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Procedía Engineering 127 (2015) 901 - 908

Procedía Engineering

www.elsevier.com/locate/proeedia

International Conference on Computational Heat and Mass Transfer-2015

Transient MHD Nanofluid Flow and Heat Transfer due to a Moving Vertical Plate with Thermal Radiation and Temperature

Oscillation Effects

V. Rajesh1*, M. P. Mallesh2 and Ch. Sridevi3

2Department of Engineering Mathematics, GITAM University Hyderabad Campus, Rudraram, Patancheru Mandal, Medak Dist. -502 329,

Telangana, India. E-mail: v.rajesh.30@gmail.com 3Department of Engineering Mathematics, Vidya Jyothi Institute of Technology,Aziz Nagar Gate, C.B. Post, Hyderabad-500075, Telangana,

India.

Abstract

We investigate the transient free convection flow and heat transfer of a viscous, electrically conducting, and incompressible nanofluid past a moving semi-infinite vertical plate subject to hydro magnetic, radiation and temperature oscillation effects. The fluid is water-based nanofluid containing nanoparticles of copper (Cu). The dimensionless governing partial differential equations are solved by using a robust, well-tested, implicit finite-difference method of Crank-Nicolson type. Numerical results are obtained for the local values of skin friction coefficient, Nusselt number as well as for the velocity and temperature profiles for selected values of the governing parameters, such as the nanoparticle volume fraction ^, the magnetic parameter M, the radiation parameter N, the phase angle cot, the thermal Grashof number Gr. The effect of significant parameters on the flow and heat transfer characteristics is presented. The present study is relevant to magnetic nanomaterials thermal flow processing in the chemical engineering and metallurgy industries.

© 2015 The Authors. PublishedbyElsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICCHMT - 2015

Keywords: Free convection, nanofluid; implicit finite difference numerical method; MHD, thermal radiation; transient flow; temperature oscillation.

* Corresponding author. Tel.: +91-9441146761; fax: +0-000-000-0000 . E-mail address: v.rajesh.30@gmail.com

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license

(http://creativecommons.org/licenses/by-nc-nd/4.0/).

Peer-review under responsibility of the organizing committee of ICCHMT - 2015

doi:10.1016/j.proeng.2015.11.358

Nomenclature

T' w Temperature of the plate Knf thermal conductivity of the nanofluid

P density p thermal expansion coefficient

M dynamic viscosity Pnf thermal expansion of the nanofluid

Pnf effective density of the nanofluid t ' time

^nf effective dynamic viscosity of the nanofluid g acceleration due to gravity

n empirical shape factor for the nanoparticle. T non dimensional temperature

V kinematic viscosity t non dimensional time

$ solid volume fraction of nanoparticles u non dimensional velocity along X-axis.

T ' V non dimensional velocity along Y-axis

Temperature of the fluid u velocity along x-axis

T' œ Temperature of the fluid far away from the plate V velocity along y-axis

K thermal conductivity

1. Introduction

Nanofluids fluids research have attracted considerable attention in recent years due to its exceptional applications in electronics, communication, computing technologies, optical devices, lasers, high-power X-rays, scientific measurement, material processing, medicine and material synthesis. Nanofluids augment thermal conductivity of the base fluid significantly, which are also very stable and have no additional problems, such as sedimentation, erosion, additional pressure drop and non-Newtonian behaviour, due to the tiny size of nanoelements and the low volume fraction of nanoelements required for conductivity enhancement. These suspended nanoparticles can change the transport and thermal properties of the base fluid. Because of multiple applications, several studies have been published on the modelling of natural convection heat transfer in nanofluids. Loganathan et al. [1] studied the effects of radiation on an unsteady natural convective flow of a nanofluid past an infinite vertical plate. Rajesh et al. [2] presented a mathematical model for the unsteady free convective flow and heat transfer of a viscous nanofluid from a moving vertical cylinder in the presence of thermal radiation. Also Rajesh and Anwar Beg [3] numerically studied the effects of MHD on the transient free convection flow of a viscous, electrically conducting, and incompressible nanofluid past a moving semi-infinite vertical cylinder with temperature oscillation. Later Rajesh et. al. [4] developed a mathematical model for the nanofluid flow and heat transfer due to the impulsive motion of an infinite vertical porous plate in its own plane in the presence of a magnetic field and viscous dissipation. Recently Chamkha et al. [5] presented review of MHD Convection of Nanofluids in various geometries and applications.

Transient free convective flow of nanofluids under the influence of a magnetic field have attracted the interest of many researchers in view of their applications in modern materials processing where magnetic fields are known to achieve excellent manipulation and control of electrically conducting materials. Thermal radiation processing of nanomaterials is a very key development in recent years. Numerous studies have identified the beneficial effects of radiative flux on nanomaterials fabrication. This includes reinforcing nanofluids with radiation damage-tolerant characteristics. In view of multiple applications, the present study considers the problem of transient free convection flow and heat transfer of Cu-water nanofluid past a moving semi-infinite vertical plate which is subject to hydro magnetic, radiation and temperature oscillation effects. A robust, well-tested, implicit finite-difference numerical method has been used to find the numerical solutions of the problem. The effect of significant parameters such as the nanoparticle volume fraction ^ , the magnetic parameter M, the radiation parameter N, the phase angle at, the

thermal Grashof number Gr on the thermal and hydrodynamics response of the system such as the skin friction and heat transfer coefficients of physical importance can be determined by investigation.

2. Mathematical Model

The unsteady laminar two-dimensional boundary layer flow of a viscous incompressible electrically conducting nanofluid past a moving vertical plate in the presence of an applied magnetic field and thermal radiation is under consideration. A schematic representation of physical model and coordinate system is depicted in Figure. 1. Initially, both the plate and the nanofluid are stationary at the same temperature Taj. They are maintained at this condition for all t' < 0 . At time t' > 0 , the plate starts moving in the vertical direction with the uniform velocity Uq The temperature on the surface of the plate is raised to Tw and a periodic temperature is assumed to be superimposed on this mean constant temperature of the surface. We assume that the uniform magnetic field with intensity of B0 acts in the normal direction to the plate and the effect of induced magnetic field is negligible, which is valid when the magnetic Reynolds number is small. The viscous dissipation, Ohmic heating, ion slip, and Hall effects are neglected as they are also assumed to be small.The thermo physical properties of the nanoparticles Oztop and Abu-Nada,[6] are given in Table l.Under the above assumptions and following the nanofluid model proposed by Tiwari and Das [7], with the usual Boussinesq approximation(Schlichting and Gersten, [8]) the governing equations in the presence of magnetic field and thermal radiation are:

du + dv _ q

dx dy (1)

du du du d u

T-7 + " — + V — = — TT +--s(T -T„)--(2)

at dx dy pnf dy pnf pnf

ST' 8T' 8T' Knf S2T' 1 8q (-y\

—r+ u-+ v-= -,-^----r--^ \->J

8t' 8x 8y (pCp )nf 8y2 (pCp 8y

(Equations (l)-(3) are subjected to the following initial and boundary conditions: ^ 0 " = 0 v = 0 f = f^or all x and y

t'>0 "="» v =0 t'=r;+(r;-r:)cos(®'i') at y = 0

u = 0 T' = Tl at x _ Q

u ^ o T'^ K as —> & ^

The effective dynamic viscosity of the nanofluid was given by Brinkman [9] as _

il A\25

Where ^ is the solid volume fraction of nanoparticles. The effective density pnj, thermal expansion coefficient (pP) capacitance ^ are given by

Pnf =(l-<fr)pf + <frp.

(P/3)nf =(i -</>){Pp)f + *{pP), )„ = (! )f + 1

(6) (V) (8)

The thermal conductivity of nanofluid restricted to spherical nanoparticles is approximated by the Maxwell-Garnett model [10]& Guerin etal.[ll],

Knf + 2Kf -2— Ks)

Kf K, + 2Kf + t{Kf-Ks) (9)

In Eqs. (5)-(9), the subscripts nf, f and s denote the thermophysical properties of the nanofluid, base fluid and nano-solid particles, respectively.

Table 1. Thermo-physical properties of water and nanoparticles

p(Kgrn-3 ) cp (jKg1 ) K(Wm-1K "') ß xl0~5 [K"')

H 2o 997.1 4179 0.613 21

Cu 8933 385 401 1.67

To simulate uni-directional radiative heat flux, we implement the Rosseland approximation (Brewster, [12]), which leads to the following expression for radiative heat flux qr: 4ct, 8T

3 ke 8y

where us is the Stefan-Boltzmann constant and ke is the mean absorption coefficient, respectively. It should be

noted that by using the Rosseland approximation we limit our analysis to optically thick nanofluids. If temperature differences within the flow are sufficiently small such that Tmay be expressed as a linear function of the temperature, then the Taylor series for Tabout , after neglecting higher order terms, is given by

y '4 _ 4j i 371 ' ^

In view equations (10) and (11), equation (3) reduces to

5T ' 5T' 5T'

32T ' 16 o,T" 32T '

-T ~ M--- v--7-\--t—1--7-\--T-

St' dx 8y ( pCp 8y2 3k, (pCp 8y2 By introducing the following non-dimensional variables

U = —, V = -

T = IldlL G= gßfVf TL TL -T' ' ' ul

p, =—, M--

af pfu-0

Eqs. (1), (2) and (12) take the following non-dimensional form

dU dV „ -+-= 0

^ + u ™ + (7 =

St SX ÔY

1 -é + q

' 1 + i!GrT -MU

(1 SY2

(Pß),

?L+v K+(7 ?L

4 ) ^ <¥r_

dt " SX SY ipc Kf 3N IP SY2

With the initial and boundary conditions

for all X and Y

t < 0; U = 0, V = 0, T = 0,

U = 0,

at Y = 0

at X = 0

3. Numerical technique

An implicit finite difference scheme of Crank-Nicolson type (Carnahan et. al. [13]) is used to solve the governing non-dimensional equations (14), (15) and (16) under the initial and boundary conditions (17). This technique has been employed very successfully in a number of studies in complex thermofluid flows in recent years including viscoelastic flows, transient nanofluid flows and magnetic induction flows. It is one of the most reliable procedures for solving partial differential equation systems. The method of solving the finite difference equations using Thomas algorithm has been discussed by Ramachandra Prasad et al. [14].The region of integration is considered as a rectangle with sides x = o to Xmax = 1 and ¥ = o to Ynlax = 14 ,where corresponds to y = « which lies very well outside the momentum and energy boundary layers. After experimenting with a few sets of mesh sizes to access grid independence, the time and spatial step sizes at = 0.01, AX = 0.05 and ay = 0.25 were found to give accurate results. The Crank-Nicolson implicit finite difference scheme is always stable and convergent.

4. Exact Solution and Validation of Numerical Solution

In the absence of radiation and inertial terms, equation (16) becomes

1 -6 +

(■PC,

Pr dY2

The analytical solution of Eq. (18), subject to the boundary conditions (17) (when mt=— ) by using Laplace

transforms method is given by

T = erfc

- if ;

In order to verify the accuracy of the numerical results, the temperature profiles of the present study (when mt = L )

in absence of radiation parameter when compared with the analytical solution given by equation (19) in Figure 2 is found to be compatible. Confidence in the implicit finite-difference numerical solutions is therefore high. 5. Engineering quantities

In engineering materials processing, several physical quantities are of practical interest, for example the skin fHntinn nnpffinipnt r and thp Wal Nusselt number Nux , which are defined, respectively, as follows:

The derivatives involved in equation (21) are evaluated by using a five-point approximationformula. 6. Results and Discussion

In order to get a physical insight into the problem, the influence of significant parameters such as magnetic parameter (M), radiation parameter (N), phase angle ( 0t ), nanoparticle volume fraction ($), thermal Grashof number (Gr) and time (f) on the transient velocity and temperature profiles in the boundary layer region are presented in figures 3-6 with Gr = 10, 15 (Which correspond to cooling of the plate by natural convection due to the temperature gradient). We consider Cu-water nanofluid with nanoparticle volume fraction ^ in the range of 00.04, as sedimentation takes place when the nanoparticle volume fraction exceeds8%. In this study, we have considered spherical nanoparticles with dynamic viscosity and thermal conductivity shown in equations (5) and (9) respectively. The Prandtl number, Pr of the base fluid is kept constant at 6.2. When (p = 0 the model contracts to the governing equations for a regular viscous fluid i.e. nanoscale characteristics are eliminated.

The effects of different parameters M, Gr, mt and N on the transient velocity and temperature profiles of Cu-water nanofluid are presented in figures 3 and 4 respectively. From the figures, it is found that the velocity of Cu-water nanofluid decreases with an increase in M. This is because the application of a transverse magnetic field will result in a resistive type of force (Lorentz force) similar to drag force, which tends to resist the flow, thus reducing its velocity. Therefore, an increase in M leads to an increase in the momentum boundary layer thickness. But Temperature of Cu -water nanofluid is observed to increase with the increase in M. This is because the supplementary work done in dragging the fluid against the retarding action of the magnetic field is dissipated as heat and this elevates temperatures. This also implies that thermal boundary layer thickness increases with an increase in M. From the figures, it is observed that the velocity of Cu-water nanofluid decreases with an increase in N.

From the definition of N, when N increases conduction contribution dominates and thermal radiative contribution recedes. Therefore, for higher values of N, a lower radiative flux is present and this exerts a decelerating influence on nanofluid boundary layer flow. It is also observed that the temperature of Cu-water nanofluid decreases with increasing values of N. This is because, greater N values imply lower thermal radiative flux which effectively suppresses the rate of energy transport to the fluid and results in a cooling of the boundary layer regime and a thinner thermal boundary layer. It is also noticed in the figures that the velocity of Cu-water nanofluid increases with an increase in Gr. This is because the parameter Gr denotes the relative influence of thermal buoyancy force and viscous force in the boundary layer regime. Buoyancy dominates when there is a larger value of this parameter and when there is a smaller value viscosity dominates. Thus an increase in the value of thermal Grashof number has the tendency to induce much flow in the boundary layer due to the effect of thermal buoyanBut an increase in Gr leads to a decrease in temperature of Cu -water nanofluid, since increasing buoyancy force causes a reduction in thermal diffusion.Therefore greater buoyancy effects will decrease thermal boundary layer thickness. Moreover the velocity and temperatures of Cu-water nanofluid decreases with the increase in phase angle co t.

The transient velocity and temperature profiles of Cu -water nanofluid for different values of ^ and t are presented in figures 5 and 6. As expected, the velocity increases with progression in time. Also the velocity of Cu -water nanofluid is less than that of pure water = 0) for all values of time t. Moreover an increase in the nanoparticle volume fraction^ leads to a decrease in the velocity for all values of t (time). From the figures, the temperature is also observed to increase with a progress in time. Moreover the temperature of Cu -water nanofluid is

Using non-dimensional variables (13), we get

high than that of pure water = 0) for all values of time t. Unlike the velocity, the temperature increases with the increase in nanoparticle volume fraction ^ . Nanoparticles therefore clearly achieve thermal enhancement in the regime

The local Skin friction coefficient and the Nusselt number are presented in figures 7-10 respectively. It is found that skin friction coefficient for Cu-water nanofluid decreases with increasing M, N, at, but it increases with increasing Gr, and t. Also an increase in nanoparticle volume fraction ^ reduces the skin friction coefficient. From figures it is also found that Nusselt number for Cu-water nanofluid decreases with increasing M, at, t, but it increases with increasing Gr, and N. Moreover an increase in ^ enhances the local Nusselt number values. This confirms that use of nanofluids achieves significant enhancement in heat transfer rates, and demonstrates the importance of utilizing nanofluids for improving industrial cooling and heating processes.

7. Conclusions

The Mathematical model is developed for the transient free convection flow and heat transfer of a viscous, electrically conducting, and incompressible nanofluid past a moving semi-infinite vertical plate subject to hydro magnetic, radiation and temperature oscillation effects. The effects of various physical parameters on the nanofluid flow and heat transfer characteristics were examined numerically. The present computations have shown that the rate of heat transfer decreases with the increase in Magnetic parameter (M) and Phase angle {at) , but it increases with the increase in thermal Grashof number (Gr), and radiation parameter (N). Moreover the local Nusselt number values enhanced with an increase in^ .

Fig.l.The physical model and coordinate system

MONO -t-. ' G<-'0 MO. NO -»■< 1, Gi-tO M»1 M>S.-^t).QKlO U" NO -4-«*. G'-'O N■1 N-l 1. G"" 5

a Present result

-Analytical solution

A Present result - - Analytical solution

Fig.2.Comparison of temperature profiles

»1*1/3 Gr-! D -t"*/3. GrOO xtsi/3. Gr-10 »1»*«. Gf"'0 >.t»i/3 Gr*TS

Fig.3 .Velocity profiles for M, N, Gr, at

Fig.4. Temperature profiles for M, N,Gr,

Fig.5. Velocity profiles for ^ ,t

Fig.6.Temperature profiles for ^ ,t

Fig.7.Skin friction coefficient for M,N, Wt

Fig.8. Skin friction coefficient for ^ ,t

Cu-water nanofluid

- M-1 N-3. -t"» 3. Gr-10 M-2. N-3. »t»« 3. Gr-10 M-1. N-5. .4«* .«. Gr-10 M-1. N-3. -t«* ft. Gr-10 M-1. N-3. ..t-T t Gr-15

Fig.9.Nusselt number for M,N,G, at

Fig.lO.Nusseltnumber for ^ ,t

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