nonlinear radiative heat transfer and hall effects on a viscous fluid in a semi porous curved channel

Nonlinear radiative heat transfer and Hall effectson a viscous fluid in a semi-porous curved channel Z.. Nilore, Islamabad 44000, Pakistan Received 18 May 2015; accepted 8 October 2015;

curved channel

Z Abbas, M Naveed, , and M Sajid

Citation: AIP Advances 5, 107124 (2015); doi: 10.1063/1.4934582

View online: http://dx.doi.org/10.1063/1.4934582

View Table of Contents: http://aip.scitation.org/toc/adv/5/10

Published by the American Institute of Physics

Nonlinear radiative heat transfer and Hall effects

on a viscous fluid in a semi-porous curved channel

Z Abbas,1M Naveed,1, aand M Sajid2

1Department of Mathematics, The Islamia University of Bahawalpur,

Bahawalpur 63100, Pakistan

2Theoretical Physics Division, PINSTECH, P.O Nilore, Islamabad 44000, Pakistan

(Received 18 May 2015; accepted 8 October 2015; published online 23 October 2015)

In this paper, effects of Hall currents and nonlinear radiative heat transfer in a viscous fluid passing through a semi-porous curved channel coiled in a circle of radius R are analyzed A curvilinear coordinate system is used to develop the mathematical model of the considered problem in the form partial differential equations Similar-ity solutions of the governing boundary value problems are obtained numerically using shooting method The results are also validated with the well-known finite

difference technique known as the Keller-Box method The analysis of the involved pertinent parameters on the velocity and temperature distributions is presented through graphs and tables.C 2015 Author(s) All article content, except where otherwise noted, is licensed under a Creative Commons Attribution 3.0 Unported License [http://dx.doi.org/10.1063/1.4934582]

I INTRODUCTION

In past few decades, study of flow and heat transfer phenomena in porous channels or tubes has received considerable attention due to their number of practical applications in the fields of biomed-ical and mechanbiomed-ical engineering Applications include flow in dialysis of blood in artificial kidney,1 blood oxygenators2and flow in the capillaries.3The study of steady incompressible viscous fluid flow between two parallel porous walls was first carried out by Berman.4He provided the exact solution for the Navier-Stokes equations The research problem considered by Berman4was extended in many directions for both Newtonian and non-Newtonian fluids For the details interested readers are referred

to the articles5 11and reference therein

It is well known that in an ionized fluid with low density the intensity of magnetic field is very strong, the conductivity perpendicular to the magnetic field is low due to the spiraling of electrons and ions about the magnetic lines of force before collision occur and a current induced in the normal direction to both electric and magnetic fields This phenomenon is called the Hall effects.12The effect

of Hall currents on the fluid flow has wide range of applications in astrophysical, meteorological and plasma flow problems Flight magnetic hydrodynamics, MHD power generations, accelerator design, Hall effects sensors, construction of turbines and centrifugal machines13 – 18are few examples of such studies

The study of thermal radiation on convective heat transfer become a very vital theory in the processes involving high temperature such as nuclear power plant, hypersonic flight missile re-entry, thermal energy storage, rocket combustion chambers, solar power technology, power plant for inter planetary flight and design of pertinent equipment Pantokratoras and Fang20discussed the Sakiadis flow with nonlinear Rosseland thermal radiation In another paper Pantokratoras and Fang21studied the effects of nonlinear Rosseland thermal radiation in Blasius flow Cortell22investigated the fluid flow and nonlinear radiative heat transfer over a stretching sheet A numerical study of nonlinear radiative heat transfer in the flow of a nano fluid due to solar energy was carried out by Mushtaq

et al.23

a Corresponding Author Tel.: +92 62 9255480 e-mail address: rana.m.naveed@gmail.com (M Naveed)

The analysis of flow through narrow, curved channel gained significant importance due to its number of application in biological and engineering processes The seeping flow theory cracks and pulmonary alveolar blood flow have low Reynold number and so they can be characterized as a Stokes flow Stokes flow in a curved channel was carried out by Khuri.24In a couple of papers Nasir et al.25 , 26 investigated the different aspects of peristaltic flow in curved channel The effects of porous medium

on forced convection of a reciprocating curved channel were discussed by Shung Fu et al.27

A literature survey reveals that no study regarding to the Hall effects on flow and heat transfer

in a semi porous curved channel by taking nonlinear thermal radiation effects into account has done before So the objective of present analysis is to investigate the Hall effects on a viscous fluid in a semi porous curved channel by incorporating nonlinear thermal radiation Numerical solution for the fluid velocity, pressure and temperature distributions are obtained using shooting method The numerical results obtained through shooting method are also validated using Keller-Box method Numerical results are presented through graphs and tables

II MATHEMATICAL FORMULATION

Consider the steady, two-dimensional, incompressible flow of an electrically conducting viscous fluid in a semi-porous curved channel of width H coiled in a circle of radius R as shown in Fig.1 Further, the upper wall of channel is porous while the lower wall is impermeable It is also assumed that the wall placing along the s-axis is heated externally with temperature Twand the electrically conducting viscous fluid is injected uniformly from the other perforated wall to cool the heated wall Flow is caused by suction or blowing A strong magnetic field of uniform strength B0is applied in the radial direction r and the effects of Hall current are also considered Presence of Hall current generates a force in the z -direction, so the flow becomes three dimensional, however there will be

no effects on flow and heat transfer properties in z -direction

The generalized Ohm’s law in the absence of electric field E and considering the Hall effects can

be written as (see Ref.28)

J= σ

1+ m2



V × B − 1

eηe

J × B

 ,

where, V is the fluid velocity, B is the magnetic induction vector, J is the current density, m= σB0/eηe

is the Hall parameter, σ= e2ηeτe/meis the electrical conductivity, e is the electron charge, τeis the electron collision time, ηeis the number density of electron meis the mass of electron The influence

of electron pressure gradient and the ion slip effects can be neglected for the weakly ionized gases Implementing these assumptions, the boundary layer equations that govern the flow and heat transfer are

∂

∂r {(r+ R) v} + R∂u∂s =0, (1)

FIG 1 Geometry of the flow problem.

r+ R =

1 ρ

∂p

v∂u

∂r +

Ru

r+ R

∂u

∂s +

uv

r+ R = −

1 ρ

R

r+ R

∂p

∂s +

ν(∂2u

∂r2 + 1

r+ R

∂u

∂r −

u (r+ R)2

)

− σB2

0

ρ (1 + m2)(u+ mw) ,

(3)



v∂w

∂r +

Ru

r+ R

∂w

∂s



= ν

∂2w

∂r2 + 1

r+ R

∂w

∂r

 + σB

2 0

ρ (1 + m2)(mu − w) , (4)

ρcp



v∂T

∂r +

uR

r+ R

∂T

∂s



= k1

∂2T

∂r2 + 1

r+ R

∂T

∂r



− 1 (r+ R)

∂

∂r (r+ R) qr, (5) where u,v and w are the velocity components in s, r and z -directions, respectively, ρ is the density

of the fluid, p is the pressure, ν is the kinematics viscosity of fluid, k1is the thermal conductivity, cp

is the specific heat at constant pressure and qris the radiative heat flux

The corresponding boundary conditions for the flow and heat transfer problem are

u= U, v = 0, w = 0, T = Tw at r = 0,

u= 0, v = − v′R

r+ R, w = 0, T → T∞as r= H, (6) where v′> 0 corresponds to suction, v′< 0 corresponds to injection and T∞is the temperature of the wall lying at a distance H By using Rosseland approximation29the radiative heat flux is given by

qr= −4σ∗ 3k∗

∂T4

∂r =−

16σ∗ 3k∗ T3∂T

where σ∗is the Stefan-Boltzman constant and k∗is the mean absorption coefficient Eq (7) leads to

a highly non-linear energy equation in T and it cannot be solved easily However this problem can

be solved by assuming the small temperature difference within the flow (see Refs.17–19) For this, Rosseland approximation can be linearized about the ambient temperature T∞by replacing T3in Eq (7) with T3

∞ Therefore, Eq (5) can be written as

ρcp



v∂T

∂r +

uR

r+ R

∂T

∂s



= k1

(

1+16σ∗T∞3

k1k∗

) ∂2T

∂r2 + 1

r+ R

∂T

∂r



However, Eq (5) yields in a highly nonlinear radiation expression if we not consider the above assumption which is the aim of the present study So the energy Eq (5) in the presence of non-linear thermal radiation become

ρcp



v∂T

∂r +

uR

r+ R

∂T

∂s



= k1 (r+ R)

∂

∂r

(

1+16σ∗T3

k1k∗ (r+ R)∂T∂r

)

The non-dimensional temperature is defined as θ(η)= T − T∞/Tw− T∞, with T = T∞[1+ (θw− 1) θ] and θw = Tw/T∞is the temperature parameters So Eq (9) yield

v∂T

∂r +

uR

r+ R

∂T

∂s =

α (r + R) ∂/∂r
(1 + Rd(1 + (θw− 1) θ)3)(r + R)∂T/∂r , (10) where α is the thermal diffusivity and Rd = 16σ∗T∞3/3k1k∗23is defined as a radiation parameter For similar solution of the flow equations, we use the following dimensionless variables

u=U s

H f

′ (η) , v= −Rv′

r+ Rf(η) , η=

r

H,

p= ρU2s2

H2 P(η) , w=U s

Hg (η) ,

(11)

Using Eq (11), continuity equation is identically satisfied and Eqs (3), (4), and (10) yields

∂P

∂η =

f′2

2K P

η + K =

1 Re



f′′′+η + Kf′′ − f′

(η+ K)2+ f′

(η+ K)3

 +

K f f′′

η + K −

K f′2

η + K +

K f f′ (η+ K)2 − M2

1+ m2( f′+ mg) ,

(13)

g′′+ g′

η + K +

KRe

η + K( f g

′−g f′ )+ M2Re

1+ m2(m f − g)= 0, (14) 1

Pr(η+ K)

 (

1+ Rd(1 + (θw− 1) θ)3)(η+ K) θ′′

+ KRe (η + K)fθ′= 0, (15) where K = R/H is the dimensionless radius of curvature, Pr = µcp/k1 is the Prandtl number, M

= σB2

0H/ρU is the magnetic parameter and Re = HU/ν is the Reynold number

The corresponding boundary conditions takes the following form

f(0) = 0, f′

(0) = 1, g (0) = 0, θ(0) = 1,

f(1) = 1, f′(1) = 0, g (1) = 0, θ(1) = 0 (16) Eliminating pressure between Eqs (10)and(11), we get

f′′′′+η + K2 f′′′ − f′′

(η+ K)2+ f′

(η+ K)3+η + KKRe ( f f′′′− f′f′′)+ KRe

(η+ K)2

(

f f′′− f′2)

−KRe f f′

(η+ K)3 − M2Re

(η+ K) (1 + m2)( f

′+ mg) − M2Re

1+ m2( f′′+ mg′

)= 0,

(17)

Once the fluid velocity f(η) is obtained the pressure can be determined from Eq (13) The physical quantities of interest are the skin-friction coefficient and the rate of heat transfer along the curved wall, which are defined as

Cf = τrs

ρU2 w

, Nus= sqw

k1(Tw− T∞), (18) where τrsis the wall shear stress and qwis the heat flux at the wall along the s -directions, which are given by

τrs= µ

∂u

∂r −

u

r+ R



r =0 , qw= −k1

∂T

∂r

r =0+ (qr)w, (19) Using Eqs (11) and (19), Eq (18) becomes

Re1/2s Cf = f′′

(0) − 1

K,

Re−1/2s Nus= − 1 + Rdθ3

w



θ′ (0)

III NUMERICAL METHOD FOR SOLUTION

The non-linear differential equations (14), (15), and (17) subject to boundary conditions (16) is solved numerically by using shooting method along with the fourth order Runge-Kutta integration scheme in the following way

f′= t, t′= p, p′= z,

z′= η + K−2z + p

(η+ K)2− t

(η+ K)3− K Re

η + K ( f z − tp) −

K (η+ K)2 f p − t2
+ KRe

(η+ K)3f t+ M2Re

(η+ K) (1 + m2)(t+ mg) + M2Re

(1+ m2)(p+ ml) ,















(20)

g′= l, g′′= η + K−l − KRe

η + K( f l − gt) −

M2Re

θ′= q,

q′= ( 1

1+ Rd(1 + (θw− 1) θ)3)



−3Rd(θw− 1) (1+ (θw− 1) θ)2q2−Re Pr K

η + K f q



− q

η + K











 ,

(22) with boundary conditions

f(0)= 0, t (0) = 1, g (0) = 0, θ (0) = 1 (23)

In order to integrate (20), (21), and (22) as an initial value problem we need the value of p(0) i.e

f′′(0) , z (0) i.e f′′′

(0) , g′ (0) i.e l (0) and θ′

(0) i.e q (0), but no such values are given Choose some suitable values as a guess for f′′

(0) , f′′′

(0) , l (0) and θ′

(0) and then integration is carried out The calculated values of f′

(1) and θ (1) are compared with the given boundary conditions

f(1)= 1, f′

(1)= 0,g (1) =0 and θ (1) = 0 and the values of f′′

(0) , f′′′

(0) , g′ (0) and θ′

(0) are adjusted by Newton Raphson’s method to give better approximation for the solution The step size

is taken as ∆η = 0.005 The process is repeated until we get the results correct up to the accuracy

of 10−5level To validate the obtained numerical results through the shooting method we have also solved the governing equations using an implicit finite difference scheme known as the Keller-Box method.30

IV RESULTS AND DISCUSSION

The nonlinear boundary value problems given in Eqs (14), (15) and (17) subject to boundary conditions (16) are solved numerically using both shooting and Keller-Box methods The fluid veloc-ities f′

(η), g (η), pressure P(η) and temperature θ(η) are plotted in order to see the influence of the several physical involved parameters namely, dimensionless radius of curvature, magnetic parameter, Hall parameter, Reynolds number, Prandtl number, radiation parameter and temperature parameter in Figs.2-11 Furthermore, the magnitude of the skin friction coefficient Re1/2

s Cf and the local Nusselt number Res−1/2Nusfor different parameters are presented in tablesIandII

Fig.2illustrates the effects of dimensionless radius of curvature K on the component of velocity

f′(η) by keeping M= 1.5,Re = 3 and m = 0.8 fixed It is found that near the permeable plate fluid velocity f′(η) is decreased with an increase in the value of K, but after η= 0.4 it starts to increase This is due to the fact that the upper wall of the channel is porous and the suction through it forces the velocity to increase The effects of the magnetic parameter M and the Hall parameter m on the component of velocity f′

(η) are shown in Fig.3 It is noticed from this Fig that the fluid velocity is

FIG 2 Variation of the dimensionless radius of curvature K on the component of velocity f′(η) by keeping M = 1.2, Re

= 5.0 and m = 0.8 fixed.

TABLE I Numerical values of skin friction coe fficient Re 1/2

s C f for various values of K, M, Re and m.

decreased by increasing the value of both the parameters M and m near the permeable wall However the effects of these parameters on the velocity are opposite near the porous wall The magnetic field suppresses the fluid velocity near the permeable wall and this is the expected fact The influence of the Reynolds number Re on the velocity f′(η) by keeping other parameters fixed is displayed in Fig.4 It

is observed that the velocity of the fluid increased by increasing the value of Re and the same behavior

is noted as for K

Fig.5elucidate the variation in transverse velocity field g(η) for different values of magnetic parameter M and dimensionless radius of curvature K It can be seen from this Fig that influence

of increasing the value of M is to increase the transverse velocity g(η) However, it has the reverse behavior for K as the transverse velocity g(η) decreased by increasing the value of K The effects of Hall parameter m and Re on transverse velocity field are shown in Fig.6 It is evident from this Fig that g(η) is increased for higher values of Re and m

Fig.7depict the variation of Re and M on pressure distribution P(η) It is evident from this Fig that the magnitude of P(η) is decreased by an increase in Re However, it increases for large values

of M The variation of Hall parameter m and dimensionless radius of curvature K on the pressure

TABLE II Numerical values of local Nusselt number Re−1/2s N u s for various values of K, Re, Pr, Rd and θ w

FIG 3 Variation of the magnetic parameter M and for two values of Hall parameter m on the component of velocity f ′

(η)

by keeping K = 2 andRe = 3 fixed.

FIG 4 Variation of the Reynold number Re on the component of velocity f′(η) by keeping K = 2, M = 0.5 and m = 1.5 fixed.

FIG 5 Variation of the magnetic parameter M and for the two values of dimensionless radius of curvature K on the transverse velocity profile g (η) by keeping Re = 3 and m = 0.5 fixed.

distribution P(η) is shown in Fig.8 It is found from this Fig that the magnitude of pressure distri-bution P(η) is a decreasing function for both m and K

Fig.9illustrate the variation in the temperature profile θ(η) for various values of the Prandtl number Pr and Re From this Fig it is evident that for high values of Pr and Re, the temperature and the thickness of the thermal boundary layer are decreased with an increase in the values of Pr and Re The effects of the nonlinear radiation parameter Rd on the temperature distribution θ (η) are displayed

in Fig.10 It is observed that the temperature and thermal boundary layer thickness are increased for

FIG 6 Variation of the Reynold number Re and Hall current parameter m on the transverse velocity profile g (η) by keeping

K = 2 and M = 1.5 fixed.

FIG 7 Variation of the Reynold number Re and magnetic parameter M on the pressure distribution P (η) by keeping

K = 2 and m = 0.8 fixed.

FIG 8 Variation of the Hall current parameter m and two values of dimensionless radius of curvature K on the pressure distribution P (η) by keeping Re = 4 and M = 1 fixed.

higher values of radiation parameter Rd Fig.11demonstrates the influence of temperature parameter

θwon the temperature profile θ(η) It is noticed from this Fig that the temperature and the thermal boundary layer thickness are increased by increasing the values of θw It is also noticed from this Fig that when θr 1, the temperature profile of the nonlinear Rosseland approximation leads to a linear Rosseland approximation and for the higher values of θw, represents the higher wall temperature as compared to ambient temperature Also the temperature profile become broader and S-shaped as dis-cussed by Pantokratoras and Fang20representing the existence of the adiabatic case for higher values

of θw

FIG 9 Variation of the Prandtl number Pr and Reynold number Re on the temperature distribution θ (η) by keeping

K = 2, M = 1.2, m = 0.5, Rd = 0 and θ w = 2 fixed.

FIG 10 Variation of the radiation parameter Rd on the temperature distribution θ(η) by keeping K = 2, M = 1.3, Re

= 5, m = 0.5, Pr = 7 and θ w = 2 fixed.

FIG 11 Variation of the temperature parameter θ w on the temperature distribution θ(η) by keeping K = 2, M = 1.5, Re

= 5, m = 0.5, Pr = 7 and Rd = 0.3 fixed.

TableIis given to show the numerical values of the skin friction coefficient Re1/2

s Cf for various values of K, M, Re and m It is found that the magnitude of Re1/2s Cf is increased by increasing the value of M, Re and m, but it has the opposite trend for K TableIIis made to show the numerical values

of the local Nusselt number Re−1/2s Nusfor various values of K, Re, Pr, Rd and θw From this table it is observed that the magnitude of Re−1/2s Nusis increased by increasing the value of Re, Pr, Rd and θw but it has the opposite behavior for K as the magnitude of local Nusselt number decreased by increas-ing the value of K

in Fig.10 It is observed that the temperature... temperature and thermal boundary layer thickness are increased for

FIG Variation of the... Re−1/2s N u s for various values of K, Re, Pr, Rd and θ w

FIG Variation of