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0Ain Shams Engineering Journal (2014) xxx, xxx-xxx
Ain Shams University Ain Shams Engineering Journal
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ENGINEERING PHYSICS AND MATHEMATICS
Influence of heat source/sink on a Maxwell fluid over a stretching surface with convective boundary condition in the presence of nanoparticles
G.K. Ramesh *, B.J. Gireesha
Department of Studies and Research in Mathematics, Kuvempu University, Shankaraghatta-577 451, Shimoga, Karnataka, India Received 24 December 2013; revised 10 March 2014; accepted 6 April 2014
KEYWORDS
Maxwell fluid; Heat source/sink; Nanoparticles; Convective boundary condition;
Numerical solution
Abstract In this article, heat source/sink effects on the steady boundary layer flow of a Maxwell fluid over a stretching sheet with convective boundary condition in the presence of nanoparticles are reported. An appropriate similarity transformation is employed to transform the governing equations in partial differential equations form to similarity equations in ordinary differential equations form. The resulting equations are then solved numerically using shooting technique. Results for the velocity, temperature and concentration distributions are presented graphically for different values of the pertinent parameters. It is found that the local Nusselt number is smaller and local Sherwood number is higher for Maxwell fluids compared to Newtonian fluids.
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1. Introduction
Prandtl's boundary layer theory proved to be great use in Newtonian fluids as the Navier-Stokes equations can be converted into much simplified equations which are easier to handle. The stretching sheet concept in Crane's [1] boundary layer flow of a Newtonian fluid over a stretching sheet has been investigated and extended by several authors [2-5] for different physical situations, due to its important applications to polymer industry. These studies restrict their analyses to
non-Newtonian fluids. Flow due to a stretching sheet also occurs in thermal and moisture treatment of materials, particularly in processes involving continuous pulling of a sheet through a reaction zone, as in metallurgy, textile and paper industries, in the manufacture of polymeric sheets, sheet glass and crystalline materials. It is well known that a number of industrial fluids such as molten plastics, polymeric liquids, food stuffs or slurries exhibit non-Newtonian character. Therefore a study of flow and heat transfer in non-Newtonian fluids is of practical importance. Many models of non-Newtonian fluids exist. Maxwell model is one subclass of rate type fluids. This fluid model predicts the relaxation time effects. Such effects cannot be predicted by differential-type fluids. This fluid model is especially useful for polymers of low molecular weight. A review of non-Newtonian fluid flow problems may be found in [6-8].
The study of heat source/sink effects on heat transfer is very important because its effects are crucial in controlling the heat transfer. Postelnicu et al. [9] examined the effect of variable
* Corresponding author. Tel.: +91 9900981204.
E-mail addresses: gkrmaths@gmail.com (G.K. Ramesh), bjgireesu@
rediffmail.com (B.J. Gireesha).
Peer review under responsibility of Ain Shams University.
2090-4479 © 2014 Production and hosting by Elsevier B.V. on behalf of Ain Shams University. http://dx.doi.org/10.1016/j.asej.2014.04.003
Nomenclature
c stretching rate u, v velocity components along x and y directions
Cf skin friction Uw stretching sheet velocity
cP specific heat x coordinate along the stretching sheet
C nanoparticle volume fraction y distance normal to the stretching sheet
DB Brownian diffusion coefficient
DT thermophoresis diffusion coefficient Greek symbols
Nb Brownian motion parameter V kinematic viscosity
Nt thermophoresis parameter / rescaled nanoparticle volume fraction
Nux local Nusselt number Pf density of the base fluid
Le Lewis number Pp density of the particles
Pr Prandtl number k suction parameter
S heat source/sink parameter b Maxwell parameter
Shx local Sherwood number h dimensionless temperature
T temperature of the fluid g similarity variable
T * w temperature at the wall a thermal diffusivity
T * 1 ambient fluid temperature sw wall shearing stress
viscosity on forced convection flow past a horizontal flat plate in a porous medium with internal heat generation, but in heat generation part they considered only space dependent heat source. Abo-Eladahab and El Aziz [10] analyzed the effect of non-uniform heat source with suction/blowing, but confined to the case of viscous fluids only. Bataller [11] examined the effects of heat source/sink, radiation and work done by deformation on flow and heat transfer of a viscoelastic fluid over a stretching sheet. Abel and Nandeppanavar [12] investigated the effects of viscous dissipation and non-uniform heat source in a viscoelastic boundary layer flow over a stretching sheet. Tsai et al. [13] studied the unsteady stretching surface with non-uniform heat source. Hsiao [14] obtained the numerical solutions for the flow and heat transfer of a viscoelastic fluid over a stretching sheet with electromagnetic effects and nonuniform heat source/sink using the combination of finite difference method, Newton's method, and Gauss elimination method. In above cited papers they are shown that for effective cooling of stretching sheet, non-uniform heat source/sink should be used.
Aforementioned studies were primarily concerned with the laminar flow of a clear fluid (Newtonian/non-Newtonian fluid). In the recent past a new class of fluids, namely nano-fluids has attracted the attention of the science and engineering community because of the many possible industrial applications of these fluids. Nanotechnology is an emerging science that is finding extensive use in industry due to the unique chemical and physical properties that the nano-sized materials possess. These fluids are colloidal suspensions, typically metals, oxides, carbides or carbon nanotubes in a base fluid. The term nanofluid was coined by Choi [15] in his seminal paper presented in 1995 at the ASME Winter Annual Meeting. It refers to fluids containing a dispersion of submicronic solid particles with typical length on the order of 1-50 nm. Kuznetsov and Nield [16] analytically studied the natural convective boundary layer flow of a nanofluid past a vertical plate. In a recent paper Khan and Pop [17] first time studied the problem of laminar fluid flow resulting from the stretching of a flat surface in a nanofluid. Alsaedi et al. [18] examine the influence of heat generation/absorption
on the stagnation point flow of nanofluid toward a linear stretching surface. Rahman et al. [19] investigate the dynamics of the natural convection boundary layer flow of water based nanofluids over a wedge in the presence of a transverse magnetic field with internal heat generation or absorption with the help of Matlab software. Nandy and Mahapatra [20] analyze the effects of velocity slip and heat generation/ absorption on magnetohydrodynamic stagnation-point flow and heat transfer over a stretching/shrinking surface and obtained the solution numerically using fourth order Runga-Kutta method with the help of shooting technique. Different from a stretching sheet, it was found that the solutions for a shrinking sheet are non-unique. Makinde and Aziz [21] investigate the effect of a convective boundary condition on boundary layer flow, heat transfer and nanopar-ticle fraction over a stretching surface in a nanofluid. Nadeem et al. [22] reported the numerical solutions of non-Newtonian nanofluid flow over a stretching sheet using the Maxwell fluid model. Hady et al. [23] studied the natural convection boundary-layer flow over a downward-pointing vertical cone in a porous medium saturated with a non-Newtonian nanofluid in the presence of heat generation or absorption and they used power-law model.
The objective of this study is to extend the work of Makin-de and Aziz [21], by considering the Maxwell (non-Newtonian) fluid over a stretching sheet with the effect of heat source/sink. Similarity transforms are presented for this problem, and non-dimensionalized equations are addressed numerically. The results obtained are then compared with those of Makinde and Aziz. Graphical results for various values of the parameters are presented to gain thorough insight toward the physics of the problem. To the best of our knowledge, this problem has not been studied before.
2. Mathematical analysis
Consider a steady flow of an incompressible Maxwell fluid in the region y >0 driven by a stretching surface located at y = 0 with a fixed stagnation point at x = 0. The sheet is coinciding with the plane y = 0, with the flow being confined to
y >0. Two equal and opposite forces are applied along the x-axis, so that the sheet is stretched, keeping the origin fixed. The temperature of sheet surface (to be determined later) is the result of a convective heating process which is characterized by a temperature Tf and a heat transfer coefficient hf.
The problem under consideration is governed by the following boundary layer equations of Maxwell fluid along nano-particles in the presence of heat source/sink which are given by
du dv o
dx dy '
du du d2u ( 2 d2u 2 „ „ 2 dx dy dy2 \ dx2 dy2 d xdy
^+^ - a(dT + PQP (T - T1)
. „ fdC dn (Dt
+ D4 @c @T) + (DT
udC+vdC~D d2c
dx dy B dy2
where u and v are the velocity components along the x and y axes, respectively. Further, a, pf ppv, T and Tx are respectively the thermal diffusivity, density of the base fluid, density of the particles, kinematic viscosity of the fluid, fluid temperature and ambient fluid temperature. k0 is the relaxation time of the UCM fluid, Q0 is the dimensional heat generation/absorption coefficient, Db is the Brownian diffusion coefficient, Dt is the thermophoresis diffusion coefficient and cp is the specific heat at constant pressure. Here s is the ratio of the effective heat capacity of the nanoparticle material and the heat capacity of the ordinary fluid and C is the nanoparticle volume fraction.
The associated boundary conditions for the present problem are
u - Uw(x), v — 0, —k — — hf(Tf — T), C - Cw at y - 0, u ! 0, v ! 0, T ! Tœ, C ! CM as y ! 0,
where Uw(x) = cx is the stretching sheet velocity, c >0 this is known as stretching rate. Cw is the nanoparticles fraction at wall, and C1 ambient nanoparticle volume fraction.
The specific forms of the stretching velocity and the surface temperature and concentration are chosen to allow the coupled non-linear partial differential Eqs. (1)-(4) to be converted to a set of coupled, non-linear ordinary differential equations by the similarity transformation
g - ~ У, f(g)-
T — Ti
Tf — Ti
/(g) -
(xvUw) CC C C
where g is the similarity variable, and f, h and / are the dimen-sionless stream function, temperature and concentration respectively. The velocity components u and v in Eq. (6) automatically satisfy the continuity Eq. (1). In terms of f(g), h(g)
and /(g) the momentum Eq. (2), energy Eq. (3) and concentration Eq. (4) can be written as
f02 -ff = f'- bf2f - 2fff'),
0" + fh + SO + Nbh/ + Nt6a - 0
/ + LePrf/ + — O" - 0
(8) (9)
Here "/" represents an ordinary derivative with respect to g and the corresponding boundary conditions in the non-dimensional form are
f - 0, f - 1, O -—Bí(1 — 0(0)),
f ! 0, O ! 0, / ! 0 as g ! 0
/ - 1 at g - 0
The dimensionless parameters in Eqs. (7)-(9) are b = k0c is the Maxwell parameter, Bi = (v/a)1/2hf/k is the Biot number, S — cQCr is the heat source (S > 0) or sink (S < 0) parameter, Nb — sDB</w-/i) is the Brownian motion, Nt — sDT(T;-Tl) is the
v ' vTra
thermophoresis parameter, Le — -¡^ is the Lewis number and sT is the Prandtl number.
For practical purposes, the functions f(g), h(g) and /(g) allow us to determine the skin friction coefficient, Nusselt number and Sherwood number respectively.
(1 + b)f '(0)
Nux - —Rei/20'(0), Shx - —Rex/2/'(0)
where Rex = xU"jx) is the local Reynolds number. 3. Numerical solutions
The exact solution do not seem feasible for a complete set of Eqs. (7)-(9) with appropriate boundary conditions given in (10) because of the non-linear form. This fact forces one to obtain the solution of the problem numerically. Appropriate similarity transformation is adopted to transform the governing partial differential equations into a system of non-linear ordinary differential equations. The resultant boundary value problem is solved numerically using an efficient fourth order Runge-Kutta method along with shooting technique.
The non-linear differential equations are first decomposed into a system of first order differential equation. The coupled ordinary differential Eqs. (7)-(9) are third order inf(g) and second order in h(g) and /(g)which have been reduced to a system of seven simultaneous equations for seven unknowns. In order to numerically solve this system of equations using Runge-Kutta method, the solutions require seven initial conditions but two initial conditions in f(g)one initial condition in each of h(g) and /(g) re known. However, the values of/(g), h(g) and /(g) are known at g fi 1. These end conditions are utilized to produce unknown initial conditions at g = 0 by using shooting technique. The most important step of this scheme is to choose the appropriate finite value of g1. Thus to estimate the value of g1 we start with some initial guess value and solve the boundary value problem consisting of Eqs. (7)-(9) to obtain/'(0), &(0) and /'(0). The solution process is repeated with another larger value of g1 until two successive values of /'(0), &(0) and /'(0) differ only after desired significant
digit. The last value is taken as the finite value of the limit for the particular set of physical parameters for determining velocity, temperature and concentration, respectively, are f(g), h(g) and /(g) in the boundary layer. After getting all the initial conditions we solve this system of simultaneous equations using fourth order Runge-Kutta integration scheme. The value of is selected to 8 depending on the physical parameters governing the flow so that no numerical oscillation would occur. Thus, the coupled boundary value problem of third-order in f(g), second-order in h(g) and /(g) has been reduced to a system of seven simultaneous equations of first-order for seven unknowns as follows:
f = p,p' = q, q' = p2 -fq + b(fq - 2fpq), 0' = r, Ï r' = -Pr fr + Nbrz + Ntr2 + S0), /' = z, I,
z' = -LePrfz - fr' J
and the boundary condition becomes f(0) = 0,p(0) = 0, r(0) = -Bi(1 - 0(0)), /(0) = 0, p(gj=0, h(gi) = 0, /(gi) = 0.
In this study, the boundary value problem is first converted into an initial value problem (IVP). Then the IVP is solved by appropriately guessing the missing initial value using the shooting method for several sets of parameters. The step size is h = 0.1 used for the computational purpose. The error tolerance of 10—6 is also being used. The results obtained are presented through tables and graphs, and the main features of the problems are discussed and analyzed.
4. Result and discussions
The problem for a regular (Newtonian) fluid involves just five independent parameters, namely the Prandtl number, Lewis number, Brownian motion parameter, thermophoresis parameter, and the Biot number. The present extension involves two more independent parameters i.e., Maxwell parameter and heat source/sink parameter. For the verification of accuracy of the applied numerical scheme, a comparison of the present results corresponding to the local Nusselt number —h0(0) and local Sherwood number —/0(0) (absence of Maxwell parameter and heat source/sink) with the available published results of Makinde and Aziz [21] is made and presented in Table 1, shows a favorable agreement thus give confidence that the numerical results obtained are accurate.
Let us first concentrate on the effects of Maxwell parameter b on velocity distribution as shown in Fig. 1. Here b = 0 gives
Table 1 Comparison of the values of local Nusselt number —h0(0) and local Sherwood number —/0(0) for various values of Nt, when Le = 10, Pr = 10, Bi = 0.1 in the absence of b and S.
Nt Nb = 0.1 Makinde and Aziz [21] Present result
-h0 (0)
-ф0 (0)
-h0 (0) - ф0 (0)
0.2 0.3 0.4 0.5
0.0929 0.0927 0.0925 0.0923 0.0921
2.2774 2.2490 2.2228 2.1992 2.1783
0.09290 2.27739
0.09273 2.24893
0.09254 2.22279
0.09234 2.19917
0.09212 2.17832
Figure 1 Velocity profile for different values of b.
the result for Newtonian fluid. The effect of increasing values of b is to enhance the velocity and hence the boundary layer thickness increases. This is exactly opposite effects for regular Maxwell fluid in the absence of nanoparticles. Furthermore, the effects of Nb, Nt and Maxwell parameter b on —h0(0) related to local Nusselt number and also the local Sherwood number —/0(0) are presented in Fig. 2. Value of —h0(0) decreases with increasing Nb, Nt and also with Maxwell parameter b, but —/0(0) increases for increase in b and Nb, Nt.
The effects of heat source/sink parameter S can be found from Fig. 3. For S >0 (heat source), it can be observed that the thermal boundary layer generates the energy, and this causes the temperature in the thermal boundary layer increases with increase in S. Whereas S <0 (heat sink) leads to decrease in the thermal boundary layer. S = 0 represent the absence of heat source/sink. In Fig. 4, the variation of temperature h(g) with g for various values of the Biot number Bi is presented. It is observed that temperature field h(g) increases rapidly near the boundary by increasing Biot number. This is because of convective heat exchange at the plate surface leading to an increase in thermal boundary layer thickness. The effect of Prandtl number Pr on thermal field is shown in Fig. 5. Here we noted that the presence of b and S, temperature is higher than the absence of b and S, also it gives the comparison of the graphical result with Makinde and Aziz [21]. The graph
Figure 2 Variation of local Nusselt number and Sherwood number with b for several values of Nb, Nt.
Figure 3 Temperature profile for different values of S.
Figure 4 Temperature profile for different values of Bi.
Figure 5 Temperature profile for different values of Pr.
depicts that the temperature decreases when the values of Pr increase. The Prandtl number signifies the ratio of momentum diffusivity to thermal diffusivity. This is due to the fact that a higher Pr fluid has relatively low thermal conductivity, which reduces conduction and thereby the thermal boundary layer thickness, and as a result, temperature decreases. In heat transfer problems, the Prandtl number and heat sink parameter controls the relative thickening of the momentum and the thermal boundary layers.
Figure 6 Temperature and concentration profile for different values of Nb.
Fig. 6 exhibit the temperature and the concentration profiles for variable values of Brownian motion Nb. The fluid velocity is found to increase with increasing Nb, whereas in concentration boundary layer reduces as Nb an increase which thereby enhances the nanoparticles concentration at the sheet. This may be due to the fact that as a Brownian motion parameter Nb decreases the mass transfer of a nanofluid. The graph of thermophoresis parameter Nt on the temperature and the concentration profiles are depicted in Figs. 7. From these plots, it is observed that the effect of increasing values of Nt is to increase the temperature and concentration profiles. Fig. 8, displays the effect of Lewis number Le on concentration profiles. It is note that the concentration of the fluid decreases with increase in Le. Physically fact that the larger values of Lewis number makes the mass diffusivity smaller, therefore it decreases the concentration field. Also observe that presence of b and S is slightly increases than the regular fluid.
Figs. 9 and 10, illustrate the nature of local Nusselt number and local Sherwood number with thermophoresis parameter Nt and Brownian motion Nb. Absolute value of — h0(0) decreases with increasing Nb and also with Nt. Due to the fact
Figure 7 Temperature and concentration profile for different values of Nt.
Figure 8 Concentration profile for different values of Le.
Figure 11 Variation of local Nusselt number with Pr for several values of Le.
Figure 9 Variation of local Nusselt number with Nt for several Figure 12 Variation of local Sherwood number with Pr for values of Nb. several values of Le.
Table 2 Computations of - h'(0), and -/'(0) for different values of Biot number (Bi) with b = 0.1, Pr = 3.0, S = 0.05, Nb = 0.1, Nt = 0.1 and Le = 1.0.
Bi (0) -h0 (0)
0.1 1.1173 0.0902
0.5 0.9972 0.3215
2.0 0.8543 0.6059
5.0 0.7936 0.7304
10 0.7683 0.7829
50 0.7459 0.8300
100 0.7429 0.8362
500 0.7405 0.8412
1000 0.7402 0.8419
5000 0.7400 0.8424
10000 0.7400 0.8425
100000 0.7399 0.8425
1000000 0.7399 0.8425
5000000 0.7399 0.8425
that for a weaker thermophoretic effect, there is a significant decrease in the rate of heat transfer at the sheet with an increase in Nb. But -/'(0) increases with increasing Nb and
increases with Nt. Figs. 11 and 12 show that the variation - h'(0) and -/'(0) with the changes of Le, for different values of Pr . From these graph we observe that - h'(0) decreases with
the increase in Le and —/'(0) increases with the increase of Le. From the graphs 9-12 one can observe that, for variation of local Sherwood number is increases in the presence of Maxwell parameter where as the local Nusselt number is decreases.
5. Conclusions
In the present investigation, the influence of the different parameters on the velocity, temperature and concentration profiles are illustrated and discussed. The numerical results give a view toward understanding the response characteristics of the Maxwell fluid in the presence of nanoparticles and heat source/sink. From the Table 2, one can see that when increase in Biot number from 0.1 to 50, the local Nusselt Number —h0(0) and the local Sherwood Number —/0(0) increases significantly. Furthermore increase in Bi has only minor effect on the —h0(0), and —/0(0). When large value of Bi, no significant change is observed for the values of the —h0(0) and — /(0).
Acknowledgements
The Authors are very much thankful to the editor and referee for their encouraging comments and constructive suggestions to improve the presentation of this manuscript.
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G.K. Ramesh was born and brought up in the district of Chitradurga, Karnataka, India. He obtained the B.Sc. Honours degree from Kuvempu University and M.Sc. Degree in Mathematics from Kuvempu University. He joined as a research scholar in the Department of Mathematics. He has published 16 research papers. His field of interest covers the areas of the application of boundary layer flows of Newtonian/non-Newtonian fluids and dusty fluid flow problems.
B.J. Gireesha born in Kolalu village, Karna-taka, India on 1st June 1974 received his Master's degree in Mathematics, M.Phil., in Applied Mathematics and Ph.D. in Fluid Mechanics from Kuvempu University, Shim-oga, India, in 1997, 1999 and 2002 respectively. Currently he is working as faculty in the Department of Mathematics, Kuvempu University. He has authored and coauthored 6 books, 109 national and international journal papers, 9 conference papers and editor of 2 conference proceedings. He has attended/presented the papers in 25 International/National
conferences. He is a member of several bodies, Editorial Board member for several Journals. His research interests include the areas of Fluid Mechanics, Differential Geometry and Computer simulation. 8
students are awarded Ph.D., degree and 7 students are awarded M.Phil. degree under his supervision. Presently he is guiding 8 students for Ph.D.
