Study the Impact of Soret and Sherwood Number on stretching sheet using Homotopy Analysis Method

International Journal of Emerging Technology and Advanced Engineering

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Study the Impact of Soret and Sherwood Number on stretching

sheet using Homotopy Analysis Method

Anoop Kumar

, Phool Singh

, N S Tomer

and Deepa Sinha

GITM, Bilaspur, Gurgaon, Haryana, India.

2_{ITM University, Gurgaon, Haryana, India.} 3_{GCFGM, Adampur, Hisar, Haryana, India.} 4_{South Asian University, New Delhi, India.}

Abstract—The aim of this paper is to study the impact of Soret and Sherwood number on stretching sheet in presence of magnetic field in a porous medium. Using stream function, partial differential equations corresponding to the momentum energy and mass transfer are converted into non-linear coupled ordinary differential equations. Approximate solutions of these equations are obtained by using Homotopy analysis method. The accuracy of the present results is shown by giving a comparison between the present results and the results already published in the literature. Effects of Sherwood and Soret numbers on temperature and concentration distribution are discussed through tables and graphs.

Keywords—Soret number, Sherwood number, Heat flux, Magneto hydrodynamics, stretching sheet.

I. INTRODUCTION

Flow of incompressible viscous fluid over a stretching surface is a classical problem in fluid dynamics and important in various process. The production of sheeting material, which includes both metal and polymer, arises in a number of industrial manufacturing processes. It is used to create polymers of fixed cross-sectional profiles, analyzing cooling of metallic and glass plates. Aerodynamic shaping of plastic sheet by forcing through die and boundary layer along a liquid film in condensation processes are among the other area of applications. Extruded polymer from a die with some prescribed velocity may become sometime stretched. The stretching surface experience cooling or heating, this may cause variation in surface velocity and temperature.

Above stated issue attracted many researchers due to its applications in many areas. Crane [1] reported an exact solution for flat stretching sheet. He assumed that stretching sheet velocity varies linearly along the distance from a fixed point due to uniform stresses.

The Crane’s pioneering work is subsequently extended by many researchers (like Mahapatra and Gupta [2], Attia [3], Singh [4]-[11] to explore numerous aspects of the flow and heat transfer happening in an infinite domain of the fluid surrounding the stretching sheets.

Fourier’s law gives the relation between energy flux and temperature gradient, and Fick’s law pronounces the correlation between mass flux and concentration gradient. The mass flux could also be created by temperature gradient and this is Soret or thermal-diffusion effect. Takhar et al. [12] and Pal [13] investigated mass flux and concentration gradient on stretching surface.

Homotopy analysis method (HAM) using the idea of homotopy, a branch of Topology, was developed by Liao[14]. HAM is one of the powerful analytical techniques to solve highly nonlinear problems in science and engineering, Lioa[15]. HAM has attracted special interest of researchers as it is flexible in applying and give sufficiently accurate results with modest effort. HAM is based upon the introduction of a homotopy parameter which takes the values from 0 to 1. At , the problem under study take a simple form which possesses an analytical solution. As p is increasing from the zero to one, we obtain the solution of the original problem. The HAM also depends on other auxiliary parameters, to bring out an optimum solution.

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In this paper, steady forced convective heat and mass transfer flow of viscous, incompressible and electrically conducting fluid past a stretching surface is considered as study problem. This problem is extension of Singh et al [5].

II. FORMULATION OF THE PROBLEM

Consider the steady forced convective heat and mass transfer flow of viscous, incompressible and electrically conducting fluid past a stretching surface coinciding with the plane as shown in Fig. 1. The fluid occupies the

upper half plane i.e. Keeping the origin fixed, two

equal and opposite forces are applied along the axis, which results in stretching of the sheet. Fluid is in a porous medium in the presence of transverse magnetic field and volumetric rate of heat generation/ absorption having radiation effect. The uniform transverse magnetic field is imposed along the axis. The induced magnetic field is neglected as the magnetic Reynolds number of the flow has been very small. It is also assumed that the external electric field is zero and the electric field due to polarization of charges is negligible. Temperature and concentration of the fluid are and , and those at the stretching surface are and , respectively. It is also assumed that pressure gradient, viscous and electrical dissipation are neglected. Other fluid properties are assumed to be constant. Under the above assumptions, the governing equations of the flow can be written as

FIGURE.1PHYSICAL MODEL OF THE PROBLEM

(1)

(2)

(3)

and

(4)

Where and are velocity components along and axes respectively, is the pressure of fluid, is kinematic viscosity, is electrical conductivity, is the permeability of the medium, is the temperature, is density of the fluid, is thermal conductivity, is specific heat at constant pressure, is the mean fluid temperature, is the concentration, is the coefficient of mass diffusivity and is the thermal-diffusion ratio.

Using free stream in equation (2), we

have

(5)

Eliminating from equation (2) and using

equation (5), we get

(6)

Boundary conditions for the given model are

(7)

Where and are positive constants with dimension (time)-1.

We introduce the stream function as defined

by and , and the similarity

variable and

into the equations (3), (4) and (9), we get the following relations

(8)

(9)

and

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Where primes denote differentiation with respect to is the dimensionless stream function,

is the magnetic parameter, is the porosity

parameter, is the stretching parameter,

is the dimensionless

temperature, is the heat source / sink

parameter, is the radiation parameter

where is approximated by Rosseland approximation (as

described in Singh et al. (2011)), is the

Prandtl number, is the

dimensionless concentration,

is the Soret number

and is Schmidt number.

The corresponding boundary conditions are

(11)

The physical quantities of interest in this problem are local skin friction, Nusselt number, Sherwood number and and Soret number. It is worth menstioning here that the effect of skin friction and Nusselt number has been reported by Singh et al.[11] The rate of mass transfer in terms of the Sherwood number at the stretching sheets is

given by, , where

. Therefore

To solve equations (8-10), we follow Homotopy analysis method (HAM). For homotopy analysis method solutions, we choose the initial guesses

and auxiliary linear operators

such that

where are constants.

We construct the Zeroth-order deformation problems as

Where

denotes the embedding parameter and indicate the non-zero auxiliary convergence parameter.

For and we have

Due to Taylor’s series expansion with respect to p, we have

Where

mth –order deformation problems

Where

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632

is chosen in such a manner that these three series are

convergent at therefore we have

As indicated by Liao, the convergence rate of approximation for the homotopy analysis method solution

strongly depend on the value of These

parameter play a important role in controlling the convergence of the series solution. The range for the

admissible value of are

, . Our calculations

further shows the series converges for

.

III. RESULTS AND DISCUSSIONS

Homotopy analysis method (HAM) is used to solve equations (8), (9) and (10) for different values of and We have compare our results with Pop et al. [16] and Mahapatra and Gupta [2] for different values of stretching parameter as shown in Table 1.

TABLE I

COMPARISON OF LOCAL SKIN FRICTION FOR DIFFERENT VALUES OF

Value of

Pop et al. [16]

Mahapatra and Gupta [2]

Present Result

0.1

0.2

0.5

2.0

3.0

TABLE II

VALUES OF AND FOR DIFFERENT VALUES OF

WHEN AND

TABLE III

VALUES OF AND FOR DIFFERENT VALUES OF

WHEN AND

TABLE IV

VALUES OF AND FOR DIFFERENT VALUES OF ,

WHEN AND

TABLE V

VALUES OF AND FOR DIFFERENT VALUES OF ,

WHEN AND

TABLE VI

VALUES OF AND FOR DIFFERENT VALUES OF ,

International Journal of Emerging Technology and Advanced Engineering

633 TABLE VII

VALUES OF AND FOR DIFFERENT VALUES OF ,

WHEN AND

TABLE VIII

VALUES OF AND FOR DIFFERENT VALUES OF ,

WHEN AND

TABLE IX

VALUES OF AND FOR DIFFERENT VALUES OF ,

WHEN AND

It is observed from the Table 2 that Sherwood number decreases with increase in magnetic parameter for

whereas Sherwood number increase when as shown

in Table 3. This is due to the inverted boundary layer formulated for Table 4 and 5 show that Sherwood number increases with increase in radiation parameter for any value of stretching parameter.

It is noticed from Table 6 and 7 that Sherwood number decreases with increase in Prandtl number for any value of stretching parameter. This shows that at higher Prandtl fluid has comparatively lower species concentration.

Effects of heat generation and absorption

on Nusselt and Sherwood number are illustrated with Table

8 and 9 for and respectively.

It is observed that Nusselt number decreases with , whereas Sherwood number increases as increase. This happens because, as value of increases temperature of the fluid decrease, whereas the concentration of species increase.

The concentration distribution of the flow field in presence of foreign speices, such as water vapour is shown in Figure (2-5). It is affected by two parameters namely, Soret number and Schmidt number . In Figure (2) and (3), it is observed that concentration distribution is vastly affected by Soret number in flow field. A comparative study of the curves of Figures (2) and (3) show that the concentration distribution of the flow field increases as become larger. Thus greater Soret number leads to a faster increase in concentration of flow field. On the contrary, the thickness of the concentration boundary layer reduce with an increase of Schmidt number as shown in Figure (4) and (5). Equation (11) and (12) are independent of Soret number and Schmidt number.

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1

Sr = 0.2

Sr = 1

Sr = 2





FIGURE 2CONCENTRATION VERSUS SIMILARITY VARIABLE AT DIFFERENT VALUES OF TAKING

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0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1

Sr = 0.2

Sr = 1

Sr = 2





FIGURE 3CONCENTRATION VERSUS SIMILARITY VARIABLE AT DIFFERENT VALUES OF TAKING

AND .

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1

Sc = 0.5 Sc = 1 Sc = 1.5 Sc = 2

 

FIGURE 4CONCENTRATION VERSUS SIMILARITY VARIABLE AT DIFFERENT VALUES OF TAKING

AND .

0 0.5 1 1.5 2 2.5 3

0 0.2 0.4 0.6 0.8 1

Sc = 0.5

Sc = 1

Sc = 1.5

Sc = 2





FIGURE 5CONCENTRATION VERSUS SIMILARITY VARIABLE AT DIFFERENT VALUES OF TAKING

AND

IV. CONCLUSION

In this paper, steady two dimensional flow of a viscous incompressible electrically conducting fluid in the vicinity of a stagnation point on a stretching sheet with free stream velocity is studied. The results indicate that

1. Sherwood number decreases with increase in

magnetic paramter for stretching parameter less than one, whereas it increases when stretching parameter greater than one.

2. Sherwood number increases with increase in radiation parameter, whereas it decreases with increase in Prandtl number for any value of stretching parameter. 3. Nusselt number decreases, whereas Sherwood number

increases as heat generation/ absorption parameter increase.

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REFERENCES

[1] Crane L.J., 1970. ―Flow past a stretching plate", J. Appl. Math. Phys. (ZAMP), 21, 645–647.

[2] Gupta, A. S. and T. R.Mahapatra, T. R. 2003. ―Stagnation-point flow towards a stretching surface ", The Canadian Journal of Chemical Engineering, 81, 258-263.

[3] Attia, H.A. 2007. ―On the effectiveness of porosity on unsteady flow between parallel plates of a viscoelastic fluid under constant pressure gradient with heat transfer‖, Computational Materials Science, 38, 746–750.

[4] Singh,P.Tomar, N.S. Sinha,D.2012. ―Oblique Stagnation-Point Darcy Flow towards a Stretching Sheet‖ Journal of Applied Fluid Mechanics, Vol. 5(3), 29-37.

[5] Singh, P. Tomer, N.S. Kumar S. and Sinha, D. 2011. ―Effect of Radiation and Porosity Parameter on Magnetohydrodynamic Flow due to Stretching Sheet in Porous Media‖, Thermal Sciences, Vol. 15(2), 517-526.

[6] Singh, P. Tomer, N.S. Jangid, A. and Sinha,D. 2011. ―Study of MHD Oblique Stagnation Point Assisting Flow on Vertical Plate with Uniform Surface Heat Flux‖. World Academy of Science, Engineering and Technology, Vol. 75, 824-829.

[7] Singh, P. Tomer, N.S. Kumar S. and Sinha, D. 2010. ―MHD oblique stagnation-point flow towards a stretching sheet with heat transfer‖, International Journal of Applied Mathematics and Mechanic, 6(13), 94-111.

[8] Singh, P. Tomer, N.S. and Kumar,M. 2010.―Effect of Variable viscosity on Convective Heat Transfer along an Inclined Plate Embedded in Porous Medium with an Applied Magnetic Field‖. International Journal of Engineering and Natural Sciences, Vol. 4(3), 170-174.

[9] Singh, P. Tomer, N.S. Jangid, A. and Sinha,D. 2010. ―Effects of Thermal Radiation and Magnetic Field on Unsteady Stretching Permeable Sheet in Presence of Free Stream Velocity‖. International Journal of Information and Mathematical Sciences, Vol.6(3),63-69. [10] Kaushik,A. Singh,P.and. Choudhary, K. K.2013 ―Study of MHD

flow on a Stretching Sheet with Variable Heat Flux and Constant Suction‖, Proceedings-International Conference on Recent Trends in Mechanical Engineering (RTME-2013), 289--308, 2013, ISBN 978-93-83723-02-7.

[11] Saini,I. Singh,P. Malik,V.2013 ―Genetic Algorithm Approach for Solving the Falkner–Skan Equation‖ International Journal of Computer Science and Engineering, Vol. 7 (3), 850-853.

[12] Takhar, H.S. Chamkha A.J. and Nath, G. 2000. ―Flow and mass transfer on a stretching sheet with a magnetic field and chemically reactive species‖, International Journal of Engineering Science, 38, 1303-1314.

[13] Pal, D.2009. ―Heat and mass transfer in stagnation-point flow towards a stretching surface in the presence of boundary force and thermal radiation‖, Meccanica, 44, 145-158.

[14] Liao, S.J.1992. The proposed homotopy analysis technique for the solution of nonlinear problems. Ph.D. Thesis, Shanghai Jiao Tong University.

[15] Liao,S.J.2003. Beyond perturbation: introduction to the homotopy analysis method. Boca Raton: Chapman and Hall/CRC Press. [16] Pop, I. Pop, S.R. and Grosan, T. 2004. ―Radiation effects on the