Analytical solution to the problem of heat transfer in an MHD flow inside a channel with prescribed sinusoidal wall heat flux

Analytical solution to the problem of heat transfer in an MHD flow inside a channel with prescribed

sinusoidal wall heat flux

K. Zniber

^*

, A. Oubarra, J. Lahjomri

Groupe Energe´tique, De´partement de Physique, Faculte´ des Sciences, Universite´ Hassan II Ain Chock, Km 8 Route d’El jadida, B.P. 5366, Maaˆrif Casablanca, Morocco

Received 3 March 2004; accepted 13 June 2004 Available online 11 September 2004

Abstract

An MHD laminar flow through a two dimensional channel subjected to a uniform magnetic field and heated at the walls of the conduit over the whole length with a sinusoidal heat flux of vanishing mean value or not, is studied analytically. General expressions of the temperature distribution and of the local and mean Nusselt numbers are obtained by using the technique of linear operators in the case of negligible Joule and viscous dissipation and by taking into account the axial conduction effect. The principal results show that an increase of the local Nusselt number with Hartmann number is observed, and, far from the inlet section, the average heat transfer between the fluid and the walls shows a significant improvement at all values of Hartmann number used when the frequency of the prescribed sinusoidal wall heat flux is increas- ing in the case of vanishing mean value of the heat flux and this is true especially at low Peclet numbers.

Ó2004 Elsevier Ltd. All rights reserved.

Keywords:Laminar; MHD flow; Sinusoidal heat flux; Forced convection; Axial conduction; Linear operators; Anal- ytical solution

0196-8904/$ - see front matter Ó2004 Elsevier Ltd. All rights reserved.

doi:10.1016/j.enconman.2004.06.023

*Corresponding author. Tel.: +212 22 23 06 80.

E-mail addresses: khalidzniber@yahoo.fr(K. Zniber), a-oubarra@hotmail.com (A. Oubarra),lahjomri@hotmail.

com (J. Lahjomri).

www.elsevier.com/locate/enconman

1. Introduction

Because of its great interest in the design of practical thermal systems, the problem of laminar forced convective flow heat transfer using a large number of fluids in a circular tube or parallel plates channel is of foremost importance. As is well known, the magnetic field influences the heat transfer and, thus, the temperature distribution through the change of the velocity of the fluid or liquid metal with the value of the Hartmann number

M

[1], which represents the ratio of the elec- tromagnetic forces to the viscous forces. Periodic wall heat fluxes used in engineering applications

Nomenclature

b

half width between parallel plates channel (m)

~B0

external uniform magnetic field vector (T)

k

thermal conductivity (W m

¹

K

¹

)

M

Hartmann number,

B₀b ffiffi

r l

q Nu

local Nusselt number

Nu_as;Nu_as

asymptotic Nusselt number and its mean value

p

pressure

Pe

Peclet number,

^3U_2a^mb

q_w

(x) wall heat flux per unit area,

k^oT_oy

ðx;

bÞ

(W m

²

)

q0

amplitude of wall heat flux (W m

²

)

T(x,y) temperature field (K)

T0

uniform inlet section temperature (K)

u(g)

dimensionless velocity profile,

²₃^u_U^xðyÞ U_m

mean fluid velocity (m s

¹

)

m

u_x

(y) Hartmann velocity profile in axial direction,

U_m ^chMchM

y

ð

b

Þ

ðchM^sh_M^MÞ

(m s

¹

)

x

axial coordinate (m)

y

transverse coordinate (m)

Greek symbols

a thermal diffusivity (m

²

s

¹

) b frequency of wall heat flux (m

¹

) d dimensionless parameter

c dimensionless parameter

g dimensionless transverse coordinate,

^y_b

l dynamic viscosity (kg m

¹

s

¹

)

h dimensionless temperature distribution,

^TTbq0⁰

h

_b

dimensionless bulk temperature

k

h

_w

dimensionless wall temperature

r electric conductivity (X

¹

m

¹

)

x dimensionless frequency, bbPe

n dimensionless axial coordinate,

_bPe^x

are of great technical interest as, for instance, in the design of cooling tubes for nuclear reactors and in the analysis of heat transfer in the heat exchangers of Stirling cycle machines. Thus, the combination between magnetic field effect, fluid type, the form of the sinusoidal wall heat flux and the value of its frequency can be, in certain conditions, of great interest in heat transfer prob- lem by following the influence of these parameters, as will be seen in the present work.

The effect of axial conduction that is considered in the extended Graetz problem becomes sig- nificant for small values of Peclet number

Pe

and, then, must be taken into account in the energy equation to predict the heat transfer in the thermal entrance region. Many attempts have been made in the past in the absence of magnetic fields to solve the problem of thermal heat transfer by including or not the axial conduction term in the energy equation [2–12]. Min and Yoo [2], using the method of separation of variables, have studied the thermal developing flow of a Bing- ham plastic in a circular pipe with uniform wall heat flux, and the analytical solution has been given in terms of the yield stress, Peclet number and Brinkmann number. The resulting eigenvalue problem is solved approximately by using the method of weighted residuals. Liou and Wang [3], following the method used by Papoutsakis et al. [4] in heat conduction, treated the problem with uniform heat flux for a power model fluid flow in a duct and three choices for the entrance bound- ary conditions, including heat generation in the energy equation, and the solution has been ob- tained for temperature profiles and local Nusselt number by the use of the linear operators method. Recently, Lahjomri et al. [5] used the same formalism to solve the problem of the thermal entrance region in the case of forced convection of Hagen–Poiseuille flow in a parallel plates chan- nel taking into account a heat generation term in the energy equation and with Neumann bound- ary conditions, and it has been reported in this work that the solution obtained can be applied to any developed flow with heat generation. The boundary condition of sinusoidal wall heat flux, which represents the simplest kind of periodic heating or cooling, has been considered in previous studies [6–12]. Pearlstein and Dempsey [6] used wall heat fluxes given either by a sinusoidal dis- tribution or by a hyperbolic tangent distribution, and plots of the bulk temperature and the tem- perature profile in the thermal entrance region are presented for various Peclet numbers. Barletta and Zanchini, in the case of negligible axial conduction [7], and Barletta and Rossi di Schio, con- sidering the axial conduction effect [8], deal with sinusoidal wall heat flux with vanishing mean value or not in the case of Poiseuille flow entering a circular duct, and the authors give the solu- tion at sufficiently great distance from the inlet section that is a sum of linear function and a peri- odic function of the axial coordinate.

In the presence of magnetic field, little has been reported on the extended Graetz problem in MHD flow. Nigam and singh [13] were the first who studied the heat transfer by laminar flow be- tween a parallel plates channel under the action of a transverse magnetic field in two semi-infinite regions. Lahjomri et al. [14], using the separation of variables method for the case of the MHD problem with step change in the wall temperatures, including axial conduction and heat genera- tion and taking into account the non-uniformity of the inlet temperature, give an improved solu- tion for the temperature distribution and Nusselt number in the thermal entrance region. In Ref.

[5], the case of Hartmann flow is also considered, and the authors, as reported above, have used

the technique of linear operators, which requires determination of a set of eigenvalues with their

associated non-orthogonal eigenfunctions. The results presented concern the temperature profiles

as well as the local Nusselt number and a correlation formula is given for local Nusselt number, as

a function of the Hartmann number, valid only for large Peclet numbers.

As far as is known, there has been no work in the literature that studied thermally developing MHD flow in a duct subject to a magnetic field with sinusoidal wall heat flux and with considering the axial conduction term in the energy equation. We propose, in the present work, to give an exact analytical solution in the thermal entrance region by use of the linear operators method in the case of the extended Graetz problem under the action of a transverse magnetic field and with prescribed sinusoidal wall heat flux of the parallel plates channel by assuming a uniform inlet temeprature and neglecting both viscous and Joule dissipation, which is justified in the work of Lahjomri et al. [5].

2. Physical problem and governing equations

Consider a two dimensional channel infinite in the

x

direction whose inner boundaries are sep- arated by a distance 2b as shown in Fig. 1. Inside the channel, an incompressible electrically con- ducting fluid with constant physical properties flows under the combined action of a constant pressure gradient

^op_ox

and imposed magnetic field

~B0

in the positive

y-direction. It is assumed that

the hydrodynamic profile is fully developed and corresponds to a Hartmann profile [1]. The effects of viscous and Joule dissipation are neglected because of the very small Brinkmann number for liquid metals [5], while the axial heat conduction is taken into account.

At the inlet section of the channel, which corresponds to

x

= 0, the temperature is supposed to be uniform with value

T0

. On the other hand, a sinusoidal heat flux is present at the walls, namely

q_w

ðxÞ ¼

q₀

ðc þ d sin bxÞ ð1Þ

where d and c are dimensionless parameters. Thus, if d = 0 and c = 1, we are in the presence of a uniform wall heat flux, the case d = 1 and c = 1 leads to a periodic wall heat flux with non-van- ishing mean value

q₀

, that can be positive (fluid heating) or negative (fluid cooling) and if d = 1 and c = 0, this corresponds to a vanishing mean value of

q_w

(x) (succession of heating and cooling) for

q₀

both positive and negative.

qw

) ( y

ux T₀

x ^2b

0

y

B

0 Fig. 1. Problem description.

The dimensionless energy equation for the MHD problem, taking into account the axial con- duction term and neglecting both viscous and Joule heating, is, after using symmetry considera- tions, given by

uðgÞoh on

1

Pe² o²

h

on²

¼

o²

h

og²

for 0

<

n

<

þ1 and 0

<

g

<

1 ð2Þ where the dimensionless variables and parameters are defined in the nomenclature.

In dimensionless forms, the conditions associated with Eq. (2) due to the inlet section, symme- try and wall are, respectively, as follows:

hð0; gÞ ¼ 0 for 0

6

gg

<

1 ð3Þ

oh

og

¼ 0 for g ¼ 0 and 0

6

n

<

þ1 ð4Þ

oh

og

¼ c þ d sin xn for g ¼ 1 and 0

6

n

<

þ1 ð5Þ

3. Analysis

We turn now to the resolution of Eq. (2) subject to boundary conditions (3)–(5). The use of the linear operators method, as already done in the past in the case of no heat generation by Papou- tsakis et al. [4], or with heat generation by Lahjomri et al. [5], both treated in the case of uniform wall heat flux, is very efficient, as will be seen below. This method consists in decomposing the energy Eq. (2) into a pair of first order partial differential equations such that the problem be- comes selfadjoint, although the initial problem was not a selfadjoint one as reported in the work of Lahjomri and Oubarra [15].

Therefore, we define the dimensionless axial energy flow through a cross-sectional area of width g, concentric with the channel cross section, due to convection and conduction, as

uðn; gÞ ¼

Z g

0

uðg⁰

Þh 1

Pe²

oh on

dg

⁰

ð6Þ

To decompose Eq. (2) into a pair of first order partial differential equations, we use Eqs. (2) and (6) and the boundary condition (4), thus we have in matrix notation and after rearrangements

o~F

on

¼

L~F

ð7Þ

where the vector

~F

¼ ðh uÞ

^t

and the matrix differential operator

L

L

¼

Pe²u

Pe

²_og^o

o

og

0

!

ð8Þ

If an inner product between two vectors is defined by h

~

/; ~ wi ¼

Z 1 0

1

Pe²

/

₁

w

₁

þ /

₂

w

₂

dg ð9Þ

with the operator

L, defined in Hilbert spaceH, ranges over the domain defined by

DðLÞ ¼ f~

/ 2

H; L~

/ 2

H;

/

₂

ð0Þ ¼ /

⁰₁

ð0Þ ¼ 0; /

₂

ð1Þ ¼ /

⁰₁

ð1Þ ¼ 0g ð10Þ where the two vectors

~

/ ¼ ð /

₁

/

₂

Þ

^t

and w ¼ ð w

₁

w

₂

Þ

^t

, then the selfadjoint eigenvalue problem is obtained as

L~

/

_n

¼ l

_n~

/

_n

ð11Þ

where the quantities l

n

are the real eigenvalues [16], which can be, in principle, positive or negative, and

~

/

_n

¼ ð /

_n

1

/

_n

2

Þ

^t

are the corresponding eigenvectors of the linear oper- ator

L.

If we use Eqs. (8) and (11), the selfadjoint eigenvalue problem can be rewritten as /

⁰⁰_n

1

þ l

_n

l

_n

Pe²

u

/

_n

1

¼ 0 ð12Þ

with boundary conditions /

⁰_n

1

ð0Þ ¼ 0 ð13Þ

/

⁰_n

1

ð1Þ ¼ 0 ð14Þ

The eigenfunctions /

_n

1

which appear in the final solution (see Eq. (22)) and which are solutions of differential Eq. (12), correspond in the case of Hartmann flow to MathieuÕs functions, which depend on Hartmann number

M

and are already given in explicit manner in the work of Lahjomri et al. [14]. The use of the boundary condition Eq. (14) leads to the determination of the eigen- values l

_n

.

By using Eq. (7) and taking the inner product with the eigenvector

~

/

_n

, one obtains by the means of

DðLÞ

the following scalar inhomogeneous differential equation:

o

on ~F; ~

/

_n

D E

¼ l

_n ~F; ~

/

_n

D E

/

_n1

ð1Þuðn; 1Þ ð15Þ

The final solution can be obtained, using an expansion theorem for the vector

~F

2

H

as

~F

¼ hðn; gÞ uðn; gÞ

¼

X^þ1

n¼1

h

~F; ~

/

^þ_n

i k

~

/

^þ_n

k

²

~

/

^þ_n

þ h

~F; ~

/

_n

i k

~

/

_n

k

²

~

/

_n

" #

¼

X^þ1

n¼1

h

~F; ~

/

_n

i k

~

/

_n

k

²

~

/

_n

ð16Þ

where k

~

/

_n

k

²

¼ h

~

/

_n; ~

/

_n

i is the square of the norm of the vector

~

/

_n

. The solution of Eq. (15) for positive eigenvalues takes the form

~F; ~

/

^þ_n

D E

¼ /

^þ_n

1

ð1Þ

Z n

þ1

uðn

⁰;

1Þ exp l

^þ_n

ðn n

⁰

Þ dn

⁰

ð17Þ

and for negative eigenvalues

~F; ~

/

_n

D E

¼

C_n

exp l

_n

n /

_n

1

ð1Þ

Z n

0

uðn

⁰;

1Þ exp l

_n

ðn n

⁰

Þ dn

⁰

ð18Þ where

C_n

is a constant of integration.

The inner products Eqs. (17) and (18) needed in Eq. (16) can be given if the expression of u(n, 1) is known. This can be done by performing a heat balance over the channel from n

⁰

= 0 to n

⁰

= n and from g = 0 to g = 1. Thus, from energy Eq. (2), we have

Z 1 0

Z n 0

uoh on⁰

1

Pe² o²

h

on⁰²

dn

⁰

dg ¼

Z 1

0

Z n 0

o²

h

og²

dn

⁰

dg ð19Þ

Integration of Eq. (19) by using Eqs. (3)–(6), leads to the following result:

uðn; 1Þ ¼ cn d

x cos xn þ

C0

ð20Þ

where

C0

is given by

C0

¼ d

x 1

Pe²

Z 1 0

oh

on

ð0; gÞ dg ð21Þ

The final expression of h(n, g) permits the complete determination of

C₀

through Eq. (21).

By substituting the expression of u(n, 1) from Eqs. (20) and (21) and integrating Eqs. (17) and (18), we have, according to the expansion theorem Eq. (16) and after rearrangements, the dimen- sionless temperature as follows:

hðn; gÞ ¼

X^þ1

n¼1

A_n

expðb

²_n

nÞ/

_n

1

ðgÞ þ

S Pe²

X^þ1

n¼1

A_n

b

²_n Z 1

0

/

_n

1

dg þ

S

cn þ h

₁

ðgÞ sin xn þ h

₂

ðgÞ cos xn þ

S

d

x x

Pe²

Z 1 0

h

₁

dg c

Pe²

Z 1 0

S

dg

GðgÞ

ð22Þ

with the eigenvalues l

_n

¼ b

²_n

, where

S

is a certain serie, and where the expansion coefficients

An

are given by

An

¼

C_n

k

~

/

_n

k

²

c/

_n

1

ð1Þ

l

_n²

k

~

/

_n

k

²

þ 1

Pe²

Z 1 0

oh

on

ð0; gÞ dg d x

/

_n

1

ð1Þ

l

_n

k

~

/

_n

k

²

ð23Þ It can be shown by using Eqs. (8), (9), (11)–(14) and the expansion theorem Eq. (16) by choosing an appropriate vector, that the serie

S

in Eq. (22) converges to a constant value given by

S

¼

X^þ1

n¼1

/

_n

1

ð1Þ/

_n1

ðgÞ l

_n

k

~

/

_n

k

²

¼ 3

2 ð24Þ

In Eq. (22), the functions

G(g),

h

₁

(g) and h

₂

(g) denote the following series:

GðgÞ ¼ cX^þ1

n¼1

/

_n

1

ð1Þ/

_n1

ðgÞ l

²_n

k

~

/

_n

k

²

þ d

x

X^þ1

n¼1

b

²_n

/

_n

1

ð1Þ/

_n

1

ðgÞ

ðb

⁴_n

þ x

²

Þk

~

/

_n

k

²

ð25Þ

h

₁

ðgÞ ¼ d

X^þ1

n¼1

/

_n

1

ð1Þ/

_n1

ðgÞ

ðl

²_n

þ x

²

Þk

~

/

_n

k

²

ð26Þ

h

₂

ðgÞ ¼ d x

X^þ1

n¼1

l

_n

/

_n

1

ð1Þ/

_n

1

ðgÞ

ðl

²_n

þ x

²

Þk

~

/

_n

k

²

ð27Þ

which, now, must be determined explicitly. By substituting Eq. (22) into Eqs. (2), (4) and (5) and using Eq. (12) and the result Eq. (24) and by introducing the complex function W(g) = h

1

(g) + ih

2

(g), one obtains

G⁰⁰

þ 3 2 cu ¼ 0

G0ð0Þ ¼

0;

G⁰

ð1Þ ¼ c

W

⁰⁰

þ ix ix

Pe²

u

W ¼ 0

ð28Þ

W

⁰

ð Þ ¼ 0 0; W

⁰

ð1Þ ¼ d ð29Þ

An integration of the differential equation appearing in the boundary value problem expressed by Eq. (28) with using Eqs. (6), (20)–(22) and (24) yields

GðgÞ ¼ cðG0

ðgÞ þ

Kð1ÞÞ

9c

4Pe

²

ð30Þ

with

G0

ðgÞ ¼

chM

2

ðg

²

1Þ

_M¹2

ðch

Mg

ch

M

Þ

ch

M

^sh_M^M

ð31Þ

KðgÞ ¼

1 ðch

M

^sh_M^M

Þ

²

g

³

6 ch

²M

g

²

2M ch

MshM

g

þg ch

MgchM

M²

þ ch

²M

M²

ch

²M

2 þ 1

2M

²

þ 1

2M 3

M³

ch

MshMg

þ sh

²Mg

4M

³

ð32Þ Comparison of Eqs. (12) and (29) shows that they present the same form, and thus, as previ- ously reported for Eq. (12), the integration of Eq. (29) leads also to the MathieuÕs function, that depend on the parameters

M

and x but here are complex due to the term ix. It can be shown that the solution given explicitly in the work of Lahjomri et al. [14] can also apply in this case.

To obtain the coefficients

A_n

given by Eq. (23), we use the boundary condition given by Eq. (3), the solution Eq. (22) and the result Eq. (24), thus we have

X^þ1

n¼1

An

/

_n

1

ðgÞ þ 3b

²_n

2Pe

²

Z 1 0

/

_n

1

dg

¼

G1

ðgÞ ð33Þ

where

G1

ðgÞ ¼

GðgÞ

h

₂

ðgÞ þ 3 2

x

Pe²

Z 1 0

h

₁

dg þ 3c 2Pe

²

d

x

The determination of the expansion coefficients

An

must be performed on the set of /

_n

1

ðgÞ with the inner product h/

_n

1;

/

_m

1

i ¼

R1 0

/

_n

1

/

_m

1

dg. Thus, Eq. (33) reduces to the system of equations

X^þ1

n¼1

A_n Z 1

0

/

_n

1

/

_m

1

dg þ 3b

²_n

2Pe

²

Z 1 0

/

_n

1

dg

Z 1

0

/

_m

1

dg

¼

Z 1

0

G₁

/

_m

1

dg

m

¼ 1; 2;

. . .;N

ð34Þ A Gaussian elimination method can then be used to obtain the coefficients

An

from Eq. (34).

4. Bulk temperature,wall temperature and local Nusselt number

Two quantities of practical importance, the bulk temperature or average temperature and the Nusselt number

Nu, can be expressed in dimensionless forms as

h

b

ðnÞ ¼

R1

0

hðn; gÞuðgÞ dg

R1

0 uðgÞ

dg ð35Þ

Nu

¼ 4

h i^oh_og

g¼1

h

_w

h

_b

ð36Þ

with

h

w

ðnÞ ¼ hðn; 1Þ ð37Þ

By employing Eqs. (22), (24), (29)–(32), (35) and (37), one obtains h

_b

ðnÞ ¼ 3

2 1

Pe²

X¹

n¼1

A_n

b

²_n Z 1

0

/

_n

1

dgð1 expðb

²_n

nÞÞ þ cn x

Pe²

Z 1 0

h

₂

dg sin xn

" #

þ 3 2

x

Pe²

Z 1 0

h

₁

dg d x

cos xn þ d x x

Pe² Z 1

0

h

₁

dg

ð38Þ

h

_w

ðnÞ ¼

X^þ1

n¼1

A_n

expðb

²_n

nÞ

_n

/

_n

1

ð1Þ þ 3 2Pe

²

X^þ1

nj¼1

A_n

b

²_n Z 1

0

/

_n

1

dg þ 3

2 cn þ h

₁

ð1Þ sin xn

þ h

₂

ð1Þ cos xn þ 3d 2x 3x

2Pe

² Z 1

0

h

₁

ðgÞ dg þ cKð1Þ ð39Þ

and according to the boundary condition Eqs. (5) and (36)

Nu

¼ 4ðc þ d sin xnÞ

h

_w

h

_b

ð40Þ

with

h

w

h

b

¼

X^þ1

n¼1

An

/

_n

1

ð1Þ þ 3b

²_n

2Pe

²

Z 1 0

/

_n

1

dg

expðb

²_n

nÞ

þ h

₁

ð1Þ þ 3x 2Pe

²

Z 1 0

h

₂

dg

sin xn

þ h

2

ð1Þ 3x 2Pe

²

Z 1 0

h

1

dg þ 3d 2x

cos xn þ cKð1Þ ð41Þ

5. Asymptotic Nusselt number and its mean value

According to Eqs. (40) and (41), the asymptotic or fully developed Nusselt number, obtained for large values of n, is given by

Nuas

¼ 4ðc þ d sin xnÞ

h

1

ð1Þ þ

_2Pe^3x2

R1 0

h

2

dg

sin xn þ h

2

ð1Þ

_2Pe^3x2

R1

0

h

1

dg þ

_2x^3d

cos xn þ cK ð1Þ

ð42Þ

Eq. (42) is a periodic function of n with a period equal to

^2p_x

and can, for some values of c and x and if a certain condition is verified, present singularities depending on whether the denominator of Eq. (42) vanishes or not. If c = 0, the period is now equal to

_x^p

, and the asymptotic Nusselt num- ber is always affected by singularities.

If no singularities arise, the mean value of the asymptotic or fully developed Nusselt number is given by

Nuas

¼ x 2p

Z ^2p_x

0

Nuas

dn ð43Þ

On account of Eq. (42), calculation of the integral in Eq. (43) can be expressed as

Nuas

¼

4 1

cKð1Þ h1ð1Þþ^3x 2Pe2

R1 0 h2dg

h₁ð1Þþ^3x 2Pe2

R1 0h₂dg

2

þ h₂ð1Þ^3x 2Pe2

R1 0h₁dgþ_2x^3d

2

2 64

3 75

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

ðcKð1ÞÞ

²

h

₁

ð1Þ þ

_2Pe^3x2

R1 0

h

₂

dg

2

h

₂

ð1Þ

_2Pe^3x2

R1

0

h

₁

dg þ

_2x^3d

2

r

þ

4 h

₁

ð1Þ þ

_2Pe^3x2

R1 0

h

₂

dg

h

1

ð1Þ þ

_2Pe^3x2

R1 0

h

2

dg

2

þ h

2

ð1Þ

_2Pe^3x2

R1

0

h

1

dg þ

_2x^3d

2

ð44Þ

For the case c = 0, due to singularities, we use the principal value of the integral appearing in

Eq. (43) which is found equal to

Nuas

¼

4 h

₁

ð1Þ þ

_2Pe^3x2

R1 0

h

₂

dg

h

1

ð1Þ þ

_2Pe^3x2

R1 0

h

2

dg

2

þ h

2

ð1Þ

_2Pe^3x2

R1

0

h

1

dg þ

_2x^3d

2

ð45Þ

6. Results and discussion

We have first compared the present work with a similar problem treated by Lahjomri et al. [5]

that uses a uniform wall heat flux, i.e. d and c in our case and for obtaining the same conditions as in this work, must take the values 0 and 1, respectively. The result found is such that the general solution obtained (Eq. (22)) by using these values coincides with the one obtained in Ref. [5].

Numerical simulation of our analytical solution that has been performed in this study for dif- ferent values of Hartmann number

M

concern the evolution of the wall temperature h

_w

, the bulk temperature h

b

and the local Nusselt number

Nu

with the dimensionless axial coordinate n for d = 1 for

Pe

= 1.5 and 150, for the dimensionless frequency x = 5 and for c = 0 and 1 and also the behaviour of the mean value of the asymptotic Nusselt number as a function of the dimension- less frequency x, for

Pe

= 1.5 and 150 and for c = 0 by employing Eqs. (38)–(41) and (45).

The value

Pe

= 150 can be considered as very close to the case of

Pe

¼ 1, which represents the situation of negligible axial conduction.

In Figs. 2 and 3, plots of the wall temperature h

_w

and the bulk temperature h

_b

are presented for the cases c = 0 and c = 1 for x = 5 for four different values of the Hartmann number;

M

= 0, 6, 50

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

ξ

0 1 2 3 4 5 6

Pe = 150, ω = 5

θ_w M = 0, 6, 50, 100

θ_b

γ = 1

γ = 0

θw, θb

M M

Fig. 2. Bulk and wall temperatures with respect to dimensionless axial coordinatenfor various values of Hartmann numberM,x= 5,Pe= 150,c= 0 and 1.

and 100, and for

Pe

= 150 and 1.5, respectively. Both figures show that the wall temperature and the bulk temperature retain globally the same shape as in the absence of magnetic field and oscil- late around an increasing axial mean value for c = 1 or around a constant mean value for c = 0.

Comparison between these figures shows that the amplitude of the periodic waves is reduced by decreasing the value of Peclet number

Pe

because of an increase of the axial conduction effect.

Because of its definition, one can note that the bulk or average temperature in a given transverse section oscillates with a more reduced amplitude than the one of the wall temperature.

In the same figures, one can study the Hartmann number effects on the wall temperature and bulk temperature. The behaviour observed is such that for

Pe

= 150, corresponding to negligible axial conduction effect, the wall temperature h

w

decreases when

M

increases due to the Hartmann effect (Fig. 2), which corresponds to an acceleration of the flow near the wall by electromagnetic forces [5], but in some axial positions, nearly no effect of

M

is observed for c = 1, while for c = 0, a periodic inversion of the tendency appears in successive axial positions. Comparison between Figs. 2 and 3 shows that a reduction of the Hartmann effect is observed by decreasing the value of the Peclet number

Pe, and this is more important in the case

c = 0. It is also noted that by increasing the value of

M, a certain saturation is reached for which the effect of the magnetic field

on the wall temperature vanishes completely for any value of Peclet number, and the plots corre- sponding to

M

= 50 and

M

= 100 can be considered as very close to each other. Thus, the heat transfer reaches its saturation.

The variation of

M

has nearly no effect on the bulk temperature h

_b

for

Pe

= 150 (Fig. 2), while for small values of Peclet number

Pe

(Fig. 3), an effect, but very limited, occurs, corresponding to an increase of h

b

with

M

and the appearance of the periodic inversion of the tendency as for h

w

for

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2

ξ

0 1 2 3 4 5 6

θw, θb

Pe = 1.5, ω = 4 M = 0, 6, 50, 100

θ θ

w

b

γ = 1 γ = 0 M

M

M

Fig. 3. Bulk and wall temperatures with respect to dimensionless axial coordinatenfor various values of Hartmann numberM,x= 5,Pe= 150,c= 0 and 1.

c = 0. Note that in some axial positions, the wall temperature equals the bulk temperature only for the case c = 0. This results from the fact that the heat removed at the wall is equivalent to the heat appearing in the fluid. The saturation cited above for h

_w

appears also in the case of the bulk tem- perature h

_b

.

In Figs. 4 and 5 are represented the plots of the local Nusselt number

Nu

for

Pe

= 150 for x = 5 for

M

= 0, 6, 50 and 100 and for c = 1 and 0, respectively. Globally, one can see that the local Nusselt number

Nu

increases with the value of

M

in the whole interval considered due to the de- crease of h

_w

with

M

and the fact that h

_b

is insensible to

M

by considering the definition

Nu

¼

^4ðcþd_h ^sinxnÞ

whb

. Thus, an enhancement of heat transfer between the walls and the fluid with increasing Hartmann number is observed. At positions close to the inlet section and for c = 1, the local Nusselt number

Nu

is characterised by a fast decrease, which becomes more important as

M

rises (Fig. 4). In the fully developed regime, obtained for sufficiently large n, the local Nusselt number

Nu

is a continuous periodic function with period

^2p_x

(Fig. 4) and with minima situated at values of the dimensionless axial coordinate n ¼

_2x^3p

þ

^2kp_x ;

ðk 2

NÞ, which does not depend onM.

For the case c = 0, the local Nusselt number presents a regular shape in the whole interval, the period is now equal to

_x^p

(Fig. 5) and the principal particularity here is the appearance of singu- larities of

Nu

at values of the dimensionless coordinate

n ¼ arctg h

₂

ð1Þ

_2Pe^3x2

R1

0

h

₁

dg þ

_2x^3d

h

1

ð1Þ þ

_2Pe^3x2

R1 0

h

2

dg

!

þ

kp;

ðk 2

NÞ

0.0 0.4 0.8 1.2 1.6 2.0 2.4 2.8

ξ

0 5 10 15 20 25 30

Nu

M = 0 Pe = 150,ω = 5, γ = 1

M = 6 M = 50 M = 100

Fig. 4. Local Nusselt number with respect to dimensionless axial coordinatenfor various values of Hartmann number M,x= 5,Pe= 150, andc= 1.

0.00 0.40 0.80 1.20 1.60 2.00

ξ

-20 -10 0 10 20 30

Nu

Pe = 150, ω = 5, γ = 0

M = 6 M = 0 M = 50 M = 100

Fig. 5. Local Nusselt number with respect to dimensionless axial coordinatenfor various values of Hartmann number M,x= 5,Pe= 150, andc= 0.

0 5 10 15 20 25 30

ξ

5.0 7.5 10.0 12.5 15.0

Nuas

Pe = 150,γ = 0 PARAMETER : M

0 6 50 100

10

Fig. 6. Mean value of the asymptotic Nusselt number with respect to dimensionless frequencyxfor various values of Hartmann numberM,Pe= 150, andc= 0.

function of the parameters

M

and x through the terms h

₁

(1) and h

₂

(1), corresponding to the zeros of Eq. (41) (see Figs. 2 and 3 for c = 0), while the wall heat flux does not vanish (Eq. (40)).

The average thermal exchange between the walls and the fluid can be investigated for various

M

by following the evolution of the mean value of the asymptotic or fully developed Nusselt number

Nuas

with respect to dimensionless frequency x. The results are presented in Figs. 6 and 7, which refer to the case c = 0 for various values of Hartmann number for

Pe

= 150 and 1.5, respectively.

It can be seen that in a little domain of low frequency,

Nuas

increases with x for moderate values of

M

(M = 6, 10), while for the case of absence of a magnetic field (M = 0) or large

M

(M = 50, 100), a very slow variation is observed (Figs. 6 and 7). Then, by increasing x continu- ously, globally

Nuas

increases more significantly at

Pe

= 1.5 for the same domain of x considered.

A linear domain appears (represented only for

Pe

= 1.5) at a value x 5 for each

M. Thus, an

improvement of the average thermal exchange with increasing x is observed, especially for low Peclet numbers for all values of Hartmann number

M

in the case of vanishing mean value of the wall heat flow (c = 0). Note that it is found that this behaviour is very much reduced, i.e.

Nuas

presents a very slow variation with x (not represented here) when the wall heat flow has a non-zero mean value (c

5

0).

7. Conclusion

Thermally developing flow, including axial conduction and neglecting both viscous and Joule heating, has been investigated analytically for a laminar HartmannÕs flow in a semi-infinite

0 2 4 6 8 10

ω

5 10 15 20 25 30

Nuas

Pe = 1.5, γ = 0 PARAMETER : M

6 0 50 100

10

Fig. 7. Mean value of the asymptotic Nusselt number with respect to dimensionless frequencyxfor various values of Hartmann numberM,Pe= 150, andc= 0.

parallel plates channel subject to a transverse uniform magnetic field and with sinusoidal wall heat flux. The problem is solved by the method of linear operators that leads to a determination of a set of eigenvalues and their corresponding eigenfunctions which are here the MathieuÕs functions.

The expansion coefficients appearing in the entrance term of the solution are determined numer- ically by the means of the Gaussian elimination method.

The temperature profiles, as well as the Nusselt number, are given in terms of Peclet number, Hartmann number and dimensionless frequency. The principal numerical results presented in the present work show that for the case of non-vanishing mean value of the wall heat flux, the local Nusselt number increases with the value of Hartmann number and is dominated by the entrance term, characterized by a fast decrease for values of the dimensionless coordinate close to the inlet section, while for large values, the variation is periodic. For the case of vanishing mean value of the wall heat flux, the particularity here is the appearance of singularities of the Nusselt number at positions for which the wall and bulk temperatures are equal while the wall heat flux does not van- ish. The mean value of the asymptotic Nusselt number presents an important increase with the dimensionless frequency for each Hartmann number used, especially at low Peclet numbers, when the wall heat flux has a vanishing mean value. Thus, it is concluded that fluids with small Peclet numbers can be heated more efficiently by using sinusoidal wall heat fluxes with vanishing mean value, large magnetic fields and also dimensionless frequencies.

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