study of boundary layer flow and heat transfer over an ...

Volume 1, No. 3, March 2013 Journal of Global Research in Mathematical Archives RESEARCH PAPER Available online at http://www.jgrma.info

STUDY OF BOUNDARY LAYER FLOW AND HEAT TRANSFER OVER AN AXISYMMETRICALLY SHRINKING SHEET WITH SUCTION C. Midya Department of Mathematics, Assistant Professor, Ghatal Rabindra Satabarsiki Mahavidyalaya, Paschim Medinipur, West Bengal, India E-mail: [email protected] Abstract:The boundary layer flow and heat transfer over a permeable sheet axisymmetrically shrinking with velocity inversely proportional to the radial distance, is investigated subject to suction at the surface. The suction at the sheet is assumed to be inversely proportional to the radial distance. The governing partial differential equations for boundary layer flow and heat transfer are reduced into ordinary differential equations (ODEs) by a similarity transformation. The reduced ODEs are then solved numerically by finite element method for power-law temperature boundary conditions. It is found that radial velocity is decreased with the increase in suction at the surface. It is also observed that the thermal boundary layer thickness decreases with the increase in suction parameter and Prandtl number. For some values of power law index, temperature overshoot is observed. Some solutions involving negative non-dimensional temperature values are also noticed.

Keywords: Axisymmetrically shrinking sheet, suction, Boundary layer flow, Heat transfer, Finite element solution INTRODUCTION The problem of boundary layer behaviors of the flow and heat transfer over a moving surface is important in view of its application in many engineering problems such as plasma studies, nuclear reactors, oil exploration, geothermal energy extractions, the boundary layer control in the field of aerodynamics and many more. Since the pioneering work of Sakiadis [1, 2], various aspects of the problem have been investigated by many authors. Wang [3] studied the problem concerning fluid flow underlying the axisymmetric spreading surface theoretically. Later, the heat transfer analysis of this problem was also investigated by Wang [4]. Lin and Chen [5] studied the effects of magnetic field on the momentum and energy transport for the spreading film over an axisymmetric surface. They assumed the spreading film velocity to be inversely proportional to the radial distance. Barakat [6] extended the problem of Lin and Chen [5] for variable viscosity. Ariel [7] investigated the slip effects on an axisymmetric flow over a radially stretching sheet and obtained exact and numerical solutions. Sahoo [8] studied the influence of a partial slip on the axisymmetric flow of an electrically conducting viscoelastic fluid over a linear stretching sheet. Gupta and Gupta [9] used homotopy perturbation method to solve axisymmetric flow over a stretching sheet with stretching velocity proportional to the nth power of the radial distance. They considered n to be positive integer. Recently, Shahzad, Ali and Khan [10] obtained exact solution for axisymmetric flow and heat transfer over a stretching porous sheet embedded with porous medium. They considered the sheet velocity proportional to the 3rd power of the radial distance. On the other hand, a little is known about the fluid flow and heat transfer over a shrinking sheet. In this type of flow, the fluid is stretched towards a slot and the flow is quite different from the stretching out case. The vorticity generated due to the shrinking sheet is not confined inside the boundary layer and consequently a situation appears where some other external force is needed to confine the vorticity inside the boundary layer so that the flow becomes steady. In confining the vorticity, the most suitable external force is the suction at the sheet. Recently, Miklavcic and Wang [11] studied one direction as well as radially shrinking axisymmetric hydrodynamic flow over a shrinking sheet with mass suction. The flow, heat and mass transfer over a sheet shrinking only in one direction was studied by many authors [12-17]. The problem of MHD viscous axisymmetric flow due to radially shrinking sheet has been solved by Sajid and Hayat [18] with the use of Homotopy Analysis Method (HAM). Muhaimin et al. [19] studied heat and mass transfer of MHD boundary layer flow over a shrinking sheet in the presence of suction. Noor et © JGRMA 2013, All Rights Reserved

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C. Midya et al, Journal of Global Research in Mathematical Archives, 1(3), March 2013, 70-77

al. [20] extended the work of Sajid and Hayat [18] by including the porosity effect and heat transfer. Ali et al. [21] studied the radiation effect on an unsteady flow and heat transfer over an axisymmetrically radially shrinking sheet. Although the flow and heat transfer over an axisymmetrically shrinking sheet inversely proportional to the radial distance is of immense importance in view of the practical use, no study has been reported so far in this connection. The aim of the present study is therefore to investigate the boundary layer flow and heat transfer over an axisymmetrically shrinking permeable sheet with shrinking velocity inversely proportional to the radial distance. The suction at the sheet is taken inversely proportional to the radial distance. Power law boundary conditions for temperature are considered here. With the use of a similarity transformation the governing partial differential equations are reduced into ordinary differential equations which are then solved numerically by finite element method. Fluid flow and temperature distributions are presented and discussed for various parameters. FORMULATION OF THE PROBLEM Let us consider a steady two-dimensional laminar flow of a viscous incompressible fluid over a continuously shrinking permeable sheet, z = 0, of the axisymmetric cylindrical polar coordinates (r, z). The sheet is shrinked along the radial direction with velocity inversely proportional to radial distance. We assume that suction velocity varies inversely proportional to radial distance. The governing boundary layer equations for momentum and energy can be written as (1) (2) (3) where u and w are the components of velocity respectively in the r and z directions, T is the temperature, κ is the thermal conductivity, cp is the specific heat, is the fluid density (assumed constant), T∞ is the temperature far from the sheet, (= μ/ ) is the coefficient of fluid viscosity. The boundary conditions for the velocity components and temperature are given by (4) and (5) where Tw is the wall temperature. The interesting physical quantities of the problem are the local skin friction coefficient and Nusselt number. The shear stress at the wall denoted by τw is defined as

The skin friction coefficient Cf at the wall is obtained as

The local heat transfer on the surface is

Consequently, the Nusselt number is

SOLUTION OF THE PROBLEM Equations (1) - (3) admit self-similar solutions of the form (6) where f and θ are the dimensionless stream function and dimensionless temperature and these, Eqs. (2) and (3) become © JGRMA 2013, All Rights Reserved

is the similarity variable. Substituting 71


(7) (8) where Pr = μcp / κ is the Prandtl number. The boundary conditions then reduce to (9) where s = w0/

(10) is a non-dimensional constant corresponding to suction.

We adopt finite element method to solve the above equations. This method has the following key steps: (i) Division of the domain into linear elements, called the finite element mesh, (ii) Generation of the element equations using variational formulations, (iii) Assembly of element equations as obtained in step (ii), (iv) Imposition of the boundary conditions to the equations obtained in (iii), (v) Solution of the assembled algebraic equations. The assembled equations can be solved by any numerical techniques. The details are available in Ref. [22]. In order to solve the differential equations (7)-(8) with the boundary conditions (9) and (10), we assume

. The equations then become

(11) (12) The boundary conditions are (13) (14) The variational form associated with equations (11)-(12) over two nodded linear elements ( e,

e+1)

is given by

(15) (16) (17) where W1, W2 and W3 are arbitrary test functions and may be viewed as the variations in f, h and θ respectively. As the domain is defined into two nodded elements, finite element approximation is taken of the form (18) with W1= W2 = W3 = ξi where ξi are the shape function for element ( e, e+1) and are defined as (19) The finite element model of the equations (15)-(17) is given by (20) where (21) (22) (23) (24) (25) and (26)

© JGRMA 2013, All Rights Reserved

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After assembly of elements, a system of nonlinear equations is obtained, which is solved iteratively, and the required accuracy of 0.5 x 10-4 is obtained.

RESULTS AND DISCUSSION First of all we should verify the accuracy of the finite element numerical scheme used here. Actually, there are no results for flow and heat transfer over an axisymmetrically shrinking sheet when sheet velocity is inversely proportional to the radial distance. So in order to verify the accuracy of our computed results, we have calculated the results for axisymmetrically stretching surface case when Pr = 1 and s = 0 and have compared the results with those obtained by Lin and Chen [5]. They obtained θ/(0) = -0.70711 for p = 0 and θ/(0) = -1.16832 for p = 1 by the use of shooting method whereas our numerical scheme results θ/(0) = -0.707124949 for p = 0 and θ/(0) = -1.16840196 for p =1. Thus we see that our numerical scheme is good enough to predicts those results. Now we use the scheme for shrinking sheet problem. The variation of dimensionless stream function f for several values of s are shown on Figure 1(a) when Prandtl number Pr = 1 and power law index p = 1. The graph shows that with the increase in suction parameter s, nondimensional stream function f increases. Now we draw our attention for the velocity profile for different values of suction parameter s. The effect of suction on the velocity f/( ) is depicted in Figure 1(b). From the figure, it is noticed that with increase of M, velocity decreases for a fixed value of . Actually, suction supresses the fluid motion and producing more resistance to the velocity field. The velocity vanishes at large distances from the sheet for all values of s. The effects of suction s on temperature θ( ) have been displayed in Figure 1(c) for Pr = 1 and p = 1. The figure shows that the temperature decreases with the increase in suction parameter s indicating the usual fact that suction stabilizes the boundary layer growth. Thus sucking the decelerated fluid particles reduces the growth of the fluid as well as thermal boundary layers. Figure 2 depicts the influence of Prandtl number Pr on temperature profile for fixed values of s = 1.5 and power index p = 2. The graph displays that the increase in Prandtl number Pr results in the decrease of temperature distribution at a particular point of the flow region. This is because there would be a decrease of thermal boundary layer thickness with the increase in values of Prandtl number. We now draw our attention to the effects of power index p on the temperature distribution. Figure 3 is the graphical representation of the temperature profile for various values of power law exponent p when s = 1.5 and p = 0. It is noted from the figure that the rate of heat transfer as well as temperature boundary layer thickness increases for power law index p = 0, 1, 2. For p = 3 and 4, the non-dimensional θ( ) becomes negative after a certain distance from the surface. This fact may be explained as the fluid temperature T near the shrinking sheet becomes lower compared to the fluid temperature T∞ far from the shrinking sheet and as a result θ( ) which is equal to (T- T∞)/crp becomes negative. At a long distance from the plate, T ⟶ T∞ and hence θ( ) ⟶ 0. The skin friction coefficient f//(0) and Nusselt number - θ//(0) are calculated for various values of suction parameter s and are displayed in Table 1. From the table we see that skin friction coefficient as well as Nusselt number increase with the increase in suction parameter s for fixed values of Pr = 1 and p = 1. Table 2 presents the Nusselt numbers for various values of Prandtl number Pr when s = 1.5 and p =1. It is seen that with the increase in Prandtl number, the Nusselt number increases. Next, variation of Nusselt number for different values of power-law index p is shown in Table 3 for fixed values of Pr = 1 and s = 1.5. It is seen from the Table 3, that for increasing values of power-law index p, the Nusselt number is decreased. Table 1. The values of f//(0) and - θ//(0) for different values of suction parameter s when Pr = 1 and p = 1. s f//(0) - θ/(0) 1.5 1.50124931 0.135719568 2.0 2.00166821 1.37457526 2.5 2.50208163 2.04660106


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Table 2. The values of -θ/(0) for different values of Prandtl number Pr when s = 1.5, p = 1. Pr - θ/(0) 0.5 -0.132282332 1.0 0.135719568 2.0 0.732348263

Table 3. The values of -θ/(0) for different values of power-law index p when Pr = 1, s = 1.5 p - θ/(0) 0 1.00186992 1 0.135719568 2 -3.58101606

CONCLUSION In conclusion, boundary layer flow and heat transfer over an axisymmetrically shrinking surface subject to suction at the surface is investigated. The shrinking velocities as well as suction at the sheet are considered to be inversely proportional to the radial distance. Power-law temperature boundary conditions are assumed at the surface. Finite element numerical method is used to solve the ordinary differential equations obtained from the governing boundary layer equations by the use of similarity transformations. The radial velocity is seen to decrease with the increase in suction parameter. The temperature is found to decrease with the increasing Prandtl number and suction parameter. On the other hand, it is increased for increasing power law index (p = 0, 1, 2) for temperature. For higher power law index (p = 3, 4) values, the non-dimensional temperature becomes negative i.e. temperature near the surface becomes lower compared to the temperature far from the sheet. REFERENCES [1] B. D. Sakiadis : Boundary-layer behavior on continuous solid surface:I.The Boundary-Layer equations for two-dimensional and asymmetric flow, AIChE J, 7 (1961) 26-28. [2] B. C. Sakiadis: Boundary-layer behavior on continuous solid surface:II.The Boundary-Layer on a continuous flat surface, AIChE J, 7 (1961) 221-225. [3] C. Y. Wang: Effect of spreading of material on the surface of a fluid - an exact solution, Int. J. Non-Linear Mech. 6 (1971) 255-262. [4] C. Y. Wang: Heat transfer to an underlying fluid due to the axisymmetric spreading of material on the surface, Appl. Sci. Res., 45 (1988) 367. [5] C. R. Lin, C. K. Chen : Hydromagnetic flow and heat transfer about a fluid underlying the axisymmetric spreading surface, Int. J. Engg. Sci., 31 (1993) 257-261. [6] E. I. I. Barakat : Variable viscosity effect on hydromagnetic flow and heat transfer about a fluid underlying the axisymmetric spreading surface, 169 (2004) 195-202. [7] P. D. Ariel : Axisymmetric flow due to a stretching sheet with partial slip. Comput. Math. Appl. 54(7-8) (2007) 1169-1183. [8] B. Sahoo : Effects of partial slip on axisymmetric flow of an electrically conducting viscoelastic fluid past a stretching sheet. Cent. Eur. J. Phys. (2009) DOI 10.2478/s11534-009-0105-x


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[9] V.G. Gupta and S. Gupta : The Application of Homotopy Perturbation Method for Axisymmetric Flow over a Nonlinearly Stretching Sheet Journal of Applied Mathematics and Fluid Mechanics, 3(2) (2011) 209-219. [10] A. Shahzad, R. Ali, M. Khan : On the Exact Solution for Axisymmetric Flow and Heat Transfer over a Nonlinear Radially Stretching Sheet, Chin. Phys. Lett. 29(8) (2012) 084705. [11] M. Miklavcic, C. Y. Wang : Viscous flow due to a shrinking sheet. Quart Appl. Math. 64(2) (2006) 283-290. [12] T. Fang, J. Zhang : Closed form exact solutions of MHD viscous flow over a shrinking sheet. Communications in Nonlinear Science and Numerical Simulation. 14(7) (2009) 2853-2857. [13] T. Fang, J. Zhang : Thermal boundary layer over a shrinking sheet : an analytical solution. Acta Mech. 209 (2010) 325-343. [14] T. Fang, J. Zhang : Viscous flow over an unsteady shrinking sheet with mass transfer, Chin. Phys. Lett., 26(1) (2009) 014703-1-4. [15] S. Nadeem, and A. Hussain : MHD flow of a viscous fluid on a nonlinear porous shrinking sheet with homotopy analysis method. Appl. Math. Mech. (Engl. Ed.) 30(12) (2009) 1569-1578. [16] C. Midya : Exact solutions of chemically reactive solute distribution in MHD boundary layer flow over a shrinking surface. Chin. Phys. Lett. 29(1) (2012) 014701-1-4. [17] C. Midya : Hydromagnetic boundary layer flow and heat transfer over a linearly shrinking permeable surface. Int. J. Appl. Math. Mech. 8(3) (2012) 57-68. [18] M. Sajid, T. Hayat : The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet. Chaos, Solitons and Fractals. 39(3) (2009) 1317-1323. [19] Muhaimin, R. Kandasamy, I. Hashim : Effect of chemical reaction, heat and mass transfer on nonlinear boundary layer past a porous shrinking sheet in the presence of suction. Nuclear Engineering and Design. 240(5) (2010) 933-939. [20] N. F. M. Noor, S. A. Kechilb, I. Hashimc : Simple non-perturbative solution for MHD viscous flow due to a shrinking sheet. Communications in Nonlinear Science and Numerical Simulation. 15(2) (2010) 144-148. [21] F. M. Ali, R. Nazar, N. M. Arifin, and I. Pop : Unsteady flow and heat transfer past an axisymmetric permeable shrinking sheet with radiation effect. Int. J. Numer. Method in Fluids, (2010), DOI: 10.1002/fld.2435. [22] J. N. Reddy, An introduction to finite element method, Tata McGraw Hill, second edition, 2003.


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Figure 1(a) Variation of f for several values of s with Pr = 1, and p = 1.

Figure 1(b) Effects of suction s on the velocity f/( ) when Pr = 1 and p = 1.

Figure 1(c) The temperature profiles θ( ) for various values of s when Pr = 1 and p = 1.


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Figure 2 The effects of Prandtl number Pr on the non-dimensional temperature θ( ) when s = 1.5 and p = 0.

Figure 3 Variation of non-dimensional temperature θ( ) for several values of p with s = 1.5 and Pr = 1.


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