Transp Porous Med (2010) 81:89–103 DOI 10.1007/s11242-009-9389-y

Brinkman Flow of a Viscous Fluid Through a Spherical Porous Medium Embedded in Another Porous Medium T. Grosan · A. Postelnicu · I. Pop

Received: 6 March 2007 / Accepted: 24 March 2009 / Published online: 24 April 2009 © Springer Science+Business Media B.V. 2009

Abstract An analytical investigation for a two-dimensional steady, viscous, and incompressible flow past a permeable sphere embedded in another porous medium is presented using the Brinkman model, assuming a uniform shear flow far away from the sphere. Semianalytical solutions of the problem are derived and relevant quantities such as velocities and shearing stresses on the surface of the sphere are obtained. The streamlines inside and outside the sphere and the radial velocity √ are shown in several graphs √ for different values of the porous parameters σ1 = (µ/µ)(a/ ˜ K 1 ) and σ2 = (µ/µ)(a/ ˜ K 2 ), where a is the radius of the sphere, µ is the dynamic viscosity of the fluid, µ˜ is an effective or Brinkman viscosity, while K 1 and K 2 are the permeabilities of the two porous media. It is shown that the dimensionless shearing stress on the sphere is periodic in nature and its absolute value increases with an increase of both porous parameters σ1 and σ2 . Keywords

Permeable sphere · Porous media · Brinkman model · Analytical solution

List of Symbols a Radius of the sphere, m Ki Permeability of the porous medium, m 2 p¯ i Pressure, Pa pi Non-dimensional pressure r Radial coordinate, m r Non-dimensional radial coordinate u i , v i Radial and transversal velocities, m s −1

T. Grosan · I. Pop Faculty of Mathematics, University of Cluj, CP 253, 3400 Cluj, Romania A. Postelnicu (B) Department of Thermal Engineering and Fluid Mechanics, Transilvania University, 500036 Brasov, Romania e-mail: [email protected]

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Non-dimensional radial and transverse components of velocity Constant velocity away from the sphere, m s −1

Greek Symbols φi Porosity of the porous medium φ Ratio of the porosities θ Angular coordinate with θ = 0 measured from the axis along the direction of √the undisturbed flow U∞ ωi Notation, a/ K i σi Porous medium parameter, (µ/µ)ω ˜ i τr θ (i) , τrr (i) Non-dimensional skin friction at the surface of the sphere in the radial and azimuthal directions ψi Non-dimensional stream function Subscript i = 1, 2 Refers to the two porous media

1 Introduction Fluid flow in porous media has been an area of intensive investigation for the last decades. The growing emphasis on effective granular and fibrous insulation systems, for the successful containment of the transport of radio-nuclide from deposits of nuclear waste materials, has stimulated various studies in fluid saturated porous media and many results were obtained for the forced and convective flow in the fundamental geometries of internal (cavities, annulus, etc.) and external (over surfaces) flows. In comprehensive reviews of heat transfer mechanisms in geothermal systems, Cheng (1978, 1985), and Bejan (1987) presented the work in this field with emphasis on its applications in geothermal and energy systems research. Since then a large number of practical applications, both industrial and environmental, have caused a rapid extension of the research, and a substantial number of articles, which relate the boundary-layer flow past surfaces of various flow configuration models, have been published. Nield and Bejan (2006), Ingham and Pop (1998, 2002, 2005), Pop and Ingham (2001), Vafai (2000, 2005), and Bejan et al. (2004) identified many applications that highlight the directions where further theoretical and experimental developments and investigations are required. Rudraiah (1984) and Rudraiah et al. (2003) have shown that in many practical applications of flow in porous media which involve porous materials, such as foam metals and fibrous media of high porosity, where dimensionless particle diameter is not small, the Darcy’s law is not adequate and a non-Darcy equation is more realistic to describe the flow. The Brinkman equation, which is appropriate for high porosity porous media, is often used as a flow model in the porous media. A rigorous theoretical justification of Brinkman’s equation was given by Tam (1969) and later by Lundgren (1972). Neale and Nader (1974) considered a coupled parallel flow with a channel and a bounding porous medium, where the Brinkman model was used. Sahraoui and Kaviany (1992) performed numerical experiments and reported comparisons of the microsolution with the macrosolution with Brinkman model, in situations where the solid matrix in porous medium was constructed from regularly spaced cylinders. Further, an extensive literature review on the Brinkman model may be found in the thesis by Laptev (2003).

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The interface coupling condition between a Stokes fluid and a Darcy porous medium encountered usually in the literature is the Beavers–Joseph condition, proposed by Beavers and Joseph (1967). Although empirical, efforts have been performed by Taylor (1971) to support it theoretically, by using the statistical treatment of Saffman (1971). The relationship between the Brinkman equation and the Beavers–Joseph boundary condition was discussed by Nield (1983). Padmavathi et al. (1993), Masliyah et al. (1987), and Berman (1996) have studied the Stokes flow past a sphere embedded in a porous medium with specifying a uniform velocity far away from the sphere. Further, Yu et al. (2000) and Yu and Lee (2000) have published the analytical solution for a two scale flow using Stokes flow for the fluid region and Brinkman equation for the fluid-saturated porous region. Also, Srivastava and Srivastava (2005) have recently discussed the steady flow of an incompressible viscous fluid streaming past a porous sphere at small Reynolds number with a uniform velocity by dividing the flow in three regions. • The region I is the region inside the porous sphere in which the flow is governed by Brinkman equation with the effective viscosity different from that of the clear (non-porous) fluid. • Regions II and III delineate clear fluid flows where Stokes and Oseen solutions are, respectively, valid. In all the three regions Stokes stream function is expressed in powers of Reynolds number. Furthermore, in a recent paper, Rudraiah and Chandrashekhar (2005) have studied twodimensional steady incompressible flow past an impermeable sphere embedded in a porous medium using the Brinkman (1947a, b) model with a uniform shear instead of a uniform velocity away from the sphere. This problem occurs in a wide variety of technological applications like removing impurities in the integrated circuits used in computers, lubrication process in porous bearings, etc. (see, for instance, Rudraiah and Chandrashekhar 2005). However, it seems that results of this article are, unfortunately, inaccurate because the authors have not correctly considered the expression for the stream function of the flow outside the porous sphere. The existence and uniqueness of the solution for the two-dimensional flow with porous inclusions based on Brinkman equation has been studied by Kohr and Raja Sekar (2007). The aim of this article is to study the steady flow of a viscous fluid past a permeable porous sphere embedded in another porous medium using the Brinkman equation model for the flow inside and outside the sphere. It is important that we succeed to present closed form solutions of the non-dimensional governing equations and the evolution of the flow field is demonstrated by plotting the streamlines, velocity, and shearing stress for the flow inside and outside the sphere. These quantities are numerically computed and depicted graphically for various values of two-dimensionless porous medium parameters σ1 and σ2 , as defined in the Nomenclature.

2 Basic Equations Consider the steady flow of a viscous and incompressible fluid past a permeable sphere of radius a with the constant velocity U∞ far from the sphere. The sphere is filled with a porous medium of permeability K 1 and placed in a fluid-saturated porous medium of permeability K 2 . The flow domain is divided into two zones. Zone I is the region inside the porous sphere and Zone II is the outside region. Let the index i in the subscript of any flow property Xi , i = 1, 2 related to Zone I (i = 1) and Zone II (i = 2). The continuity and Brinkman

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equations, with inertial terms omitted, for this problem are given by (see, Nield and Bejan 2006) ∇ · v¯ i = 0

(1)

µ ∇ p¯ i = − v¯ i + µ˜ i ∇ 2 v¯ i , Ki

(2)

where pi is the intrinsic pressure, v¯ i is the superficial velocity (Darcian velocity, see Nakayama 1995) vector, µ is the dynamic viscosity of the fluid, and µ˜ i is an effective or Brinkman viscosity. According to Nield and Bejan (2006), one may assume that for high porosity cases, µ˜ can be taken equal with µ. Also Nakayama (1995) suggests that µ˜ is often taken to be equal with µ, for the first approximation. Thus, we mention that Brinkman (1947a, b) has considered that µ˜ = µ, but Lundgren (1972) showed that if • the velocity v¯ is taken as the ensemble average of the velocity field • and p¯ is considered as the mean static pressure in the fluid, then the ratio µ/µ ˜ is variable, rising slightly above 1 as the porosity decreases from unity, attaining a maximum when the porosity is about 0.8 and then decreasing as the porosity decreases (see also the discussion provided by Nield 1983). Further, we use a spherical coordinate system (r , θ, ϕ) with the origin at the center of the sphere and the axis θ = 0 along the direction of the undisturbed flow U∞ as it is shown in Fig. 1. Due to the symmetry of the problem we have ∂/∂ϕ = 0. It is convenient to use the following dimensionless variables r = r /a, u i = u i /U∞ , vi = v i /U∞ ,

pi = a pi /(µU∞ ),

(3)

where u i and v i are the radial and azimuthal components of velocity. Using the new dimensionless variables (3), Eqs. 1 and 2 can be written as follows ∂ 2 r ∂ (r u i ) + (vi sin θ ) = 0, ∂r sin θ ∂θ

u

v

r

a θ

U∞

u 2 , v2 zone II

Fig. 1 Physical model and coordinate system

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u1 , v1 zone I

(4)

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∂ pi µ˜ i ∂ 2 u i 2 ∂u i cot θ ∂u i 1 ∂ 2ui + + = ωi2 u i − + 2 2 2 ∂r µ ∂r r ∂r r ∂θ r 2 ∂θ  2u i 2vi cot θ 2 ∂vi − 2 − 2 − , r r ∂θ r2

and µ˜ i 1 ∂ pi = ωi2 vi − − r ∂θ µ

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(5)




∂ 2 vi 2 ∂vi cot θ ∂vi 1 ∂ 2 vi + + + ∂r 2 r ∂r r 2 ∂θ 2 r 2 ∂θ  2 vi cosec θ 2 ∂u i − , + 2 r ∂θ r2

(6)

√ where ωi = a/ K i are the parameters of the porous media. Permeabilities K i of the porous media are given by the relation proposed by Carman–Kozeny, see Nield (2002), Ki =

di2 φi3 180 (1 − φi )2

,

(7)

where φi are the porosities and di are the mean particle diameters for the two porous media. Thus the parameters of the porous media are given by √ 180(1 − φi ) ωi = γi , (8) 3/2 φi where γi = a/di . In Eqs. 4–6 variables r and θ vary in zone I in the ranges 0 ≤ r ≤ 1, −180◦ ≤ θ ≤ 180◦ and in zone II in the ranges 1 ≤ r < ∞, −180◦ ≤ θ ≤ 180◦ . The matching conditions at the surface of the sphere can be written as (see Merrikh and Mohamad 2002) u 1 = u 2 at r = 1 v1 = v2 at r = 1 τr θ (1) = τr θ (2) at r = 1 τrr (1) = τrr (2) at r = 1,

(9)

where τr θ (i) and τrr (i) are the dimensionless shear stress components on the surface of the sphere. It is worth mentioning that there is a large body of literature which investigates conditions at the interface between a porous medium and a homogeneous fluid (see Ochoa-Tapia and Whitaker 1998; Alazmi and Vafai 2001; Goyeau et al. 2003). We introduce now the stream function ψi defined in the usual way to satisfy the continuity equation 1 ∂ψi 1 ∂ψi , vi = − . r 2 sin θ ∂θ r sin θ ∂r Outside the porous sphere the stream function ψ2 is given by   1 2 1 ψ2 (r, θ ) = r − sin2 θ for all r > 1. 2 r ui =

(10)

(11)

Using (10) and (11), the boundary conditions for the velocity components in zone II are u 2 ∼ cos θ, v2 ∼ − sin θ as r → ∞.

(12)

The dimensionless expression of τr θ (i) and τrr (i) are given by τr θ (i) =

1 ∂u i ∂vi vi + − r ∂θ ∂r r

(13)

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and τrr (i) = − pi + 2

∂u i . ∂r

(14)

3 Mathematical Analysis Eliminating the pressure pi between Eqs. 5 and 6 and using 10, we get  2 2    − σi2 ψi = 0,  where σi = ωi µ/µ˜ i and the operator 2 is defined as   1 ∂ ∂2 sin θ ∂ . 2 = 2 + ∂r r 2 ∂θ sin θ ∂θ

(15)

(16)

It is important  to point-out that, having in view the definition of the porous medium parameters σi = a µ/(µ˜ i K i ), it is not necessary to consider µ/µ˜ i as a new parameter. The boundary conditions far away from the sphere (12) in terms of ψ2 become ψ2 (r, θ ) ∼

r2 sin2 θ as r → ∞. 2

(17)

The boundary condition (17) suggests the following similarity solution to Eq. 15 ψi (r, θ ) = f i (r ) sin2 θ.

(18)

Substituting (18) into Eq. 15, we obtain the following ordinary differential equation   4 8 8 2 f iiv − 2 f i + 3 f i − 4 f i − σi2 f i − 2 f i = 0, (19) r r r r where primes denote differentiation with respect to r. If we consider the transformation √ 2 f i = r gi (r ), r2 then the fourth order Eq. 19 reduces to the second order equation   3 2 2 2 + (σi r ) gi = 0. r gi + rg − 2 f i −

(20)

(21)

This is the modified Bessel differential equation and hence its general solution is given by gi (r ) = Ai I3/2 (σi r ) + Bi K 3/2 (σi r ),

(22)

where I3/2 (σi r ) and K 3/2 (σi r ) are the modified Bessel functions of first and second kind of order 3/2, respectively, and Ai and Bi are arbitrary constants of integration. Thus, using (22), Eq. 20 becomes 2 f i = Ai I3/2 (σi r ) + Bi K 3/2 (σi r ), r2

(23)

√ √ r r Ci + Di r 2 + Ai 2 I3/2 (σi r ) + Bi 2 K 3/2 (σi r ), r σi σi

(24)

f i − and has the general solution f i (r ) =

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where Ci and Di are also arbitrary constants of integration. Using Eqs. 18 and 24 we can describe the flow in both the two zones I and II with proper determination of the constants Ai , Bi , Ci , and Di . For the flow in zone I where r < 1, and which contains the origin (r = 0), the solution is finite thus the constants C1 = B1 = 0 and the expression for f 1 (r ) can be written as √ r 2 f 1 (r ) = D1 r + A1 2 I3/2 (σ1 r ). (25) σ1 On the other hand, for the flow in zone II where r > 1 and f 2 (r ) → 21 r 2 as r → ∞, we get D2 = 1/2 and A2 = 0 and the expression for f 2 (r ) is given by √ r 1 C2 + r 2 + B2 2 K 3/2 (σ2 r ). f 2 (r ) = (26) r 2 σ2 To determine the constants A1 , B2 , C2 , and D1 we have to use the matching and boundary conditions, Eq. 9. To do so, we have to determine the expression for τr θ (i) and τrr (i) given by Eq. 13 and for the pressure pi as well. Substituting Eqs. 10 and 18 into Eqs. 5 and 6, and after integration we get   µ˜ i 2 4 pi = −σi2 f i + f i − 2 f i + 3 f i cos θ, (27) µ r r where the constant of integration has been taken to be zero. Thus, the relations given by Eq. 13 become 

τr θ (i) = r22 f i − r23 f i − r1 f i sin θ



(28) ˜i 4 2 − 8 f + 4 f cos θ. τrr (i) = ωi2 f i − µµ f − f + f i i 3 2 3 2 i r r i r r i Now, the four matching conditions (9) in terms of f i (r ) can be written as f 2 (1) = f 1 (1) f 2 (1) = f 1 (1)

(29)

f 2 (1) = f 1 (1)

σ22 f 1 (1) − f 1 (1) = σ22 f 1 (2) − f 1 (2). These relations can be used to determine the values of the constants A1 , B2 , C2 , and D1 for some given values of the parameters σ1 and σ2 . We mention that for large arguments (x |α 2 − 1/4|) both modified Bessel functions Iα (x) and K α (x) reduce to, see Abramowitz and Stegun (1972),  π −x 1 x e , K α (x) = (30) Iα (x) = √ e . 2x 2π x Therefore, we have to consider the following four cases: (i) σ1 ≤ 500, σ2 ≤ 500 f 1 (r ) = D1 r

2

√ r + A1 2 I3/2 (σ1 r ), σ1

√ r C2 1 2 f 2 (r ) = + r + B2 2 K 3/2 (σ2 r ) r 2 σ2 (31)

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(ii) σ1 ≤ 500, σ2 ≥ 500 f 1 (r ) = D1 r

2

√ r + A1 2 I3/2 (σ1 r ), σ1

(iii) σ1 ≥ 500, σ2 ≤ 500 f 1 (r ) = D1 r 2 ,

f 2 (r ) =

1 C2 + r2 r 2

(32)

√ 1 C2 r + r 2 + B2 2 K 3/2 (σ2 r ) r 2 σ2

(33)

f 2 (r ) =

(iv) σ1 ≥ 500, σ2 ≥ 500 1 C2 + r2 (34) r 2 Thus, we have the stream functions ψi (r, θ ) in both the two zones I and II given by f 1 (r ) = D1 r 2 ,

f 2 (r ) =

ψ1 (r, θ ) = f 1 (r ) sin2 θ, ψ2 (r, θ ) = f 2 (r ) sin2 θ,

(35)

where f 1 (r ) and f 2 (r ) are given by Eqs. 31–34. From Eqs. 10 and 31 to 34, we obtain the normal and tangential components of velocities in zones I and II as f (r ) f i (r ) cos θ, vi = − i sin θ. (36) 2 r r The case when the sphere is impermeable corresponds to u 2 = 0 and v2 = 0 at the surface of the sphere. Thus, the values of B2 and C2 are given by f 2 (1) = 0 and f 2 (1) = 0 from Eq. 29. Using Eq. 24 and the properties of the Bessel functions the calculation gives the following expressions for B2 and C2 ui = 2

B2 =

3K 3/2 (σ2 ) 3σ2 1 , C2 = − − . 2K 1/2 (σ2 ) 2 2σ2 K 1/2 (σ2 )

(37)

Thus, f 2 (r ) is given by f 2 (r ) =

√   3K 3/2 (σ2 ) 1 3 r K 3/2 (σ2 r ) 1 2 1 r − 1 + + . 2 2 σ2 K 1/2 (σ2 ) r 2σ2 K 1/2 (σ2 )

(38)

This expression is exactly the same as that reported by Pop and Ingham (1996) for the problem of flow past an impermeable sphere embedded in a porous medium using the Brinkman model if we replace in (38) σ2 by 1/σ2 . Thus, using (28) and (38) we get that the dimensionless shearing stress at any point on the surface of the sphere (i.e., r = 1) is given by 3 (39) τr θ (2) = − (1 + σ2 ) sin θ. 2 Also this expression is identical with that found by Pop and Ingham (1996) if we change σ2 by 1/σ2 . Therefore, we can conclude that the present analysis is correct.

4 Results and Discussion Based on the governing equations of motion it is seen that the parameters affecting the flow in the present problem are the porous media parameters σ1 and σ2 given by Eq. 8. Values of the porosities for different porous materials can be found in the book by Nield and Bejan (2006). For a high porosity porous media a typical value is φi = 0.8 so that Eq. 8 reduces to   σi = ωi µ/µ˜ i = 3.5γi µ/µ˜ i . (40)

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In most physical problems of interest γi is large and hence σi is large. However, we considered also small values of σi (for a 1) because such cases can occur in some problems related to vascular and tissues flow and tumor growth. In this article, calculations have been carried out using the symbolic calculus software Mathematica. For the case given by Eqs. 34 it is possible to obtain directly the values of the coefficients D1 and C2 as follows:   3 σ2 − 2 σ22 − σ12  , C2 = D1 =  2 2 2 . (41) 2 2σ1 + σ2 − 6 2σ12 + σ22 − 6 Figures 2, 3, 4, 5, and 6 present the streamlines computed from Eq. 35, drawn for some values of the parameters of the porous media σ1 and σ2 , namely the cases of σ1 σ2 , σ1 < σ2 , σ1 = σ2 , σ1 > σ2 , and σ1 σ2 . In order to illustrate the effect of the values of σ1 and σ2 for a given ratio σ1 /σ2 = 0.5 (say) we considered the following cases (σ1 , σ2 ) = {(50, 100); (500, 1000); (5000, 10000)}. The streamlines for these cases are presented in Figs. 7, 8, and 9. We notice below several conclusions drawn from these figures. • The pattern of the streamlines is symmetric between the upstream and the downstream directions. • The meandering of the streamlines near the surface of the sphere, which became parallel to the direction of the constant velocity U∞ of external flow far away from the sphere for both σ1 < σ2 and σ1 > σ2 . • In addition, in the case σ1 > σ2 a boundary layer near the surface of the sphere can be observed, see Figs. 5 and 6. The boundary layer when σ1 σ2 is very similar with the boundary layer obtained for an impermeable sphere. This is shown also in Fig. 12 where the radial velocity obtained for σ1 = 500, 1, 000, σ2 = 100, and θ = π/4 is compared with the case of impermeable sphere reported by Pop and Ingham (1996). We notice that both solutions are very close for large σ1 (σ1 = 1, 000, for example).

Fig. 2 Streamlines for σ1 = 10 and σ2 = 1,000

Fig. 3 Streamlines for σ1 = 100 and σ2 = 500

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Fig. 4 Streamlines for σ1 = 500 and σ2 = 500

Fig. 5 Streamlines for σ1 = 500 and σ2 = 100

Fig. 6 Streamlines for σ1 = 1,000 and σ2 = 10

Fig. 7 Streamlines for σ1 = 50 and σ2 = 100

• Further, we can notice from Figs. 2, 3, 7, 8, and 9 that the streamlines are distorted toward the center of the porous sphere in the cases σ1 < σ2 and σ1 σ2 . This fact is in agreement with our physical expectation concerning the passage between two porous media of different permeabilities.

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Fig. 8 Streamlines for σ1 = 500 and σ2 = 1, 000

Fig. 9 Streamlines for σ1 = 5, 000 and σ2 = 10, 000

Fig. 10 Radial velocities u 1 (r ) and u 2 (r ) for θ = π/6

• In the special case of σ1 = σ2 (see Fig. 4), the flow is not perturbed by the presence of the porous sphere, again in accordance with the physics of the phenomenon (flow from one porous medium to another one having same physical properties). • For the constant ratio σ1 /σ2 the distribution of the streamlines does not change when σ1 and σ2 are very large as it can be seen in Figs. 8 and 9. This is because for a constant ratio σ1 /σ2 the coefficients D1 and C2 go asymptotically to constant values.

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Fig. 11 Radial velocities u 1 (r ) and u 2 (r ) for θ = π/4

Fig. 12 Radial velocity u 2 (r ) near the spherical surface for σ1 = 500, 1, 000, σ2 = 100, and θ = π/4

The variation of dimensionless radial velocity u i (r )(i = 1, 2) with radius r for different values of σ1 and σ2 is presented in Figs. 10 and 11 when θ = π/6 and π/4. The dot lines represent the radial velocity calculated using the asymptotic approximation given by Eq. 34. It can be seen that for σ1 and σ2 larger than 500 there is a very good agreement between the analytical solution obtained from Eq. 31 and the asymptotic approximation (Fig. 12). The variation of the dimensionless shearing stress τr θ (2) at any point on the surface of the sphere (i.e., r = 1) with θ is illustrated in Figs. 13 and 14 for several values of the porous parameters σ1 and σ2 . We can see that the absolute value of the shearing increases with an increase in both σ1 and σ2 . It is also seen that at the front and rear points of the

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Fig. 13 Variation of tangential shear stress along the surface of the sphere for σ1 = 10 and σ2 = 1, 50, 100

Fig. 14 Variation of tangential shear stress along the surface of the sphere for σ1 = 100 and σ2 = 1, 10, 50

sphere, θ = 0◦ and θ = 180◦ , the shearing stress vanishes, whereas it attains the maximum value at θ = ± 90◦ . It should also be pointed out that there is no flow separation occurring for the flow past a sphere which is embedded in a constant porosity medium based on the Brinkman equation model. The positive or negative value of the shear stress τr θ (2) depends on the velocity direction at the porous media interface.

5 Conclusion The flow of a viscous fluid past a permeable sphere embedded in another porous medium has been investigated using the Brinkman equation model. A semi-analytical solution of the governing equations for the flow inside and outside the sphere has been obtained. It has been

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found that for σ1 and σ2 larger than 500 one can use with great confidence the asymptotic solution given by Eq. 34. In addition it is shown that the velocity field does not depend on the parameters µ˜ i /µ while the pressure and the radial stress given by Eqs. 27 and 28 depend on these parameters. The presented figures show the whole range of influence of the considered porous parameters starting from the limiting case of nearly impermeable sphere and ending at the opposite end of range with the spherical cavity filled only with clear fluid. We believe that the present analytical solution can be used for testing numerical methods for problems involving layered porous media. Acknowledgments The first author’s contribution was supported from the grants CEEX ET90 2006–2008 and PN-II-ID 525/2007. The authors would like to express their very sincerely thanks to all reviewers for their valuable many comments and suggestions.

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