MIXED CONVECTION HEAT TRANSFER FOR BINGHAM FLUID IN SQUARE CAVITY WITH PARTITIONS

T. BENMALEK*, F. SOUIDI, B. YSSAAD

Laboratory of applied fluid mechanics, Faculty of Physics, USTHB University, Bab ezzouar 16111, Algeria

Abstract

Two-dimensional steady mixed convection flow in an enclosure with partitions filled with a Bingham fluid is considered. The vertical walls are maintained at different constant temperatures and they are moving in opposite direction. The upper and the bottom walls are fixed and thermally insulated. The governing equations are normalized and solved numerically by the finite volume control. A parametric investigation is performed and a set of isotherms and streamlines are presented. The results shown that the decrease of Richardson number increase the fluid flow and enhanced the heat transfer. The effect of the yield-stress in the presence of the partition is to drop the fluid motion and to augment heat transfer where the conduction mode is dominant.

Keywords: mixed convection, heat transfer, Bingham fluid, lid-driven cavity

1. Introduction The problem of fluid flow and heat transfer in lid-driven enclosures has been studied extensively because of its wide engineering applications such as cooling of electronic devices, float glass production, and food processing, lubrication technologies, etc. In these problems, mixed convection is the outcome of interaction between forced convection induced by moving walls and the natural convection due to temperature gradients.

Many experimental [1], [2] and numerical investigations [3], [4], [5], [6], [7] focus on this type of flow. The effects of aspect ratio, Reynolds number and imposed temperature gradients are analyzed. E. Mitsoulis and Th.Zisis [8] conducted a study on the purely dynamic flow of a plastic Bingham fluid in lid driven cavity.

More recently, Doh and Muthtamilselvan [11] have studied heat generating of nanofluid in a lid-driven cavity with diferent heating of the bottom wall. Abu-nada and Chamkha[12] have analyzed mixed convection of a nanofluid in a lid driven enclosure with a wavy wall. Karimipour et al [13] have used the lattice Boltzmann method to investigate mixed convection nanofluid in a inclined cavity with moving lid . Sahin et al [14] have analyzed the effects of aspect ratio on natural convection of Bingham fluid in rectangular cavity heated from bellow for different parameters as Ra and Bn numbers. They found that heat transfer decreases with the increase of Bingham number and aspect ratio.

In this study, a numerical investigation is conducted to analyze the steady state two-dimensional mixed convection in square cavity with partitions, filed by a Non Newtonian Bingham fluid.

The Bingham fluid is one of the simplest models that describe materials with yield stress which must

be exceeded before noteworthy deformation can arise. It is characterized by a flow curve (shear stress (τ)

as a function of shear rate (��)) which is a straight line having an intercept (0τ ) on the shear stress axis,

and it is this yield stress that must be exceeded before flow is possible, the rate of deformation being proportional to the excess of the stress over the yield condition. The general form of the Bingham model is presented and is summarized as:

� γ� = 0τ < τ��τ = τ� + μ�γ� τ > τ�

To avoid the discontinuity on the shear stress, we used the following equation proposed by Papanastasiou [9]:

η �� = µ� + τ�γ� �1 − exp�−mγ� ��, (1)

TOPICAL PROBLEMS OF FLUID MECHANICS 37_______________________________________________________________________DOI: https://doi.org/10.14311/TPFM.2017.006

where ‘m’ is a constant which depends on µp and τ0. This relation is representative of an ideal Bingham fluid as reported by several authors [8,10].

2. Problem Description The physical model is presented in Fig.1. A two-dimensional square enclosure of height H is filled with a non-newtonian Bingham fluid. The left wall and the right wall are maintained at hot and cold constant temperatures respectively, and they move in opposite direction. The other walls and partitions are fixed and thermally insulated.

It is assumed that the flow is two-dimensional, steady and laminar, the fluid is incompressible. The thermophysical properties of the fluid are assumed to be constant except for the body force term in the momentum equation, where the Boussinesq approximation is considered.

3. Mathematical Model With the assumptions mentioned above, the dimensionless form of the governing equations can be written as:

0u v

x y

∂ ∂+ =∂ ∂

(2)

1 12

Re Reapp app

u u p u u vu v

x y x x x y y xη η

∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂ + = − + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(3)

2

1 12

Re Re Reapp app

v v p u v v Gru v

x y y x y x y yη η θ ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂+ = − + + + + ∂ ∂ ∂ ∂ ∂ ∂ ∂ ∂

(4)

2 2

2 2

1

RePru v

x y x y

θ θ θ θ ∂ ∂ ∂ ∂+ = + ∂ ∂ ∂ ∂

(5)

Where .

.1 1 expapp

Bnmη γ

γ

= + − −

(6)

The following non-dimensional variables are defined as:

' ' ' ' ' '

20 0 0 0

, , , , , , ,f

c f

T Tx y u v px y u v p

L L U U U T T

µµ θρ µ

−= = = = = = =−

The boundary conditions read:

0,0,0,10,1

0,0,0,10,0

1,1,0,10,1

0,1,0,10,0

=∂∂===

=∂∂===

=====−===

yvuxy

yvuxy

vuyx

vuyx

θ

θθ

θ

pp

pp

pp

pp

Figure 1: physical model

�� ��⁄ = 0

�� ��⁄ = 0

� = 0 � = 1

��

��

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for partitions surfaces ∂θ/∂y = 0 Nusselt number is given by � ! = "#$ = − %&

%'(')�

The average Nusselt number is calculated by integrating the local Nusselt number along the cold wall

dyNuNu yy ∫=1

0 (7)

4. Numerical Solution and Code Validation The discretization of the governing equations is based on a finite volume method with the power law scheme, using the SIMPLER algorithm to achieve the coupling between pressure and velocity [16]. The convergence criteria were set to 10-5 for all the relative (scaled) residuals.

4.1 Grid Independency Test To obtain a grid independency, an analysis of grid sensitivity is performed for the mixed convection in the cavity.

Five grids were tested as shown in table.1 which depicts the correspondent average Nusselt numbers. Table.1 shows clearly that the difference between the results of the average Nusselt number obtained for the 81x81 and the 101x101 grid size is negligible. For this reason, the grid of 81x81 nodes is adopted for all computations reported in this study.

Table 1: grid independent test Mesh 21*21 41*41 61*61 81*81 101*101

Nu (Ri = 10) 2.8573 2.8602 2.8612 2.8614 2.8615

Nu (Ri = 1) 4.0756 4.8487 4.8597 4.860 4.8605

Nu (Ri = 0.1) 8.0153 8.7861 8.8696 8.87 8.8706

4.2. Code Validation The present code has been subject to two validation tests: The mixed convection in two-sided lid driven differentially heated square cavity [6], and Flow of Bingham fluid in a lid-driven square cavity [8]. The streamline and isotherm plots for the case two, velocity profiles and pressure distribution presented in Fig.2, Fig.3, Fig.4 respectively present good agreement comparing with the previous results.

Figure 2: Comparison of streamline and isotherm plots of the present work and of Hakan Oztop for cas.II

TOPICAL PROBLEMS OF FLUID MECHANICS 39_______________________________________________________________________

(a)

(b) RI=0.1 RI=1 RI=10

Figure 5: Streamlines (a) and isotherms (b) plots for different Ri numbers at Bn =2

5. Analysis Mixed convection heat transfer in square cavity with two moving lids of a Bingham non-Newtonian fluid is examined. The governing parameters in this problem are Richardson number (Ri = Gr/Re2) which characterizes the convection regime and Bingham number which characterizes the rheological fluid behavior.

5.1 Richardson Number Effect Richardson number varies from 10-2 to102 through the Reynolds number. Grashof and Bingham numbers are kept constant at 104 and 5 respectively.

Streamlines and isotherms are presented in fig.5. For Ri = 0.1, where forced convection is dominant. The flow is made of two contra-rotating cells evolving in the direction imposed by the moving walls (the forced momentum diffuses up to the central axis). The isotherms show that the heat depth penetration is

0.1

0.20.3

0.4

0.5

0.70.8

0.9

0.6

0.10.2

0.3

0.40.5

0.6

0.7

0.8

0.9

0.04

0.25

0.35

0.55

0.15

0.070.450.65

0 .80

0. 90

-1.0 -0.5 0.0 0.5 1.00.0

0.2

0.4

0.6

0.8

1.0

PRESENT RESULTS

H.F. Oztop

Ri=0.01 Ri=0.01 Ri=0.1 Ri=0.1 Ri=1 Ri=1

Y

U

0.0 0.2 0.4 0.6 0.8 1.0

-400

-300

-200

-100

0

100

200

300

400

Bn=0 Bn=0 Bn=2 Bn=2 Bn=20 Bn=20

P

X /H

RESULTATS of MITSOULIS

PRESENT RESULTS

Figure 3: Comparison Velocity profiles at the vertical centreline for different Ri numbers

Figure 4: Comparison of pressure distribution along the lid for different values of Bn number

40 Prague, February 15-17, 2017_______________________________________________________________________

not significant, and its effect is felt only near the walls where the left upper corner and the right lower corner are the seat of intense thermal activity. As Richardson number increases the flow becomes flatter. For Ri = 10, natural convection is dominant, the main cell is formed due to the heat depth penetration. The effects of the moving walls are felt only in their immediate vicinity. Parallel isotherms show that the mode of conduction is dominant, and the thermal activity spread all over the walls.

Fig. 6 shows that as Richardson number increases, heat transfer decreases. This is estimated since for high Richardson number, natural convection becomes dominant and the flow motion is generally subdued. As a result conduction mode is dominant for heat transfer.

Figure 6: Local Nusselt number along the cold wall for different Ri numbers at Bn = 5

(a)

(b) Bn = 0 Bn = 5 Bn = 10

Figure 7: streamlines (a) and isotherms (b) plots for different Bn numbers at Ri = 1

0.10.2

0. 5

0.4

0.7

0.8

0.9

0.6

0.4

0.3

0.10.2

0.3

0.40.5

0.6

0.7

0.8

0.9

0.10.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

0,0 0,2 0,4 0,6 0,8 1,00

20

40

60

Ri=100Ri=10Ri=1

Ri=0.1

Nu

Y

Bn=5

Ri=0.01

TOPICAL PROBLEMS OF FLUID MECHANICS 41_______________________________________________________________________

5.2 Bingham Number Effect Streamlines and isotherms for Bingham number varying from 0 to 10 at Ri = 1 are shown in Fig. 7. In this case, forces due to moving lid and buoyancy are comparable. For law Bn number (Bn = 0) witch the case of Newtonian fluid, the flow is formed by three cells. The centre one is due to depth penetration of the buoyancy effect where the others are formed due to the moving walls. By increasing Bn number, the flow becomes flatter and made of two contra-rotating cells evolving in the direction imposed by the moving walls .The main cell disappeared in the center of the cavity, and the two contra rotating cells merged for height Bn number, this is due to the fact that the velocity of the fluid flow decreases with the increase of Bn number as seen in Fig. 8.

In fig.8 which represents velocity profile for different Bn number at Ri = 1, it is clear that velocity magnitude decreases when Bn number is increased, where the unyielded areas increase and the fluid moves as a solid for high plasticity.

The variation of Nusselt number is presented in Fig. 9 for different Bingham numbers. In this figure we can see that the heat transfer rate is low for Bn = 0 where the conduction mode is dominate (parallel isotherms in the centre). For high Bingham number the main cell disappears and the convection mode dominate. As a result, the temperature gradient near the active walls increases as presented by isotherms in Fig.7 and the heat transfer enhanced.

Figure 8: Velocity profiles at the vertical centerline for different Bn numbers (Ri = 1)

Figure 9: Local Nusselt number along the cold wall for different Bn numbers at Ri = 1

0,0 0,2 0,4 0,6 0,8 1,0-1,0

-0,8

-0,6

-0,4

-0,2

0,0

0,2

0,4

0,6

0,8

1,0

Vy

X

Bn=0 Bn=2 Bn=5 Bn=10

Ri=1

0,0 0,2 0,4 0,6 0,8 1,00

1

2

3

4

5

6

Nu

Y

Bn=0 Bn=2 Bn=5 Bn=10

Ri=1

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6. Conclusion A numerical investigation was conducted to analyze the steady state two-dimensional mixed convection in a two sided lid-driven square cavity with partitions filled by a Non-Newtonian Bingham fluid. The elliptic equations are solved using the finite volume control technique.

The results show that heat transfer as given by the Nusselt number and the velocity of the fluid flow increase with the decrease of the Richardson numbers; and they decrease with the increase of Bingham number, where the fluid moves as a solid for high plasticity.

NOMENCLATURE

Bn Bingham number c specific heat of fluid g gravity h local heat transfer coefficient k thermal conductivity L the length of the cavity m regularization parameter Nu Nusselt number P pressure Pr Prandtl number Re Reynolds number Ri Richardson number T temperature Vp plate velocity u velocity in x direction v velocity in y direction

Greek letters β thermal expansion coefficient µp plastic viscosity *app apparent or effective viscosity �� rate of strain tensor α thermal diffusivity ρ density + extra tensor stress +0 fluid yield stress θ dimensionless temperature Subscripts C cold H hot x,y cartesian coordinates

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[16] Patankar, S.V.: Numerical Heat transfer and fluid flow, Hemisphere, Washington, DC, 1980

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