Dissipation Of Energy In Viscous Liquid Through Porous Region

Dissipation Of Energy In Viscous Liquid Through Porous Region

Abstract

In the paper, the flow of a viscous liquid is considered through a cylinder containing porous region. The mechanical energy dissipated in the fluid is calculated. The boundary of the tube performs harmonic oscillations. The effect of the permeability coefficient on the flow and the dissipation of energy are examined and discussed completely.

Key words and phrases: Newtonian Fluid, dissipation of energy, Porous medium, Permeability.

developed and the internal stress generates in the fluid due to its viscous nature and produces distortions in the velocity field. In the case of highly porous medium such as fiberglass, papus of dandelion the flow occurs even in the absence of the pressure gradient.

Modifications of the classical Darcys law were considered by the Beverse and Joseph [1], saffman

O p

K

2

1. Introduction

   In the paper, the study of flow through porous medium has many interesting applications in the diverse fields of science, engineering and technology. The particular applications which are well-known include the percolation of water through soil, extraction and filtration of oils from wells, the

   Where and K are co-efficient of viscosity of the fluid and permeability of the porous medium respectively.

   The generalized equation of momentum for the flow through the porous medium is

   drainage of water, irrigation and sanitary engineering

   and also in the inter-disciplinary fields such as

   t

   . p 2

   K

   medical and bio-medical engineering etc. The lung

   alveolar is an example that finds application in the animal body. The classical Darcys law musakat (3) states that the pressure gradient pushes the fluid against the body forces exerted by the medium which can be expressed as.

   The classical Darcys law helps in studying flows through porous medium. In the case of highly porous medium such as papus of dandelion etc., The Darcys law fails to explain the flow near the surface in the absence of pressure gradient. The non-Darcian

   K

   p

   (with usual notion)

   approach is employed to study the problem of flow through highly porous medium by several

   investigators. Narsimha charyulu and pattabhi Rama Charyulu [4, 5] Narsimha Charyulu [6] and singh [7]

   The classical Darcys law gives good results in the situations when the flow is uni-directional or at low speed. In general, the specific discharge in the medium need not be always low. As the specific discharge increases, the convective forces get

   etc, studied the flow employing Brinkman law [2] for the flow through highly porous medium.

   The problem of flow of the Newtonian fluid in the presence of transverse magnetic field, find

   application in nuclear engineering and other fields.

   u(a,t) U cos pt ,

   t 0

   (2.4)

   The rotatory flow of the fluid has special applications in various engineering field such as mechanical, petroleum and chemical in addition to the geophysical fluid dynamics, which helps in explaining the phenomena like oceanic circulation [8]. Several investigations are made in the study of flow of viscous fluid in the presence of transverse magnetic field under the assumption that the induced magnetic effect is negligible on the flow of the fluid

   e.g. see Greenspan [10] and Herbut [10] etc. But some investigations are made by considering the effect of the induced magnetic field on the flow; e.g. see somdalgekar [12] and pop [13] etc.

   The flow through porous medium in the presence of transverse magnetic field is studied in the past by several investigators. The non-Darcian flow in the presence of transverse magnetic field is investigated by Nassimha charyulu [14]. Venkat Raman and

   where, U is the amplitude of the velocity fluctuation and p the frequency of the motion of the tube.

   Flow model

   r-axis

   0 2a 0 x-axis

   Figure 1. Flow of Newtonian fluid through porous region

   By applying Laplace transform on equations (2.2), (2.3) and (2.4), we get the transformed equations as

   Narsimha Charyulu [15, 16, 17, and 18] have studied the flow employing Brinkmans law [2] for the flow through a rotating porous duct and highly porous medium.

   and

   d 2u dr 2

   • 1 du r dr

   2

   u 0

   (2.5)

   In the paper, we considered the flow of a viscous

   liquid through a cylinder containing porous region.

   u (a, s)

   Us

   s 2 p 2

   (2.6)

   The mechanical energy dissipated in the fluid is

   calculated. The boundary of the tube performs

   where,

   2

   s 1 .

   harmonic oscillations. The effect of the permeability k

   coefficient on the flow and the dissipation of energy are examined and discussed completely.

   The solution of equation (2.5) satisfying the condition at the origin and the boundary condition (2.6) is

2. Formulation and solution of the

   u (r, s) Us

   I0 (r)

   (2.7)

   0

   problem

   s2 p2 I

   (a)

   The inverse Laplace transform of u(r, s) , gives

   Consider the flow of an incompressible, viscous liquid through porous region contained by an infinite circular tube of radius a. Let (r, , x) be the co-

   u(r,t)

   i

   U

   s

   i

   2i s2 p

   I0 (r)

   0

   2 I (a)

   exp(st)ds

   (2.8)

   ordinate system such that the x co-ordinate is along the axis of the tube. Let the velocity of the fluid is

   where, is a constant such that all poles lie to its left.

   It can easily be verified that the integrand is a single- valued function of s. the inversion may be performed

   given by V (u,0,0)

   continuity

   which satisfies the equation of

   by summing the residues at the simple poles s = ± ip, and s + ( /k) = – 2 where ± (n = 1, 2, 3 ) are

   n n

   V 0 . (2.1)

   The physical quantities are independent of x and also independent of because of symmetry of the flow.

   the roots of J0(n a) = 0. It can be shown that there are no branch points. Evaluating the residues and using Cauchys integral theorem gives,

   The Navier-Stokes equation for the flow problem

   U I [

   (l)r]

   U I [

   (l)r]

   will be

   u(r,t) 0 exp(ipt ) 0

   exp(ipt )

   u 2u

   t r2

   1 u

   r r k u

   (2.2)

   2 I0[

   (l)a]

   2 I0[

   (l)a]

   2 2U

   3 J ( r) 2

   where, the kinetic viscosity, t is the time and k is

   • n

   0 n exp(n t)

   (2.9)

   a 4 p2 J ( a)

   the permeability of the medium.

   n 1 n 1 n

   The boundary conditions are given by

   where,

   l i p 1 . (2.10)

   u(a,t) 0 ,

   t 0

   p/>

   k

   (2.3)

   1. Dissipation of Energy

      The time rate of energy dissipated per unit

      The energy dissipation per unit length of the tube at the end of the mth cycle is obtained by integrating the equation (2.13) over m cycles. The resulting expression is

      2U 2

      r 6

      4r 2 m

      length along the axis of the tube in viscous flow is

      obtained from

      E n exp n 1

      p (r 4 2 )2

      dE a u 2

      (2.11)

      n

      dt 2 r r dr

      . 2 2

      0

      U m i

      (i )I0[ (i )]I1[ (i )] (i )I0[ (i )]I1[ (i )]

      where, denotes the coefficient viscosity and the negative sign is inserted because the integral on the right represents a loss of energy. The velocity gradient obtained from the equation (2.9) as

      p

      I0[ (i )]I0[ (i )]

      (2.15)

      u U

      r 2

      2 2U

      (l) I1[

      I0[

      n

      (l)r] exp(ipt) U

      (l)a] 2

      4 J ( r)

      (l) I1[

      I0[

      (l)r] exp(ipt) (l)a]

      2

3. SPECIAL CASES

   1. Case (i). For the highly porous medium (i.e. k is very large)

      n 1 n exp( t) . (2.12)

      a 4 p2 J ( a)

      The energy dissipation is given by

      n1 n 1 n

      Using the equations (2.11), (2.12) and carrying out the integrations gives,

      U 2m 2 8

      r

      E n

      1 2

      192

      2 4

      (3.1)

      p n1 (r4 2 )2 81

      1. dE U 2 (i ) exp(i2 pt)

         0

         n 32

         4096

      2. dt

      8 I 2[ (i )]

   2. Case (ii). For the flow through clear

      I 2 (i ) 2 I0[ (i )]I1[ (i )] I 2[ (i )]

      medium (i.e. k)

      0 (i )

      1

      The dissipation energy is given by

      8

      I 2[ (i )]

      U 2 (i ) exp(i2 pt)

      2 4

      2

      0

      E n 192

      2

      (3.2)

      U 2m 2

      r8

      I [ (i )]I [ (i )]

      p (r4 p2 a4 )2

      1 p a

      81 p2a4

      p4a8

      0

      I 2 (i ) 2 0 1

      I 2[ (i )]

      n1 n 2

      32 2

      4096 4

      1

      (i )

      r 2 pr 2t U 2 i

   3. Case (iii). When the permeability of the

      2U 2 n exp n

      n

      n1 (r 2 2 )2

      (i )I0[ (i )]I1[

      (i )]

      4 I0[ (i )]I0[ (i )]

      (i )I0[ (i )]I1[ (i )]

      medium is very small (i.e. k0)

      The energy dissipation is given by

      2U 2 (i ) exp(ipt ) pr 2t

      exp n

      I0 [

      (i )]

      n1

      E

      i i

      (i )J1 (rn )I2 [

      (i )] rn I1[

      4

      rn

      2

      4

      (rn

      • i )(rn

      (i )]J 2 (rn )

      2 )

      1

      3

      64

      1 8(

      i i )

      (3.3)

      2U 2 (i ) exp(ipt ) pr 2t

      1 1 8( i i ) 1

      exp n

      64

      I0 [ (i )]

      n1

      r 4 (i )J (r )I [

      (i )] r I [ (i )]J (r )

      • n 1 n 2

      n 1 2 n

      (2.13)

      (r 2 i )(r 4 2 )

      where,

      n n

      rn n a and

      p 1 a2 . (2.14)

      k

      5

      4

      3

      2

      1

      0

      0 200 400 600 800 1000 1200

      -1

      -2

      E

      Figure.1 Variation of E for different values of . (rn=1)

4. Results and Conclusion

   In the present problem, an attempt is made to estimate the mechanical energy dissipated in the fluid through a circular tube whose boundary performs harmonic motion. Expression for the energy is obtained when the tube is filled with highly porous medium. The energy dissipated per unit length of the tube is obtained in terms of which involves the permeability coefficient. The graph depicting the variation of E for different values of the permeability coefficient is drawn in fig.1.

5. References

1. Beavers, S.G. and Joseph D.D. 1967, Boundary Conditions at a natural wall, Jr. of fluid mechanics 30: 197-207.

2. Brinkman H.C. 1947. The calculation of viscous force exacted by a flowing fluid on a dense swerf of particles, Jr. of Applied science Research, 27. A1 : 27-34

3. Musakat 1937. Flow of Homogeneous fluid through porous medium, Mc Graw Hill Inc., New York 1937.

4. Narsimha Charyulu. V and Pattabhi Ramacharyulu. N Ch., 1978. Steady flow through a porous region contained between two cylinders, Journal of the Indian Institution of Sciences, 60, No. 2.3 37-42.

5. Narsimha Charyulu. V and Pattabhi Ramacharyulu. N. Ch., 1978 Steady flow of a viscous liquid in a porous elliptic tube, Pro, Indian Acad. Sci., 87A, No. 2: 79-83.

6. Narsimha Charyulu. V. 1997. Magnetic Hydro flow through a straight porous tube of an arbitrary cross-section, Indian Journal of Mathematical 1997, Vol 1. 39, No. 3 PP 267 274.

7. Singh, A.K. 202. MHD free convective flow through a porous medium between two vertical parallel plates, Indian Journal of pure and applied physics, Vol. 40: 709-713.

8. H.P. Green span., theory of rotating fluids, can bridge University press 1969.

9. Saffman, P.G. 1971. on the boundary condition at the surface of porous medium, Studies of applied math., 50: 93-101.

10. H.P. Greenspan, theory of Rotating Fluids, Cambridge University ress 1969.

11. R. Herbert, Two-dimensional, ax Symmetric rotational flow of a viscous fluid, ZAAM, 55 (1975), 443-445.

12. V.M. Somdalgekar, Hydro Magnetic flow near an accelerated plate in the presence of magnetic field, Appl Sci, Res. 12B (1965), 151-156.

13. I. POP, on the hydro magnetic flow near an accelerated plate, ZAAM 48 (1968) 69.70.

14. V. Narsimha Charyulu, Magnetic Hydro Dynamic flow through straight porous tube of an arbitrary cross section, Indian J. Math. 39 (197), 267-274.

15. V. Narsimha Charyulu & P.Venkatraman, Non- Darcian MHD flow through A Rotating porous Duct, Far East Journal of Applied Mathematics, March 10(3) (2003.

16. P.Venkat Raman and V. Narasimha Charyulu, Flow of A Newtonian Fluid Between Parallel Plate with Porous Lining, Bulletin of pure and Applied Science. Vol. 26E(No.1)2007.

17. P.Venkat Raman and V. Narasimha Charyulu, Study of flow through a straight Porous tube of Arbitrary Cross-Section under the Influence of Uniform Transverse Magnetic Field. International Journal of Mathematical Sciences. June 2006, Volume 5, No. 1, pp. 47-62.

18. P.Venkat Raman and V. Narasimha Charyulu, Un-Steady flow of a Non-Newtonian fluid Through Porous Medium. International Journal of Scientific Computing 2(1) January-June 2008; pp. 73-81.

19. G.S.S Ludford, Rayleighs Problem in Hydro Magnetic: The Impulsive nation of a pole-picece, Arch. Rat. Mech. Anal 3 (1959) 14-27.