Convective heat transfer and MHD effects on Casson nanofluid flow ...

Cent. Eur. J. Phys. • 12(12) • 2014 • 862-871 DOI: 10.2478/s11534-014-0522-3

Central European Journal of Physics

Convective heat transfer and MHD effects on Casson nanofluid flow over a shrinking sheet Research Article

Rizwan Ul Haq1,2∗ , Sohail Nadeem1 , Zafar Hayyat Khan3 , Toyin Gideon Okedayo4 1 Department of Mathematics, Quaid-I-Azam University 45320, Islamabad 44000, Pakistan 2 Mechanical and Materials Engineering, Spencer Engineering Building, Room3055, University of Western Ontario, London, Ontario, Canada 3 Department of Mathematics, University of Malakand, Dir (Lower), Khyber Pakhtunkhwa, Pakistan 4 Department of Mathematics, Ondo State University of Science and Technology, Okitipupa, Nigeria

Received 1 March 2014; accepted 7 July 2014

Abstract: Current study examines the magnetohydrodynamic (MHD) boundary layer flow of a Casson nanofluid over an exponentially permeable shrinking sheet with convective boundary condition. Moreover, we have considered the suction/injection effects on the wall. By applying the appropriate transformations, system of non-linear partial differential equation along with the boundary conditions are transformed to couple non-linear ordinary differential equations. The resulting systems of non-linear ordinary differential equations are solved numerically using RungeKutta method. Numerical results for velocity, temperature and nanoparticle volume concentration are presented through graphs for various values of dimensionless parameters. Effects of parameters for heat transfer at wall and nanoparticle volume concentration are also presented through graphs and tables. At the end, fluid flow behavior is examined through stream lines. Concluding remarks are provided for the whole analysis. Keywords: Casson nanofluid • exponentially shrinking sheet • convective heat transfer • numerical solution © Versita sp. z o.o.

1.

Introduction

Non-Newtonian fluids cannot be described due to nonexistence of single constitutive relationship between stress and rate of strain. In the recent year, non- Newtonian fluids have become more and more important due to its industrial applications. In fact, the interest in boundary layer flows of non-Newtonian fluid is increasing substantially due to its large number of practical applications in indus∗

862

E-mail: ideal [email protected], [email protected]

try, manufacturing processing and biological fluids. Few of main examples related to applications are plastic polymer, drilling mud, optical fibers, paper production, hot rolling, metal spinning and cooling of metallic plates in a cooling bath and many others. Since no single non-Newtonian model predicts all the properties of non-Newtonian fluid therefore investigations proposed various non-Newtonian fluid models. These models mainly classified into three categories namely differential, rate and integral type fluids. In non-Newtonian fluid, shear stresses and rates of strain/deformation are not linearly related. Such fluid under consideration which does not obey the Newton’s law of viscosity is known as Casson fluid. Casson fluid model

Rizwan Ul Haq, Sohail Nadeem, Zafar Hayyat Khan, Toyin Gideon Okedayo

is a simple non-Newtonian fluid model of differential type. In 1959, Casson presented this model for the flow of viscoelastic fluids. This model has a more gradual transition from Newtonian to the yield region. This model is used by petroleum engineers in the characterization of cement slurry and is better for predicting high shear-rate viscosities when only low and intermediate shear-rate data are available. The Casson model is more accurate at both very high and very low shear rates. The Casson model has been used in other industries to give more accurate representation of high shear rate viscosities when only low and intermediate shear-rate data are available [1]. At the beginning Nadeem et al. [2] present the concept of Casson fluid model over an exponentially stretching sheet. Many investigations related to viscoelastic properties of fluid are under consideration [3–5]. No doubt, study of boundary layer flow for various types of fluid models over a stretching sheet has received much attention because of their extensive applications in the field of metallurgy and engineering. For examples, polymer processing industries, environmental pollution, biological process, aerodynamic extrusion of plastic sheets, glass fiber production of the boundary layer along a liquid film and condensation process, the cooling, and/or drying of paper and textiles. Also the study of heat transfer of viscous fluids plays a very important role in various industrial applications such as petroleum production and metallurgical process. Initially, Sakiadis [6, 7] presented the boundary layer flow on a continuously stretching surface with a constant speed. His work was further extended experimentally by Tsou et al. [8] for the flow of heat transfer in the boundary layer on a uniform moving surface. No doubt, Crane contributes his pioneer work in the study of stretching sheet. Crane [9] studied the fluid flow problem over a linearly stretched surface and presented the concept that the elastic sheet moves in its own plane with a velocity varying linearly with the distance from a fixed point. Later on, abundant amount of literature is available regarding the flow caused by a stretching sheet for both Newtonian and non-Newtonian fluids [10–16]. In the past few years, the study of nanofluids has achieved lot of importance. Nanofluids are heat transfer liquids with disported nanoparticles. Recent studies have shown that they are capable of improving the thermal conductivity, thermal properties of heat transport of base fluid, promote energy efficiency, and may have potential applications in the field of heat transfer enhancement. The effectiveness of heat transfer enhancement has been found to be dependent on the amount of dispersed particles, material type and particle shape so on. It is expected that nanofluids can be utilized in airplanes, cars, micro machines in micro reactors and many others. So far no

general mechanisms have been formulated to explain the strange behavior of the nanofluids including the highly improved effective thermal conductivity, although many possible factors have been considered, including Brownian motion, liquid-solid interface layer and surface charge state. Currently, there is no reliable theory to predict the anomalous thermal conductivity of nanofluids. From the experimental results of many researchers, it is known that thermal conductivity of nanofluids depends on parameters including the thermal conductivities of the base fluid and the nanoparticles, the volume fraction of the nanoparticles, the surface area, the shape of the nanoparticle and the temperature [17]. Actually nanofluids are the homogenous mixture of base fluid and nanoparticles. There are number of common base fluids including water, organic liquids (e.g. ethylene, tri-ethylene-glycols, refrigerants, etc.), oil and lubricants, bio-fluids, polymeric solution and other common liquids. Initially, Choi [18] presented the concept of nanofluids for suspension of liquids containing ultra-fine particles (diameter less than 50 nm). Then Khan and Pop [19] presented the concept of boundary layer nanofluid flow past a stretching surface. Makinde and Aziz [20] studied the influence of convective boundary condition on the flow of nanofluid past a stretching surface. After this development of nanofluids, Nadeem and Lee [21] presented the concept of nanofluid over an exponentially stretching surface. Later on many authors discussed the effects of nanoparticles for boundary layer flow over a stretching/shrinking surface [22–30].

The main emphasis of the present study is to discuss the nanoparticles analysis for the Casson fluid model assuming the convective surface boundary conditions. By applying the similarity transformation we reduce the system of nonlinear partial differential equations into the system of nonlinear ordinary differential equations. Nondimensional physical parameters namely Casson fluid parameter γ, Hartmann number M, Biot number Bi, Prandtl number Pr, Thermophoresis parameter Nt , Brownian motion parameter Nb and Lewis number Le appear after applying the similarity transformations along with the system of coupled ordinary differential equations which governs the behavior of fluid. Coupled equations are then tackled numerically and then physical behaviors of each of the parameter are shown graphically. Reduced Nusselt and Sherwood number also computed numerically and discussed through graphs. Comparison of the present article with the available literature is presented in Table 1. Numerical results for Nusselt and Sherwood numbers with thermophoresis parameter Nt and Brownian motion parameter Nb presented in Tables 2. 863

Convective heat transfer and MHD effects on Casson nanofluid flow over a shrinking sheet

2.

Mathematical model

We adjust geometry of the problem in the coordinate system such that x-axis is taken horizontally and y-axis is perpendicular to it. The steady situation of two dimensional flow past a shrinking sheet is considered. Moreover, we have considered the Casson nanofluid flow in the presence of magneto-hydrodynamic that is normal to the nanofluid and the flow is placed at y ≥ 0, where y is the coordinate measured normal to the shrinking surface. We assumed that sheet is shrank exponentially with velocity u(x) = Uw exp( xl ), where Uw is constant and x is coordinate along the shrinking surface. The temperature T and the nanoparticle volume concentration C on the boundaries are taken to be Tf and Cw , at the wall, T∞ and C∞ , respectively are far away from the wall.

is the thermal diffusivity, νnf is kinematic viscosity, DB the Brownian diffusion coefficient, DT is the thermophoretic (ρc) diffusion coefficient, τ = (ρc)p is the ratio between the f effective heat capacity of the nanoparticle material and heat capacity of the fluid, in which ρf is the density of fluid and ρp is the density of the particles. It is also consider that the magnetic field B(x) is of the form B = B0 exp

x l

(5)

,

where B0 is the constant magnetic field. The corresponding boundary conditions are uw (x) = Uw exp

x

, vw (x) = V0 exp

x , 2l

l ∂T −kf = hf (Tf − T ), C = Cw at y = 0, ∂y

u → 0, v → 0, T → T∞ , C = C∞ as y → ∞. (6)

Figure 1.

In the above expression uw (x) is stretching/shrinking velocity of the fluid with Uw (stretching/shrinking constant), here it is notice that for vw (x) is mass transfer velocity with (V0 > 0 for mass injection and V0 < 0 for mass suction), Kf is the thermal conductivity of the fluid, hf is the convective heat transfer coefficient, Tf is the convective fluid temperature below the shrinking sheet. Introducing the following similarity transformations

Geometry of the problem.

Moreover, convective condition is taken into account at the wall when fluid is passing over the shrinking surface. The boundary layer equations for two dimensional incompressible Casson nanofluid are stated as ∂u ∂v + = 0, ∂x ∂y

(1)

∂u ∂u 1 ∂2 u σ B 2 +v =ν 1+ − u u ∂x ∂y β ∂y2 ρf

(2)

∂T ∂T u +v = ∂x ∂y ( 2 2 ) ∂ T ∂C ∂T DT ∂T α + τ DB + , (3) ∂y2 ∂y ∂y T∞ ∂y

u

∂C ∂C +v = DB ∂x ∂y

2

∂ C ∂y2

+

DT T∞

2

∂ T ∂y2

Making use of Eq. (7), equation of continuity is identically satisfied and Eqs. (2) to (5) along with (7) take the following form (1 +

1 000 )f − M 2 f 0 − 2(f 0 )2 + ff 00 = 0, β

θ 00 + Pr((fθ 0 − f 0 θ) + Nb θ 0 φ0 + Nt (θ 0 )2 ) = 0, φ00 + Le Pr(fφ0 − f 0 φ) +

,

Nt 00 θ = 0, Nb

(8) (9) (10)

(4)

where u and v denote the velocities in the x−and y− directions respectively, β is the Casson fluid parameter, α 864

x u = U0 exp f 0 (η) l r x νU0 and v = − {f(η) + ηf 0 (η)}, exp 2l 2l r x U0 T − T∞ C − C∞ η=y exp ,θ = , φ= . 2νl 2l Tw − T∞ Cw − C∞ (7)

f(η) = s, f 0 (η) = λ, θ 0 (η) = −Bi[1 − θ(0)] and φ(η) = 1 at η = 0,

(11)


f 0 (η) = 0, θ(η) = 0, φ(η) = 0 at η −→ ∞. In these expressions M 2 =

2 0

2σ B l ρU0

(12)

is the magnetic parame-

ter, s is suction/injection parameter, λ = UUw0 is the stretching/shrinking parameter with (λ > 0 for stretching surface case, λ < 0 is for shrinking surface and λ = 0 is for flat B (Cw −C∞ ) surface), Pr = ανD is Prandtl number, Nb = (ρc)P Dν(ρc) P

DT (Tf −T∞ ) is the theris the Brownian motion, Nt = (ρc)P ν(ρc) P αD mophoresis parameter, Le = DB is the Lewis number and q Bi = ( hk f ) 2lν is Biot number. Expressions for skin fricU0 f tion coefficient Cf , local Nusselt number Nux and the local Sherwood number Sh are

τwx xqw xqm , Nu = , Sh = , ρuw k(Tf − T∞ ) DB (Cw − C∞ ) (13) where τwx is the wall shear stresses along x−direction. Here qw and qm are the heat flux and the mass flux respectively, define as. Cfx =

qw = −k

∂T ∂y

, qm = −DB y=0

∂C ∂y

.

(14)

4.

Dual solutions

In this section dual solutions have been constructed for various values of emerging parameter involved in the momentum equations for skin friction coefficient. Fig. 2 shows that it is possible to get dual solutions of the boundary layer equations for shrinking parameter λ(< 0). These dual solutions are in the range λc < λ < 0 and no solution exists for λ < λc < 0, where λc is the minimum value of λ for which solution exists. It is found that decreasing the values of |λc | promotes a gradual increase in the skin friction coefficient with respect to increasing values of both the magnetic parameter M and the non-Newtonian fluid parameter β. This shows that the higher values of and decrease the range of the existence of the solution to the proposed boundary value problem. The procedure for describing the stability of dual solutions is extensively available in the literature [31] and [32], so it is not worth repeating here. In the above mentioned literature it was discussed that in dual nature solution, probably one of the solution is stable while the other is physically not stable. So in the present study we only analyze the stable case with Casson nanofluid and just discard the unstable case.

y=0

Dimensionless form of Eqs. (14) take the form Re1/2 Cfx = (1 + x

1 00 )f (0) , β

Re−1/2 Nu = −θ 0 (0) , x

Re−1/2 Sh = −φ0 (0) . x

(15)

where Rex = (xU0 ex/l )/ν is local Reynolds number.

3.

Figure 2.

Numerical technique

The systems of nonlinear differential equations (8-10) parallel to the boundary conditions (11) and (12) are solved numerically using Runge-Kutta-Fehlberg method with a shooting technique. The step size ∆η = 0.001 is used to obtain the numerical solution with ηmax , with an accuracy to the fifth decimal place is chosen as the criterion of convergence. The asymptotic boundary conditions given by Eq. (12) were replaced by using a value similarity variable ηmax = 12 as follows: f 0 (ηmax ) = 0,

θ(ηmax ) = 0

and φ(ηmax ) = 0

(16)

The choice of ηmax = 12 ensures that all numerical solutions approached the asymptotic values correctly.

5.

Dual solutions for various values of MHD parameter M and Casson fluid parameter β.

Results and discussions

In the present section we have discussed the velocity profile f 0 (η), temperature profile θ(η) and nanoparticle volume concentration φ(η) for various physical parameters such as Casson fluid parameter β, Hartmann number M, suction/injection parameter s, Biot number Bi, Prandtl number Pr, Thermophoresis parameter Nt, Brownian motion parameter Nb and Lewis number Le. It is noticeable that all the graphical results are constructed for shrinking case λ = −1. It can be found through Table 1, for infinitely large values of Casson fluid parameter (β → ∞) our problem reduces to Newtonian fluid. Similarly we can observe 865


Table 1.

Variation of local Nusselt number Re−1/2 Nux for several values of Prandtl number and magnetic parameter in the absence of nanofluid x when s = 0, λ = 1, Bi → ∞ and β → ∞ (Newtonian fluid).

Pr M Magyari and Keller [13] Bidin and Nazar [14] El-Aziz [15] Ishak [16] Present study 1

0

0.9548

0.9547

0.9548

0.9548

0.9548

2

0

−

1.4714

−

1.4715

1.4714

3

0

1.8691

1.8691

1.8691

1.8691

1.8691

5

0

2.5001

−

2.5001

2.5001

2.5001

10 0

3.6604

−

3.6604

3.6604

3.6604

−

−

−

0.8611

0.8611

1

Table 2.

1

Variation of local Nusselt number and Sherwood number with Nb and Nt for S = 3, M = 2, β = 3, Bi = 0.3, λ = −1, Pr = 6.2 and Le = 1.

Nb = 0.3 Nt

−θ(0)

−φ(0)

Nb = 0.4 −θ(0)

−φ(0)

Nb = 0.5 −θ(0)

−φ(0)

0.0 0.285867 18.001155 0.278889 18.001155 0.268004 18.001155 0.2 0.285633 17.794769 0.278355 17.844328 0.266763 17.873235 0.4 0.285388 17.588241 0.277782 17.687275 0.265386 17.744930 0.6 .285130 17.381564 0.277166 17.529980 0.263844 17.616200 0.8 0.284860 17.174730 .276501 17.372425 0.261709 17.228003

Figure 3.

Effect of MHD parameter M on velocity, temperature and nanopartilce fraction.

that present problem more reducible for constant wall temperature when it is considerd infinitely large value of Biot number(Bi → ∞). In addition we can see through Table 1, when we discard the nanoparticles and suction/injection effects our problem gives the excellent comparison with the results provided by different authors. From Fig. 3, it is observed that for higher values of M it reduces both boundary layer thickness and the magnitude 866

Figure 4.

Effect of suction/injection parameter s on velocity, temperature and nanoparticle volume concentration.

of the velocity. This phenomena occurs when magnetic field induced current in the conductive fluid, then it creates a resistive-type force on the fluid in the boundary layer which slow down the motion of the fluid. Exactly the same behavior is shown for temperature and nanoparticle volume concentration. So finally, it is concluded that magnetic field is used to control boundary layer separation. Same sort of behavior appears when we compare Fig. 4 with Fig. 3 for higher values of s. Fig. 5, depicts


Figure 5.

Effect of Casson fluid parameter β on velocity, temperature and nanopartilce fraction.

Figure 6.

Effect of Prandtl number Pr on temperature and nanoparticle volume concentration.

the effects of non-Newtonian parameter β on the velocity profile f 0 (η). Here it is observed that for higher values of non-Newtonian parameter β it produce resistance in the motion of fluid. That is the fact as we can see in Fig. 5, velocity profile f 0 (η) and the boundary layer thickness decreases for higher values of β. From Fig. 6, it can observed that for increasing non-Newtonian parameter will reduce the temperature profile. Moreover, for higher values of non-Newtonian parameter β boundary layer thickness decreases (see Fig. 4). It is noticed when we increase non-Newtonian parameter β indefinitely, the problem in

Figure 7.

Effect of Biot number Bi on temperature profile.

Figure 8.

Effect of Biot number Bi on nanoparticle volume concentration.

the given case reduces to Newtonian fluid. Figs. 6 shows the behavior of Prandtl number on temperature profile and nanoparticle volume concentration due to Prandtl number is a ratio of kinematic viscosity to thermal diffusivity. Consequently, for higher values of Prandtl number it reduces the thermal diffusivity (see Fig. 6). Same behavior can be observed for nanoparticle volume concentration against Prandtl number when we compare temperature profile with nanoparticle volume concentration. Effects of Biot number Bi on temperature and nanoparticle volume concentration are mentioned in (Fig. 7 and 8). 867


Figure 9.

Figure 10.

Effects of Brownian motion parameter Nb on temperature and nanopartilce fraction.

Figure 11.

Effects of Lewis number Le on temperature and nanopartilce fraction.

Effects of Brownian motion parameter Nt on temperature and nanoparticle volume concentration.

Figure 12.

Effect of s, M and β on skin friction coefficient.

Physically, Biot number is expressed as the convection at the surface of the body to the conduction within the surface of the body. When thermal gradient applied to the surface then the ratio governs the temperature inside a body varies significantly, while the body heats or cools over a time. Normally, for uniform temperature field inside the surface it is considered Bi > 1 depicts that temperature field inside the surface is not uniform. In Fig. 6, we have discussed the effects of Biot number Bi on temperature profile θ(η) in three 868

ways. The first one is the case when Bi < 0.1. It is observed from Fig. 6 that for smallest values of the Biot number (Bi < 0.1), the variation of temperature within the body is slight and can reasonably be approximated as being uniform. While in the second case for Bi > 0.1 depicts that the temperature within the body not performing uniform behavior (see Fig. 7). The last case relates when we consider very large value of Biot number corresponds to the case of constant wall temperature (see Fig. 7). Same sort of behavior can be seen for nanoparticle volume concentration φ(η) against Biot number Bi,


Figure 13.

Effect of Nb and Nt on dimensionless heat transfer rates.

Figure 14.

Effect of Nb and Nt on dimensionless concentration rates.

Figure 15.

Stream line behavior for various values of suction/injection parameter s.

when it is compare Fig. 8 with the Fig. 7. The effect of Brownian motion parameter Nb on the temperature profile θ(η) is presented in Fig. 9. Here we can observe that with an increase in the motion of Brownian parameter Nb , temperature profile θ(η) increases. This phenomena presented that enhanced thermal conductivity of a nanofluid is mainly due to Brownian motion which producing micromixing, whereas for large value of Brownian parameter Nb , it reduces the nanoparticle fraction φ(η) (see Fig. 9). Same behavior can be observed when we compare Fig. 10 with Fig. 9 for temperature profile. While nanoparticle fraction φ(η) increases with an increase of thermophoresis parameter Nt . Finally, from Figs. 9 – 10, it is observed that both Brownian motion and thermophoresis parameter present same sort of behavior for temperature profile while it presents opposite behavior for nanoparticle volume concentration. From Fig. 11, it depicts that with an increase in Lewis number Le temperature profile increases while the profile of nanopartilce fraction decreases.

Effects of physical parameters on skin friction coefficient (1 + β1 )f 00 (0), reduced Nusselt number θ 0 (0) and reduced Sherwood number φ0 (0) are presented in Figs. 12 – 14. Fig. 12 present the variation of skin friction along with the physcial parameters s and M both Newtonian (β = ∞) and non-Newtonian (β = 6 ∞) cases. It is found that in Newtonian fluid has less friction with wall at compare to the non-Newtonian fluid, due to viscosity effects. It is observed from Fig. 13, for higher values of Brownian motion parameter Nb it reduces the Nusselt number θ 0 (0). In both cases either Pr < Le or Pr > Le, same decreasing behavior can be observed for reduced Nusselt number θ 0 (0) against Nt for increasing values of Brownian motion parameter Nb , while rest of the parameters are fixed. Fig. 14 depicts the trend of the reduced Sherwood number φ0 (0) against thermophoresis parameter Nt , while for increasing values of Brownian parameter Nb Sherwood number φ0 (0) gives same increasing behavior. It can be seen from Fig. 14, in both cases either Pr < Le or Pr > Le, Sherwood number φ0 (0) present similar behavior. Finally, we observed from Figs. 13 and 14, because of higher Prandtl number it produces opposite trend for both heat transfer rate and mass transfer rate. The 2-D view of variation of the stream lines are presented in Fig. 15, for s =-0.5, s = 0 and s = 0.5. Similarly 3-D view of variation of the stream function Ψ along with the dependent variables is presented in Fig. 16. Tables 2, present the numerical values of the local Nusselt number and Sherwood number for various values of Brownian parameter and thermophoresis parameter while rest of parameters are fixed. 869


Figure 16.

6.

(a)

(b)

(c)

(d)

3-D view of streamline behavior for various values of suction/injection parameter s.

Conclusions • MHD boundary layer flow of a nanofluid over a exponentially shrinking sheet for Casson model subject to the convective boundary condition is solved numerically. Moreover, effects for various values of existing parameters are discussed for velocity, temperature and concentration. The main results of present analysis can be listed below. • Trend of velocity is identical for MHD, Casson fluid and shrinking parameters. • The same behavior of temperature profile is found for both Brownian motion parameter and thermophoresis parameter.

870

• Opposite behavior of nanoparticle volume concentration is found for both Brownian motion and thermophoresis parameter . • The temperature profile and concentration profile decreases when both and increases. • Skin friction for non-Newtonian fluid is comparatively higher than Newtonian fluid. • The magnitude of the local Nusselt and Sherwood number shows the opposite trend for higher values of Brownian motion parameter.


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