- Email: [email protected]
Free convection in a trapezoidal cavity filled with a micropolar fluid
International Journal of Heat and Mass Transfer 99 (2016) 831–838
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Free convection in a trapezoidal cavity filled with a micropolar fluid Nikita S. Gibanov a, Mikhail A. Sheremet a,b,⇑, Ioan Pop c a
Laboratory on Convective Heat and Mass Transfer, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Applied Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b
a r t i c l e
i n f o
Article history: Received 27 February 2016 Received in revised form 13 April 2016 Accepted 19 April 2016 Available online 29 April 2016 Keywords: Free convection Micropolar fluid Trapezoidal cavity Numerical results
a b s t r a c t The present investigation deals with the study of steady laminar natural convective flow and heat transfer of micropolar fluids in a trapezoidal cavity. The bottom wall of the cavity is kept at high constant temperature, the inclined walls is kept at low constant temperatures while the top horizontal wall is adiabatic. Governing equations formulated in dimensionless stream function and vorticity variables has been solved by finite difference method of the second order accuracy. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. Computations have been carried out to analyze the effects of Rayleigh number, Prandtl number and vortex viscosity parameter both for weak and strong concentration cases. Obtained results have been presented in the form of streamlines, isotherms and vorticity profiles as well as the variation of the average Nusselt number and fluid flow rate. It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Micropolar fluid is a subject of microphoric fluid theory. The detailed description and the modeling of micropolar fluids were initially introduced by Eringen [1–3]. This theory has become very important to engineers and scientists working with hydrodynamic-fluid problems and phenomena for the last few decades. The potential importance of micropolar fluids in industrial applications has motivated many researchers to extent the study in numerous ways to include various physical effects. The essence of the theory of micropolar fluid flow lies in the extension of the constitutive equations for Newtonian fluids so that more complex fluids such as particle suspensions, liquid crystals, animal blood, lubrication, colloidal suspensions, turbulent shear flows, etc. can be described by this theory. In practice, the theory of micropolar fluids requires that one must add a transport equation representing the principle of conservation of local angular momentum to the usual transport equations for the conservation of mass and momentum, and additional local constitutive parameters are also introduced. The special features of micropolar fluids were discussed in two comprehensive review papers of the subject and application of this theory by Ariman et al. [4,5] and in the books by Eringen [6] and Łukasewicz [7]. ⇑ Corresponding author at: Laboratory on Convective Heat and Mass Transfer, Tomsk State University, 634050 Tomsk, Russia. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.04.056 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
The convective motion driven by the buoyancy forces is a well-known natural phenomenon and has attracted interest of many researchers. In particular, the topic of natural convection in cavities has received much attention in the past since many practical heat transfer situations can be modeled as flows in cavities. There have been numerous investigations of natural convective heat transfer that occurs in an enclosure (Sathiyamoorthy et al. [8], Kandaswamy and Nithyadevi [9], etc.). Sathiyamoorthy et al. [8] presented the numerical study of steady natural convection in a closed square cavity under different boundary conditions. They showed that for small Rayleigh numbers the average Nusselt number was almost constant due to heat conduction and increased steadily as Ra increased. Natural convection heat transfer and fluid flow were studied for trapezoidal enclosures filled with a viscous (Newtonian fluid) mostly at differentially heated temperature boundary conditions, see Moukalled and Darwish [10,11], Boussaid et al. [12], Moukalled and Darwish [13], Kuyper and Hoogendoorn [14], Sadat and Salagnac [15]. Analysis of convective heat transfer and fluid flow of micropolar fluid in a vertical channel, lid-driven cavity and square cavity has been conducted in [16–19]. We mention also to this end, the paper by Hsu and Hong [20] on natural convection of micropolar fluids in an open cavity, which consists by two adiabatic horizontal walls and one heated vertical wall, while the open end has several different geometric features. However, to our best knowledge, trapezoidal enclosures filled with a micropolar fluid
832
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Nomenclature g H j K L N N
gravitational acceleration height of the cavity micro-inertia density, j = L2 vortex viscosity parameter, K ¼ j=l length of the bottom hot wall dimensionless microrotation dimensional microrotation local Nusselt number average Nusselt number pressure Prandtl number, Pr ¼ m=a Rayleigh number, Ra ¼ gbðT h T c ÞL3 =ðamÞ temperature of the fluid temperature of the cooled walls temperature of the hot wall coordidimensional velocity components along x and y nates dimensionless velocity components along x and y coordinates
Nu Nu p Pr Ra T Tc Th ; v u u,
v
have not been considered yet. Therefore, the main objective of this paper is to examine the natural convection in a trapezoidal cavity filled with a micropolar fluid. Streamlines, isotherms, average Nusselt number and fluid flow rate are presented and discussed in details.
x y x, y
dimensional coordinate measured along the bottom wall of the cavity dimensional coordinate measured along the vertical direction of the cavity dimensionless Cartesian coordinates
Greek symbols thermal diffusivity b volumetric expansion coefficient of the fluid c spin-gradient viscosity, c ¼ l þ j2 j h dimensionless temperature j vortex viscosity l dynamic viscosity q fluid density u inclination angle of the inclined cooled walls w dimensionless stream function x dimensionless vorticity
a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = gbðT T c ÞL; v ¼ v = gbðT T c ÞL; u¼u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ ðT T c Þ=ðT h T c Þ; N ¼ N= gbðT T c Þ=L
x ¼ x=L;
=L; y¼y
ð6Þ and also stream function w u ¼ @w ; v ¼ @w and vorticity @y @x
x ¼ @@xv @u . Therefore the governing Eqs. (1)–(5) using the dimen@y
2. Basic equations
sionless variables (6) can be written as follows The domain of interest filled with a micropolar fluid is presented in Fig. 1 with dimensional Cartesian coordinates x and . The trapezoidal enclosure is bounded by isothermal inclined y cooled walls of temperature Tc, isothermal bottom hot wall of temperature Th (Th > Tc) and adiabatic top wall. All four walls of the cavity are assumed to be rigid and impermeable. The micropolar fluid is considered to satisfy the Boussinesq approximation and the flow regime is laminar. Taking into account the theory of Eringen [1–3] for the micropolar fluid flow the governing equations can be written in dimensional Cartesian coordinates as follows
@ v @u þ ¼0 @ x @ y
ð1Þ
! @2u @u @u @p @2u @N q u þ v ¼ þ ðl þ jÞ 2 þ 2 þ j @x @y @x @y @x @y
@ v
@ v
@ 2 v
@p
@ 2 v
!
ð2Þ
@N
! @2N @N @N @2N @ v @ u qj u þ v ¼ c 2 þ 2 þ j 2jN @x @y @x @y @x @y @T @T @2T @2T þ v ¼a þ 2 u @ x @y @ x2 @ y
ð7Þ
! rffiffiffiffiffiffi 2 @w @ x @w @ x Pr @ x @ 2 x þ ¼ ð1 þ KÞ @y @x @x @y Ra @x2 @y2 ! rffiffiffiffiffiffi 2 Pr @ N @ 2 N @h þ K þ Ra @x2 @y2 @x
ð8Þ
! rffiffiffiffiffiffi rffiffiffiffiffiffi 2 @w @N @w @N K Pr @ N @ 2 N Pr þK ¼ 1þ ðx 2NÞ þ @y @x @x @y 2 Ra @x2 @y2 Ra ð9Þ
q u þ v ¼ þ ðl þ jÞ 2 þ 2 j @y @x @x @y @x @y þ qbðT T c Þg
@2w @2w þ ¼ x @x2 @y2
@w @h @w @h 1 @2h @2h þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @y @x @x @y Ra Pr @x2 @y2
! ð10Þ
with the following boundary conditions
ð3Þ
ð4Þ
! ð5Þ
In order to analyze the fluid flow and heat transfer in general scale we introduce the following dimensionless variables
w ¼ 0;
@2w @2w ; @x2 @y2 h ¼ 0 on left and right inclined walls
x¼
N ¼ n x; w ¼ 0;
x¼
@2w ; @y2
N ¼ n x;
h ¼ 1 on bottom wall
w ¼ 0;
x¼
@2w ; @y2
N ¼ n x;
@h ¼ 0 on top wall @y
ð11Þ
Here nð0 6 n 6 1Þ is a micro-gyration parameter with n = 0 corresponding to the case where the particle density is sufficiently great that microelements close to the wall are unable to rotate
833
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Fig. 1. Physical model and coordinate system (a), and utilized finite-difference grid (b).
(strong concentration), n = 0.5 is indicative of weak concentration, and when n = 1, it corresponds to turbulent flow (see Rees and Bassom [21]). The physical quantities of interest are the local Nusselt number Nu along the hot wall and average Nusselt number Nu, that are defined as
N, h, see Eq. (11)). Distances between nodes are given by hx and hy in the x- and y-directions, respectively. It should be noted that for definition of vorticity along all rigid walls we used Jenson’s polynomial [27]. For example, for the left inclined wall the vorticity has been found in following form:
@h Nu ¼ ; @y y¼0
xNyj;j ¼
Z
1þd=L
Nu ¼
Nu dx
wNyjþ2;j 8wNyjþ1;j
ð12Þ
d=L
3. Numerical method The partial differential Eqs. (7)–(10) with corresponding boundary conditions (11) were solved using the finite difference method with the second order central differencing schemes. The diffusive terms have been approximated by central differences. The convective terms have been discretized applying the second order Samarksii monotonic difference scheme. The parabolic equations have been solved on the basis of Samarskii locally onedimensional scheme (see Sheremet and Pop [22] and Sheremet et al. [23,24]). The obtained systems of algebraic equations have been solved by Thomas algorithm. The partial differential equation for the stream function (7) has been discretized by means of the five-point difference scheme on the basis of central differences for the partial derivatives. The obtained linear discretized equation has been solved by the successive over relaxation method. The developed numerical technique has been verified comprehensively (see Sheremet and Pop [22] and Sheremet et al. [23,24]). A regular grid distribution is used as shown in Fig. 1b. The inclined walls were approximated with staircase-like zigzag lines. This technique is based on the earlier studies [25,26]. In Fig. 1b, the bold nodes are coincided with the inclined boundary of the cavity and they have Dirichlet boundary conditions for all functions (w, x,
2 2hx
þ
wNyj;jþ2 8wNyj;jþ1 2
2hy
where Ny + 1 is a number of nodes along y-axis. The present models, in the form of an in-house computational fluid dynamics (CFD) code, have been validated successfully against the works of Aydin and Pop [28] and Zadravec et al. [29] for the steady-state natural convection of micropolar fluid in a square cavity with isothermal vertical and adiabatic horizontal walls. Table 1 shows the values of the average Nusselt number computed for various Rayleigh numbers and vortex viscosity parameter in comparison with other authors. The utilized numerical technique also has been validated successfully against the commercial fluid dynamics software (FLUENT) for steady-state natural convection of clear fluid in a trapezoidal cavity (Fig. 1) for Pr = 0.7, Ra = 105 and Ra = 106, K = 0,
Table 1 Comparison of the average Nusselt number of the hot wall for n = 0. Ra
K
Aydin and Pop [28]
Zadravec et al. [29]
Present results
104
0 0.5 2.0 0 0.5 2.0 0 0.5 2.0
2.234 1.947 1.545 4.486 4.033 3.314 8.945 7.984 6.673
2.263 1.986 1.578 4.540 4.067 3.377 8.742 8.229 6.714
2.245 1.977 1.566 4.529 4.081 3.348 8.836 8.034 6.700
105
106
834
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Fig. 2. Streamlines w and isotherms h at Ra = 105: FLUENT data (a), in-house computational code data (b).
Fig. 3. Streamlines w and isotherms h at Ra = 106: FLUENT data (a), in-house computational code data (b).
We have conducted also the grid independent test, analyzing the steady-state natural convection in a trapezoidal cavity filled with a micropolar fluid at Ra = 105, Pr = 7.0, K = 0.5, n = 0.5, u = p/4. Three cases of the uniform grid are tested: a grid of 240 80 points, a grid of 300 100 points, and a much finer grid of 450 150 points. Fig. 5 shows the effect of the mesh parameters on the profiles of temperature and vertical velocity along crosssection y = 0.5. Taking into account the conducted verifications the uniform grid of 300 100 points has been selected for the further investigation. 4. Results and discussion
Fig. 4. Profiles of local Nusselt number along the hot bottom wall for Ra = 105 in comparison with numerical data of Sathiyamoorthy et al. [8]
u = p/4. Figs. 2 and 3 show computed streamlines and isotherms using the developed computational code and commercial FLUENT software. Fig. 4 shows a good comparison of the local Nusselt number along the hot bottom wall with numerical data of Sathiyamoorthy et al. [8] for the problem presented in Fig. 2 at Ra = 105.
Numerical analysis has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 104, 105, 106), Prandtl number (Pr = 0.01, 0.1, 0.7, 7.0), the vortex viscosity parameter (K = 0, 0.1, 0.5, 2.0), micro-gyration parameter (n = 0, 0.5), u = p/4 and aspect ratio L/H = 1. Particular efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms, vorticity profiles, local and average Nusselt number, fluid flow rate for different values of governing parameters mentioned above are illustrated in Figs. 6–12. Regardless of the Rayleigh number values two convective cells are formed inside the cavity illustrating an appearance of ascending flows in the central part of the cavity and descending flows close to the cold inclined walls (Figs. 6 and 7). Such fluid flow behavior characterizes an evolution of the thermal plume over the hot wall. An increase in Ra leads to an intensification of both
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
835
Fig. 5. Profiles of vertical velocity (a) and temperature (b) along the middle cross-section y = 0.5 for different mesh parameters.
Fig. 6. Streamlines w, isotherms h and vorticity contours x, for Pr = 0.7, n = 0, K = 2: Ra = 104 a, Ra = 106 b.
Fig. 7. Streamlines w, isotherms h and vorticity contours x, for Ra = 105, Pr = 0.7, n = 0: K = 0 a, K = 2 b.
convective flow with a displacement of the convective cell cores along the vertical axis to the upper part of the cavity and heat transfer with a decrease in the thermal layers thickness near the inclined cold walls. Also for high values of the Rayleigh number a thickness of the thermal plume decreases. It should be noted that vorticity contours reflect a distribution of hydrodynamic disturbance from the rigid walls into the cavity. A high density of vorticity isolines near the walls illustrates a formation of velocity boundary layers. In the case of Ra = 105 (Fig. 7) location of vorticity isolines in the central part of the cavity characterizes an evolution of the thermal plume. An increase in the vortex viscosity parameter reflects an increase in the fluid viscosity (see Eqs. (8) and (9)) that leads to
an attenuation of the convective flow and heat transfer. For example, in the case of Ra = 105 one can find a downward displacement of the convective cell cores with more thick thermal plume for K = 2 (Fig. 7b). Distribution of the vorticity isolines illustrates a decrease in the convective circulations intensity. In the case of weak concentration (n = 0.5) for Ra = 105, Pr = 0.7, K = 2 one can find small displacement of the convective cells cores from the vertical centerline in opposite sides along horizontal axis (Fig. 8). At the same time an increase in micro-gyration parameter leads to both less intensive heating of the cavity and nonsignificant changes of the vorticity. Figs. 7b and 9 show the effect of Prandtl number on contours of stream function, temperature and linear vorticity for Ra = 105,
836
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Fig. 8. Streamlines w, isotherms h and vorticity contours x, for Ra = 105, Pr = 0.7, K = 2, n = 0.5.
Fig. 9. Streamlines w, isotherms h and vorticity contours x, for Ra = 105, K = 2, n = 0: Pr = 0.1 (a), Pr = 7.0 (b).
Fig. 10. Profiles of local Nusselt number along the hot wall: for different values of Ra at Pr = 0.7, K = 0.5, n = 0 (a) and for different values of K at Ra = 105, Pr = 0.7, n = 0 (b).
K = 2, n = 0. An increase in Prandtl number leads to following changes in profiles of w, h and x: a displacement of the convective cells cores along the vertical direction close to the centerline and less intensive heating of the cavity. The latter reflects an increase in the temperature differences inside the cavity. Fig. 10 shows profiles of the local Nusselt number along the hot wall for different values of the Rayleigh number and vortex viscosity parameter. Regardless of the Rayleigh numbers the local Nusselt number has maximum values at the ends of the hot wall and minimum value at the center of this wall. An increase in x-coordinate from 1.0 to 1.5 and a decrease in x-coordinate from 2.0 to 1.5 lead to a decrease in Nu along the wall that can be explained by a formation of thermal layers over the hot wall with the origin of these thermal layers at the wall ends. It should be noted that these thermal boundary layers widen to the center of the wall and interflow into one thermal plume that leads to a decrease in the local Nusselt number from the left and right wall ends to the center. An increase in Ra leads to an increase in the local Nusselt number. Weak increase in Nu with Ra at x = 1.5 in comparison with other points can be explained by more intensive
convective upward motion in the central part of the cavity that leads to an essential heating of this part and decrease in the temperature gradient. An increase in the vortex viscosity parameter from K = 0 to K = 2 (Fig. 10b) for Ra = 105 leads to a decrease in the local Nusselt number mainly in the sections 1.05 < x < 1.3 and 1.7 < x < 1.95 where the boundary layers evolve. Fig. 11 demonstrates variations of the average Nusselt number at the hot wall and fluid flow rate with Ra and K. As has been mentioned above an increase in the vortex viscosity parameter leads to a decrease in the heat transfer rate and fluid flow intensity that is caused by a motion of more viscous fluid in comparison with Newtonian case (K = 0). Also for high values of Ra (Ra = 106) an increase in K leads to a weak attenuation of the convective flow, because a convective flow intensity for the constant value of the Rayleigh number is defined also by the geometry of the cavity. The effect of the Prandtl number on the heat transfer rate and fluid flow intensity is presented in Fig. 12. An increase in Pr leads to a heat transfer enhancement and intensification of fluid flow. Attenuation of the convective flow and heat transfer with the
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
837
Fig. 11. Variations of the average Nusselt number along the hot wall (a) and maximum absolute value of the stream function (b) with Rayleigh number and vortex viscosity parameter for Pr = 0.7, n = 0.
Fig. 12. Variations of the average Nusselt number along the hot wall (a) and maximum absolute value of the stream function (b) with Prandtl number and vortex viscosity parameter for Ra = 105, n = 0.
vortex viscosity parameter is more essential for high values of the Prandtl number. It is worth noting that for high values of K an increase in Pr leads to less intensive increase in Nu and jwjmax . At the same time for low values of the Prandtl number a variation of K does not lead to essential changes in the heat transfer rate. 5. Conclusions Free convection of a micropolar fluid in a trapezoidal cavity with cold inclined walls and hot bottom wall has been studied. Analysis of the fluid flow and heat transfer has been conducted in a wide range of the Rayleigh number, Prandtl number and vortex viscosity parameter On the basis of the obtained results we can conclude that 1. There are two convective cells inside the cavity for any value of Ra. The average Nusselt number and fluid flow rate are the increasing functions of Ra. 2. An increase in the vortex viscosity parameter illustrates the heat transfer reduction and attenuation of convective flow. An increase in K leads to a weak attenuation of the convective flow for high values of Ra (Ra = 106).
3. An increase in the Prandtl number leads to the heat transfer enhancement and fluid flow intensification. At the same time for low values of the Prandtl number a variation of K does not lead to essential changes in the heat transfer rate 4. An increase in the micro-gyration parameter from 0 to 0.5 does not lead to modification of the analyzed parameters for the considered range of Ra, Pr and K.
Acknowledgement This work of Nikita S. Gibanov and Mikhail A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/ K. The authors also wish to express their thank to the very competent Reviewers for the valuable comments and suggestions. References [1] A.C. Eringen, Simple microfluids, Int. J. Eng. Sci. 2 (1964) 205–217. [2] A.C. Eringen, Theory of micropolar fluid, J. Math. Mech. 16 (1966) 1–18. [3] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972) 480– 496.
838
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
[4] T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics – a review, Int. J. Eng. Sci. 11 (1973) 905–930. [5] T. Ariman, M.A. Turk, N.D. Sylvester, Applications of microcontinuum fluid mechanics, Int. J. Eng. Sci. 12 (1974) 273–293. [6] A.C. Eringen, Microcontinuum Field Theories, vol. I,II, Springer, New York, 2001. [7] G. Łukaszewicz, Micropolar Fluids: Theory and Application, Birkhäuser, Basel, 1999. [8] M. Sathiyamoorthy, T. Basak, S. Roy, I. Pop, Steady natural convection flows in a square cavity with linearly heated side walls, Int. J. Heat Mass Transfer 50 (2007) 766–775. [9] P. Kandaswamy, M. Nithyadevi, Buoyancy driven convection of water near its density maximum with partially active vertical walls, Int. J. Heat Mass Transfer 50 (2007) 942–948. [10] F. Moukalled, M. Darwish, Natural convection in a partitioned trapezoidal cavity heated from the side, Numer. Heat Transfer, Part A 43 (2003) 543–563. [11] F. Moukalled, M. Darwish, Natural convection in a trapezoidal enclosure heated from the side with a baffle mounted on its upper inclined surface, Heat Transfer Eng. 25 (2004) 80–93. [12] M. Boussaid, A. Djerrada, M. Bouhadef, Thermosolutal transfer within trapezoidal cavity, Numer. Heat Transfer, Part A 43 (2003) 431–488. [13] F. Moukalled, M. Darwish, Buoyancy-induced heat transfer in a trapezoidal enclosure with offset baffles, Numer. Heat Transfer, Part A 54 (2007) 337–355. [14] R.A. Kuyper, C.J. Hoogendoorn, Laminar natural convection flow in trapezoidal enclosures, Numer. Heat Transfer, Part A 28 (1995) 55–67. [15] H. Sadat, P. Salagnac, Further results for laminar natural convection in a two dimensional trapezoidal enclosure, Numer. Heat Transfer, Part A 27 (1995) 451–459. [16] A.J. Chamkha, T. Grosan, I. Pop, Fully developed free convection of a micropolar fluid in a vertical channel, Int. Commun. Heat Mass Transfer 29 (2002) 1119– 1127. [17] A.J. Chamkha, T. Grosan, I. Pop, Fully developed mixed convection of a micropolar fluid in a vertical channel, Int. J. Fluid Mech. Res. 30 (2003) 251– 263.
[18] A.J. Chamkha, M.A. Mansour, S.E. Ahmed, Unsteady mixed convection of a micropolar fluid in a lid-driven cavity: effects of different micro-gyration boundary conditions, Int. J. Energy Technol. 2 (2010) 1–10. [19] S.K. Jena, L.K. Malla, S.K. Mahapatra, A.J. Chamkha, Transient buoyancyopposed double diffusive convection of micropolar fluids in a square enclosure, Int. J. Heat Mass Transfer 81 (2014) 681–694. [20] T.H. Hsu, K.Y. Hong, Natural convection of micropolar fluids in an open cavity, Numer. Heat Transfer, Part A 50 (2006) 281–300. [21] D.A.S. Rees, A.P. Bassom, The blasius boundary-layer flow of a micropolar fluid, Int. J. Eng. Sci. 34 (1996) 113–124. [22] M.A. Sheremet, I. Pop, Thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms, Transp. Porous Media 103 (2014) 191–205. [23] M.A. Sheremet, T. Grosan, I. Pop, Free convection in shallow and slender porous cavities filled by a nanofluid using Buongiorno’s model, ASME J. Heat Transfer 136 (2014) 082501. [24] M.A. Sheremet, T. Grosan, I. Pop, Free convection in a square cavity filled with a porous medium saturated by nanofluid using Tiwari and Das’ nanofluid model, Transp. Porous Media 106 (2015) 595–610. [25] P.M. Haese, M.D. Teubner, Heat exchange in an attic space, Int. J. Heat Mass Transfer 45 (2002) 4925–4936. [26] Y. Varol, H.F. Oztop, I. Pop, Maximum density effects on buoyancy-driven convection in a porous trapezoidal cavity, Int. Commun. Heat Mass Transfer 37 (2010) 401–409. [27] V.G. Jenson, Viscous flow around a sphere at low Reynolds numbers (640), Proc. R. Soc. London, Ser. A 249 (1959) 346–366. [28] O. Aydin, I. Pop, Natural convection in a differentially heated enclosure filled with a micropolar fluid, Int. J. Therm. Sci. 46 (207) (2007) 963–969. [29] M. Zadravec, M. Hribersek, L. Skerget, Natural convection of micropolar fluid in an enclosure with boundary element method, Eng. Anal. Boundary Elem. 33 (2009) 485–492.
Contents lists available at ScienceDirect
International Journal of Heat and Mass Transfer journal homepage: www.elsevier.com/locate/ijhmt
Free convection in a trapezoidal cavity filled with a micropolar fluid Nikita S. Gibanov a, Mikhail A. Sheremet a,b,⇑, Ioan Pop c a
Laboratory on Convective Heat and Mass Transfer, Tomsk State University, 634050 Tomsk, Russia Institute of Power Engineering, Tomsk Polytechnic University, 634050 Tomsk, Russia c Department of Applied Mathematics, Babesß-Bolyai University, 400084 Cluj-Napoca, Romania b
a r t i c l e
i n f o
Article history: Received 27 February 2016 Received in revised form 13 April 2016 Accepted 19 April 2016 Available online 29 April 2016 Keywords: Free convection Micropolar fluid Trapezoidal cavity Numerical results
a b s t r a c t The present investigation deals with the study of steady laminar natural convective flow and heat transfer of micropolar fluids in a trapezoidal cavity. The bottom wall of the cavity is kept at high constant temperature, the inclined walls is kept at low constant temperatures while the top horizontal wall is adiabatic. Governing equations formulated in dimensionless stream function and vorticity variables has been solved by finite difference method of the second order accuracy. Comprehensive verification of the utilized numerical method and mathematical model has shown a good agreement with numerical data of other authors. Computations have been carried out to analyze the effects of Rayleigh number, Prandtl number and vortex viscosity parameter both for weak and strong concentration cases. Obtained results have been presented in the form of streamlines, isotherms and vorticity profiles as well as the variation of the average Nusselt number and fluid flow rate. It has been shown that an increase in the vortex viscosity parameter leads to attenuation of the convective flow and heat transfer inside the cavity. Ó 2016 Elsevier Ltd. All rights reserved.
1. Introduction Micropolar fluid is a subject of microphoric fluid theory. The detailed description and the modeling of micropolar fluids were initially introduced by Eringen [1–3]. This theory has become very important to engineers and scientists working with hydrodynamic-fluid problems and phenomena for the last few decades. The potential importance of micropolar fluids in industrial applications has motivated many researchers to extent the study in numerous ways to include various physical effects. The essence of the theory of micropolar fluid flow lies in the extension of the constitutive equations for Newtonian fluids so that more complex fluids such as particle suspensions, liquid crystals, animal blood, lubrication, colloidal suspensions, turbulent shear flows, etc. can be described by this theory. In practice, the theory of micropolar fluids requires that one must add a transport equation representing the principle of conservation of local angular momentum to the usual transport equations for the conservation of mass and momentum, and additional local constitutive parameters are also introduced. The special features of micropolar fluids were discussed in two comprehensive review papers of the subject and application of this theory by Ariman et al. [4,5] and in the books by Eringen [6] and Łukasewicz [7]. ⇑ Corresponding author at: Laboratory on Convective Heat and Mass Transfer, Tomsk State University, 634050 Tomsk, Russia. http://dx.doi.org/10.1016/j.ijheatmasstransfer.2016.04.056 0017-9310/Ó 2016 Elsevier Ltd. All rights reserved.
The convective motion driven by the buoyancy forces is a well-known natural phenomenon and has attracted interest of many researchers. In particular, the topic of natural convection in cavities has received much attention in the past since many practical heat transfer situations can be modeled as flows in cavities. There have been numerous investigations of natural convective heat transfer that occurs in an enclosure (Sathiyamoorthy et al. [8], Kandaswamy and Nithyadevi [9], etc.). Sathiyamoorthy et al. [8] presented the numerical study of steady natural convection in a closed square cavity under different boundary conditions. They showed that for small Rayleigh numbers the average Nusselt number was almost constant due to heat conduction and increased steadily as Ra increased. Natural convection heat transfer and fluid flow were studied for trapezoidal enclosures filled with a viscous (Newtonian fluid) mostly at differentially heated temperature boundary conditions, see Moukalled and Darwish [10,11], Boussaid et al. [12], Moukalled and Darwish [13], Kuyper and Hoogendoorn [14], Sadat and Salagnac [15]. Analysis of convective heat transfer and fluid flow of micropolar fluid in a vertical channel, lid-driven cavity and square cavity has been conducted in [16–19]. We mention also to this end, the paper by Hsu and Hong [20] on natural convection of micropolar fluids in an open cavity, which consists by two adiabatic horizontal walls and one heated vertical wall, while the open end has several different geometric features. However, to our best knowledge, trapezoidal enclosures filled with a micropolar fluid
832
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Nomenclature g H j K L N N
gravitational acceleration height of the cavity micro-inertia density, j = L2 vortex viscosity parameter, K ¼ j=l length of the bottom hot wall dimensionless microrotation dimensional microrotation local Nusselt number average Nusselt number pressure Prandtl number, Pr ¼ m=a Rayleigh number, Ra ¼ gbðT h T c ÞL3 =ðamÞ temperature of the fluid temperature of the cooled walls temperature of the hot wall coordidimensional velocity components along x and y nates dimensionless velocity components along x and y coordinates
Nu Nu p Pr Ra T Tc Th ; v u u,
v
have not been considered yet. Therefore, the main objective of this paper is to examine the natural convection in a trapezoidal cavity filled with a micropolar fluid. Streamlines, isotherms, average Nusselt number and fluid flow rate are presented and discussed in details.
x y x, y
dimensional coordinate measured along the bottom wall of the cavity dimensional coordinate measured along the vertical direction of the cavity dimensionless Cartesian coordinates
Greek symbols thermal diffusivity b volumetric expansion coefficient of the fluid c spin-gradient viscosity, c ¼ l þ j2 j h dimensionless temperature j vortex viscosity l dynamic viscosity q fluid density u inclination angle of the inclined cooled walls w dimensionless stream function x dimensionless vorticity
a
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi = gbðT T c ÞL; v ¼ v = gbðT T c ÞL; u¼u pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h ¼ ðT T c Þ=ðT h T c Þ; N ¼ N= gbðT T c Þ=L
x ¼ x=L;
=L; y¼y
ð6Þ and also stream function w u ¼ @w ; v ¼ @w and vorticity @y @x
x ¼ @@xv @u . Therefore the governing Eqs. (1)–(5) using the dimen@y
2. Basic equations
sionless variables (6) can be written as follows The domain of interest filled with a micropolar fluid is presented in Fig. 1 with dimensional Cartesian coordinates x and . The trapezoidal enclosure is bounded by isothermal inclined y cooled walls of temperature Tc, isothermal bottom hot wall of temperature Th (Th > Tc) and adiabatic top wall. All four walls of the cavity are assumed to be rigid and impermeable. The micropolar fluid is considered to satisfy the Boussinesq approximation and the flow regime is laminar. Taking into account the theory of Eringen [1–3] for the micropolar fluid flow the governing equations can be written in dimensional Cartesian coordinates as follows
@ v @u þ ¼0 @ x @ y
ð1Þ
! @2u @u @u @p @2u @N q u þ v ¼ þ ðl þ jÞ 2 þ 2 þ j @x @y @x @y @x @y
@ v
@ v
@ 2 v
@p
@ 2 v
!
ð2Þ
@N
! @2N @N @N @2N @ v @ u qj u þ v ¼ c 2 þ 2 þ j 2jN @x @y @x @y @x @y @T @T @2T @2T þ v ¼a þ 2 u @ x @y @ x2 @ y
ð7Þ
! rffiffiffiffiffiffi 2 @w @ x @w @ x Pr @ x @ 2 x þ ¼ ð1 þ KÞ @y @x @x @y Ra @x2 @y2 ! rffiffiffiffiffiffi 2 Pr @ N @ 2 N @h þ K þ Ra @x2 @y2 @x
ð8Þ
! rffiffiffiffiffiffi rffiffiffiffiffiffi 2 @w @N @w @N K Pr @ N @ 2 N Pr þK ¼ 1þ ðx 2NÞ þ @y @x @x @y 2 Ra @x2 @y2 Ra ð9Þ
q u þ v ¼ þ ðl þ jÞ 2 þ 2 j @y @x @x @y @x @y þ qbðT T c Þg
@2w @2w þ ¼ x @x2 @y2
@w @h @w @h 1 @2h @2h þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi @y @x @x @y Ra Pr @x2 @y2
! ð10Þ
with the following boundary conditions
ð3Þ
ð4Þ
! ð5Þ
In order to analyze the fluid flow and heat transfer in general scale we introduce the following dimensionless variables
w ¼ 0;
@2w @2w ; @x2 @y2 h ¼ 0 on left and right inclined walls
x¼
N ¼ n x; w ¼ 0;
x¼
@2w ; @y2
N ¼ n x;
h ¼ 1 on bottom wall
w ¼ 0;
x¼
@2w ; @y2
N ¼ n x;
@h ¼ 0 on top wall @y
ð11Þ
Here nð0 6 n 6 1Þ is a micro-gyration parameter with n = 0 corresponding to the case where the particle density is sufficiently great that microelements close to the wall are unable to rotate
833
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Fig. 1. Physical model and coordinate system (a), and utilized finite-difference grid (b).
(strong concentration), n = 0.5 is indicative of weak concentration, and when n = 1, it corresponds to turbulent flow (see Rees and Bassom [21]). The physical quantities of interest are the local Nusselt number Nu along the hot wall and average Nusselt number Nu, that are defined as
N, h, see Eq. (11)). Distances between nodes are given by hx and hy in the x- and y-directions, respectively. It should be noted that for definition of vorticity along all rigid walls we used Jenson’s polynomial [27]. For example, for the left inclined wall the vorticity has been found in following form:
@h Nu ¼ ; @y y¼0
xNyj;j ¼
Z
1þd=L
Nu ¼
Nu dx
wNyjþ2;j 8wNyjþ1;j
ð12Þ
d=L
3. Numerical method The partial differential Eqs. (7)–(10) with corresponding boundary conditions (11) were solved using the finite difference method with the second order central differencing schemes. The diffusive terms have been approximated by central differences. The convective terms have been discretized applying the second order Samarksii monotonic difference scheme. The parabolic equations have been solved on the basis of Samarskii locally onedimensional scheme (see Sheremet and Pop [22] and Sheremet et al. [23,24]). The obtained systems of algebraic equations have been solved by Thomas algorithm. The partial differential equation for the stream function (7) has been discretized by means of the five-point difference scheme on the basis of central differences for the partial derivatives. The obtained linear discretized equation has been solved by the successive over relaxation method. The developed numerical technique has been verified comprehensively (see Sheremet and Pop [22] and Sheremet et al. [23,24]). A regular grid distribution is used as shown in Fig. 1b. The inclined walls were approximated with staircase-like zigzag lines. This technique is based on the earlier studies [25,26]. In Fig. 1b, the bold nodes are coincided with the inclined boundary of the cavity and they have Dirichlet boundary conditions for all functions (w, x,
2 2hx
þ
wNyj;jþ2 8wNyj;jþ1 2
2hy
where Ny + 1 is a number of nodes along y-axis. The present models, in the form of an in-house computational fluid dynamics (CFD) code, have been validated successfully against the works of Aydin and Pop [28] and Zadravec et al. [29] for the steady-state natural convection of micropolar fluid in a square cavity with isothermal vertical and adiabatic horizontal walls. Table 1 shows the values of the average Nusselt number computed for various Rayleigh numbers and vortex viscosity parameter in comparison with other authors. The utilized numerical technique also has been validated successfully against the commercial fluid dynamics software (FLUENT) for steady-state natural convection of clear fluid in a trapezoidal cavity (Fig. 1) for Pr = 0.7, Ra = 105 and Ra = 106, K = 0,
Table 1 Comparison of the average Nusselt number of the hot wall for n = 0. Ra
K
Aydin and Pop [28]
Zadravec et al. [29]
Present results
104
0 0.5 2.0 0 0.5 2.0 0 0.5 2.0
2.234 1.947 1.545 4.486 4.033 3.314 8.945 7.984 6.673
2.263 1.986 1.578 4.540 4.067 3.377 8.742 8.229 6.714
2.245 1.977 1.566 4.529 4.081 3.348 8.836 8.034 6.700
105
106
834
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Fig. 2. Streamlines w and isotherms h at Ra = 105: FLUENT data (a), in-house computational code data (b).
Fig. 3. Streamlines w and isotherms h at Ra = 106: FLUENT data (a), in-house computational code data (b).
We have conducted also the grid independent test, analyzing the steady-state natural convection in a trapezoidal cavity filled with a micropolar fluid at Ra = 105, Pr = 7.0, K = 0.5, n = 0.5, u = p/4. Three cases of the uniform grid are tested: a grid of 240 80 points, a grid of 300 100 points, and a much finer grid of 450 150 points. Fig. 5 shows the effect of the mesh parameters on the profiles of temperature and vertical velocity along crosssection y = 0.5. Taking into account the conducted verifications the uniform grid of 300 100 points has been selected for the further investigation. 4. Results and discussion
Fig. 4. Profiles of local Nusselt number along the hot bottom wall for Ra = 105 in comparison with numerical data of Sathiyamoorthy et al. [8]
u = p/4. Figs. 2 and 3 show computed streamlines and isotherms using the developed computational code and commercial FLUENT software. Fig. 4 shows a good comparison of the local Nusselt number along the hot bottom wall with numerical data of Sathiyamoorthy et al. [8] for the problem presented in Fig. 2 at Ra = 105.
Numerical analysis has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 104, 105, 106), Prandtl number (Pr = 0.01, 0.1, 0.7, 7.0), the vortex viscosity parameter (K = 0, 0.1, 0.5, 2.0), micro-gyration parameter (n = 0, 0.5), u = p/4 and aspect ratio L/H = 1. Particular efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms, vorticity profiles, local and average Nusselt number, fluid flow rate for different values of governing parameters mentioned above are illustrated in Figs. 6–12. Regardless of the Rayleigh number values two convective cells are formed inside the cavity illustrating an appearance of ascending flows in the central part of the cavity and descending flows close to the cold inclined walls (Figs. 6 and 7). Such fluid flow behavior characterizes an evolution of the thermal plume over the hot wall. An increase in Ra leads to an intensification of both
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
835
Fig. 5. Profiles of vertical velocity (a) and temperature (b) along the middle cross-section y = 0.5 for different mesh parameters.
Fig. 6. Streamlines w, isotherms h and vorticity contours x, for Pr = 0.7, n = 0, K = 2: Ra = 104 a, Ra = 106 b.
Fig. 7. Streamlines w, isotherms h and vorticity contours x, for Ra = 105, Pr = 0.7, n = 0: K = 0 a, K = 2 b.
convective flow with a displacement of the convective cell cores along the vertical axis to the upper part of the cavity and heat transfer with a decrease in the thermal layers thickness near the inclined cold walls. Also for high values of the Rayleigh number a thickness of the thermal plume decreases. It should be noted that vorticity contours reflect a distribution of hydrodynamic disturbance from the rigid walls into the cavity. A high density of vorticity isolines near the walls illustrates a formation of velocity boundary layers. In the case of Ra = 105 (Fig. 7) location of vorticity isolines in the central part of the cavity characterizes an evolution of the thermal plume. An increase in the vortex viscosity parameter reflects an increase in the fluid viscosity (see Eqs. (8) and (9)) that leads to
an attenuation of the convective flow and heat transfer. For example, in the case of Ra = 105 one can find a downward displacement of the convective cell cores with more thick thermal plume for K = 2 (Fig. 7b). Distribution of the vorticity isolines illustrates a decrease in the convective circulations intensity. In the case of weak concentration (n = 0.5) for Ra = 105, Pr = 0.7, K = 2 one can find small displacement of the convective cells cores from the vertical centerline in opposite sides along horizontal axis (Fig. 8). At the same time an increase in micro-gyration parameter leads to both less intensive heating of the cavity and nonsignificant changes of the vorticity. Figs. 7b and 9 show the effect of Prandtl number on contours of stream function, temperature and linear vorticity for Ra = 105,
836
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
Fig. 8. Streamlines w, isotherms h and vorticity contours x, for Ra = 105, Pr = 0.7, K = 2, n = 0.5.
Fig. 9. Streamlines w, isotherms h and vorticity contours x, for Ra = 105, K = 2, n = 0: Pr = 0.1 (a), Pr = 7.0 (b).
Fig. 10. Profiles of local Nusselt number along the hot wall: for different values of Ra at Pr = 0.7, K = 0.5, n = 0 (a) and for different values of K at Ra = 105, Pr = 0.7, n = 0 (b).
K = 2, n = 0. An increase in Prandtl number leads to following changes in profiles of w, h and x: a displacement of the convective cells cores along the vertical direction close to the centerline and less intensive heating of the cavity. The latter reflects an increase in the temperature differences inside the cavity. Fig. 10 shows profiles of the local Nusselt number along the hot wall for different values of the Rayleigh number and vortex viscosity parameter. Regardless of the Rayleigh numbers the local Nusselt number has maximum values at the ends of the hot wall and minimum value at the center of this wall. An increase in x-coordinate from 1.0 to 1.5 and a decrease in x-coordinate from 2.0 to 1.5 lead to a decrease in Nu along the wall that can be explained by a formation of thermal layers over the hot wall with the origin of these thermal layers at the wall ends. It should be noted that these thermal boundary layers widen to the center of the wall and interflow into one thermal plume that leads to a decrease in the local Nusselt number from the left and right wall ends to the center. An increase in Ra leads to an increase in the local Nusselt number. Weak increase in Nu with Ra at x = 1.5 in comparison with other points can be explained by more intensive
convective upward motion in the central part of the cavity that leads to an essential heating of this part and decrease in the temperature gradient. An increase in the vortex viscosity parameter from K = 0 to K = 2 (Fig. 10b) for Ra = 105 leads to a decrease in the local Nusselt number mainly in the sections 1.05 < x < 1.3 and 1.7 < x < 1.95 where the boundary layers evolve. Fig. 11 demonstrates variations of the average Nusselt number at the hot wall and fluid flow rate with Ra and K. As has been mentioned above an increase in the vortex viscosity parameter leads to a decrease in the heat transfer rate and fluid flow intensity that is caused by a motion of more viscous fluid in comparison with Newtonian case (K = 0). Also for high values of Ra (Ra = 106) an increase in K leads to a weak attenuation of the convective flow, because a convective flow intensity for the constant value of the Rayleigh number is defined also by the geometry of the cavity. The effect of the Prandtl number on the heat transfer rate and fluid flow intensity is presented in Fig. 12. An increase in Pr leads to a heat transfer enhancement and intensification of fluid flow. Attenuation of the convective flow and heat transfer with the
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
837
Fig. 11. Variations of the average Nusselt number along the hot wall (a) and maximum absolute value of the stream function (b) with Rayleigh number and vortex viscosity parameter for Pr = 0.7, n = 0.
Fig. 12. Variations of the average Nusselt number along the hot wall (a) and maximum absolute value of the stream function (b) with Prandtl number and vortex viscosity parameter for Ra = 105, n = 0.
vortex viscosity parameter is more essential for high values of the Prandtl number. It is worth noting that for high values of K an increase in Pr leads to less intensive increase in Nu and jwjmax . At the same time for low values of the Prandtl number a variation of K does not lead to essential changes in the heat transfer rate. 5. Conclusions Free convection of a micropolar fluid in a trapezoidal cavity with cold inclined walls and hot bottom wall has been studied. Analysis of the fluid flow and heat transfer has been conducted in a wide range of the Rayleigh number, Prandtl number and vortex viscosity parameter On the basis of the obtained results we can conclude that 1. There are two convective cells inside the cavity for any value of Ra. The average Nusselt number and fluid flow rate are the increasing functions of Ra. 2. An increase in the vortex viscosity parameter illustrates the heat transfer reduction and attenuation of convective flow. An increase in K leads to a weak attenuation of the convective flow for high values of Ra (Ra = 106).
3. An increase in the Prandtl number leads to the heat transfer enhancement and fluid flow intensification. At the same time for low values of the Prandtl number a variation of K does not lead to essential changes in the heat transfer rate 4. An increase in the micro-gyration parameter from 0 to 0.5 does not lead to modification of the analyzed parameters for the considered range of Ra, Pr and K.
Acknowledgement This work of Nikita S. Gibanov and Mikhail A. Sheremet was conducted as a government task of the Ministry of Education and Science of the Russian Federation, Project Number 13.1919.2014/ K. The authors also wish to express their thank to the very competent Reviewers for the valuable comments and suggestions. References [1] A.C. Eringen, Simple microfluids, Int. J. Eng. Sci. 2 (1964) 205–217. [2] A.C. Eringen, Theory of micropolar fluid, J. Math. Mech. 16 (1966) 1–18. [3] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972) 480– 496.
838
N.S. Gibanov et al. / International Journal of Heat and Mass Transfer 99 (2016) 831–838
[4] T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics – a review, Int. J. Eng. Sci. 11 (1973) 905–930. [5] T. Ariman, M.A. Turk, N.D. Sylvester, Applications of microcontinuum fluid mechanics, Int. J. Eng. Sci. 12 (1974) 273–293. [6] A.C. Eringen, Microcontinuum Field Theories, vol. I,II, Springer, New York, 2001. [7] G. Łukaszewicz, Micropolar Fluids: Theory and Application, Birkhäuser, Basel, 1999. [8] M. Sathiyamoorthy, T. Basak, S. Roy, I. Pop, Steady natural convection flows in a square cavity with linearly heated side walls, Int. J. Heat Mass Transfer 50 (2007) 766–775. [9] P. Kandaswamy, M. Nithyadevi, Buoyancy driven convection of water near its density maximum with partially active vertical walls, Int. J. Heat Mass Transfer 50 (2007) 942–948. [10] F. Moukalled, M. Darwish, Natural convection in a partitioned trapezoidal cavity heated from the side, Numer. Heat Transfer, Part A 43 (2003) 543–563. [11] F. Moukalled, M. Darwish, Natural convection in a trapezoidal enclosure heated from the side with a baffle mounted on its upper inclined surface, Heat Transfer Eng. 25 (2004) 80–93. [12] M. Boussaid, A. Djerrada, M. Bouhadef, Thermosolutal transfer within trapezoidal cavity, Numer. Heat Transfer, Part A 43 (2003) 431–488. [13] F. Moukalled, M. Darwish, Buoyancy-induced heat transfer in a trapezoidal enclosure with offset baffles, Numer. Heat Transfer, Part A 54 (2007) 337–355. [14] R.A. Kuyper, C.J. Hoogendoorn, Laminar natural convection flow in trapezoidal enclosures, Numer. Heat Transfer, Part A 28 (1995) 55–67. [15] H. Sadat, P. Salagnac, Further results for laminar natural convection in a two dimensional trapezoidal enclosure, Numer. Heat Transfer, Part A 27 (1995) 451–459. [16] A.J. Chamkha, T. Grosan, I. Pop, Fully developed free convection of a micropolar fluid in a vertical channel, Int. Commun. Heat Mass Transfer 29 (2002) 1119– 1127. [17] A.J. Chamkha, T. Grosan, I. Pop, Fully developed mixed convection of a micropolar fluid in a vertical channel, Int. J. Fluid Mech. Res. 30 (2003) 251– 263.
[18] A.J. Chamkha, M.A. Mansour, S.E. Ahmed, Unsteady mixed convection of a micropolar fluid in a lid-driven cavity: effects of different micro-gyration boundary conditions, Int. J. Energy Technol. 2 (2010) 1–10. [19] S.K. Jena, L.K. Malla, S.K. Mahapatra, A.J. Chamkha, Transient buoyancyopposed double diffusive convection of micropolar fluids in a square enclosure, Int. J. Heat Mass Transfer 81 (2014) 681–694. [20] T.H. Hsu, K.Y. Hong, Natural convection of micropolar fluids in an open cavity, Numer. Heat Transfer, Part A 50 (2006) 281–300. [21] D.A.S. Rees, A.P. Bassom, The blasius boundary-layer flow of a micropolar fluid, Int. J. Eng. Sci. 34 (1996) 113–124. [22] M.A. Sheremet, I. Pop, Thermo-bioconvection in a square porous cavity filled by oxytactic microorganisms, Transp. Porous Media 103 (2014) 191–205. [23] M.A. Sheremet, T. Grosan, I. Pop, Free convection in shallow and slender porous cavities filled by a nanofluid using Buongiorno’s model, ASME J. Heat Transfer 136 (2014) 082501. [24] M.A. Sheremet, T. Grosan, I. Pop, Free convection in a square cavity filled with a porous medium saturated by nanofluid using Tiwari and Das’ nanofluid model, Transp. Porous Media 106 (2015) 595–610. [25] P.M. Haese, M.D. Teubner, Heat exchange in an attic space, Int. J. Heat Mass Transfer 45 (2002) 4925–4936. [26] Y. Varol, H.F. Oztop, I. Pop, Maximum density effects on buoyancy-driven convection in a porous trapezoidal cavity, Int. Commun. Heat Mass Transfer 37 (2010) 401–409. [27] V.G. Jenson, Viscous flow around a sphere at low Reynolds numbers (640), Proc. R. Soc. London, Ser. A 249 (1959) 346–366. [28] O. Aydin, I. Pop, Natural convection in a differentially heated enclosure filled with a micropolar fluid, Int. J. Therm. Sci. 46 (207) (2007) 963–969. [29] M. Zadravec, M. Hribersek, L. Skerget, Natural convection of micropolar fluid in an enclosure with boundary element method, Eng. Anal. Boundary Elem. 33 (2009) 485–492.