Research Article Comparison of HPM and PEM for the Flow of a Non-newtonian Fluid between Heated Parallel Plates

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1 Research Journal of Alied Sciences, Engineering and Technology 7(): , 4 DOI:.96/rjaset ISSN: ; e-issn: Maxwell Scientific Publication Cor. Submitted: November, 3 Acceted: January 5, 4 Published: May, 4 Research Article Comarison of HPM and PEM for the Flow of a Non-newtonian Fluid between Heated Parallel Plates A.M. Siddiqui, Saira Bhatti, 3 H.A. Wahab and 4 Muhammad Naeem Deartment of Mathematics, York Camus, Pennsylvania State University, York, USA Deartment of Mathematics, COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan 3 Deartment of Mathematics, 4 Deartment of Information Technology, Hazara University, Manshera, Pakistan Abstract: The resent aer studies the heat transfer flow of a third grade fluid between two heated arallel lates for the two models: constant viscosity model and Reynold's model. In each case the nonlinear momentum equation and the energy equation have been solved using HPM and PEM. Grahs for the velocity and temerature rofiles are resented and discussed for the various values of arameters entering the roblem. The dominant effect is governed by whether or not the fluid is non-newtonian, the temerature effects being relegated to a less dominant role. Keywords: Constant viscosity model, homotoy erturbation mathod, momentum equation, PEM, Reynold's model, third grade fluid INTRODUCTION In general, the flowing mixtures consist of solid articles in a fluid such as coal based slurries exhibit non-newtonian characteristics. These mixtures are imortant in a variety of industrial alications and heat transfer lays an imortant role in handling and rocessing of these mixtures.there are roerties of fluid behavior which cannot be exlained on the basis of the classical, linearly viscous models. Several constitutive equations have been suggested to characterize such non-newtonian behavior. Amongst these are fluid of the different tye of grade n (Truesdell and Noll, 965), the incomressible and homogeneous fluid of grade being the linearly viscous Newtonian fluid. For examle, it has been shown that the substantial erformance benefits can be obtained if coal-water mixture is re-heated (Massoudi and Christie, 995; Tsai et al., 988). In this stu, we consider the model described by Szeri and Rajagoal (985) for heat transfer flow of a third grade fluid between two arallel lates maintained at different temerature located in the y = and y = h lanes, resectively, of an orthogonal cartesian coordinate system. Szeri and Rajagoal (985) examined this model for the two cases via a similarity transformation and concluded that the temerature deendence was not imortant for third grade fluid for the considered arameters and the variable viscosity solutions were not too distinct from that of constant velocity. With a slight modification, we consider this model and comare our results roduced by PEM and HPM with those roduced by Szeri and Rajagoal (985). In each case the nonlinear momentum equation and the energy equation have been solved using HPM and PEM. Grahs for the velocity and temerature rofiles are resented and discussed for the various values of arameters entering the roblem. There have been several studies involving heat and transfer in the non-newtonian fluids, but most of them seem to lack a systematic and rational treatment of the thermonamics of the roblem: while the stress constitutive equation is altered to account for non- Newtonian behavior, the constitutive equation for the secific Helmholtz free energy or the heat flux vector are left unchanged. Although this might be correct for a articular fluid, it does not seem roer to assume the same a riori. In this stu, we attemt a thermonamically consistent stu of the heat transfer roblem under consideration. We consider two models in our aroach: constant viscosity model and Reynolds model (Szeri and Rajagoal, 985). The governing differential equations for the velocity and temerature are non-linear whose exact solutions are not available. Therefore, asymtotic methods rove a owerful tool to obtain aroximate solution of these equations. An excellent review of some of these methods is given in (Siddiqui et al., 8). Of various asymtotic methods, the one is the homotoy erturbation method rovided by He (998a, Corresonding Author: H.A. Wahab, Deartment of Mathematics, Hazara University, Manshera, Pakistan This work is licensed under a Creative Commons Attribution 4. International License (URL: htt://creativecommons.org/licenses/by/4./). 46

2 Res. J. Al. Sci. Eng. Technol., 7(): , 4 999,, 3) which is couling of the traditional erturbation method and the homotoy concet used in the toology. In a series of aers by Abbasbandi (6) and He (4a, b, 5a, 998b, 5b) and several others have not only alied this method successfully to obtain the solution of currently imortant roblems in science and technology, but have also shown its effectiveness and reliability. Siddiqui et al. (6a, b, 7, 8) have alied this method to analyze flow roblems of non-newtonian fluid mechanics. In this stu we have used HPM and PEM to solve the roblem. We have also given a comarison of different erturbation techniques with illustrative examles of Shakil et al. (3) and Wahab et al. (3a, b, c). METHODOLOGY Basic equations: The basic laws of the conservation of mass, conservation of momentum and the conservation of energy for an incomressible fluid are given by:. v =, () dv = f divt, () Fig. : The x coordinate direction that is chosen arallel the external ressure gradient T = I ( ) A ( ) A ( ) A ( )( tra ) A. (6) Governing equations: With reference to the model by Szeri and Rajagoal (985), consider Poiseuille flow of a thermonamically comatible third grade fluid between arallel lates located at y = and y = h, resectively (Fig. ), such that we seek velocity field of the form: v =[ u( y ),,], (7) In the absence of bo forces, the balance of linear momentum: dv divt b =. (8) d = TL. divq r, (3) where, v = The velocity field = The bo force T = The stress tensor ρ = The constant density ε = The internal energy r = The radiant energy Using velocity rofile (7), Eq. (8) reduces to: d du d du x 3 ( ) ( )( ) =, d du y =. z [ ( ) ( )]( ) =, (9) () () The constitutive equation for a third grade fluid is: T = I ( ) A ( ) A ( ) A ( ) A ( ) AA A A ( )( tra ) A. (4) 3 3 Here -I denotes the indeterminate art of stress due to the constraint of incomressibility, μ (θ) is the coefficient of viscosity and α (θ), α (θ) are material moduli, usually referred as normal stress coefficients. The Rivlin-Ericksen tensor, A n are defined as: A = l, the identity tensor and: Since the fluid is incomressible and is hence constrained to satisfy: divv =. () The modified ressure P is defined through: P du (3) = [ ( ) ( )]( ). Equation (9) to () take the simler form: DA n T An = An ( v) An ( v ). n (5) Dt d 3 ( ) du d 3( )( du ) = P x (4) In our analysis, we assume that the fluid is thermonamically comatible; hence the stress constitutive relation reduces to: 47 as P = P( x ) only. The aroriate form of the energy equation roduced as follows:

3 Res. J. Al. Sci. Eng. Technol., 7(): , 4 d = TL. divq r, (5) where, ε denotes the internal energy and r is the radiant energy, both er unit mass and: d TL. = ( ) A ( ) A 4 ( ) ( ) 3 4 tra 3( ) A, (6) Which after simlification gives: du du 4 TL. = ( )( ) 3( )( ). (7) The heat flux vector q is given by Fourier's law: q = k grad (8) where, k = k (θ) denotes the thermal conductivity, but secify that k is constant. We now consider the role of internal energy to the energy equation. Since the internal energy is related to the secific Helmholtz free energy through: =, (9) where η is the entroy, leads to: ( ) = (,) A. () 4 For the thermonamically comatible fluid, the secific entroy is related to the secific Helmholtz free energy through: =, () where, the suffix denotes differentiation with resect to that variable. It follows from () and () that: ( ) = (,) A. () 4 The material roerties of the fluid are allowed to be temerature deendent, but require the temerature to satisfy the constraint: d =. (4) Substituting (9) and () into (5), in the absence of radial heating, the balance of energy imlies that: du du 4 d ( )( ) 3 ( ) k =. (5) The following variables and arameters introduced to non-dimensionalize (4) and (5): y = y h, u = u, V T = =, * * A = V, * V * h k = ( ) where, * = ( *); * = 3 ( *) and the characteristic velocity is given by, but with relacing * μ, = () and = ( h) are the wall temerature at the lower and the uer late, resectively. The non-dimensional form of Eq. (4) and (5) are: ' ' u 6 A( u ) = ' ' ' T ( u ) A( u ) =. (6) (7) Equation (6) and (7) can be solved subjected to the boundary conditions: u() =, u () =. (8) T ()=, T ()=. (9) The viscosity is temerature deendent in (6) and (7) and the recise form of the equation of motion deends on the viscosity model chosen to reresent the fluid. We refer here two viscosity models and the corresonding form of the non-dimensional equation of motion: Constant viscosity model: ( T ) = (3) = ( y), (3) Then by the virtue of () and (3): ' ' u 6 A( u ) = (3) 48

4 Res. J. Al. Sci. Eng. Technol., 7(): , 4 u() =, u () =. (3) Reynolds's model: so that: ( )= ex( m) (33) ( T) = ex( MT) (34) ' ' ( u ) ' u 6 A MTu = where, M = m( ) (35) u()=, u ()=. (36) Our aim here is to comare the results of this model (Szeri and Rajagoal, 985) with the results roduced by HPM and PEM. Constant viscosity model by HPM: In this case, we have two non-linear equations: ' ' ' T ( u ) A( u ) =, ' ' u 6 A( u ) =, (37) (38) T() =, T () =. (46) ' ' ' 8 ' ' =, (47) 3 T uu A uu T() =, T () =. (48) And the system of linear equations for velocity rofile: u '=, (49) u() =, u () =. (5) ' 6 ' ' =, (5) u Au u u() =, u () =. (5) u ' Au ' u ' u ' 6 Au ' u ' =, (53) u() =, u () =. (54) T () =, T () = (39) u() =, u () =. (4) We have to solve these two equations simultaneously. By alying the basic definition of HPM and exanding T and u as: T = T T T... (4) u = u u u... (4) We will get system of linear equations for temerature distribution as: T '=, (43) T() =, T () =. (44) ' ' ' =, (45) 4 T u A u 49 Solving these systems of linear equations we get the temerature distribution and velocity rofile: y y y y 8y 8y 4 4y y y T = y( ) A( y ) (55) 3 4 u = y y A(8y 4y 6y y). (56) Constant viscosity model by PEM: In this case, we have two non-linear equations: ' ' ' T ( u ) A( u ) =, ' ' u 6 A( u ) =, (57) (58) T() =, T () = (59) u() =, u () =. (6) We have to solve these two equations simultaneously. Here we have to exand the arameters as:

5 Res. J. Al. Sci. Eng. Technol., 7(): , 4 T = T T T... (6) u = u u u... (6) Solving these system of linear equations we get the temerature distribution and velocity rofile: =... (63) A = A A... (64) Using these exanded arameters in above equations, we will get system of linear Equations for temerature distribution as: T '=, (65) T() =, T () =. (66) T u ' ' =, (67) T() =, T () =. (68) ' ' ' ' ' =, (69) 4 T uu u A u T()=, T ()=. (7) And the system of linear equations for velocity rofile: u '=, (7) u() =, u () =. (7) u' 6 Au ' u ' =, (73) u() =, u () =. (74) u ' Au' u ' u ' 6 Au' u ' 6 Au ' u ' =, (75) (77) 3 4 u = y y A( 8y 4y 6y y). (78) Reynolds's model by HPM: In this case, we have two non-linear equations: ' ' ' 4 T ( u ) ex( MT) A( u ) =, ' ' ' u A u MT MTu MT (79) 6 ( ) ex( ) = ex( ) (8) where, M = m( ) T() =, T () = (8) u() =, u () =. (8) We have to solve these two equations simultaneously. By alying the same rocedure as for constant viscosity model, we will get system of linear equations for the temerature distribution as: T '=, (83) T()=, T ()=. (84) M 4 T' u' Mu' T u' T A u' =, (85) T() =, T () =. (86) And the system of the linear equations for velocity rofile: u '=, (87) u() =, u () =. (76) u() =, u () =. (88) 43

6 Res. J. Al. Sci. Eng. Technol., 7(): , 4 M u' 6 A u' u' Mu' u' T u' u' T MT' u' MT M T =, (89) u() =, u () =. (9) Solving these system of linear equations we get the temerature distribution and velocity rofile: AM y AM y 8 A M 3 T = y M y M ( ) A( ) A M 6 6 AM M A 6 My M ( ) A( ) My ( A) (8 ) M 5 M M 6 My ( A) ( A ) A( ) M A M A M A M My M ( ) ( ) ( ) A( ) M 4 5 ( A ) M ( ) A My A M AM AM M 3 ( ) A My A M (9) AM y 9AM y AM y M 7 AM y M 7 u = ( M) ( M) AM y M 9 AM y M 9 AMy M M 33M ( 4 M) ( 4 M) ( M 4) M M M 4 M AMy ( 7) M 8A 3 M 8 5 M M 6 AMy ( M ) AMy ( ) M A 3 A 6 M MA A AM 7AM 9AM 3AM M AM y( A ). (9) Reynolds's model by PEM: In this case, we have two non-linear equations: ' ' ' 4 T ( u ) ex( MT) A( u ) =, (93) ' ' ' u A u MT MTu MT 6 ( ) ex( ) = ex( ) (94) where M = m( ) T() =, T () = (95) u() =, u () =. (96) Alying the same rocedure as for the constant viscosity model, we will get system of linear equations for temerature distribution as: T '=, (97) 43

7 Res. J. Al. Sci. Eng. Technol., 7(): , 4 T() =, T () =. (98) ' ' =, (99) T u T()=, T ()=. () T ' u ' u ' u ' M u ' T A u ' =, () 4 T() =, T () =. () And the system of linear equations for the velocity rofile: u '=, (3) u() =, u () =. (4) u u MTu MT ' 6 ' =, (5) u() =, u () =. (6) Solving these system of linear equations we get the temerature distribution and velocity rofile: y y y y y y y y y 4y 4 8y 8y y T = y y y y 7 y y y 3 y 63 y y y y y M M y y y y y A y (7) y y y 3y u y y y y y ym = (8) DISCUSSION OF RESULTS In both the cases: Constant viscosity model and Reynold's model, we obtained velocity rofiles and temerature distributions for different values of M and A keeing the γ fixed. For dimensionless velocity and temerature distributions for Constant viscosity model and Reynolds's model by HPM keeing γ =, it is clear from the Fig. that for M =, and 5, dearture from symmetry is slight. To investigate the effects of M on the temerature 43

8 Res. J. Al. Sci. Eng. Technol., 7(): , 4 Fig. : Dimensionless velocity and temerature distributions for constant viscosity model and Reynolds's model by HPM keeing γ = Fig. 3: Dimensionless velocity and temerature distributions for constant viscosity model and Reynolds's model by PEM keeing γ = and γ = distribution, we include viscous heating γ =. At this moderate rate of viscous heating, the temerature shows strong deendence on M for the Newtonian fluid only. 433 For the non-newtonian fluid, the temerature rofiles obtained at different values of M coalesce almost into single curve. For dimensionless velocity

9 Res. J. Al. Sci. Eng. Technol., 7(): , 4 and temerature distributions for Constant viscosity model and Reynolds's model by PEM keeing γ = and γ =, Fig. 3 indicates that at this moderate rate of viscous heating the temerature shows strong deendence on M for the Newtonian fluid only. For the non-newtonian fluid the temerature rofiles obtained at different values of M coalesce almost into a single curve. It is clear from both the figures that the temerature and velocity distributions remain sensibly invariant with resect to the viscosity index M in non-newtonian fluids if the viscosity-temerature law for these fluids is given by Reynolds' formula. The Fig. 3 indicates that even slight dearture from Newtonian behavior laces the fluid in the regime where deendence on the velocity index is only slight. CONCLUSION Here we observed the flow of third grade fluid between heated arallel lates using HPM and PEM. It is clear from the figures that results obtained by HPM (Fig. ) and PEM (Fig. 3) are very close to the numerical results obtained by Szeri and Rajagoal (985). Thus, an imortant conclusion is that the dominant effect is governed by whether or not the fluid is non-newtonian, the temerature effects being relegated to a less dominant role. We conclude that from oint of view of velocity and temerature distribution in oiseuille flow, the temerature deendence is not imortant for the third grade fluid for the considered arameters and the variable viscosity solutions were not too distinct from that of constant velocity even if the fluid is slightly non-newtonian. REFERENCES Abbasbandi, S., 6. Alication of He's homotoy erturbation method for lalace transform. Chaos Soliton. Fract., 3(5): 6-. He, J.H., 998a. An aroximation solution technique deending uon an artificial arameter: A secial examle. Commun. Nonlinear Sci., 3(): He, J.H., 998b. Newton-like iteration method for solving algebraic equations. Commun. Nonlinear Sci., 3(): 6-9. He, J.H., 999. Homotoy erturbatin technique. Comut. Methods Al. Mech. Eng., 78(3-4): He, J.H.,. A couling method of a homotoy technique and ertubation technique for non-linear roblems. Int. J. Nonlinear Mech., 35(): He, J.H., 3. Homotoy erturbation method: A new nonlinear analytical technique. Al. Math. Comut., 35(): He, J.H., 4a. Asymtotology by homotoy erturbation method. Al. Math. Comut., 5(3): He, J.H., 4b. The homotoy erturbation method for nonlinear oscillators and discontinuities. Al. Math. Comut., 5(): He, J.H., 5a. Alication of homotoy erturbation method to nonlinear wave equations. Chaos Soliton. Fract., 6(3): He, J.H., 5b. Homotoy erturbation method for bifurcation of nonlinear roblems. Int. J. Nonlinear Sci., 6(): 7. Massoudi, M. and I. Christie, 995. Effects of varibale Dissiation on the flow of a third grade fluid in a ie. Int. J. Nonlinear Mech., 3(5): Shakil, M., T. Khan, H.A. Wahab and S. Bhatti, 3. A comarison of Adomian Decomosition Method (ADM) and Homotoy Perturbation Method (HPM) for nonlinear roblems. IMPACT: Int. J. Res. Al. Nat. Soc. Sci., (3): Siddiqui, A.M., M. Ahmed and Q.K. Ghori, 6a. Couette and Poiseuille flows for non-newtonian fluids. Int. J. Nonlinear Sci., 7(): 5. Siddiqui, A.M., R. Mahmood and Q.K. Ghori, 6b. Homotoy erturbation method for thin film flow of a fourth grade fluid down a vertical cylinder. Phys. Lett., 35(4-5): Siddiqui, A.M., M. Ahmed and Q.K. Ghori, 7. Thin film flow of non-newtonian fluids on a moving belt. Choas Soliton. Fract., 33(3): 6-6. Siddiqui, A.M., R. Mahmood and Q.K. Ghori, 8. Homotoy erturbation method for thin film flow of a third grade fluid down an inclined lane. Choas Soliton. Fract., 35(): Szeri, A.Z. and K.R. Rajagoal, 985. Low of non- Newtonian fluid between heated arallel lates. Int. J. Nonlinear Mech., (): 9-. Truesdell, C. and W. Noll, 965. The Non-linear Field Theories of Mechanics. nd Edn., In: Flugge (Ed.), Handbuch der Physik (German). Sringer-Verlag, Berlin. Tsai, C.Y, M. Novack and G. Roffle, 988. Rheological and heat transfer characteristics of flowing coalwater mixtures. Final Reort, DOE/ MC/ Wahab, H.A., M. Shakil, T. Khan, S. Bhatti and M. Naeem, 3a. A comarative stu of a system of Lotka-Voltera tye of PDEs through erturbation methods. Comutat. Ecol. Software, 3(4): -5. Wahab, H.A., T. Khan, M. Shakil, S. Bhatti and M. Naeem, 3b. Analytical treatment of system of KdV equations by Homotoy Perturbation Method (HPM) and Homotoy Analysis Method (HAM). Comut. Ecol. Software, 4(): Wahab, H.A., S. Bhatti, M. Naeem, M.T. Qureshi and M. Afzal, 3c. A mathematical model for the rods with heat generation and convective cooling. J. Basic Al. Scientific Res., ISSN: (Print), ISSN: 9-44x (Online), (Acceted). 434