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In: International Journal of Microscale and Nanoscale Thermal… ISSN: 1949-4955 Volume 2, Number 1 © 2011 Nova Science Publishers, Inc.

LAMINAR MHD MIXED CONVECTION FLOW OF A NANOFLUID ALONG A STRETCHING PERMEABLE SURFACE IN THE PRESENCE OF HEAT GENERATION OR ABSORPTION EFFECTS Ali J. Chamkha1*, Abdelraheem M. Aly2 and Humood F. Al-Mudhaf 3 1

Manufacturing Engineering Department, The Public Authority for Applied Education and Training, Shuweikh 70654, Kuwait 2 Civil Engineering Department, Faculty of Engineering, Kyushu University, Japan 3 Chemical Engineering Department, The Public Authority for Applied Education and Training, Shuweikh 70654, Kuwait

ABSTRACT The boundary-layer flow of a nanofluid on a linearly moving permeable vertical surface in the presence of magnetic field, heat generation or absorption, thermopherosis, Brownian motion and suction or injection effects is studied. Similarity solutions are obtained for the boundary-layer equations subject to power-law wall temperature, nanoparticles volume fraction and velocity variations. The obtained equations are solved numerically by an efficient, iterative, tri-diagonal, implicit finite-difference method. A detailed parametric study is performed to access the influence of the various physical parameters on the longitudinal velocity, temperature and nanoparticle volume fraction profiles as well as the local skin-friction coefficient, local Nusselt number and the local Sherwood number and the results are presented in both graphical and tabular forms.

Keywords: MHD; Nanofluid; similarity solutions; free convection; suction or injection; Brownian motion; thermophoresis; heat generation or absorption.

Nomenclature

*

B(x)

magnetic field strength

Cf

local skin-friction coefficient

DB

Brownian diffusion coefficient

DT

thermophoresis diffusion coefficient

Email: [email protected]

52

Ali J. Chamkha, Abdelraheem M. Aly and Humood F. Al-Mudhaf fw

g

suction or injection parameter acceleration due to gravity

M N

magnetic field parameter

Nb Ns Nt Nu k

Brownian motion parameter

P Pr Q(x)

base fluid Pressure

Sc

u

Schmidt number local Sherwood number temperature velocity in x-direction

uw

stretching speed of the plate

Sh T

V v

buoyancy ratio parameter local buoyancy parameter thermophoresis parameter local Nusselt number base fluid thermal conductivity Prandtl number heat generation or absorption coefficient

velocity vector velocity in y-direction

Greek Symbols 

nanoparticles concentration

f

density of the base fluid

p

density of the nanoparticles material

 C  f  C  p    μ





heat capacity of the base fluid heat capacity of the nanoparticles material base fluid thermal diffusivity ratio of heat capacities electrical conductivity of the base fluid base fluid dynamic viscosity volumetric volume expansion coefficient of the nanofluid heat generation or absorption parameter

Laminar MHD Mixed Convection Flow of a Nanofluid…

53

Superscript ′

Differentiation with respect to η

Subscripts f p w 𝑎? ?

1.

base fluid nanoparticles material wall free stream

Introduction

Nanofluids are prepared by dispersing solid nanoparticles in fluids such as water, oil, or ethylene glycol. These fluids represent an innovative way to increase thermal conductivity and, therefore, heat transfer. Unlike heat transfer in conventional fluids, the exceptionally high thermal conductivity of nanofluids provides for exceptional heat transfer, a unique feature of nanofluids. Advances in device miniaturization have necessitated heat transfer systems that are small in size, light mass, and highperformance. Several authors have tried to establish convective transport models for nanofluids. Nanofluid is a two-phase mixture in which the solid phase consists of nanosized particles. In view of the nanoscale size of the particles, it may be questionable whether the theory of conventional two-phase flow can be applied in describing the flow characteristics of nanofluid [1]. Since the size of the particles is less than 100 nm, nanofluids behave like a fluid than a mixture [1–3]. Xuan and Roetzel [1] proposed homogeneous flow model where the convective transport equations of pure fluids are directly extended to nanofluids. This means that all traditional heat transfer correlations (e.g. Dittus–Boelter) could be used for nanofluids provided the properties of pure fluids are replaced by those of nanofluids involving the volume fraction of the nanoparticles. The homogeneous flow models are however in conflict with the experimental observations of Maliga et al. [3], as they under predict the heat transfer coefficient of nanofluids. Xuan et al. [4] have examined the transport properties of nanofluid and have expressed that thermal dispersion, which takes place due to the random movement of particles, takes a major role in increasing the heat transfer rate between the fluid and the wall. This requires a thermal dispersion coefficient, which is still unknown. Brownian motion of the particles, ballistic phonon transport through the particles and nanoparticle clustering can also be the possible reason for this enhancement [5]. Das et al. [6] has observed that the thermal conductivity for nanofluid increases with increasing temperature. They have also observed the stability of Al2O3–water and CuO–water nanofluid. Experiments on heat transfer due to natural convection with nanofluid have been studied by Putra et al. [7] and Wen and Ding [8]. They have observed that heat

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Ali J. Chamkha, Abdelraheem M. Aly and Humood F. Al-Mudhaf

transfer decreases with increase in concentration of nanoparticles. The viscosity of this nanofluid increases rapidly with inclusion of nanoparticles as shear rate decreases. The problem of steady hydromagnetic flow and heat transfer over a stretching surface could be very practicable in many applications in the polymer technology and metallurgy. In particular, many metallurgical processes involve the cooling of continuous strips or filaments by drawing them though a quiescent fluid and that in the process of drawing, these strips are sometimes stretched. In the case of annealing and thinning of copper wires, the properties of the final product depend to a great extent on the rate of cooling. By drawing such strips in an electrically conducting fluid subject to a magnetic field, the rate of cooling can be controlled and final products of desired characteristics might be achieved [9]. And also, in several engineering processes, materials manufactured by extrusion processes and heat treated materials traveling between a feed roll and a wind up roll on convey belts possess the characteristics of a moving continuous surface. The steady flow on a moving continuous flat surface was first considered by Sakiadis [10] who developed a numerical solution using a similarity transformation. Chiam [11] reported solutions for steady hydromagnetic flow over a surface stretching with a powerlaw velocity with the distance along the surface. Tsou et al. [12] studied a wide ranging analytical and experimental investigation of the flow and heat transfer characteristics of the boundary layer on a continuous moving surface. The two-dimensional flow caused solely by a linearly stretching sheet in an otherwise quiescent incompressible fluid which has a very simple closed from exponential solution was established by Crane [13]. Gorla et al. [14] studied the MHD effect on a vertical stretching surface with suction and blowing. Anjali Devi and Kandasamy [15] studied the steady MHD laminar boundary layer flow over a wall of the wedge with suction or injection in the presence of species concentration and mass diffusion. Seddeek [16] studied the effects of heat generation or absorption on heat and mass transfer of a viscoelastic fluid with a magnetic field over a stretching sheet. The study of heat generation has several physical problems such as those concerned with dissociating fluids. Possible heat generation effects may change the temperature distribution and, therefore, the particle deposition rate. This may occur in such applications related to nuclear reactor cores, fire and combustion modeling, electronic chips and semi conductor wafers. Representative studies dealing with heat generation or absorption effects have been reported previously by such authors as Acharya and Goldstein [17], Vajravelu and Nayfeh [18] and Chamkha [19]. The objective of this paper is study mixed convection MHD flow of a nanofluid past a stretching permeable surface in the presence of magnetic field, heat generation or absorption, thermopherosis, Brownian motion and suction or injection effects.

2.

Mathematical Analysis

Consider steady, two-dimensional flow of a nanofluid consisting of a base fluid and small nanoparticles due to the stretching of a vertical permeable surface in the presence of magnetic field, heat generation or absorption, thermopherosis, Brownian motion and

Laminar MHD Mixed Convection Flow of a Nanofluid…

55

suction or injection effects. The Oberbeck–Boussinesq approximation is employed. The governing equations are based on the balance laws of total mass, momentum, thermal energy and nanoparticles modified to include the various effects stated above. These equations can be written respectively as:

.V  0,





(1)





f V.V    2 V  P    p  1  f 1   T  T  g 











 T 



 B2 ( x) V, f

(2)

 Cf V.T  k 2 T   Cp D B.T   DT  T.T  Q(x)T  T , D  V.  D B  2    T   2 T.  T 

(3)

(4)

where V  u,  is the velocity vector with u and  being the x- and y- components of velocity, T is temperature and  is the nanoparticles concentration.  f is the density of the base fluid and  p is the density of the nanoparticles material. DB and DT are the Brownian diffusion coefficient and the thermophoresis diffusion coefficient respectively. P is the pressure,  is the electrical conductivity of the fluid, B(x) is the strength of magnetic field, Q(x) is the heat generation parameter such that Q>0 corresponds to heat generation while Q