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Natural convection of micropolar fluid in a wavy differentially heated cavity
Natural convection of micropolar fluid in a wavy differentially heated cavity Nikita S. Gibanov, Mikhail A. Sheremet, Ioan Pop PII: DOI: Reference:
S0167-7322(16)30656-0 doi: 10.1016/j.molliq.2016.06.033 MOLLIQ 5940
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
16 March 2016 14 April 2016 9 June 2016
Please cite this article as: Nikita S. Gibanov, Mikhail A. Sheremet, Ioan Pop, Natural convection of micropolar fluid in a wavy differentially heated cavity, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.06.033
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Natural convection of micropolar fluid in a wavy differentially heated cavity
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Nikita S. Gibanov1, Mikhail A. Sheremet1,2, Ioan Pop3
Department of Theoretical Mechanics, Tomsk State University, 634050, Tomsk, Russia
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Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, 634050,
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1
Tomsk, Russia
Department of Applied Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
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ABSTRACT
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An analysis of natural convective flow and heat transfer of a micropolar fluid in a wavy differentially heated cavity has been performed. Governing partial differential equations formulated in non-dimensional variables have been solved by finite difference method of second
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order accuracy. The effects of Rayleigh number (Ra = 104, 105, 106), Prandtl number (Pr = 0.1, 0.7, 7.0), vortex viscosity parameter (K = 0, 0.1, 0.5, 2.0) and undulation number (k = 1, 2, 3) on
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flow patterns, temperature fields and average Nusselt number at hot wavy wall have been studied. It is found that microrotation increases as the vortex viscosity parameter K increases.
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However, the fluid velocity decreases as K increases. It is observed that the form of streamlines is dependent on the value of vortex viscosity parameter. An increase in the undulation number leads to a decrease in the heat transfer rate at wavy wall.
Keywords: Wavy cavity, Natural convection, Micropolar fluid, Numerical results
1. Introduction Over the relatively many years, fluid flow of micropolar fluids has received considerable attention because of its important applications in engineering. The interest in micropolar fluids, which exhibit the microrotational effects and microrotational inertia, started very soon after the pioneering studies by Eringen [1, 2]. These fluids cannot be explained on the basis of Newtonian fluid flow theory. Since the publication of this micropolar fluid theory, many authors have investigated various flow and heat transfer problems. According of this theory, the deformation of the fluid microelements is ignored: nevertheless microrotational effects are still present and 1
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surface and body couple are permitted (Agarwal et al. [3], Kelson and Farrell [4], Bhargava et al. [5], Ishak et al. [6, 7], Lok et al. [8], Tetbirt et al. [9], Fakour et al. [10] etc.). Examples of
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industrially relevant flows that can be studied using micropolar theory include the flow of low
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concentration suspensions, liquid crystals, animal blood, colloidal fluids, polymeric fluids, lubrication, turbulent shear flow, etc. Extensive reviews of the theory and applications can be
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found in the review articles by Ariman et al. [11, 12] and the books by Łukaszewicz [13] and Eringen [14]. The common comment in respect of micropolar fluids was that there are no experiments whatsoever in which any of the material moduli could be measured. Hoyt and
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Fabula [15] has shown experimentally that the fluids containing minute polymeric additives indicate considerable reduction of the skin friction (about 25–30%), a concept which can be well
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explained by the theory of micropolar fluids. Power [16] has shown that the fluid flowing in brain (Cerebrospinal fluid) is adequately modelled by micropolar fluids. The works of Migun [17] and Kolpashchikov et al. [18] demonstrated an experimental method of determining
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parameters characterizing the microstructure of such fluids and seems to have laid to rest many unanswered questions on the theory. Sheikholeslami et al. [19] analyzed theoretically micropolar
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fluid flow in a channel subject to a chemical reaction. Mosayebidorcheh [20] investigated twodimensional micropolar fluid flow in a porous channel with expanding or contracting walls. It
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was found that the profiles of the streamwise velocity and microrotation are asymmetric, while the normal velocity is symmetric. Zheng et al. [21] analyzed flow structures and radiative heat transfer of a micropolar fluid over stretching/shrinking sheet using an analytical technique [22, 23].
On the other hand, it should be stated that during the past several decades, extensive studies on heat transfer in regular cavities and enclosures filled with a viscous (Newtonian) fluid have been done and various extensions of the problem have been reported in the literature (see Vahl Davis [24]). However, it is necessary to study the heat transfer for more complex geometries because the prediction of heat transfer for irregular surfaces is a topic of great importance and irregular surfaces often occur in many applications (see Peterson and Ortega [25]). Recently, several studies have been performed on the convective flow in iregular (wavy) cavities filled with nanofluids (see Sheremet and Pop [26], Sheremet et al. [27–29], Sheikholeslami et al. [30]). It is also worth mentioning the recent book by Shenoy et al. [31] on
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convective flow and heat transfer from vawy surfaces: viscous fluids, porous media and nanofluids.
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Motivated by the practical importance of the micropolar fluids, the main objective of this
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paper is to understand the fundamentals of various heating and cooling strategies, and to achieve a high performance for the free convection in a partially heated wavy cavity filled with a
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micropolar fluid using the mathematical micropolar fluid model proposed by the pioneering papers of Eringen [1, 2]).
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2. Basic equations
We consider laminar natural convection flow and heat transfer in a differentially heated
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wavy cavity of length L and height H filled with a micropolar fluid. The geometry and coordinate system are shown in Fig. 1, where x axis is measured along the bottom wall of the cavity and y axis is measured along the vertical wall of the cavity. The upper and lower walls of
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the enclosure are thermally insulated and the fluid is isothermally heated and cooled from the left wavy wall and right flat wall at uniform temperatures of Th and Tc, respectively. All four walls of
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the cavity are supposed to be rigid and impermeable. It is considered that the left wavy wall and right flat wall of the cavity are described by the relations x1 = L - L éëa + b cos ( 2pky H )ùû and
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x2 = L , respectively.
Fig. 1. Physical model and coordinate system. 3
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The physical properties of the micropolar fluid are supposed to be constant except for the
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density in the buoyancy force term of the momentum equation that is approximated by the
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Boussinesq law. The governing equations for the micropolar fluid flow and heat transfer can be written on the basis of the conservation laws for the mass, linear momentum, angular momentum
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and energy in dimensional Cartesian coordinates as follows:
(1)
æ ¶ 2u ¶ 2u ö æ ¶u ¶u ö ¶p ¶N m k +v = + + ( ) ç 2 + 2 ÷ +k ÷ ¶y ø ¶x ¶y ø ¶y è ¶x è ¶x
(2)
æ ¶ 2v ¶ 2v ö æ ¶v ¶v ö ¶p ¶N m k +v = + + + rb (T - Tc ) g ( ) ç 2 + 2 ÷ -k ÷ ¶y ø ¶y ¶y ø ¶x è ¶x è ¶x
(3)
æ ¶2 N ¶2 N ö æ ¶N æ ¶v ¶u ö ¶N ö +v = g ç 2 + 2 ÷ +k ç ÷ ÷ - 2k N ¶y ø ¶y ø è ¶x è ¶x ¶y ø è ¶x
(4)
æ ¶ 2T ¶ 2T ö ¶T ¶T +v =a ç 2 + 2 ÷ ¶x ¶y ¶y ø è ¶x
(5)
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¶u ¶v + =0 ¶x ¶y
r çu
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r çu
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r j çu
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u
Further, we introduce the following dimensionless variables
y = y L, u = u
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x = x L,
g b (T - Tc ) L , v = v
q = (T - Tc ) (Th - Tc ) , N = N
g b (T - Tc ) L ,
g b (T - Tc ) L ,
(6)
æ ¶y ¶y ö ¶v ¶u ,v=stream function ψ ç u = . Therefore the governing Eqs. ÷ and vorticity w = ¶y ¶x ø ¶x ¶y è (1)–(5) using the dimensionless variables (6) can be written as follows
¶ 2y ¶ 2y + = -w ¶x 2 ¶y 2
(7)
Pr æ ¶ 2w ¶ 2w ö Pr æ ¶ 2 N ¶ 2 N ö ¶q ¶y ¶w ¶y ¶w K = (1 + K ) + + ç ÷ ç ÷+ Ra è ¶x 2 ¶y 2 ø Ra è ¶x 2 ¶y 2 ø ¶x ¶y ¶x ¶x ¶y
(8)
Pr ¶y ¶N ¶y ¶N æ K ö Pr æ ¶ 2 N ¶ 2 N ö = ç1 + ÷ (w - 2 N ) ç 2 + 2 ÷+ K 2 ø Ra è ¶x Ra ¶y ¶x ¶x ¶y è ¶y ø
(9)
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¶y ¶q ¶y ¶q = ¶y ¶x ¶x ¶y
1 Ra × Pr
æ ¶ 2q ¶ 2q ö ç 2+ 2÷ ¶y ø è ¶x
x = x1 = 1 - a - b × cos ( 2pk y A )
N = n × w , q = 1 on
N = n × w , q = 0 on
x =1
¶q = 0 on ¶y
N = n × w,
y = 0 and
(11)
y=A
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¶ 2y y = 0, w = - 2 , ¶x ¶ 2y y = 0, w = - 2 , ¶y
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¶ 2y ¶ 2y , ¶x 2 ¶y 2
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y = 0, w = -
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with the following boundary conditions
(10)
Here n (0 £ n £ 1) is a micro-gyration parameter.
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The physical quantities of interest are the local Nusselt number Nu along the hot wall and average Nusselt number Nu , that are defined as
æ ¶q ö Nu = - ç ÷ = - ( n,Ñq ) ì è ¶n ø x = x1 ï n =1 Nu ds S sò
s y üï 1 i+ jý 2 1+s y 1+s y2 ïþ
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Nu =
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í ïî
=
¶q 2 ¶x 1+ s y 1
x = x1
sy
¶q 1 + s ¶y 2 y
x = x1
(12)
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Here s ( y ) = 1 - a - b × cos ( 2pk y A) , s y = ds dy and S = ò 1 +
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-
0
4p 2k 2b2 æ 2pk y ö sin 2 ç ÷dy . 2 A è A ø
In order to solve the formulated boundary-value problem (7)–(11) we introduce new independent variables x and h in the following form:
x=
x - x1 x - 1 + a + b × cos ( 2pk y A) = , h=y a + b × cos ( 2pk y A) D
(13)
Therefore Eqs. (7)–(10) are rewritten as follows:
éæ ¶x ö2 æ ¶x ö2 ù ¶ 2y ¶x ¶ 2y ¶ 2y ¶ 2x ¶y + + = -w êç ÷ + ç ÷ ú 2 + 2 ¶y ¶x¶h ¶h 2 ¶y 2 ¶x êëè ¶x ø è ¶y ø úû ¶x
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(14)
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2 2 ¶x ¶y ¶w ¶x ¶y ¶w Pr ïì éæ ¶x ö æ ¶x ö ù ¶ 2w = (1 + K ) + í êç ÷ + ç ÷ ú ¶x ¶h ¶x ¶x ¶x ¶h Ra ï êëè ¶x ø è ¶y ø úû ¶x 2 î
(15)
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¶x ¶ 2 N ¶ 2 N ¶ 2x ¶N ü ¶x ¶q +2 + + ý+ ¶y ¶x¶h ¶h 2 ¶y 2 ¶x þ ¶x ¶x
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2 2 ¶x ¶ 2w ¶ 2w ¶ 2x ¶w ü Pr ìï éæ ¶x ö æ ¶x ö ù ¶ 2 N +2 + + + ý- K í êç ÷ + ç ÷ ú ¶y ¶x¶h ¶h 2 ¶y 2 ¶x þ Ra ï ëêè ¶x ø è ¶y ø ûú ¶x 2 î
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2 2 ¶x ¶y ¶N ¶x ¶y ¶N æ K ö Pr ì ï éæ ¶x ö æ ¶x ö ù ¶ 2 N = ç1 + ÷ + í êç ÷ + ç ÷ ú ¶x ¶h ¶x ¶x ¶x ¶h è 2 ø Ra ï ëêè ¶x ø è ¶y ø ûú ¶x 2 î
(16)
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¶x ¶ 2 N ¶ 2 N ¶ 2x ¶N ü Pr +2 + + 2 (w - 2 N ) ý+ K 2 ¶y ¶x¶h ¶h ¶y ¶x þ Ra 2 2 ïì éæ ¶x ö æ ¶x ö ù ¶ 2q í êç ÷ + ç ÷ ú 2 + ïî êëè ¶x ø è ¶y ø úû ¶x
(17)
¶x ¶ q ¶ q ¶ x ¶q ü + 2+ 2 ý ¶y ¶x¶h ¶h ¶y ¶x þ 2
2
2
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+2
1 Ra × Pr
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¶x ¶y ¶q ¶x ¶y ¶q = ¶x ¶h ¶x ¶x ¶x ¶h
The corresponding boundary conditions for these equations are given by
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éæ ¶x ö 2 æ ¶x ö2 ù ¶ 2y y = 0, w = - êç ÷ + ç ÷ ú 2 , êëè ¶x ø è ¶y ø úû ¶x
N = n × w , q = 1 on x = 0
2
2 æ ¶x ö ¶ y , N = n × w , q = 0 on x = 1 ÷ 2 è ¶x ø ¶x ¶ 2y ¶q = 0 on h = 0 and h = A y = 0, w = - 2 , N = n × w , ¶h ¶h
y = 0, w = - ç
(18)
The local and average Nusselt numbers will be defined as follows
æ ¶x ¶x ö ¶q Nu = -s y ÷ 2 ç ¶x ¶y ø ¶x 1+ s y è 1
1 4p 2k 2b2 æ 2pk y ö , Nu = ò Nu 1 + sin 2 ç ÷dy 2 S0 A è A ø x =0 A
(19)
The partial differential equations (14)–(17) with corresponding boundary conditions (18) were solved using the finite difference method of the second order accuracy. Detailed description of the used numerical technique is presented by Sheremet and Trifonova [32], Sheremet and Pop [26, 33], Sheremet et al. [27–29].
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For the purpose of obtaining grid independent solution, a grid sensitivity analysis is performed. The grid independent solution was performed by preparing the solution for unsteady
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free convection in a square wavy cavity filled with a micropolar fluid at Ra = 105, Pr = 0.7,
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K = 0.5, n = 0.0, k = 2, a = 0.9, A = 1. Three cases of the uniform grid are tested: a grid of 50´50 points, a grid of 100´100 points, and a much finer grid of 200´200 points. Figure 2 shows an
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effect of the mesh parameters on the average Nusselt number of the left vertical wavy wall (see
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Eq. (19)).
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Fig. 2. Variation of the average Nusselt number of the left vertical wavy wall versus the dimensionless time and the mesh parameters.
Taking into account the conducted verifications the uniform grid of 100´100 points has been selected for the further investigation.
4. Results and discussion Numerical analysis has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 104, 105, 106), Prandtl number (Pr = 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), aspect ratio parameter (A = 1), undulation number (k = 1, 2, 3), wavy contraction ratio (a = 0.9). Particular efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms, linear and angular vorticity profiles, average Nusselt
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number for different values of governing parameters mentioned above are illustrated in Figs. 3–10.
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Figure 3 shows an effect of the undulation number on isolines of stream function,
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temperature, linear vorticity and angular vorticity at Ra = 106, Pr = 0.7, K = 0.5, n = 0. Convective flow appearing inside the cavity due to a presence of horizontal temperature drop illustrates a formation of an ascending flow near the hot wavy wall and descending flow close to
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the right cold wall regardless of the undulation number value. An increase in k leads to modifications of both convective flow shape and position of isotherms inside the cavity. Taking
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into account the maximum absolute value of stream function it is possible to note insignificant changes of the fluid flow rate with the undulation number. Fields of linear vorticity characterize
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a distribution of hydrodynamic perturbations from solid walls to the cavity. Moreover one can find maximum values of w along the right wall and fluidward side of the left wavy wall. Distributions of angular vorticity illustrate a circulation of micropolar fluid with two cores.
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Variation of undulation number affects only the left core of N. Isotherms reflect an interaction between ascending and descending thermal boundary layers formed along the isothermal walls.
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It should be noted that an increase in undulation number leads to more essential heating of the
number.
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wavy troughs and as a result we have the heat transfer reduction (see Fig. 4) with undulation
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Fig. 3. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Ra = 106, Pr = 0.7, K = 0.5, n = 0: k = 1 – a, k = 2 – b, k = 3 – c. Figure 4 shows the effect of the dimensionless time and undulation number on the rate of heat transfer. As has been mentioned above we have the heat transfer reduction with k. Also before the steady-state regime one can find a formation of transient regimes due to a transition from heat conduction mode at initial time to oscillation heat convection mode and after that to steady heat convection mode.
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Fig. 4. Variations of the average Nusselt number with dimensionless time and undulation number for Ra = 106, Pr = 0.7, K = 0.5, n = 0. In the case of micropolar fluid one of the governing parameters is a vortex viscosity
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parameter that characterizes an increase in the micropolar fluid viscosity due to an internal orientation of each particles [1, 2]. Figure 5 demonstrates an effect of this parameter on fluid
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flow and heat transfer. In the case of K = 0 (Fig. 5a) we have conventional Newtonian fluid. An increase in the vortex viscosity parameter leads to an attenuation of the convective flow with a
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decrease in the linear vorticity and an increase in angular vorticity. Essential influence of K on streamlines and isotherms can be found in the case of high vortex viscosity parameter values. For K = 2.0 (Fig. 5c) we have an essential modification of shapes of streamlines, linear and angular vorticity. Analysis of temperature fields shows an increase in the thermal boundary layers thickness with K that reflects less intensive convective heat transfer inside the enclosure. Figure 6 shows variations of the average Nusselt number at hot wavy wall with dimensionless time, vortex viscosity parameter and undulation number. As has been mentioned above an increase in K leads to a reduction of the heat transfer rate. Due to less intensive fluid flow for high values of K we have an obtaining of steady state with less time in comparison with Newtonian fluid.
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Fig. 5. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Ra = 106, Pr = 0.7, k = 2, n = 0: K = 0.0 – a, K = 0.1 – b, K = 2.0 – c.
Fig. 6. Variations of the average Nusselt number with dimensionless time and vortex viscosity parameter for k = 2 (a) and with undulation number and vortex viscosity parameter (b) for Ra = 106, Pr = 0.7, n = 0.
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The effect of Rayleigh number on streamlines, isotherms and vorticity patterns is presented in Figs. 7 and 3b. An increase in the buoyancy force leads to an intensification of
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convective flow and heat transfer with a decrease in the thermal boundary layers thickness.
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Small value of Ra ( = 104) illustrates weak convective flow with dominating of heat conduction. An increase in Ra leads to an intensification of circulation inside the cavity with linear and
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angular vorticity enhancement. It is worth noting that a presence of two linear vorticity cores having negative values of w near the vertical wall defines a formation of convective flow with two clockwise cores close to isothermal walls. Such fluid flow behavior can be found at Ra = 104
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and 105.
Fig. 7. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Pr = 0.7, K = 0.5, k = 2, n = 0: Ra = 104 – a, Ra = 105 – b. Intensification of convective heat transfer with Ra is shown in Fig. 8. An increase in the buoyancy force magnitude leads to an increase in time for an obtaining of the steady state regime. Also an effect of vortex viscosity parameter is more essential for high values of Rayleigh number.
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Fig. 8. Variations of the average Nusselt number with dimensionless time and Rayleigh number for K = 0.5 (a) and with vortex viscosity parameter and Rayleigh number (b) for Pr = 0.7, k = 2, n = 0. Figure 9 presents an effect of Prandtl number of micropolar fluids flow and heat transfer.
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Low values of Pr ( = 0.1) reflects a formation of complex internal hydrodynamic structure with three convective cores taking into account the linear and angular vorticity patterns. Such
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behavior of fluid flow reflects low heat transfer rate. An increase in Pr leads to an intensification
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of convective flow and heat transfer (see Fig. 10).
Fig. 9. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Ra = 106, K = 0.5, k = 2, n = 0: Pr = 0.1 – a, Pr = 7.0 – b.
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Figure 10 shows the heat transfer enhancement with Prandtl number. More intensive increase in Nu occurs for an increase in Pr from 0.1 to 0.7. Also it should be noted that an effect
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of the vortex viscosity parameter is more essential for high values of Prandtl number.
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Fig. 10. Variations of the average Nusselt number with dimensionless time and Prandtl number for K = 0.5 (a) and with vortex viscosity parameter and Prandtl number (b) for Pr = 0.7, k = 2, n = 0. 5. Conclusion
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Natural convection of micropolar fluid in a wavy-walled differentially heated cavity of unit aspect ratio has been studied numerically. Mathematical model has been formulated in dimensionless variables stream function and vorticity taking into account the conservation laws for mass, linear momentum, angular momentum and energy. Analysis of the fluid flow and heat transfer in terms of streamlines, isotherms, linear and angular vorticity patterns and average Nusselt number has been conducted in a wide range of the Rayleigh number, Prandtl number, vortex viscosity parameter and undulation number. On the basis of the obtained results we can conclude that 1. An increase in Rayleigh number leads to an intensification of circulation inside the cavity with linear and angular vorticity enhancement. At the same time one can find an increase in time for an obtaining of the steady state regime for high values of Ra. The effect of vortex viscosity parameter is more essential for high values of Rayleigh number. 2. Low values of Prandtl number illustrate an appearance of complex internal hydrodynamic structure with several convective cores. The fluid flow rate and average Nusselt
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number are the increasing functions of Prandtl number. An effect of the vortex viscosity parameter is more essential for high values of Prandtl number
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3. Thermal boundary layers thickness increases with the vortex viscosity parameter. The
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fluid flow and heat transfer rates decrease with K. Due to less intensive fluid flow for high values of K we have an obtaining of steady state with less time in comparison with Newtonian fluid.
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4. An increase in undulation number leads to the heat transfer reduction at hot wavy wall.
Acknowledgement
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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
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13.1919.2014/K.
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[28] M.A. Sheremet, I. Pop, N. Bachok, Effect of thermal dispersion on transient natural
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convection in a wavy-walled porous cavity filled with a nanofluid: Tiwari and Das’ nanofluid model, Int. J. Heat Mass Transfer 92 (2016) 1053–1060.
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[29] M.A. Sheremet, I. Pop, N.C. Rosca, Magnetic field effect on the unsteady natural convection in a wavy-walled cavity filled with a nanofluid: Buongiorno’s mathematical
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model, J. Taiwan Inst. Chem. Eng. 61 (2016) 211–222. [30] M. Sheikholeslami, S. Soleimani, D.D. Ganji, Effect of electric field on hydrothermal behavior of nanofluid in a complex geometry, J. Mol. Liq. 213 (2016) 153–161. [31] A. Shenoy, M. Sheremet, I. Pop, Convective flow and heat transfer from vawy surfaces: viscous fluids, porous media and nanofluids, CRC Press, Taylor & Francis Group, New York, 2016 (in press).
[32] M.A. Sheremet, T.A. Trifonova, Unsteady conjugate natural convection in a vertical cylinder partially filled with a porous medium, Num. Heat Transfer, Part A 64 (2013) 994– 1015. [33] M.A. Sheremet, I. Pop, Conjugate natural convection in a square porous cavity filled by a nanofluid using Buongiorno’s mathematical model, Int. J. Heat Mass Transfer 79 (2014) 137–145.
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Highlights Natural convective flow and heat transfer of a micropolar fluid is studied
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A wavy differentially heated cavity has been consiidered
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Governing partial differential equations have been solved by finite difference method
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It is found that microrotation increases as the vortex viscosity parameter K increases
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S0167-7322(16)30656-0 doi: 10.1016/j.molliq.2016.06.033 MOLLIQ 5940
To appear in:
Journal of Molecular Liquids
Received date: Revised date: Accepted date:
16 March 2016 14 April 2016 9 June 2016
Please cite this article as: Nikita S. Gibanov, Mikhail A. Sheremet, Ioan Pop, Natural convection of micropolar fluid in a wavy differentially heated cavity, Journal of Molecular Liquids (2016), doi: 10.1016/j.molliq.2016.06.033
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Natural convection of micropolar fluid in a wavy differentially heated cavity
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Nikita S. Gibanov1, Mikhail A. Sheremet1,2, Ioan Pop3
Department of Theoretical Mechanics, Tomsk State University, 634050, Tomsk, Russia
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Department of Nuclear and Thermal Power Plants, Tomsk Polytechnic University, 634050,
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1
Tomsk, Russia
Department of Applied Mathematics, Babeş-Bolyai University, 400084 Cluj-Napoca, Romania
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ABSTRACT
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An analysis of natural convective flow and heat transfer of a micropolar fluid in a wavy differentially heated cavity has been performed. Governing partial differential equations formulated in non-dimensional variables have been solved by finite difference method of second
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order accuracy. The effects of Rayleigh number (Ra = 104, 105, 106), Prandtl number (Pr = 0.1, 0.7, 7.0), vortex viscosity parameter (K = 0, 0.1, 0.5, 2.0) and undulation number (k = 1, 2, 3) on
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flow patterns, temperature fields and average Nusselt number at hot wavy wall have been studied. It is found that microrotation increases as the vortex viscosity parameter K increases.
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However, the fluid velocity decreases as K increases. It is observed that the form of streamlines is dependent on the value of vortex viscosity parameter. An increase in the undulation number leads to a decrease in the heat transfer rate at wavy wall.
Keywords: Wavy cavity, Natural convection, Micropolar fluid, Numerical results
1. Introduction Over the relatively many years, fluid flow of micropolar fluids has received considerable attention because of its important applications in engineering. The interest in micropolar fluids, which exhibit the microrotational effects and microrotational inertia, started very soon after the pioneering studies by Eringen [1, 2]. These fluids cannot be explained on the basis of Newtonian fluid flow theory. Since the publication of this micropolar fluid theory, many authors have investigated various flow and heat transfer problems. According of this theory, the deformation of the fluid microelements is ignored: nevertheless microrotational effects are still present and 1
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surface and body couple are permitted (Agarwal et al. [3], Kelson and Farrell [4], Bhargava et al. [5], Ishak et al. [6, 7], Lok et al. [8], Tetbirt et al. [9], Fakour et al. [10] etc.). Examples of
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industrially relevant flows that can be studied using micropolar theory include the flow of low
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concentration suspensions, liquid crystals, animal blood, colloidal fluids, polymeric fluids, lubrication, turbulent shear flow, etc. Extensive reviews of the theory and applications can be
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found in the review articles by Ariman et al. [11, 12] and the books by Łukaszewicz [13] and Eringen [14]. The common comment in respect of micropolar fluids was that there are no experiments whatsoever in which any of the material moduli could be measured. Hoyt and
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Fabula [15] has shown experimentally that the fluids containing minute polymeric additives indicate considerable reduction of the skin friction (about 25–30%), a concept which can be well
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explained by the theory of micropolar fluids. Power [16] has shown that the fluid flowing in brain (Cerebrospinal fluid) is adequately modelled by micropolar fluids. The works of Migun [17] and Kolpashchikov et al. [18] demonstrated an experimental method of determining
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parameters characterizing the microstructure of such fluids and seems to have laid to rest many unanswered questions on the theory. Sheikholeslami et al. [19] analyzed theoretically micropolar
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fluid flow in a channel subject to a chemical reaction. Mosayebidorcheh [20] investigated twodimensional micropolar fluid flow in a porous channel with expanding or contracting walls. It
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was found that the profiles of the streamwise velocity and microrotation are asymmetric, while the normal velocity is symmetric. Zheng et al. [21] analyzed flow structures and radiative heat transfer of a micropolar fluid over stretching/shrinking sheet using an analytical technique [22, 23].
On the other hand, it should be stated that during the past several decades, extensive studies on heat transfer in regular cavities and enclosures filled with a viscous (Newtonian) fluid have been done and various extensions of the problem have been reported in the literature (see Vahl Davis [24]). However, it is necessary to study the heat transfer for more complex geometries because the prediction of heat transfer for irregular surfaces is a topic of great importance and irregular surfaces often occur in many applications (see Peterson and Ortega [25]). Recently, several studies have been performed on the convective flow in iregular (wavy) cavities filled with nanofluids (see Sheremet and Pop [26], Sheremet et al. [27–29], Sheikholeslami et al. [30]). It is also worth mentioning the recent book by Shenoy et al. [31] on
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convective flow and heat transfer from vawy surfaces: viscous fluids, porous media and nanofluids.
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Motivated by the practical importance of the micropolar fluids, the main objective of this
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paper is to understand the fundamentals of various heating and cooling strategies, and to achieve a high performance for the free convection in a partially heated wavy cavity filled with a
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micropolar fluid using the mathematical micropolar fluid model proposed by the pioneering papers of Eringen [1, 2]).
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2. Basic equations
We consider laminar natural convection flow and heat transfer in a differentially heated
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wavy cavity of length L and height H filled with a micropolar fluid. The geometry and coordinate system are shown in Fig. 1, where x axis is measured along the bottom wall of the cavity and y axis is measured along the vertical wall of the cavity. The upper and lower walls of
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the enclosure are thermally insulated and the fluid is isothermally heated and cooled from the left wavy wall and right flat wall at uniform temperatures of Th and Tc, respectively. All four walls of
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the cavity are supposed to be rigid and impermeable. It is considered that the left wavy wall and right flat wall of the cavity are described by the relations x1 = L - L éëa + b cos ( 2pky H )ùû and
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x2 = L , respectively.
Fig. 1. Physical model and coordinate system. 3
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The physical properties of the micropolar fluid are supposed to be constant except for the
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density in the buoyancy force term of the momentum equation that is approximated by the
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Boussinesq law. The governing equations for the micropolar fluid flow and heat transfer can be written on the basis of the conservation laws for the mass, linear momentum, angular momentum
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and energy in dimensional Cartesian coordinates as follows:
(1)
æ ¶ 2u ¶ 2u ö æ ¶u ¶u ö ¶p ¶N m k +v = + + ( ) ç 2 + 2 ÷ +k ÷ ¶y ø ¶x ¶y ø ¶y è ¶x è ¶x
(2)
æ ¶ 2v ¶ 2v ö æ ¶v ¶v ö ¶p ¶N m k +v = + + + rb (T - Tc ) g ( ) ç 2 + 2 ÷ -k ÷ ¶y ø ¶y ¶y ø ¶x è ¶x è ¶x
(3)
æ ¶2 N ¶2 N ö æ ¶N æ ¶v ¶u ö ¶N ö +v = g ç 2 + 2 ÷ +k ç ÷ ÷ - 2k N ¶y ø ¶y ø è ¶x è ¶x ¶y ø è ¶x
(4)
æ ¶ 2T ¶ 2T ö ¶T ¶T +v =a ç 2 + 2 ÷ ¶x ¶y ¶y ø è ¶x
(5)
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¶u ¶v + =0 ¶x ¶y
r çu
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r çu
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r j çu
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u
Further, we introduce the following dimensionless variables
y = y L, u = u
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x = x L,
g b (T - Tc ) L , v = v
q = (T - Tc ) (Th - Tc ) , N = N
g b (T - Tc ) L ,
g b (T - Tc ) L ,
(6)
æ ¶y ¶y ö ¶v ¶u ,v=stream function ψ ç u = . Therefore the governing Eqs. ÷ and vorticity w = ¶y ¶x ø ¶x ¶y è (1)–(5) using the dimensionless variables (6) can be written as follows
¶ 2y ¶ 2y + = -w ¶x 2 ¶y 2
(7)
Pr æ ¶ 2w ¶ 2w ö Pr æ ¶ 2 N ¶ 2 N ö ¶q ¶y ¶w ¶y ¶w K = (1 + K ) + + ç ÷ ç ÷+ Ra è ¶x 2 ¶y 2 ø Ra è ¶x 2 ¶y 2 ø ¶x ¶y ¶x ¶x ¶y
(8)
Pr ¶y ¶N ¶y ¶N æ K ö Pr æ ¶ 2 N ¶ 2 N ö = ç1 + ÷ (w - 2 N ) ç 2 + 2 ÷+ K 2 ø Ra è ¶x Ra ¶y ¶x ¶x ¶y è ¶y ø
(9)
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¶y ¶q ¶y ¶q = ¶y ¶x ¶x ¶y
1 Ra × Pr
æ ¶ 2q ¶ 2q ö ç 2+ 2÷ ¶y ø è ¶x
x = x1 = 1 - a - b × cos ( 2pk y A )
N = n × w , q = 1 on
N = n × w , q = 0 on
x =1
¶q = 0 on ¶y
N = n × w,
y = 0 and
(11)
y=A
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¶ 2y y = 0, w = - 2 , ¶x ¶ 2y y = 0, w = - 2 , ¶y
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¶ 2y ¶ 2y , ¶x 2 ¶y 2
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y = 0, w = -
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with the following boundary conditions
(10)
Here n (0 £ n £ 1) is a micro-gyration parameter.
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The physical quantities of interest are the local Nusselt number Nu along the hot wall and average Nusselt number Nu , that are defined as
æ ¶q ö Nu = - ç ÷ = - ( n,Ñq ) ì è ¶n ø x = x1 ï n =1 Nu ds S sò
s y üï 1 i+ jý 2 1+s y 1+s y2 ïþ
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Nu =
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í ïî
=
¶q 2 ¶x 1+ s y 1
x = x1
sy
¶q 1 + s ¶y 2 y
x = x1
(12)
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Here s ( y ) = 1 - a - b × cos ( 2pk y A) , s y = ds dy and S = ò 1 +
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-
0
4p 2k 2b2 æ 2pk y ö sin 2 ç ÷dy . 2 A è A ø
In order to solve the formulated boundary-value problem (7)–(11) we introduce new independent variables x and h in the following form:
x=
x - x1 x - 1 + a + b × cos ( 2pk y A) = , h=y a + b × cos ( 2pk y A) D
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Therefore Eqs. (7)–(10) are rewritten as follows:
éæ ¶x ö2 æ ¶x ö2 ù ¶ 2y ¶x ¶ 2y ¶ 2y ¶ 2x ¶y + + = -w êç ÷ + ç ÷ ú 2 + 2 ¶y ¶x¶h ¶h 2 ¶y 2 ¶x êëè ¶x ø è ¶y ø úû ¶x
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(14)
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2 2 ¶x ¶y ¶w ¶x ¶y ¶w Pr ïì éæ ¶x ö æ ¶x ö ù ¶ 2w = (1 + K ) + í êç ÷ + ç ÷ ú ¶x ¶h ¶x ¶x ¶x ¶h Ra ï êëè ¶x ø è ¶y ø úû ¶x 2 î
(15)
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¶x ¶ 2 N ¶ 2 N ¶ 2x ¶N ü ¶x ¶q +2 + + ý+ ¶y ¶x¶h ¶h 2 ¶y 2 ¶x þ ¶x ¶x
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2 2 ¶x ¶ 2w ¶ 2w ¶ 2x ¶w ü Pr ìï éæ ¶x ö æ ¶x ö ù ¶ 2 N +2 + + + ý- K í êç ÷ + ç ÷ ú ¶y ¶x¶h ¶h 2 ¶y 2 ¶x þ Ra ï ëêè ¶x ø è ¶y ø ûú ¶x 2 î
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2 2 ¶x ¶y ¶N ¶x ¶y ¶N æ K ö Pr ì ï éæ ¶x ö æ ¶x ö ù ¶ 2 N = ç1 + ÷ + í êç ÷ + ç ÷ ú ¶x ¶h ¶x ¶x ¶x ¶h è 2 ø Ra ï ëêè ¶x ø è ¶y ø ûú ¶x 2 î
(16)
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¶x ¶ 2 N ¶ 2 N ¶ 2x ¶N ü Pr +2 + + 2 (w - 2 N ) ý+ K 2 ¶y ¶x¶h ¶h ¶y ¶x þ Ra 2 2 ïì éæ ¶x ö æ ¶x ö ù ¶ 2q í êç ÷ + ç ÷ ú 2 + ïî êëè ¶x ø è ¶y ø úû ¶x
(17)
¶x ¶ q ¶ q ¶ x ¶q ü + 2+ 2 ý ¶y ¶x¶h ¶h ¶y ¶x þ 2
2
2
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+2
1 Ra × Pr
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¶x ¶y ¶q ¶x ¶y ¶q = ¶x ¶h ¶x ¶x ¶x ¶h
The corresponding boundary conditions for these equations are given by
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éæ ¶x ö 2 æ ¶x ö2 ù ¶ 2y y = 0, w = - êç ÷ + ç ÷ ú 2 , êëè ¶x ø è ¶y ø úû ¶x
N = n × w , q = 1 on x = 0
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2 æ ¶x ö ¶ y , N = n × w , q = 0 on x = 1 ÷ 2 è ¶x ø ¶x ¶ 2y ¶q = 0 on h = 0 and h = A y = 0, w = - 2 , N = n × w , ¶h ¶h
y = 0, w = - ç
(18)
The local and average Nusselt numbers will be defined as follows
æ ¶x ¶x ö ¶q Nu = -s y ÷ 2 ç ¶x ¶y ø ¶x 1+ s y è 1
1 4p 2k 2b2 æ 2pk y ö , Nu = ò Nu 1 + sin 2 ç ÷dy 2 S0 A è A ø x =0 A
(19)
The partial differential equations (14)–(17) with corresponding boundary conditions (18) were solved using the finite difference method of the second order accuracy. Detailed description of the used numerical technique is presented by Sheremet and Trifonova [32], Sheremet and Pop [26, 33], Sheremet et al. [27–29].
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For the purpose of obtaining grid independent solution, a grid sensitivity analysis is performed. The grid independent solution was performed by preparing the solution for unsteady
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free convection in a square wavy cavity filled with a micropolar fluid at Ra = 105, Pr = 0.7,
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K = 0.5, n = 0.0, k = 2, a = 0.9, A = 1. Three cases of the uniform grid are tested: a grid of 50´50 points, a grid of 100´100 points, and a much finer grid of 200´200 points. Figure 2 shows an
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effect of the mesh parameters on the average Nusselt number of the left vertical wavy wall (see
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Eq. (19)).
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Fig. 2. Variation of the average Nusselt number of the left vertical wavy wall versus the dimensionless time and the mesh parameters.
Taking into account the conducted verifications the uniform grid of 100´100 points has been selected for the further investigation.
4. Results and discussion Numerical analysis has been conducted at the following values of the governing parameters: Rayleigh number (Ra = 104, 105, 106), Prandtl number (Pr = 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), aspect ratio parameter (A = 1), undulation number (k = 1, 2, 3), wavy contraction ratio (a = 0.9). Particular efforts have been focused on the effects of these parameters on the fluid flow and heat transfer inside the cavity. Streamlines, isotherms, linear and angular vorticity profiles, average Nusselt
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number for different values of governing parameters mentioned above are illustrated in Figs. 3–10.
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Figure 3 shows an effect of the undulation number on isolines of stream function,
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temperature, linear vorticity and angular vorticity at Ra = 106, Pr = 0.7, K = 0.5, n = 0. Convective flow appearing inside the cavity due to a presence of horizontal temperature drop illustrates a formation of an ascending flow near the hot wavy wall and descending flow close to
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the right cold wall regardless of the undulation number value. An increase in k leads to modifications of both convective flow shape and position of isotherms inside the cavity. Taking
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into account the maximum absolute value of stream function it is possible to note insignificant changes of the fluid flow rate with the undulation number. Fields of linear vorticity characterize
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a distribution of hydrodynamic perturbations from solid walls to the cavity. Moreover one can find maximum values of w along the right wall and fluidward side of the left wavy wall. Distributions of angular vorticity illustrate a circulation of micropolar fluid with two cores.
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Variation of undulation number affects only the left core of N. Isotherms reflect an interaction between ascending and descending thermal boundary layers formed along the isothermal walls.
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It should be noted that an increase in undulation number leads to more essential heating of the
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wavy troughs and as a result we have the heat transfer reduction (see Fig. 4) with undulation
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Fig. 3. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Ra = 106, Pr = 0.7, K = 0.5, n = 0: k = 1 – a, k = 2 – b, k = 3 – c. Figure 4 shows the effect of the dimensionless time and undulation number on the rate of heat transfer. As has been mentioned above we have the heat transfer reduction with k. Also before the steady-state regime one can find a formation of transient regimes due to a transition from heat conduction mode at initial time to oscillation heat convection mode and after that to steady heat convection mode.
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Fig. 4. Variations of the average Nusselt number with dimensionless time and undulation number for Ra = 106, Pr = 0.7, K = 0.5, n = 0. In the case of micropolar fluid one of the governing parameters is a vortex viscosity
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parameter that characterizes an increase in the micropolar fluid viscosity due to an internal orientation of each particles [1, 2]. Figure 5 demonstrates an effect of this parameter on fluid
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flow and heat transfer. In the case of K = 0 (Fig. 5a) we have conventional Newtonian fluid. An increase in the vortex viscosity parameter leads to an attenuation of the convective flow with a
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decrease in the linear vorticity and an increase in angular vorticity. Essential influence of K on streamlines and isotherms can be found in the case of high vortex viscosity parameter values. For K = 2.0 (Fig. 5c) we have an essential modification of shapes of streamlines, linear and angular vorticity. Analysis of temperature fields shows an increase in the thermal boundary layers thickness with K that reflects less intensive convective heat transfer inside the enclosure. Figure 6 shows variations of the average Nusselt number at hot wavy wall with dimensionless time, vortex viscosity parameter and undulation number. As has been mentioned above an increase in K leads to a reduction of the heat transfer rate. Due to less intensive fluid flow for high values of K we have an obtaining of steady state with less time in comparison with Newtonian fluid.
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Fig. 5. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Ra = 106, Pr = 0.7, k = 2, n = 0: K = 0.0 – a, K = 0.1 – b, K = 2.0 – c.
Fig. 6. Variations of the average Nusselt number with dimensionless time and vortex viscosity parameter for k = 2 (a) and with undulation number and vortex viscosity parameter (b) for Ra = 106, Pr = 0.7, n = 0.
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The effect of Rayleigh number on streamlines, isotherms and vorticity patterns is presented in Figs. 7 and 3b. An increase in the buoyancy force leads to an intensification of
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convective flow and heat transfer with a decrease in the thermal boundary layers thickness.
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Small value of Ra ( = 104) illustrates weak convective flow with dominating of heat conduction. An increase in Ra leads to an intensification of circulation inside the cavity with linear and
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angular vorticity enhancement. It is worth noting that a presence of two linear vorticity cores having negative values of w near the vertical wall defines a formation of convective flow with two clockwise cores close to isothermal walls. Such fluid flow behavior can be found at Ra = 104
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and 105.
Fig. 7. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Pr = 0.7, K = 0.5, k = 2, n = 0: Ra = 104 – a, Ra = 105 – b. Intensification of convective heat transfer with Ra is shown in Fig. 8. An increase in the buoyancy force magnitude leads to an increase in time for an obtaining of the steady state regime. Also an effect of vortex viscosity parameter is more essential for high values of Rayleigh number.
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Fig. 8. Variations of the average Nusselt number with dimensionless time and Rayleigh number for K = 0.5 (a) and with vortex viscosity parameter and Rayleigh number (b) for Pr = 0.7, k = 2, n = 0. Figure 9 presents an effect of Prandtl number of micropolar fluids flow and heat transfer.
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Low values of Pr ( = 0.1) reflects a formation of complex internal hydrodynamic structure with three convective cores taking into account the linear and angular vorticity patterns. Such
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behavior of fluid flow reflects low heat transfer rate. An increase in Pr leads to an intensification
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of convective flow and heat transfer (see Fig. 10).
Fig. 9. Streamlines y, isotherms q, linear vorticity w and angular vorticity N at Ra = 106, K = 0.5, k = 2, n = 0: Pr = 0.1 – a, Pr = 7.0 – b.
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Figure 10 shows the heat transfer enhancement with Prandtl number. More intensive increase in Nu occurs for an increase in Pr from 0.1 to 0.7. Also it should be noted that an effect
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of the vortex viscosity parameter is more essential for high values of Prandtl number.
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Fig. 10. Variations of the average Nusselt number with dimensionless time and Prandtl number for K = 0.5 (a) and with vortex viscosity parameter and Prandtl number (b) for Pr = 0.7, k = 2, n = 0. 5. Conclusion
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Natural convection of micropolar fluid in a wavy-walled differentially heated cavity of unit aspect ratio has been studied numerically. Mathematical model has been formulated in dimensionless variables stream function and vorticity taking into account the conservation laws for mass, linear momentum, angular momentum and energy. Analysis of the fluid flow and heat transfer in terms of streamlines, isotherms, linear and angular vorticity patterns and average Nusselt number has been conducted in a wide range of the Rayleigh number, Prandtl number, vortex viscosity parameter and undulation number. On the basis of the obtained results we can conclude that 1. An increase in Rayleigh number leads to an intensification of circulation inside the cavity with linear and angular vorticity enhancement. At the same time one can find an increase in time for an obtaining of the steady state regime for high values of Ra. The effect of vortex viscosity parameter is more essential for high values of Rayleigh number. 2. Low values of Prandtl number illustrate an appearance of complex internal hydrodynamic structure with several convective cores. The fluid flow rate and average Nusselt
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number are the increasing functions of Prandtl number. An effect of the vortex viscosity parameter is more essential for high values of Prandtl number
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3. Thermal boundary layers thickness increases with the vortex viscosity parameter. The
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fluid flow and heat transfer rates decrease with K. Due to less intensive fluid flow for high values of K we have an obtaining of steady state with less time in comparison with Newtonian fluid.
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4. An increase in undulation number leads to the heat transfer reduction at hot wavy wall.
Acknowledgement
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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
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13.1919.2014/K.
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[1] A.C. Eringen, Theory of micropolar fluids, J. Math. Mech. 16 (1966) 1–18. [2] A.C. Eringen, Theory of thermomicrofluids, J. Math. Anal. Appl. 38 (1972) 480–496.
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[3] R. S. Agarwal, R. Bhargava, A.V. S. Balaji, Finite element solution of unsteady threedimensional micropolar fluid flow at a stagnation-point, Int. J. Engng. Sci. 28 (1990) 851–
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[11] T. Ariman, M.A. Turk, N.D. Sylvester, Microcontinuum fluid mechanics – a review, Int. J. Engng. Sci. 11 (1973) 905–930.
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J. Engng. Sci. 12 (1974) 273–293.
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[14] A.C. Eringen, Microcontinuum Field Theories. II: Fluent Media, Springer, New York, 2001. [15] J.W. Hoyt, A.G. Fabula, The effect of additives on fluid friction, Tech. rep., US Naval Ordnance Test Station Report, 1964.
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[16] H. Power, Micropolar fluid model for the brain fluid dynamics, Int. Conf. Bio-Fluid
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Highlights Natural convective flow and heat transfer of a micropolar fluid is studied
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A wavy differentially heated cavity has been consiidered
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Governing partial differential equations have been solved by finite difference method
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It is found that microrotation increases as the vortex viscosity parameter K increases
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