Ain Shams Engineering Journal (2015) xxx, xxx-xxx

Ain Shams University Ain Shams Engineering Journal

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ENGINEERING PHYSICS AND MATHEMATICS

MHD natural convection flow in a vertical micro-concentric-annuli in the presence of radial magnetic field: An exact solution

Basant K. Jhaa, Babatunde Ainab'*, Sani Isaa

a Department of Mathematics, Ahmadu Bello University, Zaria, Nigeria b Department of Mathematics, Bingham University, Abuja, Nigeria

Received 14 November 2014; revised 15 June 2015; accepted 19 July 2015

KEYWORDS

Micro-concentric annuli; Velocity slip; Temperature jump; Hartmann number; Velocity profile; Volume flow rate

Abstract This paper deals with a theoretical investigation of steady fully developed MHD natural convection flow of viscous, incompressible, electrically conducting fluid in micro-concntric-annuli in the presence of radial magnetic field. The velocity slip and temperature jump at the micro-concentric annuli surface are taken into account. Exact solutions are derived for energy and momentum equations under relevant boundary conditions. The solution obtained is graphically represented and the effects of various controlling parameters such as the radius ratio (g), Hartmann number (M), rarefaction parameter (bvKn), and fluid-wall interaction parameter (F) on the flow formation are discussed. The significant result from the study is that as rarefaction parameter (bvKn) increases the velocity slip on the surface of cylinders increases while fluid wall interaction parameter (F) decreases the velocity inside the micro-concntric-annuli. Furthermore, it is found that increase in radius ratio leads to increase in the volume flow rate.

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1. Introduction

One of the major themes in science and technology during the past half century has been miniaturization down to the micro- and nanoscale. The area of micro and nanofluidics is

* Corresponding author. Tel.: +234 8063178775.

E-mail addresses: basant777@yahoo.co.uk (B.K. Jha), ainavicdydx@

gmail.com (B. Aina), sanimath@yahoo.com (S. Isa).

Peer review under responsibility of Ain Shams University.

fundamentally important due to the necessity of understanding the nature of fluid flow at this scale [1]. Free-convective gas micro flow, encountered in many engineering fields, i.e., micro electrochemical cell transport, micro heat exchanging, and microchip cooling, is an attractive branch of micro fluidics [2] due to its reliability, simplicity, and cost effectiveness in flow and heat transfer mechanism. Chen and Weng [3] analytically studied the fully developed natural convection in open-ended vertical parallel plate micro-channel with asymmetric wall temperature distribution in which the effect of rarefaction and fluid wall interaction was shown to increase the volume flow rate and decrease the heat transfer. This result is further extended by taking into account suction/injection on the micro-channel walls by Jha et al. [4]. They concluded that skin

http://dx.doi.org/10.1016/j.asej.2015.07.010

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This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

Nomenclature

Ho constant strength of applied magnetic field U dimensionless axial velocity

F fluid-wall interaction parameter, bt/bv w k2-kl

g gravitational acceleration rt, ffv thermal and tangential momentum accommoda-

ki radius of the inner cylinder tion coefficients, respectively

k2 radius of the outer cylinder

Kn Knudsen number, k/w Greek letters

M Hartmann number b coefficient of thermal expansion

q volume flow rate b, bv dimensionless variables

Q dimensionless volume flow rate c ratio of specific heats (Cq/Cv)

Pr Prandtl number h dimensionless temperature

r dimensional radial coordinate q density

R dimensionless radial coordinate le magnetic permeability

R specific gas constant v fluid kinematic viscosity

T temperature of fluid g ratio of radii (kl/k2)

To reference temperature k molecular mean free path

Ti temperature at outer surface of the inner cylinder k thermal conductivity

u axial velocity r electrical conductivity of the fluid

frictions as well as the rate of heat transfer are strongly dependent on suction/injection parameter. In an another work, Weng and Chen [5] studied the impact of wall surface curvature on steady fully developed natural convection flow in an open-ended vertical micro-annulus with an asymmetric heating of annulus surface. Recently, Jha et al. [6] further extended the work of Weng and Chen [5] by taking into account suction/ injection on vertical annular micro-channel. In a related article, Avci and Aydin [7] studied the fully developed mixed convective heat transfer of a Newtonian fluid in a vertical micro-annulus between two concentric micro-tubes. Recently, Jha and Aina [8] further extended the work of Avci and Aydin [7] to the case when the vertical micro-annulus formed by two concentric micro-tubes is porous, i.e. where there is suction or injection of fluid through the annulus surfaces.

The annular geometry is widely employed in the field of heat exchangers. Its typical application is the gas cooled nuclear reactors in which the cylindrical fissionable fuel elements are placed axially in vertical coolant channel within the graphite moderators and the cooling gas is flowing along the channel parallel to the fuel elements [9]. The practical application of flow in annular space can also be found for example in drilling operation of oil and gas wells. The MHD flow problem in an annulus was first discussed by Globe [10] who considered fully developed laminar MHD flow in an annular channel. Jain and Mehta [11] extended the problem by imposing suction/injection on the walls. An exact solution of electrically conducting viscous incompressible flow in an annulus with porous wall under an external radial magnetic field was obtained by Nandi [12]. Antimirov and Kolyshkin [13] studied the unsteady magneto hydrodynamic flow in an annular channel with radial magnetic field while Takhar and Ali [14] examined the stability of MHD Couette flow in a narrow gap annulus. In a recent study, Jha et al. [15] investigated the influence of externally applied transverse magnetic field on steady natural convection flow of conducting fluid in a vertical micro-channel while Jha et al. [16] obtained an exact solution of steady fully developed natural convection flow of viscous, incompressible, electrically conducting fluid in a vertical

annular micro-channel with the effect of transverse magnetic field. Sheikholeslami et al. [17] investigated the magnetic field effect on nanofluid flow and heat transfer in a semi-annulus enclosure via control volume based finite element method. Khan and Ellahi [18] observed the effects of magnetic field and porous medium on some unidirectional flows of a second grade fluid. Farhad et al. [19] examined the slip effect on hydromagnetic rotating flow of viscous fluid through a porous space. In another work, Farhad et al. [20] investigated the effects of slip condition on the unsteady magnetohydrodynam-ics (MHD) flow of incompressible viscoelastic fluids in a porous channel under the influence of transverse magnetic field and Hall current with heat and mass transfer. An analysis to investigate the combined effects of heat and mass transfer on free convection unsteady magnetohydrodynamics (MHD) flow of viscous fluid embedded in a porous medium is presented by Farhad et al. [21].

Some recent works related to the present investigation are found in the literature [22-27]. Avci and Aydin [22] analyzed analytically the forced convection heat transfer in fully developed flows of viscous dissipating fluids in concentric annular ducts. In [23], the hydrodynamically and thermal fully developed flows of viscous dissipating gases in annular micro-duct between two concentric cylinders are analyzed analytically. In another work, El-Shaarawi and Al-Nimr [24] considered the fully developed natural convection in open-ended vertical concentric annuli. Analytical solutions for transient fully developed natural convection in open-ended vertical concentric annuli are presented by Al-Nimr [25]. Also, Al-Nimr [26] carried out analytical solutions for fully developed MHD natural-convection flow in open-ended vertical concentric porous annuli. Al-Nimr and Darabseh [27] presented the closed forms on transient fully developed free convection solutions, corresponding to four fundamental thermal boundary conditions in vertical concentric annuli.

The objective of this present work is to investigate the influence of wall surface curvature on steady fully developed MHD natural convection flow of viscous, incompressible, electrically conducting fluid in micro-concentric annuli under a radial

magnetic field. In fact, the present work is further extension of Weng and Chen [5] for conducting fluid in the presence of radial magnetic field.

2. Mathematical analysis

The flow considered is a fully developed steady (time independent) natural convection flows of viscous, incompressible, electrically conducting fluid in micro-concentric annuli under the effect of radial magnetic field. The x-axis is parallel to the gravitational acceleration g but in the opposite direction while the radii of the inner and outer cylinder walls are k1 and k2 respectively as presented in Fig. 1. Also a magnetic field is directed radial outward and is of the form H^, where H0 is constant strength of the applied magnetic field and r is the radial distance from the axis of the micro-annulus. Such form of magnetic field was first discussed by Globe [10]. It is assumed that the magnetic Reynolds number is very small, which corresponds to negligibly induced magnetic field compared to the externally applied one. The outer surface of the inner cylinder is heated to a temperature (T1) greater than that of the surrounding fluid having temperature (T0) and the inner surface of the outer cylinder is maintained at temperature (T0). Due to this temperature difference, natural convection occurs in the vertical micro-concntric-annuli. Since the flow is fully developed and cylinders are of infinite length, the flow depends only on radial coordinate (r).

Using the following dimensionless quantities

r — k1

w — k2 — ki, U — —, в —

gß(Tw — To)w2

PT = Kn — - F = ß k ' w' F ßm'

ki a 2 g — T' ßm — k2

ßt —

r c + 1 Pr

k — yj %RTo/2l M2 — rH2i2w2 P ' Pm '

Щкг

Figure 1 Flow configuration and coordinate system.

the mathematical model representing present physical situation in dimensionless form in the presence of velocity slip and temperature jump on micro-concntric-annuli surfaces is as follows:

[g +(1 — g)R] dR

[g + (1 — g)R] dR

[g+(i—g)R] dUR

[g+(i — g)R] dR

[g + (1 — g)R]2

г+в — o

The boundary conditions which describe velocity slip and temperature jump conditions at the fluid-wall interface are [5]

U(0)— ßKndUu

в(0) — 1 + ßKnF—

u(1)——ßKdR

n ' в(1) — —ßvKnFdR

The physical quantities used in the above equations are defined in the nomenclature.

By using the transformation Z = g + (1 — g)R, the Eqs. 25 can be transformed and written as follows:

1 d ZdZ

1 d Z~dZ

zdU dZ

(1 — g)2Z2 (1 — g)2

Subject to the boundary conditions

U(g) — ßvKn(1 — g)dZ

Z— g

U(1) — —ßmKn(1 — g)dZ

h(g) — 1 + ßvKnF(1 — g)—

,в(1) — —ß„KnF(1 — g)dZ

Integrating Eq. (7) and applying the boundary conditions (9) gives:

h(Z) = Ao + Ai ln(Z) (10)

where:

Ai =--1-;-v, Ao = —pvKnF(1 — g)Ai

ln(g) ßv KnF (1 — g) 1

The expression for the temperature in Eq. (10) is exactly the same as those given by Weng and Chen [5].

Substituting Eq. (10) into the momentum Eq. (6) and solving it using the boundary condition (8) gives

, , , -M t Z2[A0 + Ajln(Z)] 4Aj(1 - g)2Z2

U(Z) = C(Z)c g) + C2(Z) (■ g) - h [ 0 + ' ( - + -----J

[4(1 - g)2 - M2\ [4(1 - g)2 - M2]

Using the expression for velocity field, the expression for volume flow rate, and skin-frictions is obtained. The dimen-sionless volume flow rate is as follows:

2nw2uc (1 — g)2

ZU(Z)dZ

Tw — T o

Substituting Eq. (12) into Eq. (13), one obtain

Cifi C2(l -(g)

(1 - g)2

— + 2

(i-g) + 2

(2- -M.) (2 (i-g))

Ai(1 - g)2(1 - g4) [4(1 - g)2 - M2]2

[(4Ao - A1)(1 - g4) - 4g4 ln(g)A:]

[4(1 - g)2 - M2]

The skin-frictions (s) at the cylinder walls are obtained by differentiating the velocity as follows:

= (1 - g)

- C(g)-

S1 = dR

4A1 (1 - g)32g (1 - g)[2gAo + Af2gln(g)+g}]

S1 = M(C1 - C2

[4(1 - g)2 - M2]2 [4(1 - g)2 - M2]

(16) (17)

8A1 (1 - g)3 (1 - g)[2Ao + A1]

n \dU = (1 - g)dZ

[4(1 - g)2 - M2]2 [4(1 - g)2 - M2]

where Ci, C2,a0, ai, di,...,d14 are all constants given in Appendix A.

In order to verify the accuracy of the present work, we have computed the numerical value for the velocity for small value of M. Table 1 gives a comparison of the numerical values of

Table 1 Comparison of the values of velocity obtained in the

present work with those obtained by Weng and Chen [5].

g R Velocity (F = 1.64)

Weng and Chen [5] Present work (M ? 0)

o.2 o.2 0.05272817 0.05272817

o.4 0.05378645 0.05378645

o.6 0.04313245 0.04313245

o.8 0.02699288 0.02699288

o.5 o.2 0.06511190 0.06511190

o.4 0.07304118 0.07304118

o.6 0.06244542 0.06244542

o.8 0.04085297 0.04085297

o.8 o.2 0.07125042 0.07125042

o.4 0.08388324 0.08388324

o.6 0.07459799 0.07459799

o.8 0.05048406 0.05048406

the velocity obtained in the present work when M ? 0 with those obtained by Weng and Chen [5] for F = 1.64. As can be seen from Table 1, the solutions of the present work perfectly agree with those of Weng and Chen [5].

3. Results and discussion

In this study, the interactive effects of the radius ratio (g), Hartmann number (M), rarefaction parameter (pvKn), and fluid-wall interaction parameter (F) on heat and fluid flow are investigated. To examine the influence of these controlling parameters, the variations of velocity, volume flow rate and skin-friction are presented in Figs. 2-10. The present parametric study has been performed in the continuum and slip flow regimes (Kn 6 0.1). Also, for air and various surfaces, the values of bv and bt range from near 1 to 1.667 and from near 1.64 to more than 10, respectively. So, this study has been performed over the reasonable ranges of 0 6 bvKn 6 0.1 and

0 6 F 6 10. The selected reference values of pvKn, and F for the present analysis are 0.05 and 1.64 respectively as given in the study by Weng and Chen [5]. In addition, the values of Hartmann number (M) are taken over the range of

1 6 M 6 2 with a reference value of M = 2 while the selected reference values of g are 0.2, 0.5, 0.8.

The effects of the rarefaction parameter (pvKn), and fluid-wall interaction parameter (F) on the temperature profile and rate of heat transfer which is expressed as the Nusselt number are exactly the same as those given by Weng and Chen [5].

Fig. 2 shows the velocity distribution for different values of rarefaction parameter (bvKn) with fixed values of M = 2.0 and F = 1.64. It is observed from Fig. 2 that the increase in rarefaction parameter (pvKn) leads to increase in the velocity for large and moderate values of radius ratio (i.e. g = 0.8 and g = 0.5). However, it is noticed that for small value of radius ratio g = 0.2, increasing rarefaction parameter (pvKn) increases the velocity slips at the surfaces of cylinders while velocity decreases with increase of rarefaction parameter

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R

Figure 2 Velocity profile as a function R for different value bvKn for F = 1.667, M = 2.0.

Figure 3 Velocity profile as a function R for different value F for bvKn = 0.05, M = 2.0.

Figure 4 Velocity profile as a function R for different value M for bvKn = 0.05, F = 1.667.

(bvKn). This is due to the fact that as radius ratio (g) decreases or rarefaction parameter (pvKn) increases (in both cases) temperature gradient decreases which causes weak convection far from the surfaces of cylinders. This result is consistent with the findings of Weng and Chen [5]. The slip induced by rarefaction effect increases as radius ratio (g) increases. In addition, the numerical rate of velocity increase from no-slip (Kn = 0) to slip (Kn = 0.05) is 10.5% while from Kn = 0.05 to Kn = 0.1 is 7.5% at the point R = 0.4 for fixed value of g = 0.8.

Fig. 3 shows the velocity distribution for different values of fluid-wall interaction parameter (F) with fixed values of bvKn = 0.05 and M = 2.0. It is clearly seen from Fig. 3 that, the increase in fluid-wall interaction parameter (F) leads to the decrease in fluid velocity near the outer surface of inner

Figure 5 Volume flow rate versus bvKn for different values of F.

Figure 6 Volume flow rate versus bvKn for different values of M.

cylinder and increase in velocity slip near the inner surface of the outer cylinder. Also, it is observed that there exist points of inflection inside the micro-concntric-annuli where velocity field is independent of fluid wall interaction parameter and these strongly depend on radius ratio parameter (g). In addition, the slip induced by fluid-wall interaction parameter (F) increases as radius ratio parameter (g) decreases, although the impact of fluid-wall interaction parameter (F) on the slip is more visible for smaller radius ratio parameter (g).

Fig. 4 depicts the velocity distribution for different values of Hartmann number (M) with fixed values of pvKn = 0.05 and F = 1.64. It is observed that as Hartmann number (M) increases, there is decrease in the fluid velocity and it is found that the numerical rate of velocity decrease from M = 1.0 to M = 1.5 is 16% while from M = 1.5 to M = 2.0 is 20% at the point R = 0.4 for fixed value of g = 0.8. It is also observed that there is higher slip at outer surface of inner cylinder compared to inner surface of outer cylinder. In addition, for fixed

Figure 7 Variation of skin friction (s0) for different values of F (R = 0).

Figure 8 Variation of skin friction (s1) for different values of F

(R = 1).

Figure 9 Variation of skin friction (s0) for different values of M

(R = 0).

Figure 10 Variation of skin friction (s1) for different values of M

(R = 1).

Hartmann number (M), as radius ratio parameter (g) increases there is an increase in the velocity slip.

Fig. 5 shows the variation of volume flow rate (Q) against the rarefaction parameter (bvKn) for different values of fluid-wall interaction parameter (F). It is interesting to note that the volume flow rate (Q) is a decreasing function of fluid-wall interaction parameter (F). Furthermore, it is found that increase in radius ratio (g) and rarefaction parameter (bvKn) leads to increase in the volume flow rate. The reason behind it is as rarefaction parameter (bvKn) increases, the velocity slip at the cylindrical surfaces increases which reduces the retarding effect of the boundaries. This yields an observable increase in the volume flow rate.

Fig. 6 reveals the volume flow rate (Q) plotted against the rarefaction parameter (bvKn) for different values of Hartmann number (M). It is observed from Fig. 6 that, the volume flow rate (Q) decreases as Hartmann number (M) and rarefaction parameter (bvKn) increase. The physical fact behind it is that fluid velocity decreases as the Hartmann number (M) increases, which reduces the volume flow rate.

Fig. 7 depicts the variation of skin-friction at outer surface of inner cylinder (R = 0). It is clear from Fig. 7 that, the skin-friction decreases with the increase of fluid-wall interaction parameter (F) and rarefaction parameter (bvKn). This is physically true because as fluid-wall interaction parameter (F) increases velocity gradient decreases.

Fig. 8 shows variation of skin-friction at inner surface of outer cylinder (R = 1). It is obvious from Fig. 8 that increase in radius ratio (g), rarefaction parameter (bvKn) and fluid-wall interaction parameter (F) leads to the increase in the skin-friction at inner surface of outer cylinder. This is due to the fact that velocity gradient at the inner surface of outer cylinder increases as radius ratio (g), rarefaction parameter (pvKn) and fluid-wall interaction parameter (F) increases. It is interesting to note that the impact of fluid-wall interaction parameter (F) is significant for higher value of radius ratio (g).

Figs. 9 and 10 present variation of skin-friction at outer surface of inner cylinder (R = 0) and inner surface of outer cylinder (R = 1), respectively for different values of Hartmann number (M). It is observed from these Figures that, skin friction decreases as Hartmann number (M) increases. The physical fact behind it is that fluid velocity decreases as the Hartmann number (M) increases.

4. Conclusions

Exact solution is obtained for steady fully developed natural convection flow of viscous, incompressible, electrically conducting fluid in vertical micro-concntric-annuli in the presence of radial magnetic field. The influence of the radius ratio (g), Hartmann number (M), rarefaction parameter (bvKn), and fluid-wall interaction parameter (F) on the fluid velocity, volume flow rate and skin-friction is analyzed. It is worthy to note that the results for M ? 0 perfectly agree with the results of Weng and Chen [5]. The main conclusions of present work are as follows:

I. It is found that as Hartmann number (M) increases, there is a decrease in the fluid velocity.

II. The slip induced by fluid-wall interaction parameter (F) increases as radius ratio (g) decreases.

III. The increase in radius ratio (g) and rarefaction parameter (bvKn) leads to increase in the volume flow rate.

IV. The increase in Hartmann number (M) leads to decrease in the volume flow rate.

V. Finally, the skin friction decreases as Hartmann number (M) increases.

Appendix A

Constants used in the present work.

[d11d13 - d14d1o]

a1 = -

[d9d13 - d1od12]

[dnd12 — d^dp] [d1od12 - d9d13] '

4A1(1 - g)2 [4(1 - g)2 - M2]2

[4(1 - g)2 - M2] '

d1 = -

8A1g(1 - g)3 (1 - g)[2gAo + A1g(2In(g) + 1)]

8A1(1 - g)3 _ (1 - g)[2Ao + A1] [4(1 - g)2 - M2]2 [4(1 - g)2 - M2] '

g2[Ao + A1In(g)] 4A1g2(1 - g)2

[4(1 - g)2 - M2] [4(1 - g)2 - M2]'

4A1 (1 - g)2

[4(1 - g)2 - M2] [4(1 - g)2 - M2]2

/ , M d5 = (g)(1-g>,

d6 = (g) (1-g),

, ■ M 1 d7 = (g)1^ 1,

d8 = (g)

IT-g)+

[4(1 - g)2 - M2]'

[4(1 - g)2 - M2]

d9 = d5 — ßmKnMd7, dio = d6 + ßm KnMdg, d11 = ßvKnd1 — d3, d12 = 1 + ßmKnM, di3 = 1 — ßmKnM, d14 = —d4 — ßmKnd2,

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Basant K. Jha was born in Darbhanga, Bihar, India, in 1964. He is a professor in Applied Mathematics at Ahmadu Bello University, Zaria, Nigeria. He was awarded Ph.D. degree in Mathematics from Banaras Hindu University, Varanasi, India, in 1992. He has published more than 140 papers in reputed National/International journals. His research interest includes flow through porous media, magneto hydrodynamics, computational fluid dynamics, and heat and mass transfer.

Babatunde Aina completed his master degree from Ahmadu Bello University, Zaria, Nigeria. He is pursuing Ph.D. in the field of Applied Mathematics at Ahmadu Bello University, Zaria, Nigeria. He is currently working in Bingham University, Abuja, Nigeria. He has a teaching experience of 3 years and a research experience of 5 years.

Sani Isa is pursuing Ph.D. in the field of Applied Mathematics at Ahmadu Bello University, Zaria, Nigeria. He has a teaching experience of 15 years and a research experience of 5 years.