[i] · 2016. 9. 15. · bangalore university journal of science -a bi annual periodical....

[i]

VIJNANA BHARATHI Bangalore University Journal of Science - A bi-annual periodical.

ISSN-0971-6882 Website: http://bangaloreuniversity.ac.in/

JOURNAL INFORMATION

Bangalore University, Bangalore, has introduced ‘The Science

Journal’ in 1995 and has been catering to the needs of various

disciplines of Science. The journal is published as a half yearly (with

special issues on specific topics in between) and accepts scientific

research articles based on original work, complying with the

objectives and the impact on society and scientific community. It also

publishes Basic and Applied science research articles. Vijnana

Bharathi also encourages articles with interdisciplinary approach.

The articles are reviewed and published. Vijnana Bharathi holds the

copyright of articles it publishes, and any act of exact or altered

versions will be considered as copyright violation.

TYPES OF ARTICLES

Articles published in Vijnana Bharathi encompass a broad range of

research topics in

Biology Systematics and Diversity, Morphology/ Anatomy and

Physiology among plant and animal kingdoms, Cell and Molecular

biology, Evolutionary biology and Ecology, Genetics, Developmental

biology, Parasitology and Pathology, Bio-chemistry, Immunology,

Bioinformatics, Bioethics etc.,

Physics General Physics, Gravitation and Astrophysics, Elementary

Particles and Fields, Nuclear and Hadronic Physics, Atomic, Molecular,

and Chemical Physics, Optical Physics, Fluid and Plasma Physics,

Statistical and Nonlinear Physics, Condensed Matter Physics, Cross

Disciplinary Physics: Biological Physics, Quantum Information etc.,

Chemistry : Organic, Inorganic, Physical, Analytical, Biological,

Pharmaceutical, Industrial, Environmental, Agricultural & Soil,

Petroleum, Polymers, Nanotechnology, Green Chemistry, Forensic,

Photochemistry, Synthetic Drugs, Computational, as well as Chemical

Physics and Chemical Engineering.

Mathematics Analytic number theory, Functional analysis and

operator theory, Real and harmonic analysis, Complex analysis,

Numerical analysis, Applied mathematics, Partial differential

equations, Dynamical systems, Control and Optimization, Probability,

Mathematical biology, Combinatorics, Mathematical physics

[ii]

Earth and Atmospheric Sciences Atmospheric sciences,

Geochemistry, Geology, Mineralogy and Petrology, Geophysics,

Hydrology, Oceanography, Climate change, geography and

geoinformatics

Computer Sciences Algorithm and computational complexity,

Computer architecture and high performance computing, Distributed

and grid computing, Human-computer interaction, Information

security, Pattern recognition and image processing, VLSI design and

testing, Artificial intelligence, Dependable computing and any other

allied multidisciplinary articles.

The authors are encouraged to contact the subject editor or Editor-in-

Chief if there are any clarifications regarding the suitability of their

manuscript in Vijnana Bharathi. The journal will accept manuscripts

in the form of research articles, review articles, technical reports and

student articles.

Research article: Findings of the original research work must be presented in the manuscript as per the journal guidelines.

The articles must not have been published in or submitted

previously to in any scientific journal. The articles must include

all the authors (with their affiliations) and should be credited

accordingly.

Review articles: The journal publishes both original and invited review articles. The Editor-in-Chief, in consultation

with Editorial Board, identifies and invites reviews in

specialized research areas. Additionally, unsolicited review

articles can also be submitted.

Technical report: A technical report should represent a new method, technique, or procedure of interest. If earlier methods

are modified or improvised a comparative account (in terms of

results achieved) should be provided.

Student articles: Aimed to encourage and promote the participation of students, this section would also include

reports of important scientific developments that will impact

the ongoing frontier areas of research. The articles can either

be experiment based or hypothetical approaches that can be

adopted in frontier areas of research. All the content in this

section will be published subject to peer review.

Review Process

All manuscripts are reviewed by qualified external reviewers

and /or by Editor and Members of the Editorial Board. Decisions will

be made as rapidly as possible, and the journal strives to return

reviewers’ comments to authors promptly. The Editorial board will

[iii]

re-review manuscripts that are accepted pending revision. It is the

goal of the Vijnana Bharathi to publish manuscripts shortly after

submission.

Proofs and Reprints

Electronic proofs will be sent (via e-mail attachment) to the

corresponding author as a PDF file. Page proofs are considered to be

the final version of the manuscript. With the exception of

typographical or minor clerical errors, no changes will be made in the

manuscript at the proof stage. Because, Vijnana Bharathi will be

published freely online/print to attract a wide readers. Authors will

have open access to the full text (in both HTML and PDF) of the article.

Authors can freely download the PDF file from which they can print

unlimited copies of their articles.

Copyright

Submission of a manuscript implies: that the work described

has not been published before (except in the form of an abstract or as

part of a published lecture, or thesis) that it is not under consideration

for publication elsewhere; that if and when the manuscript is accepted

for publication, the authors agree to automatic transfer of the

copyright to the publisher.

[iv]

EDITORIAL BOARD

Prof. Sharath Ananthamurthy

Department of Physics

Bangalore University, Bangalore

Email id : [email protected]

Prof. Asha Devi

Department of Zoology


Email id :

[email protected]

Prof. Pradeep G. Siddheshwar

Department of Mathematics


Email id :

[email protected]

Prof. G. T. Chandrappa

Department of Chemistry


Email id :

[email protected]

Prof. Tejavathi

Department of Botany


Email id :

[email protected]

Prof. B. C. Prabhakar

Department of Geology


Email id :

[email protected]

Prof. M. Hanumanthappa

Department of Computer

Application


Email id :

[email protected]

Prof. H. P. Puttaraju Department of Life Science Bangalore University Bangalore Email id : [email protected]

Dr Sayandip Mukherjee Asst. Prof- Biological Sciences, Bangalore University Research Scientist (Virology) - SSG

Hindustan Unilever Research Centre Email id : [email protected] Dr S. Sampath Kumar

Department of Life Science


Email id : [email protected]

EDITOR IN CHIEF

ASSOCIATE EDITOR

ASSOCIATE EDITOR

http://bcprabhakar2rediffmail.com/http://portal.issn.org/cgi-bin/gw/chameleon?sessionid=2015031010320215970&skin=unrestricted&lng=en&inst=consortium&host=localhost%2b5000%2bDEFAULT&patronhost=localhost%205000%20DEFAULT&search=SCAN&function=INITREQ&sourcescreen=CARDSCR&pos=1&rootsearch=3&elementcount=1&u1=33&t1=Vignana%20bharathi&beginsrch=1

DEDICATED TO THE FOND MEMORY

OF

Padma Bhushan

Prof. Dr H. NARASIMHAIAH

Former Vice-Chancellor Bangalore University

BANGALORE UNIVERSITY Jnana Bharathi, Bengaluru-560 056, India

(Phone: 080-23213172 / 22961001, Fax: 23219295, e-mail:[email protected]) Prof. B. Thimme Gowda No. VCP/ Vice Chancellor Date: 20.1.2016

FOREWORD BY THE VICE CHANCELLOR

Bangalore University is committed to excellence in research and

teaching. The staff and students are our assets. The University views

each one of its activities as an investment that helps in achieving its

objectives.

The ultimate goal of the University, as laid down by the University

Grants Commission, is to improve academic performance and aim for

excellence. The Golden Jubilee Volume on "Forced flow and free

convective instabilities in nanofluids" is an attempt in this direction.

The volume has contributions to a cutting-edge area from researchers

within India and from countries like USA, China, Japan and Malaysia. I

understand that the compiled and reviewed papers on mathematical

modelling of thermal convection problems concern ways and means of

optimising the use of energy. I compliment the editorial team of Vijnana

Bharathi for having come up with this volume and hope that the readers

from within our University and outside benefit from this.

Prof. Dr B. Thimme Gowda Vice Chancellor Bangalore University

City Campus: Central College Campus, Palace Road, Bengaluru-560 001 (Phone: 080-22211792, Fax:

22213052)

Website: www.bangaloreuniversity.ac.in https://www.karnataka.gov.in/bangaloreuniversity

http://www.bangaloreuniversity.ac.in/https://www.karnataka.gov.in/bangaloreuniversity

Table of Contents

Sl. No.

Title of the paper

Authors

Page Nos.

1

Effects of viscous-Ohmic dissipation on MHD boundary layer flow of micropolar nanofluid past a stretching sheet with thermal radiation and non-uniform heat source/sink

Dulal Pal Gopinath Mandal Andrew Chan

1-18

2

Convective heat and mass transfer flow of a nanofluid past a vertical slender cylinder in a saturated porous medium

K. V. Prasad K. Vajravelu Hanumesh Vaidya M. Prasad

19-38

3

Stagnation-point flow of a magneto-nanoliquid over a melting stretching sheet with an induced magnetic field

B. Mahanthesh B. J. Gireesha Rama S. R. Gorla

39-54

4

Effect of temperature-dependent internal heating on thermal convection in a viscoelastic nanofluid saturated porous layer

I. S. Shivakumara M. Dhananjaya Chiu-On Ng

55-84

5

Study of heat transport in Newtonian water-based nanoliquids using single-phase model and Ginzburg-Landau approach

P. G. Siddheshwar N. Meenakshi Y. Kakimoto A. Nakayama

85-101

6

Study of heat transport in Newtonian water-based nanoliquids using two-phase model and Ginzburg-Landau approach

P. G. Siddheshwar C. Kanchana Y. Kakimoto A. Nakayama

102-119

7

Study of convective thermal instability in nanofluid saturated porous media in the presence of vertical throughflow, internal heat source and rotation

B. S. Bhadauria Vineet Kumar B. K. Singh I. Hashim

120-140

8

Buoyancy induced flow past a non-isothermal axisymmetric body immersed in a porous medium of varying permeability saturated by non-Newtonian nanofluids

Shobha Bagai Chandrashekhar Nishad

141-158

`1

VIJNANA BHARATHI-The frontier journal in SCIENCE

EFFECTS OF VISCOUS-OHMIC DISSIPATION

ON MHD BOUNDARY LAYER FLOW OF

MICROPOLAR NANOFLUID PAST A

STRETCHING SHEET WITH THERMAL

RADIATION AND NON-UNIFORM HEAT

SOURCE/SINK

Dulal Pal*, Gopinath Mandal † and Andrew Chan ††

† Siksha Satra, Visva-Bharati, Sriniketan, West Bengal-731 236, India.

††

Director of Research in Engineering and Director of Studies in Civil Engineering, University of Nottingham(Malaysia Campus), Jalan Broga, Semenyih-43500, Selangor Darul Ehsan, Malaysia .

ABSTRACT:

This paper deals with studying the effects of viscous and Ohmic dissipation on electrically conducting micropolar nanofluids flow over a stretching sheet in the presence of thermal radiation, non-uniform heat source/sink and transverse magnetic field. Governing nonlinear boundary layer equations and boundary conditions are transformed into a system of nonlinear ordinary coupled differential equations and are solved numerically by Runge–Kutta-Fehlberg method with shooting technique. Influences of the microrotation parameter on flow profiles with different emerging physical parameters are presented graphically for four types of metallic or nonmetallic nanoparticles, namely silver (Ag), copper (Cu), alumina (Al2O3) and titanium dioxide (TiO2) in water based micropolar fluid in order to show some interesting phenomena. Effects of different parameters on skin friction coefficient, couple stress and local Nusselt number are also discussed. This article is to be cited as: Pal D, Mandal G and Chan A. Effects of viscous-ohmic

dissipation on MHD boundary layer flow of micropolar

nanofluid past a stretching sheet with thermal radiation and

non-uniform heat sourc/sink, Bangalore; Vijnana Bharathi, 1

(1): 1-18, 2016.

1

ISSN : 0971-6882 ISSN-L : 0377-8487 CODEN : VBHAD6

©2016 Jnanavahini All rights reserved

Key words: Nanofluids, Micropolar, Magnetic field, Thermal radiation, Ohmic dissipation, Viscous dissipation.

Received on: 4/1/2016 Accepted on: 16/1/2016

Vol. 1 | No.2 |01 - 18 |

March | 2016

*Corresponding author:

DULAL PAL Department of Mathematics Institute of Science Visva-Bharati University Santiniketan West Bengal-731 235, India [email protected]

mailto:[email protected]:[email protected]

AdminStamp

VIJNANA BHARATHI, 1(2):1-18, 2016

D. Pal, G. Mandal and A. Chan 2

Nomenclature Latin symbols

A*,B* heat generation/absorption parameter B0 strength of magnetic field c,b constants Cp skin friction coefficient Cp specific heat at constant pressure

Cn couple stress

Ec Eckert number

H microrotation profile

M magnetic field parameter

n concentration of micro element

N microrotation vector normal to x-y plane

K micropolar/material parameter

K* Rosseland mean spectral absorption coefficient

Nr thermal radiation parameter

Nux local Nusselt number

Pr Prandtl number

qr thermal radiative heat flux

qw heat flux at wall

Rix local Reynols number

S suction parameter

T temperature of the fluid

T∞ free stream temperature

Tw temperature at the wall

u velocity component in x-direction

uw stretching/shrinking sheet velocity

v velocity component in y-direction

x, y direction along and perpendicular to the plate, respectively

Greek symbols

μnf effective dynamic viscosity of the nanofluid

μf dynamic viscosity of the fluid

νf kinematic viscosity of the fluid

ρnf effective density of the nanofluid



σ*

Stefan-Boltzmann constant (unit: W m-2 K-4 )

κf thermal conductivity of the fluid (unit: W/m-K)

κ1 vortex viscosity

θ dimensionless temperature of the fluid

ψ stream function

κnf effective thermal conductivity of the nanofluid

αnf effective thermal diffusivity of the nanofluid

σnf electrical conductivity of the nanofluid

σs electrical conductivity of the solid nanoparticles

σf electrical conductivity of the base fluid

γnf spin gradient viscosity of the fluid

αf fluid thermal diffusivity

φ ,φ,→

φ4 solid volume fraction of the nanoparticles

τw wall stress function

Subscripts

nf Nanofluid

f Liquid

s Solid

1 Introduction

Nanofluids are potential heat transfer fluids with enhanced thermo-physical properties and heat transfer performance which can be applied in many devices for better performances (i.e., energy, heat transfer and other performances). The nanoparticles are typically made of oxides such as alumina, silica, titania, copper and carbides. The nanofluid (initially introduced by Choi1) is an advanced type of fluid containing nano-meter sized particles (diameter less than 100 nm) or fibers suspended in the ordinary fluid. Nanofluids are used in different engineering applications such as microelectronics, microfluidics, transportation, biomedical, solid-state lighting and manufacturing. Specific applications of nanofluids in engine cooling, solar water heating, cooling of electronics, cooling of transformer oil, improving diesel generator efficiency, cooling of heat exchanging devices, improving heat transfer efficiency of chillers, domestic refrigerator-freezers, cooling in machining, in nuclear reactor and defense and



space have been reviewed and presented. Congedo et al.2 compared different models of nanofluid (regarded as a single phase) to investigate the density, specific heat, viscosity and thermal conductivity and discussed the water–Al2O3 nanofluid in details by using CFD. Hamad et al.3 introduced a one parameter group to represent similarity reductions for the problem of magnetic field effects on free-convective nanofluid flow past a semi-infinite vertical flat plate following a nanofluid model proposed by Buongiorno4.

Magnetohydrodynamic boundary layer flow with thermal radiation is of considerable interest due to its wide usage in industrial technology and geothermal application, high temperature plasmas applicable to nuclear fusion energy conversion, MHD power generation systems and liquid metal fluids. Viscous dissipation and heat generation/absorption are quite often a negligible effect, but its contribution might become important when the fluid viscosity is very high. It changes the temperature distributions by playing a role like an energy source, which leads to affected heat transfer rates. Based on these applications, Poornima et al.5 analyzed the simultaneous effects of thermal radiation and magnetic on heat and mass transfer flow of nanofluids over a non-linear stretching sheet. The effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a moving flat plate were studied by Motsumi and Makinde6. Very recently, Makinde and Mutuku7 investigated the hydromagnetic thermal boundary layer of nanofluids over a convectively heated flat plate with viscous dissipation and Ohmic heating effects. Pal and Mandal8 studied the magnetohydrodynamic boundary layer flow of an electrically conducting convective nanofluids induced by a non-linear vertical stretching/shrinking sheet with viscous dissipation, thermal radiation, and Ohmic heating. Sandeep and Sugunamma14 illustrated the effects of radiation and inclined magneticfield on an unsteady MHD free convection flow past a moving vertical plate in a porous medium. Kayalvizhi et al.10 discussed the velocity slip effects on the heat and mass fluxes of a viscous electrically conducting fluid flow over a stretching sheet in the presence of viscous dissipation, Ohmic dissipation and thermal radiation. Ganga et al.11 analyzed the effects of internal heat generation/absorption, viscous and ohmic dissipations on steady two-dimensional radiative MHD boundary-layer flow of a viscous, incompressible, electrically conducting nanofluid over a vertical plate.

A fluid flow theory takes into consideration the microrotation of the nanoparticles could potentially clarify the theoretically and the experimental results. This theory is called micropolar fluid theory. Generalization of the Navier-Stokes equations can be considered as a micropolar fluid which has been introduced by Eringen12. After two years comprehensive theory of micropolar fluids was also developed



by Eringen13. Micropolar fluids have applications in blood flows, suspension solutions, liquid minerals, liquids with spices, movement of colloidal elucidations etc. Basically, the micropolar fluid can contain suspension of small stiff cylindrical component such as huge dumpbell shaped. So application of micropolar theory in the presence of nanoparticles may give an important insight to control the thermal conductivity of nanofluids. Das14investigated the effect of the first-order chemical reaction and the thermal radiation on hydro-magnetic free convection heat and mass transfer flow of a micropolar fluid through a porous medium. Chen15 has presented the problem of combined heat and mass transfer in buoyancy-induced MHD natural convection flow of an electrically conducting fluid along a vertical plate is investigated with Ohmic and viscous heating. E1-Hakiem16

presented an analysis for the effect of thermal dispersion, viscous and Joule heating on the flow of an electrically conducting and viscous incompressible micropolar fluid past a semi-infinite plate whose temperature varies linearly with the distance from the leading edge in the presence of uniform transverse magnetic field. Oahimire and Olajuwon17 studied heat and mass transfer effects on an unsteady flow of a chemically reacting MHD micropolar fluid through an infinite vertical porous plate in porous medium with hall Effect and thermal radiation. Hussain et al.18 reported on the radiation effects on the unsteady boundary layer flow of a micropolar fluid over a stretching sheet. Kumar19 analysed the problem of MHD mixed convective flow of a micropolar fluid with the effect of Ohmic heating, radiation and viscous dissipation over a chemically reacting porous plate with constant heat flux. Ibrahim et al.20 deliberated the properties of a magneto-hydrodynamic micropolar fluid flow with temperature-dependent heat source over a steady vertical porous plate, when the free stream fluctuates in magnitude. Singh and Kumar21 also studied the heat and mass transfer characteristics of the free convection on a vertical plate in a porous medium with variable wall temperature and concentration in a doubly stratified and viscous dissipating micropolar fluid in presence of chemical reaction, heat generation and Ohmic heating. Nazar et al.22 investigated the stagnation-point flow of a micropolar fluid over a stretching surface. Nazar et al.23 also investigated the unsteady boundary layer flow over a stretching sheet in a micropolar fluid. Mishra et al.24 discussed steady flow of an electrically conducting incompressible viscous fluid over a vertical plate in the presence of a transverse magnetic field with variable wall temperature and concentration in a doubly stratified micropolar fluid. Perdikis and Raptis25 studied the flow of a micropolar fluid past a stationary plate in the presence of radiation. The flow over a permeable stretching sheet in micropolar nanofluids with suction has been studied by Fauzia et al.26. Hussain et al.27 studied the micropolar nanofluids flow over a stretching sheet. Haq et al.28 discussed the buoyancy and radiation effects on stagnation point flow of micropolar nanofluid



along a vertically convective stretching surface.

The aim of present work is to investigate the combined effects of viscous-Ohmic dissipation and thermal radiation on the MHD flow and heat transfer of micropolar nanofluid with non-uniform heat source/sink over a stretching sheet. The equations thus obtained have been solved numerically using Runge–Kutta–Fehlberg method with shooting technique. The effects of different parameters on velocity, microrotation, temperature and concentration are presented graphically.

2 Formulation of the problem

We consider a two-dimensional steady state boundary layer flow of electrically conducting micropolar nanofluids over a stretching sheet in the presence of uniform magnetic field, viscous-Ohmic dissipation, thermal radiation and non-uniform heat source/sink. The stretching velocity of the sheet is considered as uw(x) = cx along y-axis, with c being a constant. This fluid is assumed incompressible and the flow is assumed to be laminar. It is assumed that the temperature at the stretching sheet takes the constant values Tw while the temperature of the ambient nanofluid takes the constant value T∞ at the free stream flow.

It is also assumed that the base-fluid and the nanoparticles are in thermal equilibrium and no slip occurs between them. A uniform magnetic field of strength B0 is applied parallel to y-axis and normal to the sheet (see figure (1)). After applying the boundary layer approximations, the continuity, momentum, microrotation and heat equations can be expressed as:

0,u vx x

(1)

2

21 1 02 ,nf nf nf

u v u Nu v B u

x y y y

(2)

2

1 22 ,nf nfN N u N

j u v Nx y y y

(3)

2 22

202

1 1'''

( ) ( ) ( ) ( )nf nfr

nfp p p pnf nf nf nf

BqT T d T uu v q u

x y dy C y C C y C

.

(4)



Here u and v are the velocity components along the x- and y-directions, respectively. N is the component of the microrotation vector normal to the x- and y- plane, T is the temperature of the fluid and κ1 is the vortex viscosity. The appropriate boundary conditions for the velocity components and the temperature are given by

2

,( ) , ,

, 0

w w

w

uu u x cx v v N n

xT T T bx at y

(5)

0, 0, , .u N T T as y (6)

Here Tw and T∞ are the wall and ambient fluid temperature, and vw is the wall mass flux with vw for suctions, respectively. It is important to mention that n represents the concentration of microelements such that 0



where K=κ1/μf the micropolar or material parameter so that the micropolar fluid field can predict the correct behavior in the limiting case when the microstructure effects become negligible and the total spin N reduces to the angular flow velocity or flow vorticity.

The radiation heat flux is given by

* 4

* ,43r

Tq

yK

(9)

where σ* is the Stefan-Boltzmann constant and K *is the Rossel and mean spectral absorption coefficient. Thus using Eqn. (8), T4 can be expressed as (Pal et al.28):

4 4 4{1 ( 1) } ,wT T (10)

Where θw=Tw/T∞ is the wall temperature excess ratio parameter. The non uniform heat source/sink q''' is modeled as:

The non uniform heat source/sink q''' is modeled as:

* *( )''' [ ( ) '( ) ( ) ],wnf wf

u xq A T T F T T B

x

(11)

where A*and B* are parameters of the space-dependent and temperature-dependent internal heat generation/absorption, respectively. The case A*> 0 and B*>0 corresponds to internal heat generation while A*< 0 and B*



Substituting Eqns. (7)-(12) into Eqns. (2)-(4), we get the following nonlinear ordinary differential equations:

''' '' '2 41

2 2 2

1 1( ) ' ' 0,K F FF F KH MF

(13)

'' '1

2 2

1( / 2) ' (2 '') 0,

KK H FH F H H F

(14)

3 * * ' '

3

'' 2 ' 24 4

11 1 '' ' 2

Pr

( ) ( ) 0,

nfw

f

N A F B F F

Ec F EcM F

(15)

with the corresponding boundary condition as obtained from Eqns. (5)-(6) in the form:

'0, 1, ', 1, 0,F F H nF at (16)

' 0, ( ) 0, 0, .F H as (17)

The non-dimensional constants appearing in Eqns.(12)-(17) are parameter K, magnetic field M, Prandtl number Pr, mass flux parameter S, thermal radiation parameter Nr, Eckert number Ec are defined as

* 3 221

0 *16, ,Pr , , , ,

( )3f f w

pf f f fff

u T cK M B S Nr Ecc b CKa

(18)

where

2.51 2 3

54

.

( )(1 ) , 1 , 1

( )

( ) ( ) 1 , 1

( ) ( )

ss T

Tf f

p s sC

p Cf f

CC

(19)

Here prime denotes the differentiation with respect to η.

The important physical quantities in this study are skin-friction or shear stress coefficient Cf , couple stress Cn , and local Nusselt Number Nux , which are defined by (Hussain et al.

27)



1 1 0 2, ,ww ynf fwnf

u N Cy u

(20)

0 ,n yx N

Ca y

(21)

* 4

*0 0

43

,

( )

w nfy y

wx

wnf

T Tq

y yK

xqNu

T T

(22)

Using (12) in (20), (21) and (22), skin-friction coefficient, couple stress and local Nusselt number can be expressed as:

11/2 '' 12

1/2 '

,

1Re (0),Re '(0)

Re (1 ) (0)

x x nf

x x

n KC F C N

Nu Nr

(23)

where Rex = xuw(x)/νf is the local Reynolds number based on the stretching/shrinking velocity uw(x).

4 Results and discussion

For carrying out the numerical integration, the Eqns. (2)-(4) with boundary conditions (5)-(6) are reduced to a set of first order differential equations (12)-(14) with boundary conditions (15)-(16) using similarity transformation. Now for solving Eqns. (15)–(16), a step by step integration method i.e.,Runge–Kutta-Fehlberg method with shooting technique has been applied. The convergence criterion for the present numerical method is based on the difference between the values of the dependent variables of the current and the previous iterations, i.e., when the absolute values of the difference reaches 10-5 which showed that the solution has converged to the desired accuracy and the iteration process is stopped. The Prandtl number of the base fluid (water) is kept constant at 6.8. The values of the local Nusselt number Rex

-1/2Nux and skin-friction coefficient Rex

1/2Cf for various values of Pr, K and n are tabulated in Tables-2 and 3. The validation of the present results have been verified with the previous cases and it is found that the agreement between the previously published results with the present one is good and we believe that the present results are also accurate. Figures(2)–(4) are plotted for the velocity, microrotation and temperature profile for various values of magnetic field parameter M and material parameter K for Ag-water micropolar nanofluid over stretching sheet. Magnetic field parameter generates the opposite force



to the flow, is called Lorentz force which has a tendency to reduce the velocity and enhance the temperature profile. Those effects of magnetic field parameter are clearly seen from figure(2) and figure(4) when the value of material parameter is fixed. From figure(3) it is illustrated that the micro-rotation increases with the increase in M near the vicinity of the plate. Moreover, the ratio of vortex viscosity to absolute viscosity is known as the material parameter K. If the values of this parameter are kept small then it can be said that the viscous effect is confined in a very thin layer near the wall and the thickness of this layer is proportional to K =κ/ν and inversely proportional to F'(0). As a result, it can seen from figures (1)-(3) that as the values of K increase the velocity profile and temperature profile increases but microrotation decreases when intensity of the magnetic field is unchanged for Ag-water nanofluid. Figure (5) depict the influence of the Eckert number Ec and material parameter K on the temperature profiles for the micropolar nanofluid containing Ag-nanoparticle over stretching sheet. It is known that if the values of Ec are increased, energy is stored in the liquid due to the frictional heating and temperature of the fluid should increase. But in the present problem the effect of magnetic field is more prominent and its effect of decreasing temperature of the fluid dominates the effect of viscous dissipation on the flow of Ag-water nanofluid. Figure (6) is plotted to show the effects of thermal radiation parameter Nr and material parameter K on the temperature profiles for the micropolar Ag-water nanofluid over a stretching sheet. It can be seen from figure (6) that increase in the values of Nr enhance the temperature of the nanofluid. So, the thermal radiation has an quite effective impact on controlling the thermal boundary layers of Ag-water nanofluid. Also, it can be observed form figures (5)-(6) that increase in the material parameter K reduces the temperature profile of Ag-water nanofluid whatever may be the values of Eckert number Ec and thermal radiation parameter Nr. It is evident from these figures that an increase in the magnetic field parameter depreciates the velocity profiles and enhances the temperature profiles. This is due to the fact that a raise in the It can be seen that the skin friction coefficient, the couple stress, local Nusselt number generally decrease with the increase of φ for all types of nanofluids. It is clearly seen that that Ag-water gives lowest and TiO2-water gives higher couple stress as well as heat transfer rate if compared with the rest of the nanofluids. The effects of nanoparticle volume fraction φ on skin-friction coefficient, couple stress and local Nusselt number for four different types of nanoparticles over stretching sheet are shown in figures (7)-(9). Wall skin friction is higher for the Ag-water compared to the other nanofluids because Ag nanoparticle posses highest density as seen in table (1). Thus, the Ag-water nanofluid gives a higher drag force in opposition to the flow as compared to other nanofluid. Couple stress and local Nusselt number decreases as the value of φ increases as seen



in figures (8)-(9) for all types of nanofluids. It is clear that the wall heat transfer rate decreases with increase in the nanoparticle volume fraction parameter. In addition, the rate of heat transfer at the wall is also higher in Ag-water and lower in Cu-water nanofluid than in other nanofluids.

5 Conclusion

This paper presents effect of ohmic and viscous dissipation, thermal radiation, non-uniform heat source/sink on MHD boundary layer flow of micropolar nanofluid over a stretching sheet. Four different nanoparticles namely copper Cu, alumina AlO2O3, titanium dioxide TiO2, silver Ag with water as base fluid are considered here. Following conclusions are drawn from the figures and tables:

(i) Magnetic field profile M has tendency to reduce the velocity profile and enhance the temperature and microrotation profiles of the Ag-water nanofluid over stretching sheet.

(ii) The effect of Eckert number in presence of strong magnetic field reduces the fluid temperature of of the Ag-water nanofluid over stretching sheet.

(iii) Thermal radiation Nr enhances the temperature profile for the boundary of the flow over stretching sheet of the Ag-water nanofluid.

(iv) Micropolar/material parameter K has the effect to reduce all the velocity, temperature and microrotation profiles for Ag-water nanofluid.

(v) Nanoparticle volume fraction φ has reduces the skin friction coefficient, couple stress and local Nusselt number for all the four types of nanofluids Ag, Cu, Al2 O3, TiO2-water.

References

[1] Choi, S. (1995), “Enhancing Thermal Conductivity of Fluids with Nanoparticles, Developments and Applications of Non-Newtonian Flows,” Eds. D. A. Siginer and H.P. Wang, ASME, New York, FED-vol. 231/MD-vol. 66, pp. 99-105.

[2] Congedo, P., Collura, S. and Congedo, P. (2009), “Modeling and analysis of natural convection heat transfer in nanofluids,” Proc., ASME Summer Heat Transfer Conf., vol. 3, pp. 569–579.

[3] Hamad, M., Pop, I. and Ismail, A. (2011), “Magnetic field effects on free convection flow of a nanofluid past a semi-infinite vertical flat plate,” Nonlinear Analysis: Real World Appl, vol.12, pp.1338–1346.

[4] Buongiorno, J. (2006), “Convective transport in nanofluids,” ASME J. Heat Transfer, vol. 128, pp. 240–250.



[5] Poornima, T. and Bhaskar Reddy, N. (2012), “Radiation effects on MHD free convective boundary layer flow of nanofluids over a nonlinear stretching sheet,” Adv. Appl. Sci. Res., vol. 4, pp. 190–202.

[6] Motsumi, T. G. and Makinde, O. D. (2012), “Effects of thermal radiation and viscous dissipation on boundary layer flow of nanofluids over a permeable moving flat plate,” Phys. Scr., vol. 86 045003, pp. 1-8, 2012.

[7] Makinde, O. D. and Mutuku, W. N. (2014), “Hydromagnetic thermal boundary layer of nanofluids over a convectively heated flat plate with viscous dissipation and Ohmic heating,” UPB Sci. Bull Ser A, vol. 76, pp. 181–192.

[8] Pal, D. and Mandal, G. (2015), “Hydromagnetic convective–radiative boundary layer flow of nanofluids induced by a non-linear vertical stretching/shrinking sheet with viscous–ohmic dissipation,” Powder Technology, vol. 279, pp. 61-74.

[9] Sandeep, N. and Sugunamma, V. (2014), “Radiation and inclined magnetic field effects on unsteady hydromagnetic free convection flow past an impulsively moving vertical plate in a porous medium,” J. Appl. Fluid Mech., vol. 7, pp. 275–286.

[10] Kayalvizhi, M., Kalaivanan R., Ganesh, N. V., Ganga, B. and Hakeem, A. A. (2016), “Velocity slip effects on heat and mass fluxes of MHD viscous–Ohmic dissipative flow over a stretching sheet with thermal radiation,” Ain Shams Engg J. vol. 7, pp.791-797.

[11] Ganga, B., Mohamed Yusuff Ansari, S., Ganesh, N. V. and Abdul Hakeem, A. K. (2015), “MHD radiative boundary layer

flow of nanofluid past avertical plate with internal heat generation/absorption, viscous and Ohmic dissipation effects,” J.

Nigerian Math. Soc., vol. 34, pp. 181-194. [12] Eringen, A. C. (1969), “Simple Micropolar fluids,” Int. J. Eng.

Sci., vol. 2, pp. 205-207. [13] Eringen, A. C. (1966), “Theory of Micropolar fluids,” J. Math.

Mech., vol. 16, pp. 1-18. [14] Das, K. (2001), “Effect of chemical reaction and thermal radiation

on heat and mass transfer flow of MHD micropolar fluid in a rotating frame of reference,” Int. J. Heat Mass Transfer, vol. 54, pp. 3505-3513.

[15] Chen, C. (2004), “Combined heat and mass transfer in MHD free convection from a vertical surface with Ohmic heating and viscous dissipation,” Int. J. Engg. Sci., vol. 42, pp. 699–713.

[16] El-Hakiem, M.A. (2000), “Viscous dissipation effects on MHD free convection flow over a non-isothermal surface in a

micropolar fluid,” Int. Comm. Heat Mass Transfer, vol. 27, pp. 581-590.

[17] Oahimire, J. I. and Olajuwon, B. I. (2014), “Effect of Hall current and thermal radiation on heat and mass transfer of a chemically



reacting MHD flow of a micropolar fluid through a porous medium,” J. King Saud University–Engg. Sci., vol. 26, pp. 112– 121.

[18] Hussain, M., Ashraf, M., Nadeem, S. and Khan, M. (2013), “Radiation effects on the thermal boundary layer flow of a micropolar fluid towards a permeable stretching sheet,” J. Franklin Inst., vol. 350, pp. 194–210.

[19] Kumar, H. (2013), “Mixed convective magnetohydrodynamic (MHD) flow of a micropolar fluid with Ohmic heating radiation and viscous dissipation over a chemically reacting porous platesubjected to a constant heat flux and concentration gradient,” J. Serb. Chem. Soc., vol. 78, pp.1–17.

[20] Ibrahim, F. S., Hassanien, I. A. and Bakr, A. A. (2004), “Unsteady magneto hydrodynamic micropolar fluid flow and heat transfer over a vertical porous plate through a porous medium in the presence of thermal and mass diffusion with a constant heat source,” Can. J. Phy., vol. 82, pp.775-790. [21] Singh,K. and Kumar, M. (2015), “Effect of Viscous Dissipation on Double Stratified MHD Free Convection in Micropolar Fluid

Flow in Porous Media with Chemical Reaction, Heat Generation and Ohmic Heating,” Chemical Process Engg. Research, vol.31, pp. 75-80.

[22] Nazar, R., Amin, N., Filip, D. and Pop,I. (2004),“Stagnation point flow of a micropolar fluid towards a stretching sheet,” Int. J. Non-

linear Mech., vol. 39, pp. 1227-1235. [23] Nazar, R., Ishak, A. and Pop, I. (2008), “Unsteady boundary layer

flow over a stretching sheet in a micropolar fluid,” Int. J. Math. Phy. Engg. Sci., vol. 2, pp. 161-165.

[24] Mishra, S. R., Pattnaik, P. K. and Dash, G. C. (2013), “Effect of heat source and double stratification on MHD free convection in a micropolar fluid,” J. Egyptian Math. Soc., vol. 21, pp. 370-378.

[25] Perdikis, C. and Raptis, A. (1996), “Heat transfer of a micropolar fluid by the presence of radiation,” Heat Mass Transfer, vol. 31, pp. 381-382.

[26] Fauzia, E. L. A., Ahmada, S. and Pop, I. (2014), “Flow over a permeable stretching sheet in micro polar nanofluids with suction,” AIP Conference Proceedings, vol. 1605, pp. 428-455.

[27] Hussain, S. T., Nadeem, S. and Haq, R. U. (2014), “Model-based analysis of micropolar nanofluid flow over a stretching surface,” Eur. Phy. J. Plus, vol. 129, pp. 1-10.

[28] Pal, D., Mandal, G. and Vajravelu, K. (2014), “Flow and heat transfer of nanofluids at a stagnation point flow over a stretching/shrinking surface in a porous medium with thermal radiation,” Appl. Math. Comp., vol. 238, pp. 208–224.

[29] Kline, K. A. (1977), “A spin-vorticity relation for unidirectional plane flows of micropolar fluids,” Int. J. Eng. Sci., vol. 15, pp. 131-134.



[30] Reddy, J. V. R., Sugunamma, V., Sandeep, N. and Sulochana, C.

(2015), “Influence of chemicalreaction, radiation and rotation on MHD nanofluid flow past a permeable flat plate in porous medium,” J. Nigerian Math. Soc., vol. 35, pp. 48-65.

[31] Wang, C.Y. (1989), “Free convection on a vertical stretching surface,” J. Appl. Math. Mech., vol. 69, pp. 418-420. Table 1: Thermophysical properties of fluid and nanoparticles (Pal and Mandal8, Reddy et al.30)

Physical properties

Fluid phase (water)

Cu

Al2O3

TiO2

Ag

Cp(J/kgK) ρ(kg/m3) κ(W/m K) σ (S/m)

4179 997.1 0.613 5.5×10-6

385 8933 400 59.6×106

765 3970 40 35×106

686.2 4250 8.9538 2.6×106

235 10500

429 62.1×106

Table 2:Comparison of results for local Nusselt number Rex-1/2Nux

when n=φ=M= K=S=Nr =A*=B*=0 and power of x is zero in Eqn. (5)

Pr Wang31 Hussain et al.27

Present results

0.7 2.0 7.0 20 70

0.4539

0.9114

1.8954

3.3539

6.4622

0.4582

0.9114

1.8954

3.3539

6.4622

0.4549

0.9113

1.8954

3.3539

6.4621

Table 3:Values of the skin friction coefficient Rex

½ Cf for various values of K and n when φ=M=S =Nr =A

*=B* = 0 and power of x is zero in Eqn. (5)

m K Nazar et al.22 Fauzi et al.26 Present results 0

0 1 2 4

-1.0000 -1.3679 -1.6213 -2.0042

-1.0000 -1.3680 -1.6225 -2.0075

-1.0000 -1.3679 -1.6213 -2.0042

0.5

0 1 2 4

-1.0000 -1.2247 -1.4142 -1.7321

-1.0000 -1.2248 -1.4159 -1.7381

-1.0000 -1.2247 -1.4142 -1.7321



Figure 1: Flow configuration.

Figure 2: Effect of M and K on velocity profiles for Ag-water nanofluid.

Figure 3: Effect of M and K on microrotation profiles for Ag-water nanofluid.



Figure 4: Effect of M and K on temperature profiles for Ag-water nanofluid.

Figure 5: Effect of Ec and K on temperature profiles for Ag-water

nanofluid.

Figure 6: Effect of Nr and K on temperature profiles for Ag-water nanofluid.



Figure 7: Effect of φ on skin-friction coefficient for types of nanofluids.

Figure 8: Effect of φ on couple stress for types of nanofluids.

Figure 9: Effect of φ on local Nusselt number for types of nanofluids.

`1

VIJNANA BHARATHI-The frontier journal in SCIENCE

CONVECTIVE HEAT AND MASS TRANSFER

FLOW OF A NANOFLUID PAST A VERTICAL

SLENDER CYLINDER IN A SATURATED

POROUS MEDIUM

K. V. Prasad*, K. Vajravelu† , Hanumesh Vaidya †† ,

M. Prasad§ † Department of Mathematics, Department of Mechanical, Materials and Aerospace Engineering, University of Central Florida, Orlando, FL 32816, USA.

†† Department of Mathematics, SSA Government First Grade College, Ballari-583 101, Karnataka, India.

§ Department of Mathematics, PES College of Engineering, Mandya- 571401, Karnataka, India

ABSTRACT:

An analysis is carried out to investigate the effect of transport phenomena (heat and Mass transfer) on the axi-symmetric nanofluid flow over a vertical porous slender cylinder. Flow is driven solely by linearly stretching the cylinder where nanoparticles move randomly and increase the energy exchange rates thus incorporating the effects of Brownian motion and thermophoresis. Using suitable transformation, the governing cylindrical equations are transformed into coupled ordinary differential equations and are solved numerically by Keller-Box method. The effect of sundry parameters on the velocity, temperature and concentration profiles are presented graphically and analyzed. To validate the numerical method, comparisons are made with the available results in the literature for some special cases and the results are found to be in excellent agreement.

This article is to be cited as: Prasad K V, Vajravelu K, Vaidya H and Prasad M. Convective

heat and mass transfer flow of a nanofluid past a vertical

slender cylinder in a saturated porous medium, Bangalore;

Vijnana Bharathi, 1 (2): 19-38, 2016.

19

©2016 Jnanavahini All rights reserved

Key words: Nanofluid, porous medium, Brownian motion, Thermophoresis, Finite difference method, Slender cylinder.

Received on: 27/11/2015 Accepted on: 16/1/2016

Vol. 1 | No.2 |19 - 38 |

March | 2016

*Corresponding author:

K . V. PRASAD Department of Mathematics VSK University Vinayaka Nagar Ballari-583 105 Karnataka, India. [email protected]

ISSN : 0971-6882 ISSN-L : 0377-8487 CODEN : VBHAD6

AdminStamp

AdminStamp


K. V. Prasad, K. Vajravelu, H. Vaidya and M. Prasad 20

Nomenclature Latin symbols

a radius of the cylinder b stretching rate C concentration of the fluid

fC skin friction coefficient

wC concentration of the stretching sheet

0C constant

C concentration of the fluid for away from the wall

fc specific heat of the fluid

pc specific heat of the nano particle D mass diffusivity

BD brownian diffusion coefficient

TD thermophoretic diffusion coefficient f dimensionless stream function g acceleration due to gravity Gc mass Grashof number k thermal conductivity

1K permeability or porous parameter

l reference length Le Lewis number Nb brownian motion parameter Nt thermophoresis parameter

xNu local Nusselt number Pr Prandtl number

mq surface mass flux

wq surface heat flux

r radial coordinate Rex local Reynolds number

Shx local Sherwood number T fluid temperature

T ambient temperature wT temperature of the stretching sheet

0T constant

u axial velocity component

wU stretching velocity

v radial velocity component x axial coordinate



Greek symbols thermal diffusivity thermal expansion coefficient

* concentration expansion coefficient transverse curvature similarity variable dimensionless temperature dimensionless concentration stream function dynamic viscosity

f kinematic viscosity of the fluid

f density of the fluid

p density of the nanoparticle

fc heat capacity of the fluid

pc effective heat capacity of the nanoparticle material

free convection parameter ratio between the effective heat capacity of the nanoparticle material and heat capacity of the fluid

w surface shear stress

Subscripts and Superscripts w conditions at the stretching sheet condition at infinity ‘ differentiation with respect to

1 Introduction

In recent years, great interest has been evinced concerning to the study of nanofluids after the pioneering work by Choi1 due to its vast applications in technological industry. The innovative technique of Choi1 induces a new class of heat transfer fluids which can be derived from conventional heat transfer fluids by suspending metallic nanoparticles in to it. Masuda et al.2 used the technique of electrostatic repulsion to understand how much the thermal conductivity of a liquid can be altered by dispersing a small amount of ultra-fine particles into it by using fine powders of Al2O3, SiO2 and TiO2 and water as a base liquid. Lee et al.3 continued the work of Ref1 and analyzed Oxide nanofluids experimentally by a transient hot-wire method and suggested that not only particle shape but size also considered to be dominant in enhancing the thermal conductivity of nanofluids. Buongiorno4,5 summarized their research work in the area heat transfer characteristics of nanofluids by of evaluating the benefits for and applicability to nuclear power systems (i.e., primary coolant, safety systems, severe accident mitigation strategies) with particular emphasis to boiling behavior, including, prominently, the Critical Heat Flux limit



and quenching phenomena and analyzed the characteristics of seven slip mechanisms that can produce a relative velocity between the nanoparticles and the base fluid. Kuznetsov and Nield6 followed Buongiorno5 and analyzed the influence of nanoparticals on a natural convection flow passing a vertical plate by treating temperature and nanoparticle fraction as constant along the wall analytically. Recently, Nadeem et al.7 praposed flow of a Jeffrey fluid model in the presence of nano particles using numerical method. Many researchers proposed several models to predict the flow pattern and nature of thermal conductivity of nanofluid8-12. Above mentioned researchers restricted their analysis to flow over a stretching sheet with various physical models. The convective flow over a vertical slender cylinder has gained momentum because of its extensive applications in many branches of science and technology such as the nuclear reactors cooled during emergency shutdown, electronic devices cooled by fans, heat exchangers placed in a low velocity environment, solar central receivers exposed to wind currents, melt spinning process, wire and fiber drawing, etc. Flow over cylinders is considered to be two-dimensional if the body radius is large compared to the boundary layer thickness. For a thin cylinder, the radius of the cylinder may be of the same order as the boundary layer thickness. Therefore the flow may be considered as axisymmetric instead of two-dimensional. Taking these facts in to account, Lin and Shih 13 obtained Local similarity solutions for the laminar boundary layer heat transfer along static and moving cylinders with prescribed surface temperature or wall heat flux. Further, Wang14 has studied the steady flow in a viscous and incompressible fluid outside of a stretching hollow cylinder in an ambient fluid at rest. Ishak et al.15

presented a numerical solution of flow and heat transfer outside a stretching permeable cylinder and Vajravelu et al.16 examined the axisymmetric flow and heat transfer at a non-isothermal stretching cylinder in the presence of transverse magnetic field. Further, Zaimi et al.17 carried out an analysis to investigate the unsteady viscous flow and heat transfer due to a permeable stretching/shrinking cylinder. One of the important physical phenomenon is the case in which the difference between the surface temperature and the free stream temperature appreciably large. The findings for such a physical phenomenon will have a definite bearing on the plastic, fabric and polymer industries. Hence, it is interesting to study the effects of thermal buoyancy and the Prandtl number. In many practical situations the material moves in a quiescent fluid with the fluid flow induced by the motion of the solid material and by the thermal buoyancy. Therefore the resulting flow and the thermal fields are determined by these two mechanisms. It is well known that the buoyancy force stemming from the heating or cooling of the continuous stretching sheet alter the flow and the thermal fields and thereby, the heat transfer characteristics of the manufacturing processes. However, the buoyancy force effects were not considered in

http://www.tandfonline.com/author/Lin%2C+Hhttp://www.tandfonline.com/author/Shih%2C+Yhttp://www.tandfonline.com/author/Shih%2C+Y



the afore-mentioned studies. In view of this Mahmood and Merkin18, Ganesan, and Loganathan19, Bachok and Ishak20, Hossa21 and Rani22 have studied flow past vertical cylinder. It is quite evident from literature survey that the porous media can be used to enhance the heat transfer rate from stretching surfaces which has numerous applications in geophysics, oil recovery techniques, heat storage systems, insulation design, grain storage, geothermal systems, heat exchangers, filtering devices, metal processing, catalytic reactors etc. Realizing these applications, Prasad et al.23-27 has carried out an extensive work on fluid flow through porous media. Recently, Filip28 has studied the effect of anisotropy on fully developed mixed convection flow in a vertical porous channel. To the best of authors’ knowledge, not much attention has been paid to understand the effects of nanofluid parameters on the free convective flow of a nanofluid over a vertical slender cylinder embedded in a porous medium. Hence, the present article is arranged to investigate the effects of base-fluid and assuming it to be a two-component non-homogeneous equilibrium model with Brownian diffusion and thermophoresis along with combined buoyancy effects due to temperature difference as well as nanoparticles species diffusion in a porous medium. In this case, the governing equations contain the transverse curvature term which strongly influences the convective flow, heat and mass transfer fields. The coupled non-linear partial differential equations governing the problem are transformed to a system of coupled non-linear ordinary differential equations. The transformed equations are solved numerically via a second order finite difference scheme known as the Keller-box method. The effects of pertinent parameters on the velocity, temperature, nanoparticles concentration fields are presented graphically and the numerical values for the local skin friction coefficient, the local Nusselt number and the local Sherwood number are listed in Table III. It is expected that the results obtained from the present investigation will provide useful information for applications and will also serve as a complement to previous studies.

2 Mathematical formulation Consider a steady, two-dimensional free convective flow, heat and mass transfer of a viscous, incompressible nanofluid, past an impermeable vertical slender cylinder of constant radius a embedded in a porous medium with a linear stretching velocity. The positive r coordinate is measured along the stretching cylinder in the direction of motion and the positive x coordinate is measured normal to the sheet in the upward direction (see Figure. 1 for details). The radius of the slender cylinder is the same as the order of the boundary layer thickness and hence the flow is taken to be axisymmetric. Prescribed surface temperature and the prescribed species diffusion, which are of the following forms



( )wU x b x l ,

0( )wT x T T x l ,

0( ) .wC x C C x l

The viscous dissipation effects and the pressure gradient term along the boundary layer are negligible in the energy equation, due to the slow stretching motion of the slender cylinder. All the physical properties are assumed to be constant except for the density in the buoyancy force term which is given by the usual Boussinesq approximation. Under these assumptions, the basic boundary layer equations governing the physical equations for nanofluids can be written as

0x rru rv , (1)

*( )1vx r f rr r

f f f

g T T g C C K uuu u u u

r

, (2)

21

v ,Tx r rr r B r r rD

uT T T T D C T Tr T

(3)

1 1v .Tx r B rr r rr r

DuC C D C C T T

r T r

(4)

Here, the subscript denotes partial differentiation with respect to the independent variable. ( ) /( )p fc c is the ratio between the

effective heat capacity of the nanoparticle material and heat capacity of the fluid . The second term in right hand side of the Eqn. (2) represents the influence of thermal buoyancy force on the flow field, where ‘+ and ‘ -” signs refer to the buoyancy assisting and buoyancy opposing flow region. Figure (1) provides the necessary information of such a flow field for a vertical slender cylinder with upper half of the flow field being assisted and the lower half of the flow field being opposed by the buoyancy force. For the assisting flow, the x axis points upwards in the direction of the stretching cylinder such that the stretching induced flow and the thermal buoyant flow assist each other. For the opposing flow, the x-axis points vertically downward in the direction of the stretching cylinder. But in this case the stretching induced flow and the thermal buoyant flow oppose each other. The reverse trend occurs if the stretching cylinder is cooled below the ambient temperature. The boundary conditions for the physical problem under consideration are given by

, v 0, , at.

0, , asw w wu U T T C C r a

u T T C C r

(5)

The particular forms of , andw w wU T C have been chosen to devise a

similarity transformation which transform the governing partial differential Eqns. (2) - (4) into a set of ordinary differential equations. It is convenient to reduce the number of equations from four to three as



well as to transform them in to dimensionless form. Introducing the dimensionless functions ,f and and the similarity variable as (See Bachok and Ishak34):

1

2

12 22

, ( ) , ( ) ,

.2

f ww w

w f

T T C CxU a f

T T C C

r aU x

a

(6)

In Eqn. (6), is the stream function defined as 1 1

and v ,r xu r r which identically satisfies continuity Eqn.

(1). By defining in the above form, the boundary condition at r a reduces to the boundary condition at η = 0, which is more convenient for analytical / numerical computations. Substituting Eqn. (6) in Eqns. (2) - (4), gives:

2 1(1 2 ) ( ') '' 0,f f ff Gc K f (7)

2(1 2 ) Pr Pr(1 2 ) ' 0,f f Nb Nt (8)

(1 2 ) (1 2 ) 0.Nt

Le f fNb

(9)

Here prime denotes differentiation with respect to . The corresponding boundary conditions are:

(0) 0, 0 1, (0) 1, (0) 1.

0, 0, 0

f f

f

(10)

Here, the parameters 1, , , , , , Pr andK Gc Nt Nb Le are defined

as follows:

*0 0

12 2 2

ν, , , ,

.

,Pr and

f T w

f f f

f fB w

f B

l D T Tg T g CK K b Gc Nt

ba b b T

D C CNb Le

D

(11)

It is worth mentioning that 0 aids the flow and 0 opposes the

flow, while 0 i.e., wT T represents the case of forced

convection flow. On the other hand, if is of significantly greater order of magnitude than one, then the buoyancy forces will be predominant. Hence, combined convective flow exists when 0(1) . For all practical purpose, the physical quantities of interest are the skin



friction coefficient fC , the local Nusselt number xNu and the local

Sherwood number xSh , which are defined by:

2, ,

2w w m

f x xf w w w

xq xqC Nu Sh

U k T T D C C

, (12)

where w , wq and mq are given by:

, , .w f r w r m rr a r a r au q k T q D C (13)

Using the similarity variables (6), results in:

1 2 1 2 1 21

Re 0 , 0 and 0 ,2 Re Re

x xf x

x x

Nu ShC f (14)

where Rex w fU x is the local Reynolds number.

3 Exact solutions for some special cases Here, in certain special cases the exact solutions are presented. Such solutions are useful and serve as a baseline for comparison with the solutions obtained via numerical schemes.

3.1 No free convection currents, no mass Grash of number and no transverse curvature ( 0Gc ) In the limiting case of 0Gc , the convective flow, heat and mass transfer problem degenerates. In this case the closed form solution for the velocity field is given by

1

1where 1 ,

mef m K

m

(15)

while the solutions for the temperature field as well as the species diffusion field is the Confluent hypergeometric series, namely, the Kummer’s function, M , to wit:

1 21 0 1 1 1 1 1 1 1

1 22 00 2 10 10 2 2 10 10

2 2 2 20 00

1 0 1 0 10 00 10 00

exp , , , , , Pr

exp , , , , ,.

Pr , Pr exp , , exp

1, 1 , 1, 1

C ma a b z C a b m

C ma a b z C a b Le m

a m z m m a Le m z Le m m

a a b a a a b a

(16) In the absence of nanofluid parameters, transverse curvature and porous parameter, equations reduce to those of Vajravelu29 in the absence of species diffusion. In the presence of species diffusion,



equations reduce to those of Abel et al.30 in the absence of nanofluid particles and transverse curvature.

4. Numerical procedure

Final reduced Eqns. (7) - (9) are highly non-linear, coupled ordinary differential equations with variable coefficients of third-order in f and second-order in both and . Exact analytical solutions are not possible for the complete set of these equations and therefore, the efficient numerical method with second order finite difference scheme known as Keller-box method is used. The coupled boundary value problem (7) - (9) has been reduced to a system of seven simultaneous ordinary differential equations of first order for three unknowns following the method of superposition (See for details Na31). To solve the system of first order equations, seven initial conditions are required whilst only two initial conditions on f and one initial condition

and are known. There are three initial condition

0 , 0 and (0)f

which are not prescribed, however the

values of , andf are known at . Further,

employ numerical Keller-box scheme where these three boundary conditions are utilized to produce three unknown initial conditions at

0 . To select , begin with some initial guess value of the

unknown initial conditions and solve the boundary value problem to obtain 0 , 0 and (0)f . Let the correct values

of 3 2 20 , 0 and 0f be 0 0 0, and and integrate the system

using fourth order Runge-Kutta method. The solution process is repeated with another larger value of until two successive values of

unknown conditions differ only after desired digit signifying the limit of the boundary along . The last value of is chosen as appropriate

value for that particular set of parameters. Finally, the problem is solved numerically using a second order finite difference scheme known as Keller-box method (See for details Cebeci and Bradshaw 32, Keller33 and Vajravelu and Prasad34). For numerical calculations, a uniform step size of 0.01 is found to be satisfactory and the

shooting error was controlled with an error tolerance of 610 in all the cases. The physical domain in this problem is unbounded, whereas the computational domain has to be finite and on this reason, far field boundary conditions are applied for the similarity variable at a finite

value denoted by max . A value of max 14 is found to be sufficient to

achieve the far field boundary conditions asymptotically for all values of the parameters considered. To assess the accuracy of the present method, comparison of the skin friction and the wall temperature gradient between the present results with the available results in the literature for a special case i.e., for stretching sheet (see in tables (1) and (2)) and the comparison show a very good agreement.



5 Results and discussion In order to understand the mathematical nanofluid porous model, the numerical results are presented graphically for the axial velocity ( )f , the temperature and the species diffusion in

figures (2)-(9) for different values of pertinent parameters such as transverse curvature , permeability or porous parameter 1K ,

buoyancy or free convection parameter , mass Grashof number Gc , thermophoresis parameter Nt , Brownian motion parameter Nb , Prandtl number Pr and Lewis number Le . The computed numerical values for the skin friction (0)f , the Nusselt number (0) and the wall Sherwood number (0) are tabulated in table (3). It can be

observed from the graphs that all the profiles ( )f , and

monotonically decreases and tends asymptotically to zero as the distance increases from the boundary. Figures (2)-(3) are the graphical representation of ( )f for

increasing values of 1, ,K and Gc . It is apparent from figure (2)

that the boundary layer thickness is higher for non-zero values as compared to zero values of and for increasing values of 1K results in

flattening of ( )f . It is quite clear from the profiles that ( )f decrease with an increase in porous parameter. This is because, the larger values of porous parameter correspond to densely pack porous medium and hence the flow rate will be reduced. The effect of increasing values of Gc and on ( )f is demonstrated in figure (3). Increase in Gc and (increase in temperature gradients) leads to the enhancement of ( )f . This is due to the fact that Gc depends upon

( wC C ) which is assumed to be positive and can be interpreted

physically because the level of concentration is higher near the plate than away from the surface. Physically, 0 corresponds to an externally heated plate because the free convection currents are carried towards to the plate, 0 corresponds to an externally cooled plate and 0 corresponds to the absence of free convection currents. The behavior of for increasing values of K1, , Pr, are

plotted in figures (4) –(6). The effects of and Gc on are

presented graphically in figure 4. An increase in the value of results in an increase in the thermal boundary layer thickness and this results in an increase in the magnitude of the wall temperature gradient and hence produces an increase in the surface heat transfer rate, this is even true for non zero values of K1. The effect of K1 is to increase the wall temperature gradient. Nb and Nt are two physical parameters of utmost interest, which depends on the type of the nanoparticle present through the parameters BD and TD , while Nb depends on the nanoparticle



concentration namely wC C . An increase in either of the

parameters Nb or Nt leads to decrease in the magnitude of the wall temperature gradient and hence, increases the thermal boundary layer thickness; exhibited graphically in figure (5). The effects of Pr and

on are depicted graphically in figure (6). It is clear from the graph

that an increase in Pr leads to decrease the temperature distribution and hence thermal boundary layer thickness. The effect of increasing values of is to increase the temperature and hence enhances the thermal boundary layer thickness. Figures (7) – (9) explain the effects of different parameters on . Figure (7) illustrates the effects of 1K and Gc on . It is

noticed that the effect of 1K is to enhance . This observation

holds for non-zero values of Gc. Increasing the value of Gc results in an increase in the mass transfer gradient, and hence produces an increase in the surface mass transfer rate. The effect of Nb on is

shown in Figure (8) for different values of .Nt It can be seen that increases with an increase in Nb as well as Nt . The effects of

Le and on are elucidated in figure (9). The effect of increasing

values of Le leads to decrease in the thickness of the nanoparticle concentration boundary layer. This is due to the thinning of the nanoparticle layer with the introduction of species diffusion. This phenomenon is true even for increasing values of . The impact of all the pertinent parameters on (0)f , (0) and

(0) are tabulated in Table III. (0)f , (0) and (0) found to decrease with increase in . An increase in leads to an increase in

(0)f and decrease in (0) and (0) , but the reverse trend is found

in 1K .It is interesting to note that the surface mass transfer decreases

and surface heat transfer increases with , andLe Nb Nt .But the effect of Pr is to decrease (0) and increase (0) .

6 Conclusion The following conclusions are drawn from the computed numerical values:

The suspended nanoparticles substantially increase the heat transfer rate at any given buoyancy parameter.

An increase in permeability or porous parameter leads to an increase in the wall temperature gradient and the surface mass transfer rate where as in the case of velocity profiles reverse trend is observed.

An increase in either Brownian motion parameter or the thermophoresis parameter serves to increase the thermal as well as nanoparticle concentration profiles.



Concentration profiles enhance for increasing value of mass Grashof number and squeezes with Lewis number.

Skin-friction increases for increasing values of mass Grashof number and thermophoresis parameter. However opposite effect is observed for Brownian motion parameter and Lewis number.

Sherwood number enhance for larger values of Porous parameter and Prandtl number the reverse trend is observed in the case of Brownian motion parameter, the thermophoresis parameter and the Lewis number.

Acknowledgment One of the authors (Hanumesh Vaidya) is thankful to the University Grants Commission, New Delhi for supporting financially under Minor Research Project (Grant No. MRP(S)-0060/12-13/KAUG062/UGC-SWRO)

References [1]. Choi, S. (1995), “Enhancing thermal conductivity of fluids with nanoparticle,” In: D. A. Siginer, H. P. Wang Eds.), Developments

and Applications of Non- Newtonian Flows, ASME, MD 231 and FED, vol. 66, pp. 99-105.

[2]. Masuda, H., Ebata, A., Teramae, K. and Hishinuma, N. (1993), “Alteration of thermal conductivity and viscosity of liquid by

dispersing ultra-fine particles,” Netsu Bussei, vol. 7, pp. 227-233. [3]. Lee, S., Choi, S. U. S., Li, S. and Eastman, J. A. (1999),

“Measuring Thermal Conductivity of Fluids Containing Oxide Nanoparticles,” J. Heat Transfer, vol. 121, pp. 280-289.

[4]. Buongiorno, J. and Hu, W. (2005), “Nanofluid coolants for advanced nuclear power plants,” Paper No.5705, in: Proc. of ICAPP ’05, Seoul, pp. 15–19. [5]. Buongiorno, J. (2006), “Convective transport in nanofluids”, ASME J. of Heat Transfer, vol 128, pp. 240-245. [6]. Kuznetsov, A. V. and Nield, D. A. (2010), “Natural Convective Boundary-Layer Flow of a Nanofluid Past a Vertical Plate,” Int. J. of Thermal Sci., vol. 49, pp. 243−247. [7]. Nadeem, S., Haq, R. U. and Khan, Z. H. (2014), “Numerical

solution of non-Newtonian nanofluid flow over a stretching sheet,” Appl. Nanosci., vol. 4, pp. 625–631.

[8]. Wang, L. and Wei, X. (2009), “Heat conduction in nanofluids,” Chaos Solitons Fractals, vol. 39, pp. 2211–2215. [9]. Ghasemi, B. and Aminossadati, S. M. (2009), “Natural convection heat transfer in an inclined enclosure filled with a water-Cuo Nanofluid,” Numer. Heat Transfer, Part A, vol. 55, pp. 807−823. [10]. Vajravelu, K., Prasad, K. V., Lee, J., Lee, C., Pop, I. and Van

http://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=S.+Lee&q=S.+Leehttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=S.+U.-S.+Choi&q=S.+U.-S.+Choihttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=S.+Li&q=S.+Lihttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=J.+A.+Eastman&q=J.+A.+Eastman



Gorder, R. A. (2011), “Convective heat transfer in the flow of viscous Ag-water and Cu-water nanofluids over a stretching surface,” Int. J. Thermal Sci., vol. 50, pp. 843.

[11]. Vajravelu, K., Prasad, K. V. and Chiu-On, N. G. (2013), “The effect of variable viscosity of a viscous Ag-water and Cu-water nanfluids,” J. Hydrodynamics, Ser. B, vol. 25, pp. 1-9.

[12]. Prasad, K. V., Vajravelu, K., Vaidya, H. and Raju, B. T. (2015), “Heat Transfer in a Non-Newtonian Nanofluid Film over a Stretching Surface,” J. of Nanofluid, vol. 4, pp. 1-12.

[13]. Lin, H. T. and Shih, Y. P. (1980), “Laminar boundary layer heat transfer along static and moving cylinders,” J. of the Chinese Institute of Engineers, vol. 3.

[14]. Wang, C. Y. (1988), “Fluid flow due to a stretching cylinder,” Phys. Fluids, vol. 31, pp. 466-468, 1988. [15]. Ishak, A., Nazar, R. and Pop I. (2008), “Uniform suction/blowing effect on flow and heat transfer due to a stretching cylinder,” Appl. Mathematical Modell., vol. 32, pp. 2059–2066. [16].Vajravelu, K., Prasad, K.V. and Santhi, S. R. (2012), “Axisymmetric magneto-Hydrodynamic (MHD) flow and heat transfer at a non-isothermal stretching cylinder,” Appl. Math. and Comp, vol. 219, pp. 3993-4005. [17]. Zaimi, K., Ishak, A. and Pop, I. (2013), “Unsteady viscous flow and heat transfer due to a permeable stretching/shrinking cylinder,” AIP Conf. Proc. 1522, pp. 299-305. [18]. Mahmood, T. and Merkin, J. H. (1988), “Similarity solutions in axi symmetric mixed-convection boundary-layer flow,” J. Eng. Math. vol. 22, pp. 73-92. [19]. Ganesan, P. and Loganathan, P. (2006), “Numerical study of

double-duffusive free convective flow past moving vertical cylinder with chemically reactive species diffusion,” J. Eng. Phys. Thermo Phys., vol. 79, pp. 73–78.

[20]. Bachok, N. and Ishak, A. (2009), “Mixed convection boundary layer flow over a permeable vertical cylinder with prescribed surface heat flux,” Euro. J. of Sci. Res., vol. 34, pp. 46–54.

[21]. Md. Hossain, A., Mondal, R. K., Ahmed, R. and Ahmmed, S. F. (2015), “A Numerical Study on Unsteady Natural Convection

Flow With Temperature Dependent Viscosity past an Isothermal Vertical Cylinder,” J. of Pure Appl. and Industrial Phys., vol. 5, pp. 125-135.

[22]. Rani, H. P., Janardhan Reddy, G., Chang Nyung Kim, and Rameshwar, Y. (2015), “Transient Couple Stress Fluid Past a Vertical Cylinder with Bejan’s Heat and Mass Flow Visualization for Steady-State,” J. Heat Transfer, vol. 137, pp. 032501.

[23]. Prasad, K.V. and Subhas Abel, M. (2000), “Momentum and heat transfer in visco-elastic fluid flow in a porous medium over a non- isothermal stretching sheet,” Int. J. of Numer. Methods for Heat & Fluid Flow, vol. 10, pp. 786-801. [24]. Prasad, K.V., Subhas Abel, M. and Ambuja Joshi, (2000),

http://www.tandfonline.com/author/Lin%2C+Hhttp://www.tandfonline.com/author/Shih%2C+Yhttp://www.tandfonline.com/loi/tcie20?open=3#vol_3http://scitation.aip.org/content/contributor/AU0988837http://scitation.aip.org/content/contributor/AU0119563http://scitation.aip.org/content/contributor/AU0037128http://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=H.+P.+Rani&q=H.+P.+Ranihttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=G.+Janardhan+Reddy&q=G.+Janardhan+Reddyhttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=Chang+Nyung+Kim&q=Chang+Nyung+Kimhttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=Y.+Rameshwar&q=Y.+Rameshwarhttp://heattransfer.asmedigitalcollection.asme.org/solr/searchresults.aspx?author=Y.+Rameshwar&q=Y.+Rameshwar



“Oscillatory motion of a viscoelastic fluid over a stretching sheet in porous media”, J. of Porous Media, vol. 3, pp. 61-68.

[25]. Prasad, K.V., Subhas Abel, and Datti, P.S. (2003), “Diffusion of chemically reactive species of a non-Newtonian fluid immersed in a porous medium over a stretching sheet,” Int. J. Non-Linear Mech., vol. 38, pp. 651-657. [26]. Prasad, K. V., Vajravelu, K., Datti, P. S. and Raju, B. T. (2013),

“MHD Flow and Heat Transfer in a Power-law Liquid Film at a porous Surface in the presence of Thermal Radiation,” J. of Appl. Fluid Mech., vol. 6, pp. 385- 395.

[27]. Vajravelu, K. and Prasad, K. V. (2014), “Mixed Convection Heat Transfer in an Anisotropic Porous Medium with Oblique Principal Axes,” J. of Mech., vol. 30, pp. 327-338.

[28]. Filip, Diana Andrada, Trîmbiţaş, Radu, T. and Ioan Pop, (2015), “Fully developed mixed convection through a vertical porous

channel with an anisotropic permeability: case of heat flux,” vol. 60, pp. 341-350.

[29]. Vajravelu, K. (1994), “Convection heat transfer at a stretching sheet with suction or blowing,” J. Math. Anal. vol. 188, pp. 1002–1011. [30]. Subhas Abel, M., Khan, S. K. and Prasad, K.V. (2001), “Convective Heat and Mass Transfer in a visco-elastic fluid Flow through a Porous Medium over a Stretching Sheet,” Int. J. Numer. Methods for Heat and Fluid Flow, vol.11, pp. 779-792. [31]. Na, T. Y. (1979), Computational methods in engineering boundary value problems, Academic Press, New York, vol.

145. [32]. Cebeci, T. and Bradshaw, P. (1984), “Physical and Computational Aspects of Convective Heat Transfer,” Springer-Verlag, New York. [33]. Keller, H. B. (1992), “Numerical Methods for Two-point Boundary Value Problems,” Dover Publ., New York. [34]. Vajravelu, K. and Prasad, K.V. (2014), “Keller-box method and its application,” HEP and Walter De Gruyter GmbH, Berlin/Boston. [35]. Ishak, A. and Nazar, R. (2009) “Laminar boundary layer flow along a stretching cylinder,” Euro. J. Sci. Res., vol. 36, pp. 22–29. [36]. Grubka, L. G. and Bobba, K. M. (1985), “Heat transfer characteristics of a continuous stretching surface with variable temperature,” ASME J. Heat Transfer, vol. 107, pp. 248-250. [37]. Ali, M. E. (1994), “Heat transfer characteristics of a continuous stretching surface,” J. Heat Mass Transfer, vol. 29, pp. 227–234.

http://www.dl.begellhouse.com/journals/49dcde6d4c0809db,50f5ee6276734748,1c8dc04c4e624b3d.htmlhttp://www.dl.begellhouse.com/journals/49dcde6d4c0809db,50f5ee6276734748,1c8dc04c4e624b3d.htmlhttp://www.sid.ir/en/VEWSSID/J_pdf/112520130303.pdfhttp://www.sid.i

[i] · 2016. 9. 15. · bangalore university journal of science -a bi annual periodical....

Documents