ], (31) Θ 1 (x) = B r L 1 [D 0 ] (32) d x 2, 158

3 J. Appl. Environ. Biol. Sci., 5(2) , 2015 v 2 (x) = ηl 1 [B 1 ] ηl 1 [C 1 ] (33) Θ 2 (x) = +B r L 1 [D 1 ], (34) Using the slip boundary conditions for zero component velocity and temperature problems are v 0 (0) = 1 Λ ( dv 0 ηθ dv 0 0 ), Θ 0(0) = 0, (35) dv 0 (1) = 0, Θ 0 (1) = 1, (36) Solution of zero component problem inserting eqs.(35,36) into (28,29) are v 0 (x) = (1 + ΛS t ) (S t )x + ( S t ) 2 x2, (37) Θ 0 (x) = x, (38) Using slip boundary conditions for first component velocity and temperature problems are v 1 (0) = Λ ( dv 1 ηθ dv 1 0 ηθ dv 0 1 ), Θ 1(0) = 0, (39) dv 1 (1) = 0, Θ 1 (1) = 0, (40) Solution of first component problem inserting Eqs.(39,40) into (30,31) we get v 1 (x) = [ ηs t 2 ]x2 + S t [ η ] 3 x3, (41) Θ 1 (x) = B r S t 2 [ x x2 3 4 ], Using the slip boundary conditions for second component problems are (42) v 2 (0) = Λ[ dv 2 η (Θ dv Θ dv Θ dv 0 2 )], Θ 2 (0) = 0, (43) dv 2 (1) = 0, Θ 2 (1) = 0, (44) The solutions of second components of velocity and temperature distribution using boundary conditions in eq (42, 43) is too large, therefore its expression are mentioned graphically. V. Formulation of Drainage problem Consider a film of liquid draining down on a vertical belt. But the belt is stationary and the flow of fluid is due to the gravity. In drainage problem the Stock number is positively taken in eq (15). The geometry and assumptions of drainage problem are identical as in preceding problem but the boundary condition of velocity is changed. So momentum equation and velocity slip boundary conditions are The slip boundary conditions are d 2 v η ( dv 2 dθ v(0) = γt xy, The non-dimensional slip boundary conditions are: v n (0) = Λ( dv n ηθ dv n d 2 v + Θ(x) ) + S 2 t = 0, (45) dv(1) ), dv n (1) = 0, (46) = 0, when n = 0,1,2,3... (47) VI. The ADM Solution of Drainage problem The Adomian polynomials are same for both problems. By applying the standard form of ADM in Eq. (15) and (45), we obtained zero, first and second components problem of velocity and temperature distribution Solution of zero and first component problem of velocity and temperature distribution using the slip boundary conditions in Eq. (49) are v 0 (x) = (1 + ΛS t ) + (S t )x ( S t ) 2 x2, (48) Θ 0 (x) = x, (49) v 1 (x) = S t [ η ] 2 x2 [ ηs t ] 3 x3, (50) Θ 1 (x) = B r S 2 t [ 1 [1 + 4 S 4 3 t 2 ]x 1 2 x2 + x3 x x6 ], (51) The solutions of second components of velocity and temperature distribution are too large, therefore its expression are mentioned graphically. 159

4 Taza Gul et al., 2015 Fig 1-2: Effect of Brinkman number on the lift velocity (on left) when the other parameter are fixed S t = 50; η = 0.3; Λ = 0.6; lift temperature (on right) when the other parameter are η = 0.05; S t = 0.4; Λ = 0.08; Fig 3-4: The influence of Brinkman number on the drainage velocity (on left) when η = 0.6; S t = 0.4; Λ = 0.7; and the influence of drainage temperature (on right) when other parameter are fixed η = 0.05; S t = 0.5; Λ = 0.04; Fig 5-6: The influence of the stock number on Lift velocity (on left) when B r = 50; η = 0.4; Λ = 0.6;and temperature distribution (on right) when the other parameter are fixed B r = 50; η = 0.7; Λ = 0.6; Fig 7-8: Effect of the stock number on drainage velocity (on left) when other parameter areb r = 40; η = 0.08; Λ = 0.5; and the profile of temperature distribution (on right) when. B r = 70; η = 0.7; Λ =

5 J. Appl. Environ. Biol. Sci., 5(2) , 2015 Fig 9-10: Effect of viscosity parameter on lift velocity (on left) by keeping other fixed B r = 50; S t = 0.08; Λ = 0.4;and temperature distribution (on right) when B r = 50; S t = 0.4; Λ = 0.9; Fig 11-12: The influence of viscosity parameter on drainage velocity (on left) when B r = 80; S t = 0.4; Λ = 0.6; and temperature distribution (on right) when B r = 90; S t = 0.2; Λ = 0.5; VII. RESULTS AND DISCUSSION In current article we studied analysis of heat transfer on a thin film flow under the specific slip boundary conditions. The problem has been modeled for lift and drainage velocity and temperature distribution. The influences of model parameters such as Stock number S t, slip parameter Λ, the Brinkman number B r, and viscosity parameter η have been investigated for velocity and temperature distribution. The influences of these physical parameters have been deliberated in Figures1-12. Figure 1, 2 shows the effect of the Brinkman numberb r, for lift velocity and temperature profile. It has been seen that the velocity decreases when we increases the value of B r and temperature of liquid increasing when the value of B r increases. Figure 3, 4 shows the influence of the Brinkman numberb r, for drainage velocity and temperature profile. We see that velocity of Brinkmen number B r decreases when we increase the value ofb r. While there is increase in drainage temperature of liquid increasing when the value of B r increases. Figure 5, 6 shows the effect of Stock numbers t on lift velocity and temperature profile and we observed that the velocity and temperature increases with we increase in the value of Stock number S t. The effect of Stock number S t on drainage velocity and temperature profile as shown in figure 7, 8 and we see that the velocity and temperature increase with an increase in the value of Stock number S t. Fig 9, 10 identifies the effect of viscosity parameter η on lift velocity and temperature profile. Increase in viscosity parameter η there is decrease in velocity profile, while there is increases in the temperature of liquid when we increase the value of viscosity parameter. Fig 11, 12 recognizes the effect of viscosity parameter η on drainage velocity and temperature profile. The velocity is decreases when the value of viscosity parameter decreases when value of η increasing the temperature become maximum. VIII. Conclusion We have considered the thin film fluid flow of lifting and drainage problem with the temperature dependent viscosity. The problems have been solved analytically by using Adomian Decomposition Method (ADM).The analytical expression for velocity fields and temperature distribution has been presented and plotted graphically. Graphically the profile of model parameters such as Stock number S t, slip parameter Λ, the Brinkman number B r, and viscosity parameter η have been studied for velocity and temperature distribution. 161

6 Taza Gul et al., 2015 REFERENCES [1] T.Gul, S. Islam, R.A. Shah, I. Khan and S. Shafie, Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with Temperature Dependent Viscosity. PloS ONE 9(6): (2014) e doi: /journal.pone [2].Ali M. and Awais M. (2014) Laplace Transform Method for Unsteady Thin Film Flow of a Second Grade Fluid through a Porous Medium, Journal of Modern Physics, 5: [3] Hayat T. and Sajid M. (2007) on analytic solution for thin film flow of a fourth grade fluid down a vertical cylinder, Physics Letters A 361(5): [4] Zheng L. C., Jin X., Zheng X. X. and Zhang J. H.,(2013) Unsteady heat and mass transfer in MHD flow over an oscillating stretching surface with soret and Dofour effect, Acta Mechanica Sinica, 29(5): [5] Aiyesimi Y. M., Okedayo G.T. and Lawal O. W. (2012) Viscous Dissipation effect on the MHD flow of a third grade fluid down an inclined plane with ohmic heating, Math. Theory Model, 2(9): [6] Aiyesimi Y. M., Okedayo G.T. and Lawal O. W. (2013) Unsteady magneto-hydrodynamic (MHD) thin film flow of a third grade fluid with heat transfer and no slip boundary condition down an inclined plane, International Journal of Physical Sciences, 8(19): [7] T.Gul. R.A. Shah. S. Islam. M. Arif, MHD thin film flow of a third grade fluid on a vertical belt with slip boundary conditions. J. Appl. Math. (2013). [8] Hashmi M. S., Khan N. and Mahmood T. (2013) Optimal Homotopy Asymptotic Solution for Thin Film Flow of a Third Grade Fluid with Partial Slip, World Applied Sciences Journal 21 (12): [9] S. Nadeem, M. Awais, Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity. Phy. Let. A. 372 (2008) [10] B. Sahoo, S. Poncet, Flow and heat transfer of a third grade fluid past an exponentially stretching sheet with partial slip boundary condition, International Journal of Heat and Mass Transfer,54(2013): [11] K.Nrgis, T. Mahmood, The influence of slip condition on the thin film flow of a third order fluid. Int. J. of Nonlin. Sci. 1(13) (2012) [12] Abbas bandy S. (2003) Improving Newton Raphson method for nonlinear equations by modified Adomian decomposition method, Applied Mathematics and Computation, 145 (2-3): [13] Ramana P. V. and Prasad B. K. R. (2014) Modified Adomian Decomposition Method for Vander Pol equations, International Journal of Non-Linear Mechanics 65: [14] G. Adomian, A review of the decomposition method and some recent results for non-linear equations, Math Comp. Model. 13 (1992) [15] G. Adomian. Solving frontier problems of physics: the decomposition method, Klu. Acad. Pub [16] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Bratu-type equations, Appl. Math. Comp. 166 (2005) [17] A.M. Wazwaz, Adomian decomposition method for a reliable treatment of the Emden Fowler equation, Appl. Math. Comp. 161 (2005) [18] Noor N.F.M., Abdulaziz O. and Hashim I. (2010) MHD flow and heat transfer in a thin liquid film on an unsteady stretching sheet by the homotopy analysis method, International Journal for Numerical Methods in Fluids, 63(3): [19] Sidra. Abid, S. Islam, Taza Gul, S. Nasir, Aaiza Gul, A. Khan, Magnetic Hydrodynamic Flow of Unsteady Second Grade Fluid between Two Vertical Plates with Oscillating Boundary Conditions. J. Appl. Environ. Biol. Sci., 4(10) 1-11, 2014 [20] Taza Gul, S. Islam, K. khan, A. Khan, R.A.Shah, Aaiza Gul, MuradUlla, Thin Film Flow Analysis of a MHD Third Grade Fluid on a Vertical Belt With no -slip Boundary Conditions. J. Appl. Environ. Biol. Sci., 4(10)71-84, [21] H. Rasheed, Taza Gul, S. Islam, S. Nasir, A. Khan, Aaiza Gul, Study of Couetteand Poiseuille flows of an Unsteady MHD Third Grade Fluid.J. Appl. Environ. Biol. Sci., 4(10)12-21, 2014 [22] Sanela Jamshad, Taza Gul, S.Islam, S. Nasir, M.A. Khan, R.Ali Shah. Flow of Unsteady Second Grade Fluid between Two Vertical Plates when one of the Plate Oscillating and other is stationary. J. Appl. Environ. Biol. Sci., 4(12)41-52,