Meccanica (2011) 46:1103–1112 DOI 10.1007/s11012-010-9371-3

Effect of Hall current on MHD mixed convection boundary layer flow over a stretched vertical flat plate F.M. Ali · R. Nazar · N.M. Arifin · I. Pop

Received: 14 February 2010 / Accepted: 12 October 2010 / Published online: 30 October 2010 © Springer Science+Business Media B.V. 2010

Abstract In this paper, the steady magnetohydrodynamic (MHD) mixed convection boundary layer flow of an incompressible, viscous and electrically conducting fluid over a stretching vertical flat plate is theoretically investigated with Hall effects taken into account. The governing equations are solved numerically using an implicit finite-difference scheme known as the Keller-box method. The effects of the magnetic parameter, the Hall parameter and the buoyancy parameter on the velocity profiles, the cross flow velocity profiles and the temperature profiles are presented graphically and discussed. Investigated results indicate that the Hall effect on the temperature is small, and the magnetic field and Hall currents produce opposite ef-

F.M. Ali Department of Mathematics, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia R. Nazar () School of Mathematical Sciences, Faculty of Science & Technology, Universiti Kebangsaan Malaysia, 43600 UKM Bangi, Selangor, Malaysia e-mail: [email protected] N.M. Arifin Department of Mathematics & Institute for Mathematical Research, Universiti Putra Malaysia, 43400 UPM Serdang, Selangor, Malaysia I. Pop Faculty of Mathematics, University of Cluj, 3400 Cluj, CP 253, Romania

fects on the shear stress and the heat transfer at the stretching surface. Keywords Stretched flat plate · Hall effect · Magnetohydrodynamic · Mixed convection · Boundary layer Nomenclature a, c constants the strength of the imposed magnetic field B0 skin friction coefficient in x-direction Cf x skin friction coefficient in z-direction Cf z e electric charge (C) f dimensionless stream function g acceleration due to gravity (m s−2 ) local Grashof number Grx external magnetic field H0 m Hall parameter the mass of an electron (kg) me M magnetic parameter electron number density ne Pr Prandtl number local Reynolds number Rex T fluid temperature (K) electron collision time (s) Te surface temperature (K) Tw ambient temperature (K) T∞ u, v, w velocity components along the x, y and z directions, respectively (m s−1 ) uw (x) velocity of the stretching plate (m s−1 )

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x, y, z Cartesian coordinates along the stretching surface, normal to it, and transverse to the xy plane, respectively (m) Greek Letters α thermal diffusivity (m2 s−1 ) β thermal expansion coefficient (1/K) λ constant buoyancy or mixed convection parameter θ dimensionless temperature ν kinematic viscosity (m2 s−1 ) μ dynamic viscosity (kg m−1 s−1 ) μe magnetic permeability (H m−1 ) ρ fluid density (kg m−3 ) τw wall shear stress (Pa) Subscripts w condition at the surface ∞ ambient condition

1 Introduction Study of stretching surfaces and the several combinations of additional effects on the stretching problems are important in many practical applications because the production of sheeting material arises in a number of industrial manufacturing processes and includes both metal and polymer sheets. In the manufacture of the latter, the material is in a molten phase when thrust through an extrusion die and then cools and solidifies some distance away from the die before arriving at the collecting stage. The quality of the resulting sheeting material, as well as the cost of production, are affected by the speed of collection and the heat transfer rate, and a knowledge of the flow properties of the ambient fluid is clearly desirable (see Banks and Zaturska [1]). On the other hand, it should be pointed out that the very important practical problems of the thermal processing of sheet-like materials which is a necessary operation in the production of paper, linoleum, polymeric sheets, roofing shingles, insulating materials, fine-fiber matts are described in the excellent papers by Sparrow and Abraham [2] and Abraham and Sparrow [3]. Other very important applications of the stretching sheets are described in the papers by Lakshmisha et al. [4] and Kumari et al. [5]. We note that in addition to the polymer-sheet driven flows mentioned above, such flows can be generated in a number of other ways: (i) at an air-water interface (Taylor

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[6]), (ii) in oscillatory viscous flow resulting from the steady streaming component (Stuart [7]), and (iii) in the excitation of liquid metals when placed in a highfrequency magnetic field (Moffatt [8]). It is worth mentioning that Sakiadis [9] was the first to study the boundary layer flow on a continuously moving solid surface. Crane [10] extended this idea for the steady two-dimensional boundary layer flow due to a stretching sheet whose velocity varies linearly with the distance from a fixed point in the sheet. Recently, the study of magnetohydrodynamics (MHD) become important in engineering applications, such as in designing cooling system with liquid metals, MHD generator and other devices in the petroleum industry. For ionized gases, the conventional MHD is not valid under the strong electric field. In an ionized gas, the density is low and the magnetic field is very strong, the conductivity normal to the magnetic field is reduced due to the free spiralling of electrons and ions about the magnetic lines of force before suffering collisions, and also a current is induced in the direction normal to both electric and magnetic fields. This phenomenon is also known as the Hall effect (Sato [11], Sutton and Sherman [12], Pop [13], Raptis and Ram [14], Hossain and Rashid [15], Watanabe and Pop [16], Gupta and Takhar [17], Pop et al. [18], Ghosh and Pop [19]). The case of non-isothermal stretching flat plate in the presence of a transverse magnetic field where the Hall effect has been neglected was considered by Ezzat et al. [20] and Zakaria [21]. Yih [22] studied the effect of free convection on MHD heat and mass transfer of a continuously moving permeable vertical surface, while Abo-Eldahab [23] discovered the combined free convection heat and mass transfer effects on MHD three-dimensional flow. On the other hand, MHD stagnation-point flow towards a stretching vertical sheet with transverse uniform magnetic field has been studied by Ishak et al. [24, 25] in viscous and micropolar fluids, respectively. The study of MHD free convection flow along a vertical surface with the effects of Hall current over a porous plate has been considered by Hossain [26], Hossain and Rashid [15] and Hossain and Mohammad [27], while Ram [28] has considered the problem in porous medium. Megahed et al. [29] studied MHD free convection flow over a semi-infinite vertical flat plate with Hall effects taken into account, while Abo-Eldahab et al. [30] investigated the effect of Hall current on MHD mixed convection flow over an inclined permeable

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continuously stretching plate with internal heat generation/absorption and blowing/suction. Also, Salem and Abd El-Aziz [31] have studied the effects of Hall current and chemical reaction on the hydromagnetic flow of a stretching vertical surface. Recently, Ghosh et al. [32] have studied the Hall effects in a parallel plate channel, while Abd El-Aziz [33] has analyzed the effects of Hall currents on the flow and heat transfer of an electrically conducting fluid over an unsteady stretching surface in the presence of a strong magnetic field. The study of both MHD and Hall effect has important engineering applications such as in the problems of Hall accelerators and also in flight magnetohydrodynamics. MHD flows is also important for several practical applications, such as, improved MHD energy generators, planetary fluid dynamics, electromagnetic materials processing, control of crystal growth systems, etc. In this present work, we study the Hall effect on MHD mixed convection boundary layer flow over a stretched vertical flat plate. The partial differential equations are reduced to similarity or nonlinear ordinary differential equations which are solved numerically.

2 Analysis Consider the steady mixed convection flow of an incompressible, viscous and electrically conducting fluid past a stretching flat plate in the vertical direction with a velocity proportional to the distance from the fixed origin O of a stationary frame of reference (x, y, z), as shown in Fig. 1. The frame of reference

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(x, y, z) is chosen such that the x-axis is along the direction of motion of the surface, the y-axis is normal to the surface and the z-axis is transverse to the xy-plane. We consider that an external constant magnetic field H0 is applied in the positive y-direction. It is also assumed that the surface of the sheet has a variable temperature Tw (x), while the ambient fluid has a uniform temperature T∞ , where Tw (x) > T∞ corresponds to a heated plate and Tw (x) < T∞ corresponds to a cooled plate. The basic governing equations in vectorial form are ∇ · v = 0,

(1)

1 (v · ∇)v = − ∇p + ν∇ 2 v + (μe /ρ)j × H ρ + ρβ(T − T∞ )g,

(2)

(v · ∇)T = α∇ 2 T ,

(3)

∇ × H = j,

(4)

∇ × E = 0,

(5)

∇ · H = 0,

(6)

where v, H, j, E, g, ρ, ν, α, β and μe are the velocity vector of the fluid, the vector of the magnetic field, the vector of the electric current density, the vector of the electric field, the vector of the gravity acceleration, density, kinematic viscosity, thermal diffusivity, thermal expansion coefficient and magnetic permeability, respectively. Taking Hall effects into account and assuming that the electron pressure gradient, the ion slip and the thermo-electric effects are neglected, the generalized Ohm’s law can be written as (Cowling [34])   μe j×H (7) j = σ E + μe v × H − ene where the electron pressure gradient and ion-slip effects are neglected. In (7), ne and e stand for the electron number density and the electric charge, respectively and the electrical conductivity, σ is given by σ = e2 ne Te /me

Fig. 1 Physical model and coordinate system

(8)

where Te and me denote the electron collision time and the mass of an electron, respectively. The effect of Hall current gives rise to a force in the z-direction resulting in a cross-flow in this direction and thus the flow becomes three-dimensional. Using the boundary layer variables with the observation that the physical

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variables do not depend on the coordinate z, it can be shown that (1)–(8) can be reduced to the following equations: ∂u ∂v + = 0, ∂x ∂y u

(9)

∂u ∂u ∂ 2 u B0 +v =ν 2 − jz + gβ(T − T∞ ), ∂x ∂y ρ ∂y

∂w ∂w ∂ 2 w B0 u +v =ν 2 + jx , ∂x ∂y ρ ∂y ∂T ∂T ∂ 2T +v =α 2 , ∂x ∂y ∂y  σ  jx = Ex − B0 w + m(Ez + B0 u) , 2 1+m jy = σ Ey ,  σ  jz = Ez + B0 u − m(Ex − B0 w) , 1 + m2 u

(21)

(13)

u = cxf  (η),

(14) (15)

(17)

u

∂T ∂T ∂ 2T +v =α 2 ∂x ∂y ∂y

(19)

subject to the boundary conditions v = w = 0, (20)

v = −(cν)1/2 f (η),

w = cxh(η),

(22)

θ (η) = (T − T∞ )/(Tw − T∞ ),

η = (c/ν)1/2 y

where primes denote differentiation with respect to η. By substituting (22) into (17)–(19), we obtain the following ordinary differential equations: M (f  + mh) + λθ = 0, (23) 1 + m2 M h + f h − f  h + (mf  − h) = 0, (24) 1 + m2 1  θ + f θ  − f θ = 0 (25) Pr f  + ff  − f 2 −

and the boundary conditions (20) become f (0) = 0,

f  (0) = 1,

h(0) = 0,

θ (0) = 1,

f  (∞) = 0,

B02 σ ∂w ∂w ∂ 2w +v =ν 2 + (mu − w), (18) ∂x ∂y ∂y ρ(1 + m2 )

at y = 0,

Tw (x) = T∞ + ax

(12)

(11)

u

T = Tw (x)

as y → ∞.

where c(> 0) and a are constants. We notice that for a > 0 (Tw (x) > T∞ ), the plate is heated and for a < 0 (Tw (x) < T∞ ), the plate is cooled. We look for a similarity solution of (16)–(19) of the form

B02 σ ∂u ∂u ∂ 2u u +v =ν 2 − (u + mw) ∂x ∂y ∂y ρ(1 + m2 )

u = uw (x),

T = T∞

In order for (16)–(19) admit a similarity solution, we assume that uw (x) and Tw (x) vary linearly with the variable x as uw (x) = cx,

(16)

+ gβ(T − T∞ ),

w = 0,

(10)

where m = eB0 Te /me is the Hall parameter, B0 = μe H0 is the strength of the imposed magnetic field and (u, v, w), (0, B0 , 0), (Ex , Ey , Ez ) and (jx , jy , jz ) are the components of v, B, E and j, respectively. In this study, we assume that the magnetic Reynolds number is small (Rem  1) so that the induced magnetic field is negligible. jy is a constant due to the equation of conservation of electric charge ∇ · j = 0. Since jy = 0 at the plate, which is electrically non-conducting, it results in that jy = 0 everywhere in the flow. Further, it can be shown that Ex = Ez = 0 and Ey = E(y) (Gupta and Takhar [17]). Thus, (9)–(15) reduce to ∂u ∂v + = 0, ∂x ∂y

u = 0,

h(∞) = 0,

(26) θ (∞) = 0

where Pr is the Prandtl number, M is the magnetic parameter and λ is the constant buoyancy or mixed convection parameter which are given by M=

σ B02 , cρ

λ=

Grx Re2x

(27)

with Grx = gβ(Tw − T∞ )x 3 /ν 2 is the local Grashof number and Rex = uw (x)/ν is the local Reynolds number. It should be mentioned that λ > 0 corresponds to the assisting flow (heated plate) while λ < 0 corresponds to the opposing flow (cooled plate) and λ = 0 corresponds to the forced convection flow. The

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Table 1 Values of f  (0) and −θ  (0) for various values of M with m = 0, 2, λ = 1, Pr = 1 M

Ishak et al. (m = 0) [36]

Present results (m = 0)

(m = 2)

f  (0)

f  (0)

f  (0)

−θ  (0)

−θ  (0)

−θ  (0)

0

−0.5607

1.0873

−0.5607

1.0873

−0.5608

1.0873

0.01

−0.5658

1.0863

−0.5658

1.0863

−0.5618

1.0871

0.04

−0.5810

1.0833

−0.5810

1.0833

−0.5649

1.0865

0.25

−0.6830

1.0630

−0.6830

1.0630

−0.5878

1.0816

1

−1.0000

1.0000

−1.0000

1.0000

−0.6816

1.0591

4

−1.8968

0.8311

−1.8968

0.8311

−1.0779

0.9494

25

−4.9155

0.4702

−4.9155

0.4702

−2.7598

0.5688

quantities of physical interest are the skin friction coefficients Cf x and Cf z as well as the Nusselt number Nu, which are defined as τwx 2 τwz 2 , Cf z = , ρuw ρuw xqw Nu = k(Tw − T∞ )

(28)

where τwx and τwz are the wall shear stresses in the directions of x and z, respectively, while qw is the heat flux from the surface of the flat plate, which are given by     ∂u ∂w , τwz = μ , τwx = μ ∂y y=0 ∂y y=0 (29)   ∂T qw = −k ∂y y=0 with k being the thermal conductivity of the fluid. Using (22), we get 1/2

−1/2

NuRex

Cf z Rex = h (0),

= −θ  (0)

Pr

Grubka and

Ali [38]

Yih [22]

Present

Bobba [37]

Cf x =

Cf x Rex = f  (0),

Table 2 Values −θ  (0) for various values of Pr with m = 0, λ = 0 and M = 0

1/2

(30)

3 Results and discussion Equations (23) to (25) subject to the boundary conditions (26) are solved numerically using an implicit finite-difference scheme known as the Kellerbox method, as described in the book by Cebeci and Bradshaw [35]. Comparisons are made graphically with the figures plotted in Gupta and Takhar [17] and numerically with the values obtained in Ishak et al.

0.01

0.0197

–

0.0197

0.0198

0.72

0.8086

0.8058

0.8086

0.8086

1

1.0000

0.9961

1.0000

1.0000

3

1.9237

1.9144

1.9237

1.9237

10

3.7207

3.7006

3.7207

3.7208

100

12.2940

12.2940

12.3004

–

[36], Grubka and Bobba [37], Ali [38] and Yih [22], for the skin friction coefficient in the direction of x, f  (0) and the local Nusselt number, −θ  (0) as shown in Tables 1 and 2, for various values of M and Pr, respectively. The comparisons are found to be in excellent agreement. Therefore, we are confident that the present results are accurate. However, we have considered only the assisting flow (λ > 0) and forced convection flow (λ = 0) cases, and the value of the Prandtl number of Pr = 0.02 has been used. It should be pointed out that we have presented most of the results by taking the value of Pr = 0.02 (such as metallic alloys) due to comparison purposes with those of Gupta and Takhar [17] and for the nature of the problem itself. It is worth mentioning that for an electrically conducting fluid (like fluid metal), the Prandtl number Pr is very small. However, we have also presented results for other values of Pr such as those presented in Tables 1 to 3 and in Fig. 8. The new results for the case of m = 2 are also included in Table 1 in order to see the changes in the values of f  (0) and −θ  (0) when the Hall effect is taken into account, i.e. m = 0. It is shown in Table 1

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Table 3 Values of f  (0) and −θ  (0) for various values of M with m = 2, λ = 1, Pr = 0.02 and 6.2 M

Pr = 0.02

Pr = 6.2

f  (0)

−θ  (0)

f  (0)

−θ  (0)

0

−0.1273

0.1420

−0.7853

2.9120

0.01

−0.1287

0.1410

−0.7863

2.9117

0.04

−0.1330

0.1418

−0.7894

2.9110

0.25

−0.1659

0.1403

−0.8119

2.9054

1

−0.3101

0.1323

−0.9028

2.8815

4

−0.9141

0.0895

−1.2597

2.7803

25

−2.74137

0.03584

−2.8138

2.2644 Fig. 3 Cross flow velocity profiles h(η) for some values of M with fixed m = 1, Pr = 0.02 and λ = 5

Fig. 2 Velocity profiles for some values of M with fixed m = 1, Pr = 0.02 and λ = 5

that when M = 0, the Hall parameter has the effect of decreasing the magnitude of the skin friction coefficient in the direction of x, f  (0) and increasing the local Nusselt number, −θ  (0). On the other hand, large values of the magnetic field parameter M are taken into account, because in order that the Hall effects to become important, the applied magnetic field must be quite strong. Further, it should be noted that Table 3 presents the numerical values of f  (0) and −θ  (0) for various values of M with m = 2, λ = 1, when Pr = 0.02 and 6.2. It is observed that as the magnetic parameter M increases, the magnitude of the skin friction coefficient f  (0) also increases while the local Nusselt number, −θ  (0) decreases. It is also found that as the Prandtl number Pr increases, both the magnitude of f  (0) and −θ  (0) increase, with the increase in f  (0) for large values of M is less than for smaller values of M. Figures 2, 3 and 4 show the effects of magnetic parameter, M on the velocity f  (η), the cross flow ve-

Fig. 4 Temperature profiles for some values of M with fixed m = 1, Pr = 0.02 and λ = 5

locity h(η) and the temperature θ (η) profiles, respectively. The curve for the case M = 0 (magnetic field is absent) is also included in each figure in order to see the changes. As expected, the velocity f  (η) and the cross flow velocity h(η) profiles decrease while the temperature profiles θ (η) increase with the increase in M. When M increases, this will also increase the Lorentz force which opposes the flow and leads to enhanced deceleration of the velocity profiles. Further, Figs. 5, 6 and 7 show the effects of the Hall parameter, m on the velocity f  (η), the cross flow velocity h(η) and the temperature θ (η) profiles, respectively. The curve for the case m = 0 (Hall effect is absent) is also included in each figure in order to see the changes. The effect of increasing m has increased the velocity profiles f  (η) and the cross flow velocity profiles h(η), however it decreases the temperature profiles θ (η). It is observed that the effect of the Hall

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Fig. 5 Velocity profiles for some values of m with fixed M = 40, Pr = 0.02 and λ = 5

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Fig. 7 Temperature profiles for some values of m with fixed M = 40, Pr = 0.02 and λ = 5

Fig. 6 Cross flow velocity profiles h(η) for some values of m with fixed M = 40, Pr = 0.02 and λ = 5

Fig. 8 Temperature profiles for some values of Pr with fixed M = 40, m = 1 and λ = 5

parameter m on the temperature is small and that the magnetic and Hall parameters have opposite effects of the velocity and temperature profiles. On the other hand, Fig. 8 illustrates the temperature profiles for some values of Pr with fixed M = 40, m = 1 and λ = 5. It is shown in this figure that as the Prandtl number Pr increases, the temperature profiles decrease. At large Pr, the thermal boundary layer is thinner than at a smaller Pr. This is because for small values of the Prandtl number (Pr  1), the fluid is highly conductive. Physically, if Pr increases, the thermal diffusivity decreases and these phenomena lead to the decreasing of energy ability that reduces the thermal boundary layer. The results which incorporate the effect of buoyancy or mixed convection parameter λ on f  (η), h(η) and θ (η) are illustrated in Figs. 9, 10 and 11, respectively. The curve for the case λ = 0 (forced convection

flow) is also included in each figure in order to see the changes from forced convection to mixed convection flows. When m and M are fixed at m = 1 and M = 10, the maximum value of the velocity and the cross flow velocity profiles increase with increasing the parameter λ, while the temperature profiles decrease with increasing λ. It is shown that the thickness of the velocity boundary layer is much smaller than the thickness of the thermal boundary layer due to the small value of Pr, and this is a well-known fact for low Prandtl number fluids (Schlichting and Gersten [39]). Figures 12 and 13 display the variations of the skin 1/2 friction coefficient in the direction of x, Cf x Rex and −1/2 with λ for varithe local Nusselt number NuRex ous values of the Hall parameter m at fixed value of M = 20 and Pr = 0.02, respectively. It can be seen 1/2 −1/2 that both Cf x Rex and NuRex increase with the increase of m. On the other hand, Figs. 14 and 15 il-

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Fig. 9 Velocity profiles for some values of λ with fixed m = 1, Pr = 0.02 and M = 10

Fig. 11 Temperature profiles for some values of λ with fixed m = 1, Pr = 0.02 and M = 10

Fig. 10 Cross flow velocity profiles h(η) for some values of λ with fixed m = 1, Pr = 0.02 and M = 10

Fig. 12 Variation of the skin friction coefficient in the direction of x with λ for some values of m and fixed M = 20 and Pr = 0.02

lustrate the variations of the skin friction coefficient 1/2 in the direction of x, Cf x Rex and the local Nus−1/2 with λ for various values of the selt number NuRex magnetic parameter M at fixed value of m = 0.5 and 1/2 Pr = 0.02, respectively. It is found that both Cf x Rex −1/2 and NuRex decrease when M increases. Thus, the magnetic field and the Hall currents produce opposite effects on the skin friction coefficient and the local Nusselt number.

4 Conclusions A numerical study is performed for the problem of the steady MHD mixed convection boundary layer flow of an incompressible, viscous and electrically conducting fluid over a stretching vertical flat plate when the Hall

Fig. 13 Variation of the local Nusselt number with λ for some values of m and fixed M = 20 and Pr = 0.02

effects are considered. In this study, only the assisting flow (λ > 0) and forced convection flow (λ = 0)

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1111

the thermal diffusivity decreases and these phenomena lead to the decreasing of energy ability that reduces the thermal boundary layer. Acknowledgements The authors gratefully acknowledge the financial supports received in the form of a research university grant scheme (RUGS) from the Universiti Putra Malaysia and FRGS grant from the Ministry of Higher Education, Malaysia. The authors are also grateful to the anonymous reviewers for the valuable comments and suggestions, which led to the improvement of the paper.

Fig. 14 Variation of the skin friction coefficient in the direction of x with λ for some values of M and fixed m = 0.5 and Pr = 0.02

Fig. 15 Variation of the local Nusselt number with λ for some values of M and fixed m = 0.5 and Pr = 0.02

cases are considered. The numerical results when the Hall parameter and the buoyancy parameter are absent have been compared with previously published results and it has been found that the agreement is very good. We observed that the effect of the Hall parameter m on the temperature profile is small. The magnetic field and Hall currents produce opposite effects on the shear stress and the heat transfer at the stretching surface. On the other hand, small value of Pr produces a smaller velocity boundary layer thickness than the thermal boundary layer thickness. Large values of the magnetic field parameter M are taken into account, because in order that the Hall effects to become important, the applied magnetic field must be quite strong. It is also found that at large Pr, the thermal boundary layer is thinner than at a smaller Pr. This is because for small values of the Prandtl number (Pr  1), the fluid is highly conductive. Physically, if Pr increases,

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