Meccanica (2012) 47:1349–1357 DOI 10.1007/s11012-011-9518-x
Free convection from a truncated cone subject to constant wall heat flux in a micropolar fluid Adrian Postelnicu
Received: 16 September 2010 / Accepted: 10 November 2011 / Published online: 6 December 2011 © Springer Science+Business Media B.V. 2011
Abstract The paper studies the problem of free convection about a vertical frustum of a cone in a micropolar fluid. It is assumed that the flow is laminar, steady and the wall is subjected to a constant heat flux and the angle of the frustum of the cone is large enough so that the transverse curvature effects are negligible. Under these assumptions, the governing boundary layer equations subject to appropriate boundary conditions are transformed into a set of equations of parabolic type, that are solved using the local nonsimilarity method. The space of parameters contains the Prandtl number Pr, the micropolar parameter and the microrotation parameter n. Numerical solutions are obtained by varying Pr from 6.7 to 100, from 0 (Newtonian fluid) to 2 and considering two values of n with physical significance (0 and 0.5). Flow and heat transfer characteristics are determined and are shown in graphs. The results are discussed and compared at some extent with those reported by the present author in a previous study (Postelnicu in Int. J. Eng. Sci. 44:672–682, 2006) on the isothermal case. Keywords Free convection · Micropolar fluid · Truncated cone · Constant wall heat flux A. Postelnicu () Department of Fluid Mechanics and Thermal Engineering, Transilvania University, Brasov 500036, Romania e-mail:
[email protected]
1 Introduction The micropolar fluids are non-Newtonian fluids with microstructure, such as polymeric additives, colloidal suspensions, liquid crystals, etc., in which the microscopic effects arising from local structure and micromotions of the fluids elements are taken into account. Eringen developed the theory of micropolar fluids and extended it for thermomicropolar fluids [1, 2]. Comprehensive reviews of the subject and various applications of micropolar fluids at level of ‘70s are found in Ariman et al. [3, 4]. Recent textbooks focusing on the theory and applications of micropolar fluids are those by Lukaszewicz [5] and Eringen [6]. The boundary layer concept is useful also in micropolar fluids and we mention here Peddiesen and McNitt [7] and Wilson [8] who initiated this direction of analysis. In engineering, applications of micropolar fluids include solidification of liquid crystals, lubricants and tribology, colloidal suspensions, while in biology the animal blood may be modeled as a micropolar fluid. Free convection flow from cones constitutes an important heat transfer problem for engineering purposes. In viscous fluids, there are many papers dealing with the free convection about the vertical frustum of a cone. A seminal paper seems to be that by Na and Chiou [9], where both constant wall temperature and constant wall heat flux cases have been considered. Later on, Singh et al. [10] approached the constant wall heat flux case for Newtonian fluids, using as numerical method the local nonsimilarity method. Wavy
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frustums [11], as well as rotating truncated cones [12], have been studied for clear fluids. Such studies seem to be very rare in micropolar fluids, to the author’s best knowledge. Gorla and Takhar [13] dealt with the free convection boundarylayer flow of a micropolar fluid past slender bodies, Gorla and Nakamura [14] studied the mixed convection of a micropolar fluid from a rotating cone. AboEldahab and El Aziz [15] solved recently the problem of laminar mixed convection boundary layer flow of a electrically conducting micropolar fluid from a constantly rotating cone, taking into account the effects of Hall current and Ohmic heating. Postelnicu [16] and Cheng [17] dealt with the same geometry in natural convection of micropolar fluids, by considering isothermal wall conditions and temperature power-law variation, respectively. There are also several papers dealing with this theme for other non-Newtonian fluids than micropolar. A recent example is Cheng [18], where the doublediffusion flow over a vertical truncated cone located in a porous medium was studied. The medium is saturated with a power-law fluid and variable heat and mass fluxes are applied to the wall. There are of course other articles focusing on this configuration in porous media, but these are out of our present scope. Thus, the objective of the present paper is the study of free convection about a vertical frustum of a cone in a micropolar fluid, using the boundary layer assumptions, in the conditions of a uniform heat flux imposed to the wall. It is worth mentioning that the present study complements the paper by Postelnicu [16], called hereinafter Part 1.
Fig. 1 Physical model for a truncated cone
• momentum equation ∂u ∂ 2u ∂u ∂N +v = (μ + κ) 2 + κ ρ u ∂x ∂y ∂y ∂y + ρgβ(T − T∞ ) • angular momentum equation ∂N ∂N ∂u ρj u +v = −κ 2N + ∂x ∂y ∂y +γ
Figure 1 shows the flow model and physical coordinate system. The origin of the coordinate system is placed at the vertex of the full cone, where x is the coordinate along the surface of cone measured from the origin and y is the coordinate normal to the surface. The flow is laminar, steady and all the fluid properties are assumed to be constant except for density variations in the buoyancy force term. Introducing the boundary layer and Boussinesq approximations, the governing equations are • continuity equation ∂ ∂ (ru) + (rv) = 0 ∂x ∂y
(1)
∂ 2N ∂y 2
(3)
• energy equation u
2 Analysis
(2)
∂T ∂T ∂ 2T +v =α 2 ∂x ∂y ∂y
(4)
In the above equations x0 is the distance of the leading edge of the truncated cone measured from the origin; r the radius of the truncated cone; u and v are the velocity components in x- and y-directions; N the microrotation component in the x–y plane; β coefficient of thermal expansion; ρ the density; μ the dynamic viscosity coefficient; α the thermal diffusivity and T the temperature. Various terminologies exist in the literature for κ: microrotation parameter, coefficient of rotational viscosity, coefficient of giroviscosity, or vortex viscosity. The spin-gradient viscosity γ is given by the equation: γ = (μ + κ/2)j , where j is the microinertia density. This later quantity will be assumed constant for the
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present study, as in many other papers dealing with micropolar fluid flows. We omit intentionally to list here various references, but a personal survey of the literature reveals that approximately three quarters of the papers in the field accept this assumption. Anyway, it is not a limitation, but only a starting point, otherwise another equation of conservation for j must be added to (1)–(4). The boundary conditions of the problem are ∂u , ∂y on y = 0, x0 ≤ x < ∞
u = v = 0,
u → 0,
N = −n
N → 0,
qw = −k
∂T , ∂y (5a)
T → T∞
as y → ∞ (5b)
where qw is a constant heat flux applied to the wall. The second boundary condition in (5a) states that the microrotation vector is proportional to the fluid shear stress, which is commonly accepted in the literature. This proportionality is quantified by the microrotation parameter n which is a constant with values between 0 and 1: • n = 0 represents the case of concentrated particle flows in which the microelements close to the wall are not able to rotate, see Jena and Matkur [19]; • the case n = 1/2 indicates the vanishing of antisymmetric part of the stress tensor and denotes weak concentrations, according to Ahmadi [20]; • the case n = 1 is representative for turbulent boundary layer flows, as suggested in Peddieson [21]. We look for similarity solutions in the form y 1/5 1/5 η = ∗ Grx ∗ , ψ = νrGrx ∗ f (ξ, η), x 3/5
νGrx ∗ N= g(ξ, η), x ∗2 θ (ξ, η) =
(6)
T − T∞ 1/5
qw x ∗ /(kGrx ∗ )
,
ξ=
x∗ x0
where ψ is the stream function, defined through the usual relationships u=
1 ∂ψ , r ∂y
v=−
1 ∂ψ r ∂x
(7)
in order to satisfy the continuity equation (1). Further, gβ(qw x ∗ /k)x ∗ 3 cos δ ν2 is the local Grashoff number and x ∗ = x − x0 . Grx ∗ =
After some algebra, the governing equations (2)– (4) become 4 ξ 3 + ff − f 2 + g + θ (1 + )f + 1+ξ 5 5 ∂f ∂f =ξ f − f (8) ∂ξ ∂ξ 1 4 ξ 2 1 + g + + f g − f g 2 1+ξ 5 5 ∂g ∂f − g (9) − ξ 2/5 2g + f = ξ f ∂ξ ∂ξ 4 1 ξ 1 θ + + f θ − f θ Pr 1+ξ 5 5 ∂θ ∂f − θ (10) =ξ f ∂ξ ∂ξ where Pr = α/ν is the Prandtl number and = K/μ is a dimensionless material parameter. In the above equations, primes denote the differentiation with respect to η. In deriving (9), the microinertia density was 2/5
gβ(q x /k)x 3 cos δ
w 0 0 . taken as j = x02 /Grx0 , where Grx0 = ν2 Equations (8)–(10) must be solved along the boundary conditions
f (ξ, 0) = 0,
f (ξ, 0) = 0,
θ (ξ, 0) = −1,
g(ξ, 0) = −nf (ξ, 0)
f (ξ, ∞) = 0,
θ (ξ, ∞) = 0,
(11a)
g(ξ, ∞) = 0 (11b)
We mention that (8) and (10) reduce for = 0 (i.e. for Newtonian fluids) to those derived in [10]. On the other hand, the problem (8), (9), (10), (11a), (11b) becomes at ξ = 0 4 3 (12) (1 + )f + ff − f 2 + g + θ = 0 5 5 1 4 2 1 + g + f g − f g = 0 (13) 2 5 5 1 4 1 θ + fθ − f θ =0 (14) Pr 5 5 f (0) = 0, f (0) = 0, (15a) g(0) = −nf (0) θ (0) = −1, f (∞) = 0,
θ (∞) = 0,
g(∞) = 0
(15b)
and this in fact a similarity formulation. Obviously (12), with = 0, and (14) reduce to (3.5) from the paper by Singh et al. [10]. We remark that, in contradistinction with the Newtonian case, studied by these authors, no similarity solutions exist here for ξ → ∞, due to the presence of the term ξ 2/5 (2g + f ) in (9).
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3 Numerical analysis
Equations (16) to (21) must be solved along the boundary conditions
The method used in the present numerical analysis is the local non-similarity method, for short LNS. This method, devised for solving parabolic equations, typical for boundary layers, was introduced by Sparrow et al. [22] and then applied to thermal problems by Sparrow and Yu [23]. LNS was successfully used over the years, see for instance Hossain et al. [24], Singh et al [10]. Both 2-equations and 3-equations models were considered in these papers, and it was concluded that there is no significant difference between the results given by these two levels of truncation. To obtain the 2-equations model, (8)–(10) are rewritten as ξ 3 4 (1 + )f + ff − f 2 + g + θ + 1+ξ 5 5 = ξ f F − f F (16) 1 4 ξ 2 1 + g + + f g − f g 2 1+ξ 5 5 2/5 − ξ (17) 2g + f = ξ f G − g F 1 4 ξ 1 θ + + f θ − f θ Pr 1+ξ 5 5 =ξ f −θ F (18)
f (ξ, 0) = 0,
∂g ∂θ where F = ∂f ∂ξ , G = ∂ξ and = ∂ξ . In a second step, we perform the differentiation of (8)–(10) with respect to ξ and neglecting the second order derivatives ∂ 2 /∂ξ 2 , we get 4 1 ξ (1 + )F + + ff + 1+ξ 5 (1 + ξ )2 6 × f F + f F − f F + G + 5 (19) − f F + f F = ξ F 2 − F F 4 1 ξ 1+ G + + f g + 2 2 1+ξ 5 (1 + ξ ) 7 2 × f G + g F − f G − F g + g F 5 5 2 −3/5 2g + f − ξ 2/5 2G + F − ξ 5 = ξ F G − G F (20) 1 4 1 ξ + + f + θ F f θ − 2 Pr 1+ξ 5 (1 + ξ ) 6 1 − f − F θ + θ F = ξ F − F (21) 5 5
• Wall shear-stress ∂u τw = (μ + κ) + κN ∂y y=0
f (ξ, 0) = 0,
g(ξ, 0) = −nf (ξ, 0), F (ξ, 0) = 0,
F (ξ, 0) = 0,
G(ξ, 0) = −nF (ξ, 0), f (∞) = 0,
θ (ξ, 0) = −1
(ξ, 0) = 0
(22a)
(22b)
g(ξ, ∞) = 0,
θ (ξ, ∞) = 0,
F (ξ, ∞) = 0,
G(ξ, ∞) = 0,
(ξ, ∞) = 0
(23)
These equations can be treated as a system of differential equations, since ξ may be considered as a prescribed parameter. Consequently, they are solved numerically using an integrator for boundary value problems governed by ordinary differential equations. For this purpose, the dsolve routine from MAPLE [25] was used.
4 Results We are interested in computing some quantities of physical interest, such as:
3/4 νGrx ∗ (1 + )f (ξ, 0) + g(ξ, 0) (24) ∗2 x Consequently, one may define a reduced wall shear-stress (which is in fact a skin friction coefficient) as
=μ
τ¯w = (1 + )f (ξ, 0) + g(ξ, 0)
(25)
or, on using (11a), τ¯w = 1 + (1 − n) f (ξ, 0)
(26)
Local couple stress at the wall ∂N νGrx ∗ Mw = γ = γ ∗3 g (ξ, 0) ∂y y=0 x
(27)
• Nusselt number Nux ∗ =
qw x ∗ 1 −1/5 , Nux ∗ Grx ∗ = k(Tw − T∞ ) θ (ξ, 0)
(28)
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Table 1 Comparison between the present results for Newtonian fluids ( = 0) and Pr = 10 with those reported by Singh et al. [10]
Table 2 Comparison between the results given by the 2- and 3-equations model ξ
f (ξ, 0)
Pr
n
Model f (ξ, 0) g (ξ, 0)
10 0.7
0
1
θ(ξ, 0)
θ(ξ, 0)
Singh et al. [10]
Present
Singh et al. [10]
Present
ξ =0
0.5832
ξ = 21.75
0.4544
0.5825
1.0590
1.0581
0.4540
0.9337
0.9331
ξ = 127.75
0.4501
0.4503
0.9315
0.9319
The values of the Prandtl number, chosen as representative, were 6.7, 10 and 100 (micropolar liquids). The microrotation parameter n was set to 0 and 0.5, based on the explanations given in the Sect. 2. Finally, the material parameter was set 0, 1 and 2. One claims the convergence of the LNS method may become increasingly difficult for large ξ values. However, no such problems were encountered in the present analysis, where we found a smooth stabilization of the solution for very large ξ . We compared our numerical solutions in the case of Newtonian fluids ( = 0) and Pr = 10 with those reported by Singh et al. [10] (obtained with their 2equations model), and the results are in good agreement, as seen in Table 1. We tested also the results given by the 3-equations model, whose description is skipped here in order to conserve space. Table 2 shows a comparison between the results obtained by these two levels of truncation. One can remark that these results are practically coincident. Accordingly, we decided to use further the 2-equations model. Numerical results are displayed graphically in Figs. 1 to 3, for several combinations of the parameters n, and Pr, chosen as mentioned above. An overall view on these figures shows that the behaviors of f (ξ, 0) and θ (ξ, 0) present a tendency to acquire constant values as ξ increases. The numerical results demonstrated a value ξ = 1000 was enough for their stabilization. However, in the isothermal case, considered in Part 1, a smaller maximum value ξ = 100 was enough to achieve this behavior. A wrong conclusion would be that the flow adapts heavier when a (constant) wall heat flux is imposed than in isothermal conditions, since in Part 1, the streamwise variable was −1/2 defined as ξ = (x ∗ /x0 )2 Grx ∗ , with another (appropriate) definition of the local Grashoff number.
10 0.7
1 6.7
1 6.7
100 6.7
100 6.7
0
0
0
2
1
2
0.5 1
0.5 2
1000 100 0.5 1
1000 100 0.5 2
2 eq.
0.82098
−0.40047 1.97906
3 eq
0.82098
−0.40047 1.97906
2 eq.
0.63270
−0.47369 2.0842
3 eq
0.63270
−0.47369 2.0842
2 eq.
0.38429
−0.06948 1.22855
3 eq
0.38429
−0.06948 1.22855
2 eq.
0.29943
−0.08802 1.31859
3 eq
0.29943
−0.08802 1.31859
2 eq.
0.40179
0.25538 1.11060
3 eq
0.40179
0.25538 1.11060
2 eq.
0.33871
0.21510 1.17150
3 eq
0.33871
0.21510 1.17150
2 eq.
0.13701
0.14046 0.63728
3 eq
0.13701
0.14046 0.63728
2 eq.
0.11517
0.11991 0.67492
3 eq
0.11517
0.11991 0.67492
Other remarks on the graphs grouped in Fig. 2 are listed below. • The main variations of the skin friction occur for ξ between 0 and (500 . . . 1000), in other words this quantity acquires an almost constant value at large values of ξ . • The well-known feature of micropolar fluids to reduce drag in comparison with the viscous fluids ( = 0) is retrieved in the present analysis, as observed in Fig. 2. • The effect of the Prandtl number is to decrease the skin friction, either in viscous fluids ( = 0), either in micropolar fluids, at fixed values of and n. • The effect of n, at fixed Pr and , is to increase the skin friction as n departs from 0. For a similar conclusion, see for instance Gorla et al. [26] (their Fig. 11), in a study of free convection over a vertical plate. The local couple stress at the wall, represented by g (ξ, 0), is shown in Fig. 3. It is negative and decreases with ξ when n = 0, while when n = 0.5 it is positive
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Fig. 2 Skin friction coefficient variation in the streamwise direction: (a) Pr = 6.7, (b) Pr = 10, (c) Pr = 100
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Fig. 3 Streamwise variation of the local couple stress at the wall: (a) n = 0, (b) n = 0.5
and the curves are increasing in the downstream direction, with smaller slopes. It is worth mentioning that large Prandtl numbers produce less effect on the local couple stress at the wall. • Another important remark is about the same monotonicity of the curves of downstream variation of local couple stress at the wall and skin friction when n = 0 and n = 0.5, respectively. This behavior was found also in the isothermal case (Part 1) when n = 0. • At n = 0.5, Fig. 3b shows, that for any Prandtl number the local couple wall stress is decreased when the material parameter increases (as in Part 1). We notice that an increase of signifies a preva-
lence of the gyroviscosity coefficient over the absolute viscosity. The streamwise variation of the Nusselt number is rep−1/5 resented in Fig. 4, where Nuss denotes Nux ∗ Grx ∗ , according to (28). • We have again to discern between the cases n = 0 and n = 0.5: in the former case, the rate of heat transfer to the wall is lesser, probably due to the fact that smaller mobility of the microelements close to the wall (characteristic when n = 0) do not work in the favor of heat transfer. • At fixed n, we have Nu( = 0) > Nu( = 1) > Nu( = 2), as in the isothermal case (Part 1).
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Fig. 4 Nusselt number variation in the streamwise direction: (a) Pr = 6.7, (b) Pr = 10, (c) Pr = 100
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• At fixed and n, Nusselt number clearly increases with Prandtl for the whole range of ξ . • The well-known feature of micropolar fluids to reduce surface heat transfer with respect to the viscous fluids ( = 0) is retrieved in the present analysis, as shown in Fig. 4.
5 Conclusion This paper was intended to analyze the steady laminar free convection flow about a truncated vertical cone in a micropolar fluid. Flow and heat transfer characteristics such skin-friction, couple stress at the wall and Nusselt number have been obtained for several combinations of Prandtl number, micropolar and microrotation parameters. Qualitative features of the behavior of these quantities in streamwise direction for constant wall heat flux are shown to be similar at some extent with the isothermal wall case. Due to the fact that micropolar fluids constitute an important class of non-Newtonian fluids, this kind of study is of both theoretical and practical interest. There are of course various extensions of the present study, such as: variable microinertia density, variable heat flux applied to the wall, rotation of the cone and so one, but the concern must be where and how this kind of standard geometry may be found in practice. Acknowledgements The anonymous reviewers are gratefully acknowledged for their constructive comments and suggestions.
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