Jurnal Teknologi Full paper Unsteady Flow of a Second Grade Fluid between Two Oscillating Vertical Plates Aaiza Gul Taza Gul b Ilyas Khan c Saeed Islam b Sharidan Shafie a * a Department of Mathematical Sciences Faculty of Science Universiti Teknologi Malaysia 81310 UTM Johor Bahru Johor Malaysia b Mathematics Department Abdul Wali K.U Mardan KPK Pakistan. c College of Engineering Majmaah University P.O. Box 66 Majmaah 11952 Saudi Arabia. *Corresponding author: sharidan@utm.my Article history Received XXXX Received in revised form XXXX Accepted XXXX Graphical abstract Abstract This article deals with the study of unsteady flow of a second grade fluid between two oscillating vertical plates. Three different flow situations are discussed. The problem is modeled in terms of non-linear partial differential equation with some physical conditions. Exact analytic solutions are obtained by using the Optimal Homotopy Asymptotic Method (OHAM).This method is frequently used for solving nonlinear differential equations arise in various applied sciences and is found quite useful. The physical influence of various parameters on velocity is studied graphically and discussed. Keywords: Unsteady flow; Second grade fluid; Oscillating plates; OHAM Abstrak Artikel ini memperkatakan kajian aliran tak mantap cecair gred kedua di antara dua plat menegak berayun. Tiga keadaan aliran yang berbeza dibincangkan. Masalahnya dimodelkan dari segi persamaan pembezaan separa bukan linear dengan beberapa keadaan fizikal. Penyelesaian analitik yang tepat diperolehi dengan menggunakan Optimum homotopi asimptot Kaedah ( OHAM ) kaedah wabak sekiranya kerap digunakan untuk menyelesaikan persamaan pembezaan linear bukan timbul dalam pelbagai ilmu gunaan dan didapati agak berguna. Pengaruh fizik bagi pelbagai parameter pada halaju dikaji secara grafik dan dibincangkan. Kata kunci: Aliran tak mantap ; Cecair gred dua ; Plat berayun ; OHAM 2012 Penerbit UTM Press. All rights reserved. 1.0 INTRODUCTION The subject of non-newtonian fluids is popular and is an area of active research especially in mathematics industry and engineering problems. Examples of non-newtonian fluids include plastic manufacturing food processing movement of biological fluids wire and fiber coating paper production gaseous diffusion transpiration cooling drilling mud heat pipes etc. Several complex fluids such as polymer melts paint shampoo mud ketchup blood certain oils and greases and many emulsions are involved in the class of non-newtonian fluids. These fluids are described by a non-linear relationship between stress and the rate of deformation tensors and therefore several models have been proposed. Hence a number of fluid models have been suggested to predict the non-newtonian performance of different kinds of materials. Taza Gul et al. 1 investigated the unsteady flow of second grade fluid on a vertical and oscillating belt. Amongst these fluid models one of the very important models suggested for non-newtonian fluids is called the second- grade fluid. The unsteady thin film flow of non-newtonian fluid over a moving belt was discussed by Nemati et al 2. The unsteady Boundary Layer Flow of a Second Grade Fluid over a Stretching Sheet was investigated by Ahmed 3. For the solution differential equation HAM method are used and the effects of the physical parameters are discussed through graphs. The unsteady shearing flow of a second grade fluid between two horizontal parallel plates was observed by Chauhan et al. 4. For the solution of the problem they applied the Laplace transform method. Abbas et al. 5 discussed unsteady thin film flow of second grade fluid through stretching surface. Kumari et al 6 investigated Effects of magnetic field and thermal in Stokes problem for unsteady second grade fluid flow. The problem was solved analytically for velocity field and the temperature field. Fetecau et al 78 investigated the second grade fluid between the oscillating plates for unsteady unidirectional flow. Mahmood et al. 910 was investigated the flow of second grade fluid between two coaxial circular cylinders. Ali et al. 11 used Laplace transformation for the solution of second order fluid between two plates. The effects of different model
1213 parameters have been discussed graphically. Gul et al. investigated Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with Temperature Dependent Viscosity. For the analytical solution of problem they used the ADM and OHAM techniques. Haroon et al. 14 investigated the three different type s unsteady flows of third grade fluid between two parallel plates. The unsteady flow of Second Grade Fluid between Two Vertical Plates with MHD was investigated by Sidra et al 15. Aamer et al 16 investigated Thin Film Non-Newtonian Fluid on a Porous and Lubricating Vertical Belt. They used OHAM method for the analytical solution of the problem. Iqbal et al. 17 solved the linear and nonlinear Klein Gordon equation using HPM. Marinca et al 18 discussed OHAM method for the solution of problem. He 1920 used OHAM method for the solution non linear differential equation. Wahab et al. 21 applying the HPM 22 method on KdV equation of orders four. Elbeleze et al. calculated the solution of fractional partial differential equation using HPM method. 2.0 BASIC EQUATIONS The governing equation of the unsteady incompressible flow of second grade fluid are (1) (2) Where is the fluid density is the velocity vector of the fluid is gravity is the caushy stress tensor and the material time derivative define as The field of velocity is of the form Oscillating boundary conditions are: ω denote frequency of the oscillating belt. (7) The continuity equation is satisfied identically the momentum equation reduces to the form (8) (9) The components of the Cauchy stress tensor T as (10) Putting equation (10) in equation (8) and (9) we get (11) (12) Introducing the following non-dimensional physical quantities (13) (3) Where the non-dimensional variable. The cauchy stress tensor T for second grade fluid is given by (4) Using the above dimensionless variables in equation (13) and dropping bars we obtain Here are the material constants are the Rivlin- Ericksen tensors given by boundary conditions (14) 3.0 FIRST PROBLEM (5) (6) 3.1 Formulation of problem when one plate moving upward and oscillating Consider two vertical and parallel plates such that one of the plates oscillating and moving upward with constant velocity and other is stationary. is the total thickness of fluid between two plates. Moving and oscillating plate caries with itself a liquid of thickness. The coordinate system is selected as in which the x-axis is taken vertical and y-axis parallel to the plates. 4.0 OPTIMAL HOMOTOPY (OHAM) 4.1 Basic Idea of OHAM (15) ASYMTOPIC METHOD Here we discuss the concept of OHAM; we consider the boundary value problem as consider in [18]: u x u( x) Nux gx 0 B u 0 L Where L is a linear operator in the differential equation N is a non-linear term is an independent variable B is a boundary (16)
operator and g is a source term. According to [18] we construct a set of equation for OHAM. p L x p g x H p L x p g x N x p xp B x p 0 1 x (17) is an embedding parameter H(p) is a non-zero auxiliary function for and. Is an un known function. Obviously when it holds that: x 0 u0 x x 1 u x (18) When varies from then also varies from. Where the zero component solution is obtained from equation (18) when L u x gx B u x x 0 0 0 0 Auxiliary function is choosing as u x 0 (19) 2 H p pc1 p c2 (20) are auxiliary constants. The general governing equations for are given by k 1 i1 1 0 0 L u x L u x c N u x k k k c i L uk i x Nk 1 u0 x u x u x 1 ki x uk k 23 Buk x 0 x Here is the coefficient of in the expansion of. i 0 0 N x p c N u x m1 m N u x u x u x p 0 1 The convergence of the Series in equation (21) depend upon the auxiliary constants If it converges at then the mth order approximation is u m i m (24) (25) m x c c c u x ux c c c 1 2 m 0 1 1 2 Inserting Eq. (26) into Eq. (16) the residual is obtained as: i (26) Marinca [18] uses a special procedure to expand with respect to by using Taylor Series. i i i R x c L u x c g x N u x c i 12 m (27) x p c u x u x c p k i 12 i 0 k1 k i Inserting Eq. (21) into Eq. (17) collecting the same powers of and equating each coefficient of. The zero order problems given in equation (19) and the first order and second order given in equations (22 23). 1 1 0 0 u1 x x 0 L u x g x c N u x Bu 1 x 2 x x 2 0 (21) (22) L u2 x L u1 x c2n0 u0 x c1 L u1 x N1 u0 x u x 1 Bu u x (23) Numerous methods like Ritz Method Method of Least Squares Galerkin s Method and Collocation Method are used to find the optimal values of We apply the Method of Least Squares in our problem as given below: J c c c x c c c b 1 2 n 1 2 m dx (28) a Where and are the constant values taking from domain of the problem. Auxiliary constants can be identified from: J c 1 J c 2 0. (29) Finally from these auxiliary constants the approximate solution is well-determined.
5.0 THE OHAM SOLUTION In this unit we apply OHAM method on equation (14) with boundary condition in equation (15) and study zero first and second component problems. Zero and first component problem velocity and temperature profiles are (30) (31) (32) Solutions of zero first and second components problem using boundary conditions equation (15) in equations (30-32) are given by: (33) (34) Now we apply OHAM method on equation (40) with boundary condition in equation (41) and study zero first and second component problems Zero and first component problem velocity and temperature profiles are (42) (43) (44) Solutions of zero first and second components problem using boundary conditions from equation (41) in equations (42-44) are given by: (45) (46) (35) The solution of second component of temperature distribution is too large. So derivation are given up to first order while graphical solutions are given up to second order. Putting equations (33 34 35) in equation (36) we have (36) (47) The solution of second component of temperature distribution is too large. So derivation are given up to first order while graphical solutions are given up to second order. (48) Putting equations (45 46 47) in equation (48) we have (37) 6.0 SECOND PROBLEM 6.1 OHAM Solution of the flow in the presence of pressure gradient. In this case both of the plates is stationary. The flow is under the effect of pressure gradient in the presence of gravity. (38) Boundary conditions (39) Using the dimensionless parameters define in equation (13) the momentum equations become Boundary conditions (40) (41) 7.0 THIRD PROBLEM (49) 7.1 OHAM Solution In this case one of the plates is moving as well as oscillating. The flow is under the combined effect of pressure gradient and gravity. Boundary conditions (50) (51) Using the dimensionless parameters define in equation (13) the momentum equations become Boundary conditions (52) (53)
Now we apply OHAM method on equation (51) with boundary condition in equation (53) and study zero first and second component problems Figure 2: Effect of the pressure parameter on first problem velocity profile when Solutions of zero first and second components problem using boundary conditions in equations (53) (54) (55) (56) The solution of second component of temperature distribution is too large. So derivation are given up to first order while graphical solutions are given up to second order. (57) Figure 3: Effect of the time t on second problem velocity profile when Putting equations (54 55 56) in equation (57) we have (58) Figure 4: Effect of the pressure profile when on second problem velocity Figure 1: Effect of the time t on first problem velocity profile when Fig5: Effect of pressure gradient profile when on second problem velocity
9.0 CONCLUSION Figure 6: Effect of the time t on third problem velocity profile when The main aim of the presence article is to derive an approximate solution for the problem of second order fluid flow between two vertical and parallel plates in the existence of thermal effect. In each of the three problems (i) when one belt is moving and other as stationary (ii) when the belts are stationary and the flow is due to the pressure gradient (iii) the flow is due to combined effect of pressure gradient and when both belt is in motion the solution for velocity and temperature distributions are derived using Optimal Homotopy Asymptotic Method (OHAM). Effects of various model parameters t ω Ω on velocity and temperature distribution are also presented through graphs. References (1) Gul T Islam S Shah RA Khan I Khalid A et al. (2014) Heat Transfer Analysis of MHD Thin Film Flow of an Unsteady Second Grade Fluid Past a Vertical Oscillating Belt. PLoS ONE 9(11): e103843. doi:10.1371/journal. pone. 0103843 (2) Nemati H. Ghanbarpou M. Hajibabayi M. and Hemmatnezhad (2009) Thin film flow of non-newtonian fluids on a vertical moving belt using Homotopy Analysis Method Journal of Engineering Science and Technology Review 2 (1): 118-122 (3) Ahmad I.(2013) On Unsteady Boundary Layer Flow of a Second Grade Fluid over a Stretching Sheet Adv. Theor. Appl. Mech. 6(2): 95 105 Figure 7: Effect of when 8.0 RESULTS AND DISCUSSION on the third problem velocity profile In the present research paper the Optimal Homotopy Asymptotic Method (OHAM) used for the analytical solutions of heat transfer and velocity field of unsteady second grade fluid between two vertical plates for appropriate boundary conditions. The effects of several physical parameters on three different velocity and temperature problems have been deliberated graphically. Fig 1-7 shows the effect of physical parameters tωω on heat and velocity field of first second and third problem respectively. Figure 1-2 are plotted to indicate time and pressure for first problem. Figure 1indicates the profile of time t for velocity filed of first problem. From figure 1 we investigated that velocity decrease when the values of t increasing. The effect of pressure parameter ω as shown in figure 2 the velocity decreases when the value of ω increasing. Figure 3-5 are plotted to examine the effect of tωωfor time pressure and pressure gradient for second problem. Increasing values of t shows decrease in the velocity profile is seen figure 3. In Figure 4 we see that there as decrease in velocity as we increase the value of ω. In figure 5 we observed that as we increase the value of Ω there as decrease in velocity profile. Figure 6-7 displays the effect of tω for time and pressure discussed for third problem. Here we observed from figure 6 that velocity decreases when the value of t increases. Figure 7 are plotted to examine the effect of ω pressure distribution. We see that the velocity decreasing by increasing the values of ω. (4) Chauhan D. S. and Kumar V. (2012) Unsteady flow of a non- Newtonian second grade fluid in a channel partiallyfilled by a porous medium Advances in Applied Science Research 3 (1): 75-94 (5) Abbas Z. Hayat T. Sajid M. and Asghar S. (2008) Unsteady flow of a second grade fluid film over an unsteady stretching sheet Mathematical and Computer Modeling 48: 518 526 (6) Kumari B. A. and Parsad K. P. (2014) Effects of magnetic field and thermal in Stokes second problem for unsteady second grade fluid flow International Journal of Conceptions on Computing and Information Technology 2(1): 2345 9808 (7) FetecauC. and FetecauC. (2005)Starting solutions for some unsteady unidirectional flows of a second grade fluid International Journal of Engineering Science 43(10): 781 789 (8) Fetecau C. Hayat T. Fetecau C. and Ali N. (2008) Unsteady flow of a second grade fluid between two side walls perpendicular to a plate Nonlinear Analysis Real World Applications 9: 1236-1252. (9) Mahmood A. Khan N. A. Fetecau C. Jamil M. and Rubbab Q.(2009) exact analytic solutions for the flow of second grade fluid between two longitudinally oscillating cylinders Journal of Prime Research in Mathematics 5: 192-20. (10) M.ahmood A. Parveen S. and Khan N. A. (2011) Exact solutions for the flow of second grade fluid in annulus between torsionally oscillating cylinders Acta Mech. Sin. 27(2): 222 227 (11) Ali F Khan I Shafie S (2014) Closed Form Solutions for Unsteady Free Convection Flow of a Second Grade Fluid over an Oscillating Vertical Plate. PLoSONE 9(2): e85099. doi:10.1371/journal. pone.0085099 (12) Gul T Islam S Shah RA Khan I Shafie S (2014) Thin Film Flow in MHD Third Grade Fluid on a Vertical Belt with
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