MHD mixed convection stagnation point flow of a ...

Industrial Lubrication and Tribology MHD mixed convection stagnation point flow of a viscous fluid over a lubricated vertical surface Khalid Mahmood, Muhammad Sajid, Nasir Ali, Tariq Javed,

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MHD mixed convection stagnation point flow of a viscous fluid over a lubricated vertical surface Khalid Mahmood Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan

Muhammad Sajid Downloaded by INTERNATIONAL ISLAMIC UNIVERSITY, Mr Khalid Mahmood At 04:25 09 August 2017 (PT)

Theoretical Physics Division, PINSTECH, Islamabad, Pakistan, and

Nasir Ali and Tariq Javed Department of Mathematics and Statistics, International Islamic University, Islamabad, Pakistan Abstract Purpose – An attempt is made to study magnetohydrodynamic viscous fluid impinging orthogonally toward a stagnation point on a vertical surface lubricated with power law fluid. It has been assumed that the surface temperature varies linearly with the distance from the stagnation point. The problem is governed by system of partial differential equations for both the base fluid and the lubricant. The continuity of velocity and shear stress is assumed at the interface layer between the base fluid and the lubricant. Dimensionless variables are introduced to transform original problem into ordinary differential equations. An implicit finite-difference scheme known as the Keller-Box method is implemented to obtain the numerical solutions. The influence of various important parameters is presented in the form of graphs and tables. The limiting cases for full and no-slip conditions are deduced from the present solutions. A comparison of the present results with the existing results in the special case validates the obtained numerical solutions. The purpose of this study is to see the behaviour of flow characteristics in the presence of lubrication. Design/methodology/approach – The authors’ problem is governed by system of partial differential equations for both the base fluid and the lubricant. Dimensionless variables are introduced to transform original problem into ordinary differential equations. The obtained ordinary differential equation along with boundary conditions are highly nonlinear and coupled. An implicit finite-difference scheme known as the Keller-Box method is implemented to obtain the numerical solutions. Findings – Some findings of this study are that the lubricant increases the velocity of the base fluid inside the boundary layer. In the case of full slip, the effects of viscosity are suppressed by the lubricant. The temperature of the base fluid decreases by increase in lubrication on the surface. By increasing the slip on the surface, the skin friction decreases and local Nusselt number increases, but the rate of increase or decrease is less in magnitude for the case of opposing flow. The similarity solutions only exist for n ⫽ 1/2. A non-similar solution is obtained for the other values of the power-law index n. Originality/value – The study of flow phenomenon over a lubricated surface has important applications in machinery components such as fluid bearings and mechanical seals. Coating is another major application of lubrication including the preparation of thin films, printing, painting, etc. The authors hope that the current study will provide the roadmap for the future studies in this direction. Keywords MHD, Mixed convection, Stagnation-point flow, Keller-box method, Power law lubricant, Vertical surface Paper type Research paper

1. Introduction

convection stagnation-point flow is another prime area of significant importance. The temperature difference between the surface and the fluid having the free stream velocity generates the buoyancy forces. These buoyancy forces have remarkable effects on the temperature and velocity of the fluid. Due to which, the wall shear stress and heat transfer rate can be augmented or reduced significantly. The buoyancy forces may reinforce or oppose the fluid flow depending on the direction of flow, and, as a result, one can enhance or reduce the heat transfer rate. The problem under consideration would make possible for us to know how stagnation-point flows develop a boundary layer and how different parameters effect the boundary layer.

The flow of fluids impinging orthogonally over a vertical surface has a vital role in the various fields. The technical process such as cooling of nuclear reactors, extrusion of polymer sheets, cooling of computer and other electronic devices, manufacturing of artificial fibers and many hydrodynamic mechanisms involve the stagnation-point flows. In the stagnation-point flows, the fluid strikes the rigid or stretching surface either orthogonally or obliquely. Mixed

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Industrial Lubrication and Tribology 69/4 (2017) 527–535 © Emerald Publishing Limited [ISSN 0036-8792] [DOI 10.1108/ILT-02-2016-0025]

Received 13 February 2016 Revised 19 April 2016 Accepted 12 May 2016

527


MHD mixed convection stagnation point flow

Industrial Lubrication and Tribology

Khalid Mahmood, Muhammad Sajid, Nasir Ali and Tariq Javed

Volume 69 · Number 4 · 2017 · 527–535

The problems on magnetohydrodynamic (MHD) stagnation-point flows was studied by Chamkha (1998), and Chamkha and Issa (1999). Kumari (2001) found effects of viscosity on boundary layer flow by considering free as well as mixed convection. A similar problem has been analyzed by Prasad et al. (2010) on a non-linear stretching sheet. Because of significant importance in metal working and modern metallurgical processes, many researchers have been investigating different aspects of MHD flows of various fluids. The produced electromagnetic field generates a Lorentz force that effects the wall shear stress and heat transfer coefficient at the surface. In his investigation, Attia (2003) reported that increasing magnetic field affects velocity field and boundary layer thickness. Impact of magnetic field on Maxwell fluid for steady as well as unsteady cases has been addressed by Kumari and Nath (2009). They were of the view that magnetic parameter alters both velocity profile and heat transfer. Singh et al. (2010a, 2010b) investigated effects of magnetic and radiation parameters on stretching surface by considering steady as well as unsteady flows. Ziya et al. (2009) and Ziya and Kumar (2008) studied MHD-free convection flow on an inclined porous plate at high temperature. Mahapatra et al. (2009) carried out MHD stagnation-point flow of a power-law fluid over a stretched sheet. Mixed convection flow of a Newtonian fluid impinging orthogonally on a heated vertical permeable surface has been investigated by Abdelkhalek (2006). Aydin and Kaya (2007) presented mixed convection flow with viscous dissipation adjacent to a flat wall. Ramachandran et al. (1998) investigated a flow problem for a viscous fluid due to mixed convection stagnation point along a vertical wall. Hayat et al. (2008) extended the work of Ramachandran et al. (1998) by taking viscoelastic fluid. The problem of Ramachandran et al. (1998) was also reconsidered by Devi et al. (1991) for unsteady case. A paper explaining heat transfer and flow characteristics due to micropolar fluid impinging orthogonally over a non-isothermal vertical surface was presented by Hassanien and Gorla (1990). Lok et al. (2006) considered time-dependent mixed convective stagnation-point flow of a micropolar fluid along a vertical sheet. Aman et al. (2013) discussed slip effects on mixed convection flow near stagnation point on a vertical plate. MHD stagnation-point flow with mixed convection along a vertical wall has been incorporated by Ali et al. (2011). An analysis has been carried out by Ishak et al. (2010) by considering MHD mixed convection flow of electrically conducting viscous fluid in the stagnation zone adjacent to a vertical permeable wall. In another investigation, Hayat et al. (2010) found the analytical solution of the MHD stagnation-point flow with mixed convection over a vertically stretching porous sheet. The flow of a viscous fluid near stagnation point over a flat plate using slip boundary condition was studied by Wang (2003). Blyth and Pozrikidis (2005) investigated the flow of a viscous fluid stagnating on another viscous fluid over a vertical wall. Andersson and Rousselet (2006) considered the slip flow over a lubricated rotating disc. They used a viscous fluid as a base fluid and a power-law fluid as a lubricant and deduced slip boundary condition at the fluid-lubricant interface. Axisymmetric flow of a viscous fluid stagnated on a lubricated stationary disk was carried

out by Santra et al. (2007). He also used a power-law fluid as a lubricant. Recently, Sajid et al. (2012) examined axisymmetric flow of a viscous fluid near stagnation point for a generalized slip boundary condition formulated by Thompson and Troian (1997). More recently, Sajid et al. (2013) discussed viscoelastic fluid flow in the stagnation region produced over a lubricated sheet. In the present investigation, we are interested to analyze MHD flow of Newtonian fluid with mixed convection impinging orthogonally on a vertical wall lubricated with a thin coating of a power-law fluid. Our aim is to investigate how slip parameter effects the velocity and temperature distributions in the presence of other parameters. A well-known method called Keller-box method (Na, 1979; Cebeci and Bradshaw, 1954; Keller and Cebeci, 1972; Keller, 1970; Ahmad and Nazar, 2010; Ibrahim and Shanker, 2012; Sarif et al., 2013) is utilized to obtain the numerical solution of the transformed system of equations for certain values of the physical parameters. The manuscript is presented as follows: the mathematical formulation of the paper is discussed in the Section 2. The numerical developments have been depicted in the Section 3 in the form of graphs and tables. The influence of material parameters on the flow problem is also discussed in this section. Some concluding remarks have been explored in the Section 4.

2. Mathematical formulation Consider a steady two-dimensional mixed convection flow of a viscous fluid toward a stagnation point over a semi-infinite flat plate placed vertically. A power-law lubricant spreads on the surface of the plate by making a thin layer. It is assumed that T⬁ is the uniform ambient temperature of the bulk fluid, and Tw (x) is the temperature of the surface such that Tw (x) ⬎ T⬁ is for the heated and Tw (x) ⬍ T⬁ is for the cooled surface. A magnetic field of uniform strength B0 is applied perpendicular to the plate. Moreover, the plate is resting along x-axis and the fluid flows along y-axis toward the plate with the free stream velocity ue (x), as shown in Figure 1. Hiemenz (1911) proved that the stagnation-point flow has the same attributes regardless the shape of the figure. We assume that the wall temperature Tw and the free stream velocity ue are linearly proportional to the distance x from the center of the plate as mentioned in the following equations: Figure 1 Flowing phenomenon showing (a) assisting and (b) opposing flow ( )>

0

( )

O

∞

0

0

( )< Lubricant

(a) 528

∞

0

( )

O

∞

( )

∞

Lubricant

(b)

∞




Volume 69 · Number 4 · 2017 · 527–535

共 Lx 兲,

TW共x兲 ⫽ T⬁ ⫹ T0

共 Lx 兲

ue共x兲 ⫽ Ue

共 ⭸U ⭸y 兲

where L is the characteristic length, and T0 ⬎ 0 and Ue denote, respectively, reference temperature and reference velocity. The power-law fluid comes out from the center of the plate with the flow rate Q given as:

兰

(2)


where U (x, y) is the velocity along the plate, and h(x) is the variable thickness of the lubricant. After implementing the boundary layer approximation, the governing equations reduce to: ⭸v ⭸u ⫹ ⫽0 ⭸x ⭸y

(11)

˜ 共x兲 denotes velocity component at the interface of Here, U both fluids parallel to the plate. Using equation (2), the thickness h (x) of the lubricant can be expressed as:

0

u

˜ 共x兲 y U h共x兲

U共x, y兲 ⫽ U共x, y兲dy

(10)

In which k is dynamic coefficient of viscosity, and n is flow behavior index. If U(x, y) varies linearly from y ⫽ 0 to the interface between both the fluids y ⫽ h (x) then:

h(x)

Q⫽

n⫺1

␮L ⫽ k

(1)

h共x兲 ⫽

(3)

2Q ˜ 共x兲 U

(12)

Using equations (10) to (12) in equation (9), one attains the following slip boundary condition:

due B 20 ⭸u ⭸2u ⭸u ⫹v ⫽ ue ⫹ ␯ 2 ⫾ g␥共T ⫺ T⬁兲 ⫹ ␴ (ue ⫺ u) ⭸x ⭸y dx ␳ ⭸y (4)

k 1 n˜ 2n ⭸u ⫽ U ⭸y ␮ 2Q

⭸T ⭸2T ⭸T ⫹v ⫽␣ 2 ⭸x ⭸y ⭸y

By assuming the continuity of the velocity component at the ˜ ⫽ u. Substituting the value of U ˜ we get: interface, we have U

u

共兲

(5)

共兲

⭸u k 1 n 2n ⫽ u ⭸y ␮ 2Q

where u is the velocity along the plate and v is the velocity normal to the plate of the bulk fluid. ␳ is density, ␯ is kinematic viscosity and g is acceleration due to gravity. Furthermore ␣ is thermal diffusivity, ␥ is thermal expansion coefficient, ␴ is electrical conductivity and B0 is magnetic field, respectively. The ⫾ signs in equation (4) is for the assisting and opposing flows, respectively. To discuss the present flow situation, the boundary conditions are applied at the surface of the plate, fluid–fluid interface and at the free stream. The boundary conditions at the surface imply: U共x, 0兲 ⫽ 0 ,

V共x, 0兲 ⫽ 0

共兲

x T共x, 0兲 ⫽ Tw共x兲 ⫽ T⬁ ⫹ T0 L

∀ y ⑀ 关0, h(x)兴

v 共x, h共x兲兲 ⫽ V 共x, h共x兲兲

⭸u ⭸U ⫽ ␮L ⭸y ⭸y

(15)

By using equation (6), we get: v 共x, h共x兲兲 ⫽ 0

(16)

(6) As the lubrication layer is assumed to be very thin, therefore, we can impose boundary conditions [equations (14) and (16)] at the surface where y ⫽ 0. The same assumption has already been imposed in the articles related to lubricated surface flows (Andersson and Rousselet, 2006; Santra et al., 2007; Sajid et al., 2012, 2013). The conditions at the free stream imply:

(7)

(8)

共 Lx 兲,

u 共x, ⬁兲 ⫽ Ue

The boundary conditions at the fluid-lubricant interface are obtained by applying continuity of shear stress and velocity of both the fluids. Following Santra et al. (2007), the continuity of shear stress at the fluid-lubricant interface implies:

␮

(14)

Similarly, the normal velocity components of viscous fluid and the lubricant are also continuous at the interface. Therefore:

where V is the velocity of the lubricant along y-axis. As the lubrication film is very thin, therefore, the normal velocity component of the lubricant gives: V共x, y兲 ⫽ 0 ,

(13)

共 Ly 兲,

v共x, ⬁兲 ⫽ ⫺Ue T共x, ⬁兲 ⫽ T⬁

(17)

Defining the dimensionless variables:

冪L␯ ,

␩⫽y

(9)

Ue

共 Lx 兲f

u ⫽ Ue

⬘共

␩兲,

v⫽⫺

共 Lx 兲␪ ␩

T ⫽ T⬁ ⫹ T0

where ␮ and ␮L are the viscosities of the viscous and power-law fluids, respectively. Assuming ⭸U/⭸x ⬍⬍ ⭸U/⭸y, the viscosity of the lubricant ␮L is given by:

冪L␯f ␩ , Ue

共兲

共兲

Equations (3), (4), (6), (12), (14) and (15) yield: 529

(18)




Volume 69 · Number 4 · 2017 · 527–535

f ⬙’ ⫺ f ⬘ ⫹ ff ⬘’ ⫹ 1 ⫹ ␤␪ ⫹ M(1 ⫺ f ⬘) ⫽ 0

(19)

␪ ⬙ ⫹ Pr(f␪⬘⫺f ⬘␪) ⫽ 0

(20)

f共0兲 ⫽ 0,

f ⬙共0兲 ⫽ ␭ f ⬘共0兲2n,

␪共0兲 ⫽ 1


␪(⬁) ⫽ 0, f ⬘共⬁兲 ⫽ 1

Figure 2 Response of f ⬘共␩兲 under the influence of ␭ when M ⫽ 1, Pr ⫽ 1, ␤ ⫽ 0.1, n ⫽ 1/2

(21) (22)

where ␤ ⫽ Gr/Re2 is the mixed convection parameter, in which Gr ⫽ g␥T0L3/␯2 is Grashof and Re ⫽ UeL/␯ is Reynolds number. It is important to mention that ␤ ⬎ 0 coressponds for assisting flow, ␤ ⬍ 0 for opposing flow and ␤ ⫽ 0 for the forced convection flow. The other dimensionless parameters are magnetic parameter M ⫽ ␴B0L/␳Ue and Prandtl number Pr ⫽ ␯/␣. The parameter ␭ given in equation (21) is called slip parameter and is of the following form:

␭⫽

k兹␯ a2n x2n⫺1 ␮ a3/2 (2Q)n

Figure 3 Response of ␪ (␩) under the influence of ␭ when M ⫽ 1, Pr ⫽ 1, ␤ ⫽ 0.1, n ⫽ 1/2

(23)

Where a ⫽ Ue/L. From equation (23), we see that equations (19) and (20) possess a similar solution when n ⫽ 1/2 because the parameter ␭ becomes independent of x when n ⫽ 1/2. For n ⬍ 1, the shear thinning characteristics of power-law fluid can be observed. n ⫽ 1/2 provides non-similar solutions that are also included in the paper. Furthermore, ␭ from equation (23) can be re-written as:

␯

␭⫽

冪a

␮ 兹2Q k

⫽

Lvisc Llub

Figure 4 Response of f ⬘共␩兲under the influence of M when ␭ ⫽ 5, Pr ⫽ 1, ␤ ⫽ 0.1, n ⫽ 1/2

(24)

The case when the lubrication length Llub is small, i.e. when the flow rate Q is small and k is large (lubricant is highly viscous), the parameter ␭ becomes large. As ␭ ¡ ⬁, the no-slip boundary condition f ⬘共0兲 ⫽ 0 is achieved from equation (21). In the case when Llub ¡ ⬁, we get ␭ ¡ 0, and the full slip boundary conditions f ⬙共0兲 ⫽ 0 is obtained from equation (21). Thus, the parameter ␭ describes an inverse measure of slip.

3. Numerical results and discussions

Figure 5 Response of ␪ (␩) under the influence of M when ␭ ⫽ 5, Pr ⫽ 1, ␤ ⫽ 0.1, n ⫽ 1/2

The values of f ⬘, f ⬙, ␪ and ␪⬘ are evaluated by solving equations (19)-(22) numerically with the help of well-known implicit finite difference technique called Keller-Box method (Na, 1979; Cebeci and Bradshaw, 1954; Keller and Cebeci, 1972; Keller, 1970; Ahmad and Nazar, 2010; Ibrahim and Shanker, 2012; Sarif et al., 2013) for certain values of pertinent parameters. To see the effects of slip parameter ␭, magnetic parameter M, mixed convection parameter ␤, Prandtl number Pr and flow behavior index n on f ⬘ and ␪, Figures 2-10 have been plotted. Wall shear stress f ⬙共0兲 and local Nusselt number ⫺ ␪⬘共0兲 assessed for different values of parameters have been presented in Tables I-III. Figures 2-3 are displayed to analyze the behavior of wall slip due to lubricant on the velocity and temperature distributions. Figure 2 depicts the dependence of velocity component f ⬘ on slip parameter ␭. According to this figure,

f ⬘ increases with decrease in the magnitude of ␭. It means that the lubricant increases the fluid velocity. For the case when ␭ approaches to zero, i.e. in the case of full slip, the effects of viscosity are suppressed by the lubricant. Figure 3 demonstrates how the slip parameter ␭ effects the temperature profile ␪. We observe that the numerical value 530





Volume 69 · Number 4 · 2017 · 527–535

Figure 6 Response of f ⬘共␩兲under the influence of ␤ when ␭ ⫽ 5, Pr ⫽ 1, M ⫽ 1, n ⫽ 1/2

Figure 10 Response of ␪ (␩) under the influence of n when ␭ ⫽ 5, Pr ⫽ 1, M ⫽ 1 and ␤ ⫽ 0.1

Figure 7 Response of ␪ (␩) under the influence of ␤ when ␭ ⫽ 5, Pr ⫽ 1, M ⫽ 1, n ⫽ 1/2

Figure 4 illustrates that applied magnetic field excites the bulk motion and supports the lubrication effects. According to Figure 5, the temperature profile ␪ decreases by increase in the numerical value of M. Moreover, an increase in the numerical value of M reduces the thermal boundary layer thickness. To observe the effects of mixed convection parameter ␤ on f ⬘ and ␪ for some particular values of ␭, M and Pr, both for assisting as well as opposing flows, Figures 6 and 7 are plotted. Figure 6 depicts that velocity profile f ⬘ and mixed convection parameter ␤ are directly proportional to each other for the assisting flow and are inversely proportional for the opposing flow. Influence of ␤ on temperature ␪ is presented in Figure 7. This figure shows that by increasing ␤, the temperature of fluid decreases for the case of assisting flow, and it increases when there is opposing flow. Figure 8 elucidates how Prandtl number Pr effects the temperature ␪ in the existence of lubrication when M ⫽ 1 and ␤ ⫽ 0.1. From this figure, it is clear that as we increase the numerical value of Pr for some fixed value of ␭, the temperature profile decreases. As we move from no-slip to full slip (i.e. ␭ decreases), this decrease is more rapid. The impact of flow behavior index n on f ⬘ and ␪ is illustrated in Figures 9 and 10. Figure 9 shows that the velocity component f ⬘ is increased by increase in n. According to Figure 10, the temperature ␪ decreases by increasing flow behavior index n. The effects of slip parameter ␭ on f ⬙共0兲 and ⫺ ␪⬘共0兲 when M ⫽ 1 and Pr ⫽ 1 are presented in Table I. The cases for the assisting flow as well as opposing flow are considered. We observe that by increasing ␭, f ⬙共0兲 is increased and ⫺ ␪⬘共0兲 is decreased. However, the rate of increase or decrease is slower when there is opposing flow. The impact of M on f ⬘’共0兲 and ⫺ ␪⬘共0兲 is elucidated in Table II when ␭, Pr and ␤ are constant. It has been observed that by increasing M, both f ⬙共0兲 and ⫺ ␪⬘共0兲 increase. The effects of Pr on f ⬙共0兲 and ⫺ ␪⬘共0兲 for assisting and opposing flows are shown in Table III. A close look at this table clarifies that as the Pr is increased, f ⬙共0兲 is decreased and ⫺ ␪⬘共0兲 is increased for case of assisting flow. However, both quantities are increased for the case of opposing flow. Numerical values of f ⬙共0兲 and ⫺ ␪⬘共0兲 for the no-slip case agree well with the values already described in the literature (Ramachandran et al., 1998; Devi et al., 1991; Hassanien

Figure 8 Response of ␪ (␩) under the influence of Pr in the presence of slip when M ⫽ 1, ␤ ⫽ 0.1, n ⫽ 1/2

Figure 9 Response of f ⬘共␩兲under the influence of n when ␭ ⫽ 5, Pr ⫽ 1, M ⫽ 1 and ␤ ⫽ 0.1

of temperature profile ␪ is increased by increasing slip parameter ␭ showing that temperature of the fluid is reduced by augmenting lubrication on the plate. The impact of magnetic parameter M on f ⬘and ␪ when ␭ ⫽ 3, Pr ⫽ 1, ␤ ⫽ 0.1 is depicted in Figures 4 and 5. 531




Volume 69 · Number 4 · 2017 · 527–535

Table I Influence of slip parameter ␭ on f ⬙共0兲 and ⫺ ␪⬘共0兲 when M ⫽ 1 and Pr ⫽ 1 when ␤ ⫽ 0.1 (assisting flow) and ␤ ⫽ ⫺ 0.1 (opposing flow)


␭

f ⬘’共0兲 (assisting flow)

⫺ ␪⬘共0兲 (assisting flow)

f ⬘’共0兲 (opposing flow)

⫺ ␪⬘共0兲 (opposing flow)

0.0101121 0.0495158 0.0965228 0.3994490 0.6527577 0.9466515 1.2769618 1.4346705 1.5837308 1.6038870 1.6202570 1.6243923

1.2599325 1.2525857 1.2437203 1.1837080 1.1290828 1.0593235 0.9700909 0.9220959 0.8725170 0.8654437 0.8596270 0.8581573

0.0097796 0.0478723 0.0932823 0.3850065 0.6276409 0.9075208 1.2198865 1.3682181 1.5079733 1.5268410 1.5421597 1.5460290

1.2427839 1.2354619 1.2266314 1.1667000 1.1129564 1.0442861 0.9570234 0.9103471 0.8623061 0.8554656 0.8498433 0.8484232

0.01 0.05 0.1 0.5 1.0 2.0 5.0 10 50 100 500 ⬁

Table II Influence of magnetic parameter M on f ⬙共0兲 and ⫺ ␪⬘共0兲 when ␭ ⫽ 1 and Pr ⫽ 1 when ␤ ⫽ 0.1 (assisting flow) and ␤ ⫽ ⫺ 0.1 (opposing flow)

M 0.1 0.5 1.0 2.0 5.0 10 50 100 500 1,000 2,000 50,000 ⬁

f ⬘’共0兲 (assisting flow)


f ⬘’共0兲 (opposing flow)


0.6149554 0.6335878 0.6527577 0.6822195 0.7360880 0.7827440 0.8800347 0.9108617 0.9574739 0.9694763 0.9781892 0.9955499 0.9999998

1.1057394 1.1174154 1.1290828 1.1463037 1.1753963 1.1978075 1.2343744 1.2424560 1.2506954 1.2519452 1.2526083 1.2532860 1.2533220

0.5831787 0.6052519 0.6276409 0.6615126 0.7220656 0.7733190 0.8771720 0.9092798 0.9571130 0.9692902 0.9780941 0.9955459 0.9999987

1.0845621 1.0988763 1.1129564 1.1333866 1.1671049 1.1924990 1.2329398 1.2416976 1.2505353 1.2518645 1.2525678 1.2532844 1.2533211

Table III Influence of Prandtl number Pr on f ⬙共0兲 and ⫺ ␪⬘共0兲 when ␭ ⫽ 1 and M ⫽ 1 when ␤ ⫽ 0.1 (assisting flow) and ␤ ⫽ ⫺ 0.1 (opposing flow)

Pr

f ⬙共0兲 (assisting flow)


f ⬙共0兲 (opposing flow)


0.6593814 0.6584386 0.6547115 0.6527577 0.6507812 0.6483054 0.6466401 0.6436939 0.6428188

0.3047311 0.3913332 0.8142949 1.1290828 1.5655846 2.4160414 3.3625665 7.3144593 10.2620591

0.6208986 0.6218514 0.6256484 0.6276409 0.6296540 0.6321715 0.6338614 0.6368419 0.6377242

0.3009513 0.3852558 0.8015589 1.1129564 1.5457547 2.3911373 3.3339433 7.2785393 10.2237614

0.05 0.1 0.5 1.0 2.0 5.0 10 50 100

power-law fluid is investigated. Numerical solutions are obtained to observe the influence of slip parameter ␭, magnetic parameter M, mixed convection parameter ␤, power- law index n and Prandtl number Pr on the fluid velocity, temperature distribution, wall shear stress and local heat transfer coefficient. Results are presented in the

and Gorla, 1990; Lok et al., 2006; Aman et al.2013; Ali et al., 2011) and are presented in Tables IV-VI.

4. Conclusion In this paper, MHD mixed convection flow of a viscous fluid stagnated over a vertical wall lubricated with 532




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Table IV Comparison showing the influence of Pr on f ⬙共0兲 when ␭ ⫽ ⬁, M ⫽ 0 and ␤ ⫽ 0

Pr


0.7 1 7 10 20 40 60 80 100

Ramachandran et al. (1998)

Devi et al. (1991)

Lok et al. (2006)

Hassanien and Gorla (1990)

Aman et al. (2013)

Present work

1.7063 – 1.5179 – 1.4485 1.4101 1.3903 1.3774 1.3680

1.7064 – 1.5180 – 1.4485 – 1.3903 – 1.3680

1.7064 – 1.5180 – 1.4486 1.4102 1.30903 1.3773 1.3677

1.70632 – – 1.49284 – – – – 1.38471

1.7063 1.6754 1.5179 1.4928 1.4485 1.4101 1.3903 1.3774 1.3680

1.706333 1.675450 1.517922 1.492848 1.448492 1.410067 1.390283 1.377401 1.368043

Table V Comparison showing influence of Pr on ⫺ ␪⬘共0兲 when ␭ ⫽ ⬁, M ⫽ 0 and ␤ ⫽ 0

Pr 0.7 1 7 10 20 40 60 80 100

Ramachandran et al. (1998)

Devi et al. (1991)

Lok et al. (2006)

Hassanien and Gorla (1990)

Aman et al. (2013)

Present work

0.7641 – 1.7224 – 2.4576 3.1011 3.5514 3.9095 4.2116

0.7641 – 1.7223 – 2.4574 – 3.5517 – 4.2113

0.7641 – 1.7226 – 2.4577 3.1023 3.5560 3.9195 4.2289

0.76406 – – 1.94461 – – – – 4.23372

0.7641 0.8708 1.7224 1.9446 2.4576 3.1011 3.5514 3.9095 4.2116

0.764073 0.870788 1.722480 1.944644 2.457652 3.101146 3.551412 3.910952 4.213370

Table VI Comparison showing influence of ␤ on f ⬙共0兲 and ⫺ ␪⬘共0兲 when ␭ ⫽ ⬁/, M ⫽ 0 and Pr ⫽ 0.7

␤ ⴚ0.6 ⴚ0.8 ⴚ1.0 ⴚ1.2 ⴚ1.4 ⴚ1.6 0.1 0.5 0.9 1.6

f⬙共0兲 Ali et al. [(2011)

f⬙共0兲 Present work

⫺ ␪⬘共0兲 Ali et al. [(2011)

⫺ ␪⬘共0兲 Present work

0.9194 0.8079 0.6917 0.5696 0.4401 0.3003 1.2823 1.4755 1.6610 1.9707

0.9193778 0.8078804 0.6916868 0.5696416 0.4401302 0.3003289 1.2823140 1.4755318 1.6610101 1.9707155

0.6673 0.6510 0.6332 0.6134 0.5909 0.5574 0.7157 0.7383 0.7591 0.7916

0.6672568 0.6510393 0.6332424 0.6134485 0.5908937 0.5574276 0.7157189 0.7382642 0.7591422 0.7916414

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form of tables and figures for certain values of parameters by considering assisting as well as opposing flow situations. Some findings of this study are as follows: ● The lubricant increases the velocity of the base fluid inside the boundary layer. Moreover, the effects of viscosity are suppressed by the lubricant in the case of full slip. ● The temperature of the base fluid decreases by increase in lubrication on the surface. ● By increasing the slip on the surface, the wall shear stress, i.e. f ⬙共0兲 decreases and heat transfer coefficient, i.e.⫺ ␪⬘共0兲 increases, but the rate of increase or decrease is less in magnitude for opposing flow. ● The similarity solutions only exist for n ⫽ 1/2. A non-similar solution is obtained when n ⫽ 1/2.

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