Scholarly article on topic 'Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid'

Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid Academic research paper on "Mathematics"

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Abstract of research paper on Mathematics, author of scientific article — Umar Khan, Naveed Ahmed, Sheikh Irfan Ullah Khan, Syed Tauseef Mohyud-din

Abstract Thermodiffusion effects on stagnation point flow of a nanofluid towards a stretching surface with applied magnetic field is presented. Similarity transforms are applied to reduce the equations that govern the flow to a system of nonlinear ordinary differential equations. Runge-Kutta-Fehlberg method is applied to solve the system. Results are compared with existing solutions that are special cases to our problem. Concrete graphical analysis is carried out to study the effects of different emerging parameters such as stretching ratio A, magnetic influence parameter M, Prandtl number Pr, Lewis number Le, Brownian motion parameter Nb, thermophoresis parameter Nt, nanofluid Lewis number Ln, modified Dufour parameter Nd and Dufour solutal number Ld coupled with comprehensive discussions. Numerical effects of local Nusselt number, local Sherwood number and nanofluid Sherwood number are also discussed.

Academic research paper on topic "Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid"

Propulsion and Power Research 2014;3(3):151-158

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ORINIGAL ARTICLE

Thermo-diffusion effects on MHD stagnation point flow towards a stretching sheet in a nanofluid

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Umar Khana, Naveed Ahmeda, Sheikh Irfan Ullah Khanab, Syed Tauseef Mohyud-dina'*

aDepartment of Mathematics, Faculty of Sciences, HITEC University, Taxila Cantt, Pakistan COMSATS Institute of Information Technology, University Road, Abbottabad, Pakistan

Received 25 March 2014; accepted 3 July 2014 Available online 3 October 2014

KEYWORDS

Nanofluid; Stretching sheet; Thermodiffusion; Numerical solution; Nusselt number; Sherwood number; Nanofluid Sherwood number

Abstract Thermodiffusion effects on stagnation point flow of a nanofluid towards a stretching surface with applied magnetic field is presented. Similarity transforms are applied to reduce the equations that govern the flow to a system of nonlinear ordinary differential equations. Runge-Kutta-Fehlberg method is applied to solve the system. Results are compared with existing solutions that are special cases to our problem. Concrete graphical analysis is carried out to study the effects of different emerging parameters such as stretching ratio A, magnetic influence parameter M, Prandtl number Pr, Lewis number Le, Brownian motion parameter Nb, thermophoresis parameter Nt, nanofluid Lewis number Ln, modified Dufour parameter Nd and Dufour solutal number Ld coupled with comprehensive discussions. Numerical effects of local Nusselt number, local Sherwood number and nanofluid Sherwood number are also discussed.

© 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V.

All rights reserved.

"Corresponding author: Tel.: +92 3235577701. E-mail address: syedtauseefs@hotmail.com (Syed Tauseef Mohyud-din).

Peer review under responsibility of National Laboratory for Aeronautics and Astronautics, China.

1. Introduction

Over the years boundary layer flows have been of much interest to researchers because of their real world applications such as engineering melt spinning, manufacturing of rubber sheets, glass fiber production, and so on. One of these boundary layer flows is the stagnation point flow. Cooling of

2212-540X © 2014 National Laboratory for Aeronautics and Astronautics. Production and hosting by Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/jjppr.2014.07.006

Nomenclature Nux local Nusselt number

Shx Sherwood number

A velocity ratio parameter Shrx nanofluid Sherwood number

BQ applied magnetic field Nur reduced Nusselt number

cf skin friction coefficient Shr reduced Sherwod number

C w solutal concentration at the surface Shrn reduced nanofluid Sherwood number

T i w temperature at the surface

4> w nanoparticle volume fraction at the surface Greek symbols

C ambient solutal concentration

T ambient temperature n dimsensionless similarity variable

<j> i ambient nanoparticle volume fraction » dynamic viscosity of the fluid

Ui free stream velocity u kinematic viscosity of the fluid

u, v velocity components along x and y directions (p)f (pc)f density of the base fluid

M magnetic parameter heat capacity of the base fluid

f dimensionless stream function (pc)p V effective heat capacity of the nanoparticle

s dimensionless concentration stream function

DT thermophoresis diffusion coefficient am thermal diffusivity

DS solutal diffusivity G electric conductivity

Db Brownian motion diffusion coefficient 0 dimensionless temperature

DTC Dufour diffusivity <P dimensionless nanoparticle volume fraction

DCT Soret diffusivity T parameter defined by (pc)p/(pre-

Pr Prandtl number

Nb Brownian motion parameter Subscripts

Nt thermophoresis parameter

Nd modified Dufour paramter condition at free stream

Le Lewis number i

Ld Dofour solutal Lewis number w condition at the wall

Ln nano Lewis number

electronic devices by fans, cooling of nuclear reactors during emergency shutdowns, etc. are the applications of stagnation flows. Crane [1] first studied this problem for stagnation point flow towards a solid surface. After this seminal work, many researchers explored various aspects of boundary layer flows incorporating innumerable physical configurations [2-6].

In real world, most of the fluids such as water, kerosene oil, ethylene, glycol, and others are poor conductors of heat due to their lower values of thermal conductivity. To cope up with this problem and to enhance the thermal conductivity or other thermal properties of these fluids, a newly developed technique is used which includes, addition of nano-sized particles of good conductors such as copper, aluminum, titanium, iron and other oxides to the fluids. Choi [7] was the first one to come up with this idea. Choi et al. [8] also showed that the thermal conductivity of conventional fluids can be doubled by adding nano particles to base fluids that also incorporate other thermal properties. These enhancements can be used practically in electronic cooling, heat exchangers, double plane windows, etc. Buongiorno [9] presented a more comprehensive model for the nanotechnology based fluids that unveils the thermal properties superior to base fluids. He discussed all the convective properties of nanofluids by developing a more general model. After these developments in nanofluids, Khan and Pop [10] were the first ones to study boundary layer flow over a stretching sheet by using the model of

Nield and Kuznetsov [11,12]. Mustafa et al. presented first study on stagnation point flow of a nanofluid [13]. They presented both Brownian motion and thermophoresis effects on transport equations by reducing them to a nonlinear boundary value problem. One can easily find enough literature on nano fluid flows, however some of the studies are reported in [14-18] and references therein.

Study of electrically conducting fluids hold importance due to applications in modern metallurgy and metal working processes. Magnetic nanofluids are used to regulate the flow and heat transfer by controlling the fluid velocity. Mahaputra [19] studied MHD stagnation point flow over a stretching sheet using numerical simulations. MHD stagnation point flow for nanofluid was presented by Ibrahim et al. [20] employing fourth order Runge-Kutta technique. Some others studies regarding magneto-nanofluids are presented in [21-23] and references therein.

In all presented studies, Soret and Dufour effects were neglected. It is a well-known fact that the temperature and concentration gradients present mass and energy fluxes, respectively. Concentration gradients result in Dufour effect (diffusion-thermo) while Soret effect (thermal-diffusion) is due to temperature gradients. Such effects play a significant role when there are density differences in the flow. For the flows of mixture of gases with light molecular weights (He, H2) and moderate weights (N2, air), Soret and Dufour effects cannot be neglected. Thermo-diffusion effects on the flow over a stretching sheet are examined by Awad et al.

[24]. Same idea is extended for power law stretching sheet by Goyal et al. in a recent study [25].

To the best of our knowledge, no one has ever attempted to study the thermo-diffusion effects for stagnation point flow towards a stretching sheet. Main objective of this study is to analyze Soret and Dufour effects for stagnation point flow of electrically conducting nanofluid. In mathematical formulation of the problem, combined effects of thermo-phoresis, Brownian motion and diffusion thermo are presented. Resulting system of non-linear equations is solved by employing a well-known numerical technique called the Runge-Kutta-Fehlberg method after converting the boundary value problem into an initial value problem using shooting technique. Influence of emerging parameters on velocity, temperature, solute and nanoparticle concentration profiles are presented with the help of graphs. A comprehensive discussion is also included. Also, a comparison of current study to the previous ones is provided to validate our solutions.

2. Governing equations

In this problem, a two-dimensional stagnation point flow of a nanofluid towards a stretching sheet is taken into account. Sheet is kept at a constant temperature Tw, solutal concentration Cw and nanoparticle concentration ((w. At a large distance from the sheet, temperature, solutal concentration and nanoparticle concentration are represented by T<», CM and , respectively. A constant transvers magnetic field is applied to the flow in y-direction. Strength of magnetic field is taken to be B — B0. Induced magnetic field is assumed to be small as compared to applied magnetic field and is neglected. Fluid phase and nanopar-ticles are assumed to be in thermal equilibrium and there is no slip between them. A coordinate system is chosen such that the x-axis is along stretching sheet and y-axis is normal to it. Under assumptions mentioned above, governing equations for conservation of mass, momentum, energy, solute and nanoparticles can be written as [20]:

du dv o dx dy

D idtt dT dtt dT

DTI (dT

T™ 1 \dx

dC dC u — + v— = Ds

d2C d2C dx2 dy2

d2 C d2C

x2 y 2

+DKCT\d2 + dy2"

u di+v dl — Db( d2t+ dx dy I dx2 dy2 J

with boundary conditions

u — uw — ax, v — 0, T — Tw,

DT + —

1 1 C Cw

d2T d2T

x2 y 2

'0, T-T1,

u — U1 — bx, v-C-Ci, (( —q>œ at y —1,

(p — (pw at y — 0,

where, u and v are velocity components along x and y-axis, respectively, Uœ is the free stream velocity, pf is density of base fluid, u — p/pf is kinematic viscosity, p is pressure, a is electrical conductivity, B0 is magnetic field flux density, a is thermal diffusivity, DB is Brownian motion diffusion coefficient, DT is thermo-phoresis diffusion coefficient, DTC and DCT are Soret and Dufour diffusivities, DS is solutal diffusivity, T is fluid temperature, C is solutal concentration, tp is the nanoparticle volume fraction, (pc)f and (pc)p are heat capacity of fluid and effective heat capacity of nanoparticle material respectively, t is the parameter defined by (pc)f /(pc)p. Using boundary layer approximations, Eqs. (1)-(6) reduce to

du dv o

u— + v— — Ui —

d-u dy2

— (Ui- u),

_1dP+u dU i

Pf x 1 x

(d2u\ oB0, dx2 + — ) + — (Ui-u)'

dv dv 1 dp f d2v d2 v

x y P f y x2 y2

dT dT u— + v— — am

dx2 dy2

- ^ v, (3)

dT dT dx dy

d-T Hy-2

+ +( dT

Tldydy Til \dy

udC dC _D íd2C\ id-T

dx dy \ dy2 J C\ dy2

<rC dy1 J' (10)

d T d T

u дА+v дф—Db кф

дх ду I ду2

DT id2T

tZ\ д2

For mathematical analysis of the problem, we use following similarity transforms

n =\ ~y> V =p/avf(n); 0(n)= T T1

s(n) =

p(t]) —

p - p1 pP w - Poo

Shx —

Shx,n —

C —C

dC дУ

ppw - PP 1 V ^

у — 0

— -ReyV(0),

у — 0

— -Re^cp' (0).

Reduced form of the Nusselt, Sherwood and nanofluid Sherwood number is given by

Nur —

and Shrn —

Re1'2'

where stream function y is defined in a usual way as u — dy/dy and v — — (dy/dx).

After the implementation of Eq. (13), Eq. (8) is identically satisfied and from Eqs. (9)-(12), we get a system of following nonlinear differential equations,

f" + ff —f2 + M (A —f) + A2 — 0,

0" + PrfO' + PrNb0'p' + PrNtd'2 + Nds" — 0, s" + Lefs' + Ld0" — 0.

ф" + Lnf t + Nb 0" — 0.

Using Eq. (13), boundary conditions Eq. (7) reduce to f (0) = 0, f (0)=1, f (1)-A, 0(0)= 1; 0(l)-O, s(0)=l, s(l)-0;

<K0)=1, p(i)-0, (18)

where primes denote the differentiation with respect to n. Besides,

oB02 . b v am

M —-, A — -, Pr — —, Le — —,

Pfa a am Ds

tDB (tw - ф1)

Nt —

tDb(Tw - T1)

T1V DTC(Cw - C1)

Ln — —, Db

and Ld

DCT (Tw - T 1)

am((Tw T1) am(Cw C1)

represent the magnetic number, velocity ratio, Prandtl number, Lewis number, Brownian motion parameter, ther-mophoresis parameter, nano Lewis number, modified Dufour parameter and Dufour solutal Lewis number respectively.

The quantities of engineering interest are local Nusselt number Nux, Sherwood number Shx and nanofluid Sherwood number Shx>n. These parameters characterize the wall heat, regular and nano mass transfer rates, respectively and are defined by

Nux —

у — 0

— -Re1'20' (0),

3. Numerical solution

System of coupled nonlinear Eqs. (14)-(17) are solved numerically by using an efficient numerical technique Runge-Kutta-Fehlberg method. System of boundary value problem Eqs. (14)-(17) are first converted into an initial value problem by using shooting technique. We choose nmax such that our solution converges for all the parameters. A fifth decimal place accuracy is restricted for the sake of convergence.

4. Results and discussions

In this section, effects of all the emerging parameters on velocity, temperature, solute and nanoparticles concentration

Figure 1 Variation of f'(n) for different values of A.

Figure 2 Variation of f (n) for different values of M.

profiles are highlighted with the help of graphs and tables. Figures 1-11 are plotted to discuss the effects of these parameters. In Figure 1, behavior of A (ratio of free stream velocity to the velocity of stretching sheet) is depicted on velocity profile f(rf). It can be observed that when free velocity is less than the velocity of stretching sheet, there is a decrease in velocity of the fluid and boundary layer thickness. On the other hand, when free stream velocity is greater than the velocity of stretching sheet, there is an increase in velocity of the fluid, while boundary layer thickness decreases. Figure 2 depicts the behavior of magnetic number M on the velocity field. Rise in magnetic number M for the case when velocity of stretching

sheet is greater than the main stream velocity, has decelerating effect on main flow velocity. That is, with increasing values of M, velocity of the fluid decreases. This is because of the retardation force created by the Lorentz force that opposes the flow. Also, boundary layer thickness is a decreasing function for increasing M. However, increase in velocity is observed for the case when main stream velocity is greater than the velocity of the stretching sheet. In short, magnetic field can be used to control boundary layer separation in several physical phenomenon.

Effects of Prandtl number Pr, Brownian motion parameter Nb, thermophoresis parameter Nt and modified

A = 0.1, M=0.\,Nb=Nt=Le=Ld=Ln= I, Nb=0.2

1.0 0.8 0.6 0.4 0.2 0

—i-1-1-1-1-1-1-1-1-1-1-1-1—

Pr~ 0.4

* > \ .....Pr=0.8

i* \ ---Pr= 1.2

\ » \ - Pr= 1.6

-........7" :-t—

Figure 3 Variation of 6(rj) for different values of Pr.

Figure 6 Variation of d(n) for different values of Nd.

A = 0.1, M= 0.1, Pr=Le = Ld = 0.5, Nb = Nt =Nd= 0.2

1 1 I I II 1 1 1 1 1 1 1 1 1 1 1 ¿«-0.1 \ _

N " , \

: ^ X -----¿»-0.3

: N\ \ \ ---¿«=0.5

\\ \ \ - ¿»=0.7

v * » ' •,.

■— •.

"................. . . . i ". :

Figure 9 Variation of </>(n) for different values of Ln.

Figure 10 Variation of $(if) for different values of Nt.

Figure 11 Variation of $(if) for different values of Nb.

Dufour parameter Nd on the temperature profile are demonstrated in Figures 3-6.

From Figure 3, it can be observed that increase in Pr decreases temperature quite significantly. Since, Prandtl number is the ratio of viscous diffusion rate to the thermal diffusion rate, therefore increase in Pr results in decrease of thermal diffusivity. Figure 4 depicts the behavior of temperature profile for increasing values of thermophoresis parameter Nt. The thermophoretic force generated by temperature gradient results in a fast flow away from the stretching surface, therefore, increasing Nt results in increased temperature. In Figures 5 and 6, effects of Brownian motion

parameter Nb and modified Dufour solute parameter Nd on temperature profile are plotted. It is clear that with an increase in Nb and Nd, the temperature of the fluid increase considerably.

Effects of Lewis number Le and Dufour-solute Lewis number Ld on concentration profile are shown in Figures 7 and 8, respectively. Figure 7 depicts that the increase in Le decreases s(n) quite significantly, i.e. there is a rapid decrease in mass. Ld makes opposite effects on concentration profile as compared to Le that is increasing Ld gives increased s(n). As a result the concentration boundary layer thickness increase.

Behavior of nanofluid Lewis number Ln, thermophoresis parameter Nt, Brownian motion parameter Nb, and on nanomass volume fraction profile ^(n) are depicted in Figures 9-11. Figure 9 considers the behavior of Ln on

Ln is the ratio between thermal diffusivity and mass diffusivity; it is used to analyze the fluid flows where simultaneous effects of heat and mass transfer by convection are considered.

Increment in nanofluid Lewis number Ln decreases ^(n). This is due to the fact that Brownian diffusion decreases with an increase in Ln that forces the concentration to decrease. Influences of thermophoresis parameter Nt on ^(n) are plotted in Figure 10. Rise in Nt increases ^(n) rapidly. This happens because of the fact that increasing Nt corresponds to greater mass flux (due to temperature gradient) that increase the concentration appreciably.

Increasing values of Brownian motion parameter Nb are plotted for ^(n) in Figure 11. A considerable drop in ^(n) is observed for increasing values of Nb. Rise in Nb is due to increase in Brownian motion that increases temperature, which in result is account for the fall in concentration.

Tables 1 and 2, are drawn to compare the results obtained by Runge-Kutta-Fehlberg method with already existing results in literature. From Table 1, it can be observed that our results are in exact agreement to the ones obtained in previous studies. In Table 2, numerical values of local Nusselt number for the case of a regular Newtonian fluid are compared with the existing solutions. Again, excellent agreement is seen between the solutions.

Tables 3 and 4 are drawn to study the effects of different parameters on reduced Nusselt number, reduced Sherwood number and reduced nanofluid Sherwood number. From

Table 1 Comparison of values of f "(0) with existing solutions for M = 0.

Af f"(0)

Present results Hayat [13] Mahaputra [19] Ibrahim [13]

0.01 - 0.9980 - 0.9982 - 0.9980 - 0.9980

0.1 - 0.9694 - 0.9695 - 0.9694 - 0.9694

0.2 - 0.9181 - 0.9181 - 0.9181 - 0.9181

0.5 - 0.6673 - 0.6673 - 0.6673 - 0.6673

2.0 2.0175 2.0176 2.0175 2.0175

Table 2 Comparison of values local Nusselt number — £>'(0) with existing solutions for Nb = Nt=Nd=0.

PrJ. AJ. — £'(0)

Present results Hayat [13] Mahaputra [19] Ibrahim [20]

1.0 0.1 0.6022 0.6021 0.603 0.6022

0.3 0.6255 0.6244 0.625 0.6255

0.5 0.6924 0.6924 0.692 0.6924

1.5 0.1 0.7768 0.7768 0.777 0.7768

0.3 0.7971 0.7971 0.797 0.7971

0.5 0.8648 0.8647 0.863 0.8648

Table 3 Variation of Nur and Shrn for different values of Nt and

Nb = 0.3 Nb = 0.5 Nb = 0.7

Nt{ Nur Shrn Nur Shrn Nur Shrn

0.1 0.3430 0.3157 0.3060 0.3491 0.2713 0.3630

0.2 0.3193 0.2020 0.2839 0.2938 0.2506 0.3319

0.3 0.6968 0.1164 0.2628 0.2532 0.2309 0.3103

0.7 0.2780 0.0271 0.2456 0.2018 0.2152 0.2837

Table 4 Variation of Shr with varying Ld and Le.

Lei Ld = 0.1 Ld=0.3 Ld=0.5

Shr Shr Shr

1.0 0.1359 0.1277 0.1186

3.0 0.2676 0.2697 0.2663

Table 3, one can clearly observe that with an increase in Brownian motion parameter Nb, decrease in values of reduced Nusselt number is seen and reduced nanofluid Sherwood number is found to be an increasing function of Nb. Thermophoresis parameter Nt effects the Nusselt number and nanofluid Sherwood number in a same manner as Brownian motion parameter Nt. That is, decrease in Nusselt number and increase in nanofluid Sherwood number is observed for increasing values of Nt.

Table 4 gives us the numerical values of reduced Sherwood number for increasing values of Dufour solutal number Ld and Lewis number Le. A slight increase in Sherwood number is seen for increasing values of Le. While increase in Ld decreases the numerical values of Sherwood number.

5. Conclusions

Analysis of thermodiffusion effects on stagnation point flow of magneto nanofluid is presented. Governing equations after implementing similar transforms are converted into system of non-linear ordinary differential equations. Numerical solution using Runge-Kutta-Fehlberg method is

carried out. Emerging parameters are analyzed graphically. Comparison with existing solutions is also made to check the efficiency of numerical technique applied. It is concluded that increase in Brownian motion parameter Nb and thermophoresis parameter Nt result in decreasing values of reduced Nusselt number and increasing reduced nanofluid Sherwood number. Also, numerical values of reduced Sherwood number increase slightly with increase in Dufour solutal number Ld. Slight increase in Sherwood number is seen for increasing values of Le. Decrease in Sherwood number us seen for increasing values of Lewis number Le. This work can be extended for non-Newtonian nanofluids for the better understanding of different properties of nanofluids.

Acknowledgement

Authors are thankful to the anonymous reviews and editor for their worthy comments that really helped to improve the quality of presented work.

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