ENGINEERING PHYSICS AND MATHEMATICS
Slip effects on MHD boundary layer flow over
an exponentially stretching sheet with suction/blowing
and thermal radiation
Swati Mukhopadhyay
*
Department of Mathematics, The University of Burdwan, Burdwan 713 104, WB, India Received 12 April 2012; revised 5 October 2012; accepted 15 October 2012
Available online 21 December 2012
KEYWORDS Exponential stretching; MHD; Suction/blowing; Velocity slip; Thermal slip; Radiation; Similarity solutions
Abstract The boundary layer flow and heat transfer towards a porous exponential stretching sheet in presence of a magnetic field is presented in this analysis. Velocity slip and thermal slip are con-sidered instead of no-slip conditions at the boundary. Thermal radiation term is incorporated in the temperature equation. Similarity transformations are used to convert the partial differential equa-tions corresponding to the momentum and energy equaequa-tions into non-linear ordinary differential equations. Numerical solutions of these equations are obtained by shooting method. It is found that the horizontal velocity decreases with increasing slip parameter as well as with the increasing mag-netic parameter. Temperature increases with the increasing values of magmag-netic parameter. Temper-ature is found to decrease with an increase of thermal slip parameter. Thermal radiation enhances the effective thermal diffusivity and the temperature rises.
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1. Introduction
The study of laminar flow and heat transfer over a stretching sheet in a viscous fluid is of considerable interest because of its ever increasing industrial applications and important bear-ings on several technological processes. Examples are numer-ous and they include the cooling of an infinite metallic plate
in a cooling bath, the boundary layer along material handling conveyers, the aerodynamic extrusion of plastic sheets, the boundary layer along a liquid film in condensation processes, paper production, glass blowing, metal spinning, drawing plas-tic films and polymer extrusion, to name just a few[1]. Crane
[2]investigated the flow caused by the stretching of a sheet. Under different physical situations, many researchers extended the work of Crane [2]. Most of the available literature deals with the study of boundary layer flow over a stretching surface where the velocity of the stretching surface is assumed linearly proportional to the distance from the fixed origin.
However, realistically stretching of plastic sheet may not necessarily be linear[3]. Flow and heat transfer characteristics past an exponentially stretching sheet has a wider applications in technology. For example, in case of annealing and thinning
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of copper wires the final product depends on the rate of heat transfer at the stretching continuous surface with exponential variations of stretching velocity and temperature distribution. During such processes, both the kinematics of stretching and the simultaneous heating or cooling have a decisive influence on the quality of the final products [4]. Magyari and Keller
[5]focused on heat and mass transfer on boundary layer flow due to an exponentially continuous stretching sheet. Elbashbe-shy [6] studied the flow past an exponentially stretching surface. Khan [7] and Sanjayanand and Khan [8] discussed the viscous–elastic boundary layer flow and heat transfer due to an exponentially stretching sheet. Later, Sajid and Hayat
[9]find the analytic solution using homotopy analysis method and discussed the influence of thermal radiation on the bound-ary layer flow due to an exponentially stretching sheet. Recently, the effect of thermal radiation on the steady laminar two-dimensional boundary layer flow and heat transfer over an exponentially stretching sheet was reported by Bidin and Nazar[10]. Of late, El-Aziz[11]and Ishak [12]described the flow and heat transfer past an exponentially stretching sheet.
All the above mentioned studies continued their discussions by assuming the no slip boundary conditions. The no-slip bound-ary condition (the assumption that a liquid adheres to a solid boundary) is one of the central tenets of the Navier–Stokes the-ory. However, there are situations wherein this condition does not hold. Partial velocity slip may occur on the stretching bound-ary when the fluid is particulate such as emulsions, suspensions, foams and polymer solutions. The non-adherence of the fluid to a solid boundary, also known as velocity slip, is a phenomenon that has been observed under certain circumstances[13]. Re-cently, many researchers[14–18], etc. investigated the flow prob-lems taking slip flow condition at the boundary. The fluids that exhibit boundary slip have important technological applications such as in the polishing of artificial heart valves and internal cav-ities. For some coated surfaces, such as Teflon, resist adhesion, the no slip condition is replaced by Navier’s partial slip condi-tion, where the slip velocity is proportional to the local shear stress. However, experiments suggest that the slip velocity also depends on the normal stress. A number of models have been ad-vanced for describing the slip that occurs at solid boundaries. A new dimension is added to the above mentioned study by consid-ering the effects of partial slip at the stretching wall. The study of magnetohydrodynamics boundary layer flow of a conducting fluid is also important as it finds applications in a variety of stretching sheet problems. Representative studies dealing with such effects can be found in Turkyilmazoglu[19–21].
Suction or injection (blowing) of a fluid through the bound-ing surface can significantly change the flow field. In general, suction tends to increase the skin friction whereas injection acts in the opposite manner. Injection or withdrawal of fluid through a porous bounding wall is of general interest in prac-tical problems involving boundary layer control applications such as film cooling, polymer fiber coating, and coating of wires. The process of suction and blowing has also its impor-tance in many engineering activities such as in the design of thrust bearing and radial diffusers, and thermal oil recovery. Suction is applied to chemical processes to remove reactants. Blowing is used to add reactants, cool the surface, prevent cor-rosion or scaling and reduce the drag.
The radiative effects have important applications in physics and engineering. The radiation heat transfer effects on differ-ent flows are very important in space technology and high
temperature processes. But very little is known about the fects of radiation on the boundary layer. Thermal radiation ef-fects may play an important role in controlling heat transfer in polymer processing industry where the quality of the final product depends on the heat controlling factors to some extent. High temperature plasmas, cooling of nuclear reactors, liquid metal fluids, magnetohydrodynamics (MHD) accelera-tors, power generation systems are some important applica-tions of radiative heat transfer from a vertical wall to conductive gray fluids.
Since no attempt has been made to analyze the effects of partial slip on MHD boundary layer flow over an exponen-tially stretching surface with suction or injection, so it is con-sidered in this article. Using similarity transformations, a third order ordinary differential equation corresponding to the momentum equation and a second order differential equa-tion corresponding to the heat equaequa-tion are derived. Using shooting method numerical calculations up to desired level of accuracy were carried out for different values of dimension-less parameters of the problem under consideration for the purpose of illustrating the results graphically. The analysis of the results obtained shows that the flow field is influenced appreciably by the slip parameter in the presence of magnetic field and suction or injection at the wall. Estimation of skin friction which is very important from the industrial application point of view is also presented in this analysis. It is hoped that the results obtained will not only provide useful information for applications, but also serve as a complement to the previous studies.
2. Equations of motion
Consider the flow of an incompressible viscous electrically con-ducting fluid past a flat sheet coinciding with the plane y = 0 (seeFig. 1). The x-axis is directed along the continuous stretch-ing surface and points in the direction of motion while the y-axis is perpendicular to the surface. The flow is confined to y> 0. Two equal and opposite forces are applied along the x-axis so that the wall is stretched keeping the origin fixed. The flow is assumed to be generated by stretching of the elastic boundary sheet from a slit with a large force such that the velocity of the boundary sheet is an exponential order of the
x y O Too u v V(x) U Tw Boundary layer B(x)
flow directional coordinate x. The flow takes place in the upper half plane y > 0. A variable magnetic field BðxÞ ¼ B0e
x 2L is applied normal to the sheet, B0being a constant. The
continu-ity, momentum and energy equations governing such type of flow are written as
@u @xþ @v @y¼ 0; ð1Þ u@u @xþ t @u @y¼ m @2u @y2 rB2 q u; ð2Þ u@T @xþ t @T @y ¼ j qcp @2T @y2 1 qcp @qr @y; ð3Þ
where u and t are the components of velocity respectively in the x and y directions, m¼l
qis the kinematic viscosity, q is
the fluid density (assumed constant), l is the coefficient of fluid viscosity, r is the electrical conductivity, qris the radiative heat
flux, cpis the specific heat at constant pressure, j is the thermal
conductivity of the fluid.
In writing Eq.(2), we have neglected the induced magnetic field since the magnetic Reynolds number for the flow is assumed to be very small.
Using Rosseland approximation for radiation[22]we can write qr¼ 4r 3k @T4 @y ; ð3aÞ
where r*is the Stefan–Boltzman constant, k*is the absorption coefficient.
Assuming that the temperature difference within the flow is such that T4may be expanded in a Taylor series and expanding T4 about T1, the free stream temperature and neglecting higher orders we get T4 4T3
1T 3T 4 1. Therefore, the Eq.(3)becomes u@T @xþ t @T @y ¼ j qcp @2T @y2þ 16rT3 1 3qcpk @2T @y2: ð4Þ 2.1. Boundary conditions
The appropriate boundary conditions for the problem are gi-ven by u¼ U þ Nm@u @y; t¼ VðxÞ; T¼ Twþ D @T @y at y¼ 0; ð5Þ u! 0; T! T1 as y! 1: ð6Þ Here U¼ U0e x
L is the stretching velocity, T
w¼ T1þ T0e x 2L is the temperature at the sheet, U0, T0are the reference velocity
and temperature respectively, N¼ N1e x
2L is the velocity slip factor which changes with x, N1is the initial value of velocity
slip factor and D¼ D1e x
2Lis the thermal slip factor which also changes with x, D1is the initial value of thermal slip factor.
The no-slip case is recovered for N = 0 = D. V(x) > 0 is the velocity of suction and V(x) < 0 is the velocity of blowing, VðxÞ ¼ V0e
x
2L, a special type of velocity at the wall is consid-ered. V0is the initial strength of suction.
2.2. Method of solution
Introducing the similarity variables as g¼ ffiffiffiffiffiffiffiffi U0 2mL r e2Lxy; u¼ U 0e x Lf0ðgÞ; t¼ ffiffiffiffiffiffiffiffi mU0 2L r e2LxffðgÞ þ gf0ðgÞg; T¼ T 1þ T0e x 2LhðgÞ ð7Þ
and upon substitution of(5)in Eqs. 2, 3, 3a, 4 the governing equations reduce to f000þ ff00 2f0 2 M2 f0¼ 0; ð8Þ 1þ4 3R h00þ Prðfh0 f0hÞ ¼ 0; ð9Þ
and the boundary conditions take the following form: f0¼ 1 þ kf00; f¼ S; h¼ 1 þ dh0 at g¼ 0 ð10Þ
and
f0! 0; h ! 0 as g! 1; ð11Þ
where the prime denotes differentiation with respect to g, M¼ ffiffiffiffiffiffiffiffiffi 2rB2 0L qU0 q
is the magnetic parameter, k¼ N1
ffiffiffiffiffiffi
U0m 2L
q
is the velocity slip parameter, d¼ D1
ffiffiffiffiffiffi
U0 2mL
q
is the thermal slip param-eter and S¼ Vffiffiffiffiffi0
U0m 2L
p >0 (or <0) is the suction (or blowing) parameter, R¼4rT31
jk is the radiation parameter, Pr¼ lcp
j is
the Prandtl number.
Eq.(9)can also be written as h00þ Prmðfh0 f0hÞ ¼ 0 where
Prm¼ m=j 1 þ43R
is the modified Prandtl number. 3. Numerical method for solution
The above Eqs. (8) and (9) along with the boundary conditions are solved by converting them to an initial value problem. We set f0¼ z; z0¼ p; p0¼ ½2z2 fp þ M2z; ð12Þ h0¼ q; q0¼ 3Pr 4Rþ 3 ðfq zhÞ; ð13Þ
with the boundary conditions
fð0Þ ¼ S;f0ð0Þ ¼ 1 þ kc; f00ð0Þ ¼ c; hð0Þ ¼ 1 þ db; h0ð0Þ ¼ b:
ð14Þ In order to integrate (12) and (13) as an initial value prob-lem one requires a value for p(0), i.e. f00ð0Þ and q(0), i.e. h0(0)
but no such value is given at the boundary. The suitable guess value for f00ð0Þ and h0(0) are chosen and then integration is
carried out.
The most important factor of shooting method is to choose the appropriate finite value of g1. In order to determine g1for
the boundary value problem stated by Eqs. (12)–(14), we start with some initial guess value for some particular set of physical parameters to obtain f00ð0Þ and h0(0). The solution procedure is
repeated with another large value of g1 until two successive
values of f00ð0Þ and h0(0) differ only by the specified significant
digit. The last value of g1 is finally chosen to be the most
parameters. The value of g1 may change for another set of
physical parameters. Once the finite value of g1is determined
then the integration is carried out. We compare the calculated values for f0and h at g = g
1( = 10) (say) with the given
bound-ary conditions f0(10) = 0 and h(10) = 0 and adjust the
estimated values, f00ð0Þ and h0(0), to give a better approximation
for the solution.
We take the series of values for f00ð0Þ and h0(0) and apply the
fourth order classical Runge–Kutta method with step-size h= 0.01. The above procedure is repeated until we get the converged results within a tolerance limit of 105.
4. Results and discussion
In order to analyze the results, numerical computation has been carried out using the method described in the previous section for various values of the velocity slip parameter ðkÞ, suction (/injection) parameter (S), magnetic parameter (M), radiation parameter (R), thermal slip parameter (d). For illus-trations of the results, numerical values are plotted in theFigs. 1a–7d.
For the verification of accuracy of the applied numerical scheme, comparisons of the present results corresponding to the values of heat transfer coefficient [h0(0)] for k¼ 0; d ¼ 0
and S = 0 (i.e. in absence of velocity slip, thermal slip and suction at the boundary) are made with the available results of Magyari and Keller [5], Bidin and Nazar [10], El-Aziz
[11]and Ishak[12](for some special cases) and presented in
Table 1. The results are found in excellent agreement. As in this paper, numerical solution is obtained with the help of shooting method so our results differ slightly with the other researchers. One knows that such type of solution depends on the initial guesses which are not obviously be accurate so the slight variation in the data presented inTable 1is found.
Let me first concentrate on the effects of velocity slip parameter k on velocity and shear stress profiles in presence of suction at the wall.
InFig. 2a, velocity profiles are shown for different values of kðk ¼ 0:1; 0:3; 0:5Þ. The velocity curves show that the rate of transport decreases with the increasing distance (g) of the sheet. In all cases the velocity vanishes at some large distance from the sheet (at g = 6). With the increasing k, the horizontal velocity is found to decrease.
When slip occurs, the flow velocity near the sheet is no longer equal to the stretching velocity of the sheet. With the in-crease in k, such slip velocity inin-creases and consequently fluid velocity decreases because under the slip condition, the pulling of the stretching sheet can be only partly transmitted to the fluid. It is noted that k has a substantial effect on the solutions. Temperature profiles are presented inFig. 2bfor the variation of velocity slip parameter in presence of suction and magnetic field. With the increasing k, the temperature is found to de-crease initially but after a certain distance from the sheet it in-creases with k (Fig. 2b).
η λ λ λ = 0.1 = 0.3 = 0.5 M = 0.1, S = 0.1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 /( ) f η
Figure 2a Variation of horizontal velocity f0(g) with g for several values of velocity slip parameter k.
η λ λ λ = 0.3 = 0.5 = 0.1 S = 0.1, R = 0.1, M = 0.1, = 0.1, Pr = 0.7 δ 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10 θ (η)
Figure 2b Variation of temperature h(g) with g for several values of velocity slip parameter k.
η λ = 0.1, S = 0.1 M = 0 M = 0.3 M = 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 /( ) f η
Figure 3a Variation of horizontal velocity f0(g) with g for several values of magnetic parameter M.
θ (η) η M = 0.6 M = 0 M = 0.3 λ = 0.1, S = 0.1, R = 0.1, δ = 0.1, Pr = 0.7 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10
Figure 3b Variation of temperature h(g) with g for several values of magnetic parameter M.
Fig. 3aexhibits the nature of velocity field for the variation of magnetic parameter M. With increasing M, velocity is found to decrease (Fig. 3a) but the temperature increases in this case (Fig. 3b). The transverse magnetic field opposes the motion of the fluid and the rate of transport is considerably reduced. This is because with the increase in M, Lorentz force increases and it produces more resistance to the flow. As M increases, ther-mal boundary layer thickness increases.
Figs. 4a and 4bdepict the effects of suction parameter S on velocity and temperature profiles respectively in presence of slip at the boundary for exponentially stretching sheet. It is ob-served that velocity decreases significantly with increasing
suc-tion parameter whereas fluid velocity is found to increase with blowing (Fig. 4a). It is observed that, when the wall suction (S > 0) is considered, this causes a decrease in the boundary layer thickness and the velocity field is reduced. S = 0 repre-sents the case of non-porous stretching sheet. Opposite behav-ior is noted for blowing (S < 0).
FromFig. 4b, it is seen that temperature decreases with increasing suction parameter whereas it increases due to blow-ing (Fig. 4b). Temperature overshoot is noted for blowing (S < 0). This feature prevails up to certain heights and then
η λ = 0.1 S = −0.5 S = −0.3 S = 0 S = 0.3 S = 0.5 M = 0.1, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 0 1 2 3 4 5 6 7 /( ) f η
Figure 4a Variation of horizontal velocity f0(g) with g for several values of suction/blowing parameter S.
η S = 0.5 S = 0.3 S = 0 S = −0.5 S = −0.3 λ = 0.1, = 0.1, R Pr = 0.7 δ = 0.1, M = 0.1, 0 0.2 0.4 0.6 0.8 1 1.2 0 2 4 6 8 10 θ (η)
Figure 4b Variation of temperature h(g) with g for several values of suction /blowing parameter S.
η δ δ δ = 0.1 δ = 1.5 = 2.5 = 4.0 R=0.1,Pr=0.7 S = 0.1, = 0.1, λ M = 0.1, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 1 2 3 4 5 6 7 8 9 θ (η)
Figure 5 Variation of temperature h(g) with g for several values of thermal slip parameter d.
R = 0.1 R = 0.3 R = 0.6 η Pr = 0.7 = 0.1, δ = 0.1, S = 0.1, λ M = 0.1, 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 2 4 6 8 10 θ (η)
Figure 6 Variation of temperature h(g) with g for several values of thermal radiation parameter R.
S λ = 0.3 λ = 0.1 M = 0 M = 0.3 −1.4 −1.3 −1.2 −1.1 −1 −0.9 −0.8 −0.7 −0.4 −0.2 0 0.2 0.4 //(0) f
Figure 7a Skin-friction coefficient f00ð0Þ against suction param-eter S for two values of magnetic paramparam-eter M and velocity slip parameter k. R = 0.1, = 0.1, Pr = 0.7δ λ = 0.1 M = 0 M = 0.3 = 0.3 λ S −0.2 −0.1 0 0.1 0.2 0.3 −0.4 −0.2 0 0.2 0.4 /(0) θ
Figure 7b Heat transfer coefficient h0(0) against suction param-eter S for two values of magnetic paramparam-eter M and velocity slip parameter k.
the process is slowed down and at a far distance from the wall temperature vanishes.
Fig. 5depicts the effects of thermal slip parameter d on tem-perature. Initially the temperature decreases with thermal slip parameter d but after a certain distance g from the sheet, such feature is smeared out. With the increase of thermal slip parameter d, less heat is transferred to the fluid from the sheet and so temperature is found to decrease.
Next, the effect of thermal radiation on temperature pro-files is presented in Fig. 6. It is found that temperature in-creases as the radiation parameter R inin-creases [Fig. 6]. This is in agreement with the physical fact that the thermal bound-ary layer thickness increases with increasing R. The effect of radiation in the thermal boundary layer Eq.(4)is equivalent with an increased thermal diffusivity, i.e. Pr= 1þ4
3R
in Eq.
(9)can be considered as an effective Prandtl number which re-duces with increasing values of R.
Fig. 7aexhibits the nature of skin-friction coefficient½f00ð0Þ
with suction/blowing parameter S for two values of velocity slipðk ¼ 0:1; 0:3Þ. It is found that skin-friction coefficient de-creases with S whereas it inde-creases with increasing magnetic parameter M as well as with the higher values of slip velocity at the boundary.
Fig. 7b presents the behavior of heat transfer coefficient [h0(0)] with suction/blowing parameter S for two values of
velocity slipðk ¼ 0:1; 0:3Þ. It is very clear that heat transfer de-creases with increasing values of suction, magnetic parameter and also with increasing values of velocity slip.Fig. 7cdisplays the nature of heat transfer coefficient against thermal slip parameter d for two values of radiation parameter R. Heat transfer increases with thermal slip parameter d but decreases with radiation parameter R. Rate of heat transfer increases with Prandtl number (Fig. 7d). An increase in Prandtl number reduces the thermal boundary layer thickness. Prandtl number signifies the ratio of momentum diffusivity to thermal diffusiv-ity. Fluids with lower Prandtl number will possess higher ther-mal conductivities (and thicker therther-mal boundary layer structures) so that heat can diffuse from the sheet faster than for higher Pr fluids (thinner boundary layers). Hence Prandtl number can be used to increase the rate of cooling in conduct-ing flows.
But thermal slip parameter, Prandtl number and radiation parameter have no effects on skin friction coefficient as the momentum boundary layer equation is independent of h.
5. Conclusions
The present study gives the numerical solutions for steady boundary layer MHD flow and radiative heat transfer over an exponentially porous stretching surface in presence of
R R = 0.1 = 0.6 δ r P=0.7 S = 0.1, λ = 0.1, M = 0, −0.0275 −0.027 −0.0265 −0.026 −0.0255 −0.025 −0.0245 −0.024 −0.0235 −0.023 −0.0225 0.5 1 1.5 2 2.5 3 3.5 4 /(0) θ
Figure 7c Heat transfer coefficient h0(0) against thermal slip parameter d for two values of radiation parameter R.
δ Pr = 0.7 Pr = 0.5 = 0.1, R = 0.1 S λ = 0.1, M = 0, −0.0255 −0.025 −0.0245 −0.024 −0.0235 −0.023 −0.0225 0 0.5 1 1.5 2 2.5 3 3.5 4 /(0) θ
Figure 7d Heat transfer coefficient h0(0) against thermal slip parameter d for two values of Prandtl number Pr.
Table 1 Values of [h0(0)] for several values of Prandtl number, radiation parameter and magnetic parameter in the absence of velocity and thermal slips and suction/blowing.
Pr R M Magyari and Keller[5] Bidin and Nazar[10] El-Aziz[11] Ishak[12] Present study
1 0 0 0.9548 0.9547 0.9548 0.9548 0.9547 2 1.4714 1.4715 1.4714 3 1.8691 1.8691 1.8691 1.8691 1.8691 5 2.5001 2.5001 2.5001 2.5001 10 3.6604 3.6604 3.6604 3.6603 1 1 0.8611 0.8610 0.5 0 0.6765 1 0.5315 0.5312 0.5311 1 0.4505 0.4503 2 0.5 0 1.0735 1.0734 1 0.8627 0.8626 3 0.5 1.3807 1.3807 1 1.1214 1.1213
magnetic field and partial slips at the boundary. Based on the present investigation, the following observations are made:
(i) The effect of suction parameter on a viscous incompress-ible fluid is to suppress the velocity field which in turn causes the enhancement of the skin-friction coefficient. (ii) Due to increasing velocity slip, velocity decreases. With
the increase in thermal slip parameter, temperature decreases.
(iii) Surface shear stress increases as the magnetic parameter increases.
(iv) Magnetic parameter reduces the rate of transport. (v) Wall temperature increases with increasing magnetic
parameter.
(vi) The temperature increases with increasing values of the radiation parameter. This phenomenon is ascribed to a higher effective thermal diffusivity.
(vii) Thermal boundary layer thickness increases with the increase in magnetic parameter as well as radiation parameter.
(viii) Prandtl number reduces the thermal boundary layer thickness.
Acknowledgement
Thanks are indeed due to the reviewers for their constructive suggestions which led to a definite improvement of the quality of the manuscript.
References
[1] Ishak A, Nazar R, Pop I. Hydromagnetic flow and heat transfer adjacent to a stretching vertical sheet. Heat Mass Transfer 2008;44:921–7.
[2] Crane LJ. Flow past a stretching plate. Z Angew Math Phys 1970;21:645–7.
[3] Gupta PS, Gupta AS. Heat and mass transfer on a stretching sheet with suction or blowing. Can J Chem Eng 1977;55:744–6. [4] Pal D. Mixed convection heat transfer in the boundary layers on
an exponentially stretching surface with magnetic field. Appl Math Comput 2010;217:2356–69.
[5] Magyari E, Keller B. Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J Phys D: Appl Phys 1999;32:577–85.
[6] Elbashbeshy EMA. Heat transfer over an exponentially stretching continuous surface with suction. Arch Mech 2001;53:643–51. [7] Khan SK. Boundary layer viscoelastic fluid flow over an
expo-nentially stretching sheet. Int J Appl Mech Eng 2006;11:321–35. [8] Sanjayanand E, Khan SK. On heat and mass transfer in a
visco-elastic boundary layer flow over an exponentially stretching sheet. Int J Therm Sci 2006;45:819–28.
[9] Sajid M, Hayat T. Influence of thermal radiation on the boundary layer flow due to an exponentially stretching sheet. Int Commun Heat Mass Transfer 2008;35:347–56.
[10] Bidin B, Nazar R. Numerical solution of the boundary layer flow over an exponentially stretching sheet with thermal radiation. Eur J Sci Res 2009;33(4):710–7.
[11] El-Aziz MA. Viscous dissipation effect on mixed convection flow of a micropolar fluid over an exponentially stretching sheet. Can J Phys 2009;87:359–68.
[12] Ishak A. MHD boundary layer flow due to an exponentially stretching sheet with radiation effect. Sains Malaysiana 2011;40:391–5.
[13] Yoshimura A, Prudhomme RK. Wall slip corrections for couette and parallel disc viscometers. J Rheol 1998;32:53–67.
[14] Andersson HI. Slip flow past a stretching surface. Acta Mech 2002;158:121–5.
[15] Ariel PD, Hayat T, Asghar S. The flow of an elastico-viscous fluid past a stretching sheet with partial slip. Acta Mech 2006;187:29–35.
[16] Abbas Z, Wang Y, Hayat T, Oberlack M. Slip effects and heat transfer analysis in a viscous fluid over an oscillatory stretching surface. Int J Numer Meth Fluids 2009;59:443–58.
[17] Turkyilmazoglu M. Effects of partial slip on the analytic heat and mass transfer for the incompressible viscous fluid of a porous rotating disk flow. J Heat Transfer 2011;133:122602.
[18] Turkyilmazoglu M. Multiple solutions of heat and mass transfer of MHD slip flow for the viscoelastic fluid over a stretching sheet. Int J Therm Sci 2011;50(11):2264–76.
[19] Turkyilmazoglu M. Analytic heat and mass transfer of the mixed hydrodynamic/thermal slip MHD viscous flow over a stretching sheet. Int J Mech Sci 2011;53(10):886–96.
[20] Turkyilmazoglu M. Multiple solutions of hydromagnetic perme-able flow and heat for viscoelastic fluid. J Thermophys Heat Transfer 2011;25(4):595–605.
[21] Turkyilmazoglu M. MHD fluid flow and heat transfer due to a stretching rotating disk. Int J Therm Sci 2012;51(1):195–201. [22] Brewster MQ. Thermal radiative transfer properties. John Wiley
and Sons; 1972.
Swati Mukhopadhyay was born and brought up in the district of Burdwan, West Bengal, India. She obtained the B.Sc. Honours and M.Sc. degrees in Mathematics from the University of Burdwan. She joined M.U.C. Women’s College, Burdwan as an Assistant Professor in Mathematics in 2006. She was awarded Ph.D. degree in Fluid Mechanics by the University of Burdwan in 2007. Being awarded BOYSCAST Fellowship by the Department of Science and Technology, Govt. of India in 2007–2008, she carried out her post-doctoral research work in NTNU, Trondheim, Norway in 2008. She is serving the Department of Mathematics, the University of Burdwan, West Bengal as an Assistant Professor since 2010. Besides teaching she is actively engaged in research in the field of Fluid mechanics particularly, in bio-fluid dynamics, boundary layer flows, Newtonian/non-Newtonian fluids, heat and mass transfer in porous/non-porous media. Her research interest also covers the Nano fluid flow problems, existence and uniqueness of solutions and other related matters.
