Meccanica (2011) 46:1093–1102 DOI 10.1007/s11012-010-9368-y
Magnetohydrodynamic (MHD) flow of a second grade fluid in a channel with porous wall S.K. Parida · S. Panda · M. Acharya
Received: 16 January 2009 / Accepted: 1 October 2010 / Published online: 27 October 2010 © Springer Science+Business Media B.V. 2010
Abstract An analysis has been carried out to study the effect of magnetic field on an electrically conducting fluid of second grade in a parallel channel. The coolant fluid is injected into the porous channel through one side of the channel wall into the other heated impermeable wall. The combined effect of inertia, viscous, viscoelastic and magnetic forces are studied. The basic equations governing the flow and heat transfer are reduced to a set of ordinary differential equations by using appropriate transformations for velocity and temperature. Numerical solutions of these equations are obtained with the help of RungeKutta fourth order method in association with quasilinear shooting technique. Numerical results for velocity field, temperature field, skin friction and Nusselt number are presented in terms of elastic parameter, Hartmann number, Prandtl number and Reynolds number. Special case of our results is in good agreement with earlier published work.
S.K. Parida Dept. of Maths, NM Institute of Engineering and Technology, Bhubaneswar 751019, India S. Panda () Department of Mathematics, NIT Calicut, Calicut 673601, India e-mail:
[email protected] M. Acharya Dept. of Physics, College of Basic Sc. and Humanities, OUAT, Bhubaneswar 751003, India
Keywords Second-grade fluid · Magnetic field · Porous channel Nomenclature v Dimensional velocity vector T Dimensional temperature p Pressure c Specific heat of the fluid at constant pressure x, y Dimensional space variables k Thermal conductivity h Heat transfer coefficient M Hartmann number Pr Prandtl number F Lorentz force B Magnetic field E Electric field J Current density I Identity tensor First two Rivlin Erickson tensors A1 , A2 t Dimensional time variable K Non-dimensional viscoelastic parameter Nu Nusselt number Re Reynolds number Greek symbols ρ Density Electric conductivity σ0 η Non-dimensional space variable σ Stress tensor μ Dynamic viscosity
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ν Kinematic viscosity θ Non-dimensional temperature αi (i = 1, 2) Material constants Operators ∇ ∇2 d/dt
Nabla operator Laplacian operator Material time derivative
1 Introduction Fluid in which viscosity may depend upon deformation rate and some of them also on elastic behavior i.e. the combine characteristics of solid and fluid are denoted as non-Newtonian. The studies of laminar boundary flow of non-Newtonian fluid have received much attention because the power needed in stretching a sheet and heat transfer rate in a non-Newtonian fluid are quite different from those of a Newtonian fluid. In this case the problems on viscoelastic fluid flow have been extensively studied by various authors [1–4], which includes numerous applications in several industrial manufacturing processes. One class of viscoelastic fluid is of the differential type, namely the second grade. The first model for such a fluid is proposed by Rivlin and Ericksen [5, 6]. The range of application of this model is restricted to materials that are slightly viscoelastic. In order to accommodate flow with moderate or high deformation rate a third order model is later proposed (see, for example, [7]). A complete discussion on second and third order fluid model is given by Dunn and Rajagopal [8]. Recently the application of viscoelastic boundary layer flow extend to the area with additional effects such as heat transfer in porous medium, the effects of magnetic field and flow over a stretching sheet [9]. Various approaches have been proposed in the past and present to investigate the flow through parallel channels due to its range of applications from cooling of electronic devices to that of solar energy collectors. This explain that why the literature is so rich with regard to this kind of flow [10–16]. Further a review series is published by Goldstein et al. [17–19] on heat transfer including section on porous media and channel flow of Newtonian and non-Newtonian fluid. In addition, Ariel [20] has obtained exact analytical solution for the flow problem of second grade fluid through two parallel porous walls and axially symmetric cases. More recently, Kurtcebe and Erim [21] have
studied heat transfer of a second grade fluid flow in a channel with one porous wall. Several authors have studied the effect of cooling or heating of the plates in a small Reynolds number regime. However, they have neglected the viscous dissipation term which might have significant effect over the low range of Reynolds number because the temperature difference between plates is quite higher. Like polymeric additives, magnetic fields are also known to affect flow kinematics in many fluid flows. After the pioneering work of Hartmann and Lazarus [22], which deals with laminar flow of viscous fluids between parallel plates, literature has witnessed similar effects in many other flow geometries. Now a days MHD systems are used effectively in many applications such as power generators, pumps, accelerators, electrostatic filters and droplet purifiers etc. Thus, in order to control the cooling rate and to achieve the desired characteristic of final products, the use of electrically conducting fluids subject to a magnetic field have gained great importance. In the present study the laminar flow of a second grade electrically conducting viscoelastic fluid between two parallel plates separated by a distance H is considered. One of the impermeable plate is extremely heated and cooled by coolant injection through the other porous plate. It is assumed that the fluid is injected through a vertical porous plate with uniform velocity. A constant magnetic field strength B0 is applied transversely along the y-axis (Fig. 1). The basic governing equations of the fluid and temperature are reduced to a set of non-linear ordinary differential equations by using appropriate transformations for velocity and temperature. These equations are then solved numerically. The effect of various physical parameters like viscosity, viscoelastic parameter and magnetic field on the flow pattern and temperature are studied. The skin friction and heat transfer has been analyzed graphically. Unlike the previous heat transfer analysis of Kurtcebe et al. [21], which was restricted to the
Fig. 1 Sketch of flow geometry
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flow of second grade viscoelastic fluid for Reynolds number not greater than 10, we have been able to obtained results for the electrically conducting viscoelastic fluid of second grade for Reynolds number up to 100. Consequently, neglecting the viscous dissipation and magnetic heating effects the cooling and heating performance of the plate are analyzed.
external electric field is zero and induced electric field is negligible, the current density is related to the velocity by Ohm’s law i.e. J = σ0 (v × B). Under the condition that magnetic Reynolds number is small, the induced magnetic field is negligible compared with the applied field. This condition is usually well satisfied in terrestrial applications especially in low velocity free convection flows. Then the term due to Lorentz can be simplified as:
2 Basic equations and their solutions Let us consider steady-state laminar flow of a conducting viscoelastic fluid between two parallel flat plates as shown in Fig. 1. The wall lying along the x-axis is heated externally and from the other perforated wall electrically conducting viscoelastic fluid is injected uniformly in order to cool the heated wall and the yaxis is perpendicular to the x-axis. The magnetic field of uniform field strength B0 is applied transversely along the y-axis. Since two parallel plates are nonconducting an uniform electric field is applied along the z-direction to generate Lorentz force. Neglecting gravitational forces, the governing equations for a steady two-dimensional flow can be written as: • Continuity equation ∇ · v = 0.
(1)
(2)
• Temperature equation ρc (v · ∇) T = k∇ 2 T .
(5)
Viscoelastic fluid is termed as second grade if the Cauchy stress tensor σ is given by [5] σ = −pI + μA1 + α1 A2 + α2 A21 ,
(6)
where p is the hydrostatic pressure, μ is the coefficient of dynamic viscosity, α1 and α2 are the material moduli usually referred to as the normal stress coefficients and I is the identity tensor. The term −pI is the spherical stress due to constraint of in-compressibility. The symbols A1 and A2 are the first two Rivlin-Ericksen tensors and they are defined as A1 = (∇v) + (∇v)∗ ,
• Momentum equation ρ (v · ∇) v = ∇ · σ + F.
F = σ0 (v × B) × B.
A2 =
d A1 + A1 (∇v) + (∇v)∗ A1 , dt
(7)
d where dt is the material time derivative which is ded = ∂t∂ + v · ∇. Here ∂t∂ is the partial derivafined by dt tive with respect to time t and the symbol (*) stands for the transpose operator. The boundary conditions are:
(3) At the upper plate
The symbols v, ρ and T denote the fluid velocity, the constant density and the temperature. Here c and k are the constant heat capacity and the thermal conductivity, and F is the source term due to imposed magnetic field. The symbols σ and ∇ denote the Cauchy stress tensor and gradient operator, respectively. The external force F may be written as: F = J × B,
(4)
where J = σ0 (E + v × B) is the current density and B = (0, B0 , 0) is the transverse uniform magnetic field applied to the fluid layer. Here B0 is the y-component of B. The symbols σ0 and E are the electric conductivity and the electric field, respectively. Further, when
y = H : v = (0, −U ) ,
T = T0 .
(8)
At the lower plate y = 0 : v = 0,
T = T1 .
In the above equations, several assumptions have been made. First, the walls are non-conducting and the flow is steady and laminar. Second, the fluid is incompressible and the body force per unit mass is neglected. Third, the physical properties i.e. viscosity, heat capacity, etc., of the fluid remain invariant throughout the fluid. Fourth, the effects of radiant heating, viscous dissipation and the Hall effects and induced fields are neglected.
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The constitutive model (6) is derived by considering second order approximation of retardation parameter. Dunn and Fosdick [23] have shown that this model equation is invariant under transformation and therefore the material constants must meet the following restriction μ ≥ 0,
α1 ≥ 0,
α1 + α2 = 0.
(9)
The fluids characterized by these restrictions (9) are called second grade fluid. The fluid model represented by (6) with the relationship (9) is compatible with the hydrodynamics. The third relations of (9) is the consequence of satisfying the Clausis-Duhem inequality by fluid motion and a second relation arises due to the assumptions that specific Helmholtz free energy of the fluid takes its minimum values in equilibrium. Generally, the fluid satisfying model (6) with αi < 0 (αi = 1, 2) is termed as second order fluid and with αi > 0 is termed as second grade fluid. Although second order fluid obeying model (6) with α1 < α2 , α1 < 0, exhibits some undesirable instability characteristic (Fosdick and Rajagopal [24]). The second order approximation is valid at low shear rate (Dunn and Rajagopal [8]). In order to obtain solution of present problem, the velocity field is first determined then the solution of energy equation is obtained. The boundary layer equations presented are non-linear partial differential equations and, in general, difficult to solve. However, these equations admit of a self similar solution. Therefore transformation allow them to be reduced to a system of ordinary differential equations and relatively easy to solve numerically. Following Wang and Skalak [25], we look for solution compatible with (1), of the form u=
U xf (η) H
and v = −Uf (η),
(10)
where η = y/H and the prime denotes differentiation with respect to η. The symbols u and v are the velocity components corresponding to the x and y directions, respectively. Using (10) in (6), (7), and using condition (9), and eliminating the pressure term from (2), we arrive at the following equation Re f f − ff = f iv + K Re f f iv − ff v − M2 f ,
(11)
where the typical numbers are the dimensionless Reynolds number Re = ρU H /μ that characterizes the
relation between inertial and viscous forces and the dimensionless Hartmann number M = H B0 (σ0 /μ)1/2 that characterizes the relation between magnetic and viscous forces. The non-dimensional elastic parameter K representing the non-Newtonian character of the fluid is defined by K = α1 /ρH 2 . The corresponding boundary conditions for the velocity field are at η = 1:
f (η) = 1,
f (η) = 0,
at η = 0:
f (η) = 0,
f (η) = 0.
(12)
Equation (12) is the consequence of transformation relation (10). It is observed that (11) contains elastic terms resulting from (6). The terms resulting from A21 are automatically zero. It may be further recorded that for the non-magnetized fluid, i.e. M = 0, (11) together with boundary conditions (12) are the same as those obtained by Kurtcebe and Erim [21]. It is noticed that the presence of the elasticity term K in the fluid yields fourth-fifth order differential equation. It requires an additional boundary conditions to obtain a solution. Moreover, (11) is a non-linear equation and, in general, difficult to solve. In order to overcome this difficulty it is convenient to apply perturbation technique. In the case of viscoelastic fluid, assuming K 1, we write f = f0 + Kf1 + K 2 f2 + · · · .
(13)
Using (13) in (11), and equating the corresponding coefficient of K up to second order, the following set of perturbation relations is obtained. • Zero order Re f0 f0 − f0 f0 = f0iv − M2 f0 .
(14)
• First order Re f0 f1 + f1 f0 − f0 f1 − f1 f0 = f1iv + Re f0 f0iv − f0 f0v − M2 f1 .
(15)
• Second order Re f0 f2 + f2 f0 + f1 f1 − f0 f2 − f2 f0 − f1 f1 ) = f2iv + Re f0 f1iv + f1 f0iv − f0 f1v − f1 f0v − M2 f2 .
(16)
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From (12) and (13) it follows that the boundary conditions for (14)–(16) are:
thermal diffusivity. The corresponding boundary conditions are
f0 (0) = f0 (0) = f0 (1) = f0 (1) − 1 = 0,
θ (0) = 1 and θ (1) = 0.
fi (0) = fi (0) = f1 (1) = fi (1) = 0,
(17)
for i = 1, 2, 3, . . . . The solutions considered in this problem are valid for small value of K. Therefore, we retain up to second order term. In order to solve energy (3), we assume that the porous plate at y = H has the same temperature as that of the incoming coolant, i.e. the heated wall is assumed to have a linear variation about x = 0. So we introduce a temperature field of the following form T = T0 + (T1 − T0 ) θ (η),
(18)
where T0 and T1 are the constant temperature of the porous and impermeable plate, respectively (see (8)). Using usual non-dimensional procedure and substituting (10) and (18) into (3), we obtain θ + Pr Re f θ = 0,
(19)
k where Pr = ν/( cρ ) is the Prandtl number that measure the relative importance of kinematic viscosity to
(20)
In order to solve (19), the functions f0 , f1 and f2 are first determined from (14)–(16), then it is solved numerically.
3 Simulation and results 3.1 Note on the numerical method The numerical method used to solve differential (14)– (16) together with boundary conditions (17), is the Runge-Kutta fourth-order method. Firstly, these equations together with the associated boundary conditions are reduced to first order differential equations. Since equations to be solved are fourth order, the values of fi , fi , fi and fi are needed at the starting point η = 0. As per boundary conditions (17) the values of fi and fi are unknown at η = 0. Therefore, the shooting method (quasi-linear approach) is used to solve the boundary value problem. For the accuracy of the results the missing initial values are tabulated in Tables 1 and 2 for different flow parameters.
Table 1 Missing initial values for various values of Reynolds number with M = 0 Re
f0 (0)
f0 (0)
0.5
6.2266
−13.166
1
6.4539
−14.364
5
8.1662
−24.551 −38.069
f1 (0) −0.1185 −0.5172 −18.412 −78.199
f1 (0) 0.5687 2.5119 102.837 519.15
f2 (0)
f2 (0) −0.5374
0.1264
−4.595
1.0494 231.268 2513.616
−1253.48 −17383.15
10
10.0307
50
19.1801
−145.46
−1218.30
15248.53
308741.02
−4621959.99
100
26.235
−276.042
−3580.47
61171.091
2347756.02
−49143902.94
Table 2 Missing initial values for various values of Reynolds number with M = 6 Re
f0 (0)
f0 (0)
f1 (0)
f1 (0)
0.5
9.0296
−54.73003
−4.2118
35.6206
1
9.1601
−55.8155
−8.6303
73.4602
f2 (0)
f2 (0) 2.4619 12.852
−39.338 −181.004
5
10.161
−64.5958
−44.8014
415.7188
645.543
−7283.79
10
11.3953
−76.3180
−99.0058
998.7073
3491.995
−40195.23
50
19.1405
−176.2755
−1055.89
15469.876
261282.95
−4556315.45
100
26.239
−307.947
−3339.93
61334.86
2173524.73
−48658799.26
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Fig. 2 Leading order velocity profiles at different Hartmann numbers M keeping Reynolds number fixed at Re = 2, left: Tangential velocity, right: Normal velocity
Fig. 3 Leading order velocity profiles at different Reynolds numbers Re keeping Hartmann number fixed at M = 1, left: Tangential velocity, right: Normal velocity
3.2 Results and discussion The effect of different flow parameters like magnetic strength, viscosity, viscoelasticity on the velocity of the fluid as well on the heat transfer coefficient are analyzed. Simulations were performed for different nondimensional parameters like Hartmann (M), Reynolds (Re), Prandtl (Pr) numbers and viscoelastic parameters (K). In Figs. 2–5 the velocity components are plotted for different flow parameters. In Fig. 2 the effects of magnetic strength on the motion of the fluid by varying the Hartman number keeping the Reynolds number constant are analyzed.
When M is small the viscosity dominates the induction drag and the velocity profile is nearly parabolic and clearly for M = 0 (see left panel of Fig. 2), the fluid motion is parabolic. The viscosity is unimportant for large M and its effect is restricted to a thin boundary layer near the wall. Further, away from the wall, velocity is nearly constant for large M because induction drag just balances the pressure gradient. On the other hand, the right panel of Fig. 2 shows that the normal velocity which is independent of Hartmann number about η = 0.42. The effect of viscous force on the fluid motion is described in Fig. 3. The tangential and normal veloc-
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Fig. 4 Velocity profiles for various viscoelastic parameters with Re = 3, M = 6, left: Tangential velocity, right: Normal velocity
Fig. 5 Velocity profiles for various viscoelastic parameters with Re = 3, M = 0, left: Tangential velocity, right: Normal velocity
ities are plotted in the left and right panel of this figure for different values of the Reynolds number, while Hartmann number is kept constant. The right panel of Fig. 3 depicts normal component of velocity. Normal component of velocity increases due to decrease in viscous drag i.e. increase in Re. The variation of the tangential component of velocity vector is given in the left panel of Fig. 3 for various values of the Reynolds number. More precisely, two important observations are made here: (i) At low Reynolds number (Re = 1) the velocity profiles exhibit center-line symmetry indicating a Poiseulle flow and (ii) At high Reynolds number (Re = 100) maximum velocity point is shifted towards the solid wall where shear stress becomes larger as Reynolds number grows. Neglecting magnetic ef-
fect in the flow, the similar results are also predicted in [21, 26]. The distribution of velocity (both x-and y-direction) for various values of the elastic parameter K with Re = 3 are shown in Fig. 4 for the magnetized fluid, i.e. for the Hartman number M = 6 and in Fig. 5 for the non-magnetized fluid, i.e. for M = 0. In the studies of Kurtcebe et al. [21] the higher Reynolds numbers are shown to be responsible for the stronger influence of viscoelastic coefficient on the velocity field. The similar comparison can be made by varying Hartmann number. It is observed from Figs. 4 and 5 that the effect of viscoelastic parameter is prominent in case of higher Hartmann number M = 6 (Fig. 4), whereas for M = 0 (see Fig. 5) its
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Fig. 6 f (0) for various values of K, solid and dotted lines are for the magnetized and non-magnetized fluid, respectively
Fig. 7 Temperature profiles for different viscoelastic parameter K keeping other parameters fixed at Re = 10, M = 6 and Pr = 1
effect on the fluid motion is negligible. The effect of viscoelastic parameter K is to reduce the velocity throughout the flow field, which is quite obvious (left panel Fig. 4). Thus, in the presence of magnetic force, Poiseulle flow is observed for low viscoelastic parameter and center line symmetry in velocity point shifted towards porous wall for high viscoelastic parameter. The problem as formulated in Sect. 2 is quite suitable for solution by using the outlined numerical method in Sect. 3.1. Following numerical procedure the solutions are obtained as demonstrated in the simulations for various Hartmann and Reynolds numbers. The missing initial values for the given boundary value problems (14)–(16) are given in Tables 1 and 2 for different flow parameters. It is evident from the tabulated data that the trends of the missing initial conditions for the first and second order perturbation equations are increasing with increase of Hartmann and Reynolds number. This leads to a limitations on the value of the viscoelastic parameter K. It enables to analyze the wall friction parameter f (0), which is given by ∂u , (21) τyx = μ ∂y y=0
non-conducting fluid, the f (0) is plotted in Fig. 6 against the Reynolds number. The solid lines correspond to the magnetized viscoelastic fluid (M = 6) for various values of K and the dotted lines are associated with the non-magnetized viscoelastic one (M = 0). In this figure only the first order perturbation term is considered for f (0). One can first observe that f (0) decreases with the increasing K values for constant Reynolds number Re. As a matter of fact that an elastic property in a viscoelastic fluid reduces the frictional force. Furthermore, it is also interesting to compare the results of f (0) between the magnetized viscoelastic and non-magnetized viscoelastic fluid. It can be seen from Fig. 6 that the difference is quite significant. Physically this implies that the energy required for the coolant injection is greater in presence of a magnetic field than in the absence of magnetic field. The effect of viscoelastic parameter on temperature field is illustrated in Fig. 7. In order to cool the heated plate coolant fluid is injected from the upper plate which is perforated one. Therefore temperature gradually increases and attains the temperature of impermeable (lower plate) which is heated externally. Viscoelastic parameter causes rapid increase in temperature and attains steady value earlier than Newtonian fluid (Fig. 7). Temperature distribution is shown for various parameter like Re and Pr in Fig. 8. When inertial and viscous forces balance each other i.e. Re = 1, temperature rises steadily towards cooled porous plate in case of same momentum diffusivity and thermal
or, equivalently, in dimensionless form, (21) can be x f (η = 0). According to the tabwritten as τ = μU H2 ulated data the first order term has a decreasing effect on the skin friction whereas the second order term does modify the first order solution for all values of M. To see more clearly how the viscoelastic term influences the skin friction parameter on conducting and
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Fig. 8 Temperature profiles for different Reynolds and Prandtl numbers keeping other parameters fixed at M = 6 and K = 0.02
diffusivity (Pr = 1). Increase in inertia causes further reduction in temperature field i.e. injected coolant reduces the temperature. For laminar flow increase in momentum diffusivity reduces thermal boundary layer. The rate of heat transfer at the impermeable heated wall is discussed in terms of the non-dimensional Nusselt number. Following (18) the heat transfer coefficient h can be defined as ∂T , (22) h (T0 − T1 ) = −k ∂y and the non-dimensional Nusselt number is then obtained as Nu =
hH = −θ (η = 0). k
(23)
Nusselt number Nu is calculated for Reynolds number up to Re = 100 and for a wide range of Prandtl number. However, the maximum value of Peclet number Pe = Pr × Re is limited to Pe ≤ 100 and therefore the range of Prandtl number Pr is linked to the value of Reynolds number Re. At high Peclet numbers (Pe > 100) the numerical method was unable to converge on a solution. A possible reason for the poor convergence of the model at large Peclet numbers is that the numerical method used to solve (19) was the fourth-order Runge-Kutta method. A system of two differential equations of first order are obtained by reducing (19), and then solved. But at high Peclet number the second order term of (19) become less
Fig. 9 Nusselt number versus Reynolds number: effect of K, Pr and M, dotted and solid lines are for the magnetized and non-magnetized fluid, respectively
important since the first order term dominates due to high convection and the equation reduced to first order. The Nusselt number Nu is plotted against Re in Fig. 9 for various values of Hartmann number, Prandtl number and viscoelastic parameter. It can be clearly seen that Nusselt number is increasing with the increase of Prandtl number. However the same is not true for the Hartmann number M, which reduces the Nusselt number when compared with non-magnetized fluid (i.e. for M = 0). Another interesting observation is that within the range of Reynolds numbers (0 < Re < 20) and for small Prandtl number, nonNewtonian flow changes into Newtonian flow. Thus for small Reynolds number viscoelastic flow, the cohesion between fluid particles increases while the mobility of the fluid decreases and therefore heat transfer rate decreases. But, when the Reynolds number increases i.e. Re ≥ 20, the convection dominates viscous dissipation and elastic effect is negligible on the rate of heat transfer. Hence convective heat transfer increases with the increase of Reynolds number for all values of the elastic parameter K.
4 Conclusion Laminar flow of a incompressible second grade conducting fluid between two parallel flat plates is considered, where one of the impermeable plate is externally heated and cooled by coolant injection through
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the other vertical porous plate. It is observed in the presence of uniform magnetic field the tangential velocity component decreases with increasing induction drag and velocity remains constant away from the wall. The effect of viscoelastic parameter on the velocity field is significant for higher values of M. It is found that the skin friction is more for the higher magnetic force even though it decreases with the increasing viscoelastic parameter. The solution of the heat transfer equation reveals that Prandtl number (Pr) controls the relative thickness of thermal boundary layer. Small Pr indicates that heat diffuses quickly. Further, the heat transfer on impermeable wall is considerable higher for non-conducting viscoelastic fluid as compared with the conducting fluid. Moreover, heat transfer rate is higher for high Prandtl number fluid. On the other hand, viscoelastic parameter in the presence of strong magnetic field decreases the heat transfer on the impermeable wall within a small range of Reynolds number. It is further seen that the effect of elastic parameter is negligible for higher values of Reynolds number. Acknowledgement The authors are grateful to anonymous reviewer for constructive suggestions which help the overall improvement of the paper.
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