Advances in Fluid Mechanics VIII 67 A new class of exact solutions of the Navie Stokes equations fo swiling flows in poous and otating pipes A. Fatsis1, J. Stathaas2, A. Panoutsopoulou3 & N. Vlachakis1 1 Technological Univesity of Chalkis, Depatment of Mechanical Engineeing, Geece 2 Technological Univesity of Chalkis, Depatment of Aeonautical Engineeing, Geece 3 Hellenic Defence Systems, Geece Abstact Flow field analysis though poous boundaies is of geat impotance, both in engineeing and bio-physical fields, such as tanspiation cooling, soil mechanics, food pesevation, blood flow and atificial dialysis. A new family of exact solution of the Navie Stokes equations fo unsteady lamina flow inside otating systems of poous walls is pesented in this study. The analytical solution of the Navie Stokes equations is based on the use of the Bessel functions of the fist kind. To esolve these equations analytically, it is assumed that the effect of the body foce by mass tansfe phenomena is the poosity of the poous bounday in which the fluid moves. In the pesent study the effect of poous boundaies on unsteady viscous flow is examined fo two diffeent cases. The fist one examines the flow between two otated poous cylindes and the second one discusses the swil flow in a otated poous pipe. The esults obtained eveal the pedominant featues of the unsteady flows examined. The developed solutions ae of geneal application and can be applied to any swiling flow in poous axisymmetic otating geometies. Keywods: exact solution, Navie Stokes, poous, viscous flow, unsteady flow, lamina flow, swil flow, Bessel functions. doi:10.2495/afm100061
68 Advances in Fluid Mechanics VIII 1 Intoduction In the pevious yeas, poblems of fluid flow though poous ducts have aoused the inteest of Enginees and Mathematicians; the poblems have been studied fo thei possible applications in cases of tanspiation cooling, gaseous diffusion and dinking wate teatment, as well as biomedical engineeing. The cases whee an exact solution fo the Navie Stokes equations can be obtained ae of paticula impotance in ode to descibe the fluid motion of viscous flows. Howeve, since the Navie Stokes equations ae non-linea, thee cannot be a geneal method to solve analytically the full equations. Exact solutions on the othe hand ae vey impotant fo many easons. They povide a efeence solution to veify the accuacies of many appoximate methods, such as numeical and/o empiical. Although, nowadays, compute techniques make the complete integation of the Navie Stokes equations feasible, the accuacy of numeical esults can be established only by compaison with an exact solution [1]. The Navie Stokes equations wee extensively studied in the liteatue. Exact solutions aleady known ae one-dimensional o paallel shea flows, ectilinea motion flows, o duct flows [1 3]. The flow of fluids ove boundaies of poous mateials has many applications in pactice, such as bounday laye contol and tanspiation pocesses. Exact solutions ae geneally easy to find when suction o injection is applied to a fluid flow. In the case of flows though poous media, a simple solution of the Navie Stokes equations can be obtained fo the flow ove a poous plane bounday at which thee is a unifom suction velocity [4]. Moeove, fully developed lamina flow though poous channel with a poous pipe fo low Reynolds numbes was investigated in [5] and the flow in a duct of ectangula coss-section in [6]. This poblem was extended in [7] to high Reynolds numbes. The exact solution of the Navie Stokes equations fo the case of steady lamina flow between two poous coaxial cylindes with diffeent pemeability was obtained using the petubation technique [8]. The cylindes wee assumed to otate with diffeent angula velocities and the fluid between them was flowing with a constant axial pessue gadient. A mathematical model fo paticle motion in viscous flow between two otating poous cylindes was also pesented [9]. In that pape, a steady flow of a mixtue of fluid and paticles was assumed. The mass faction of paticles in the flow was small, so the petubations of the mean liquid flow due to the pesence of paticles wee negligible. An analytical appoximate solution fo decaying lamina swiling flows within a naow annulus between two concentic cylindes was also obtained. It was found that the swil velocity exhibits a Hagen-Poiseuille flow pofile decaying downsteam [10]. An exact solution of the Navie Stokes equation was obtained in [11] fo the lamina incompessible flow in a unifomly poous pipe with suction and injection. In this study the velocity field was expessed in a seies fom in tems of the modified Bessel function of the fist kind of ode n. Fo lage values of the non-dimensional time, the unsteady flow solution appoaches its asymptotic value of the steady state poblem. Lamina flow ove pipes with injection and suction though the
Advances in Fluid Mechanics VIII 69 poous wall was studied by means of analytic solutions fo the case of low Reynolds numbes [12]. In the pesent study the full unsteady thee-dimensional Navie Stokes equations ae consideed fo the case of incompessible poous flow. An exact solution is obtained by employing the Bessel functions fo the case of theedimensional unsteady flow between otated poous cylindes and fo the case of unsteady swil flow in otated poous pipes. 2 Mathematical and physical modelling Assuming fo the fist case study the flow of a Newtonian fluid though an annulus fomed between two otating cylindes, figue 1a, and, fo the second case study the flow within a otating cylindical pipe, figue 1b, the basic equations ae the mass consevation equation and the equations of motion (Navie Stokes), in a cylindical system of coodinates,, z whee the z axis lies along the cente of the pipe, peipheal angle. Figue 1: is the adial distance and is the (a) Flow between two otating poous cylindes; (b) flow within a otating poous pipe. 2.1 Govening equations Consideing that the flow modelling descibes the motion of a homogeneous Newtonian fluid the incompessible Navie Stokes equations ae the govening equations, while the following simplified assumptions ae made: a) the otating cylindes o the otating pipe ae consideed of finite length b) the pemeable wall bounday is teated as a `fluid medium'. c) the gavitational foces due to the fluid weight ae negligible. The continuity equation is: u u u z 1 u 0 z The system of the Navie Stokes equations can be witten: u u u 2 u 1 p u u z 2 2 u t z (1)
70 Advances in Fluid Mechanics VIII 2u 1 u u 2u 2 2 z 2 (2) u u u u u 1 1 P u u z 2 u t z u 2 1 u u 2u 2 2 z 2 (3) u z u u 1 p 2u 1 u 2u u z u z z 2z z 2z t z z z (4) Solution methodology 3 3.1 Unsteady flow between two coaxial poous otating cylindes An incompessible fluid of dynamic viscosity μ and density ρ is consideed between two otating cylindes of length L. The inne cylinde can otate with peipheal velocity Ri and the oute can otate with peipheal velocity Ro. At time level t t0 the fluid entes the cylindes gap unifomly at z 0 and exits at z * L o at non-dimensional axial distance z 1. * The following bounday conditions ae satisfied: Fo Ri u Ri ui u Ri u i Ri uz Ri Fo Ro u Ro uo u Ro u o Ro uz Ro 1 0 Fo the test case selected, the value of the axial velocity was set equal to zeo at the oute adius. Resolving the system of equations (1) to (4), it was found that the axial velocity u z, the adial velocity u and the tangential velocity u, can be expessed in tems of the functions: u z J 0 b e bz e kt A (1 2 ) u J1 b e bz e kt u C D kt e (5a) (5b) (5c)
Advances in Fluid Mechanics VIII whee A, C, D ae integation constants and J 0 b and 71 J1 b ae the Bessel functions of the Fist kind given in detail in [13]. The constant D is defined as D 1 2, so it coves the case of counteotating cylindes o cylindes otated in the counte-clockwise diection. The static pessue field is then calculated analytically as: p(, z,, t ) k 1 CJ J 0 e bz e kt J12 J 02 e 2bz e 2 kt 1 e bz e kt b 2 2A 4 J1 e bz ekt 2 ACz z AJ 0 e bz e kt AJ 0 2 e bz e kt kd e kt (6) b Re whee the Reynolds numbe is defined as: Re U L The poposed solution was validated fo the case of the lamina fully developed swiling flow in the annulus between two coaxial cylindes. Fo this case, one can find in the liteatue numeical solutions as well as analytical ones [10]. Figue 2 pesents the compaison between the axial velocity obtained by the pesent analytical method (solid line) and the analytical solution obtained in [10] (dashed line). The compaison is consideed satisfactoy to validate the pesent method, since the maximum diffeence between these esults does not exceed 5%. The solution of the Navie Stokes equations defined by equations (5) and (6) does satisfy the continuity equation (1) and momentum equations (2) to (4). Figue 3 shows the adial velocity distibution fo thee diffeent time levels, namely fo t 0, 1, 2 along the adius. It was assumed that the inne nondimensional (by the cylindes length L) adius is Ri 0.1, while the oute Figue 2: Distibution of the axial velocity fo the case of swiling flow between two cylindes. (solid line: pesent esults, dashed line: esults fom [10]).
72 Advances in Fluid Mechanics VIII Figue 3: Distibution of the adial velocity in tems of adius fo diffeent time levels. Figue 4: Distibution of the tangential velocity in tems of adius fo diffeent time levels. non-dimensional adius is Ro 0.8 fo specific values of the constants A, C, D, k. A decaying behaviou is obseved along the adial gap. The tangential velocity distibution shows in figue 4 a eduction fom the inne to the oute cylinde fo t 0, 1, 2. As the time level inceases moe fluid is moving tangentially, thus tangential velocity values incease. The axial velocity has a decaying distibution fom the inne to the oute adius, figue 5. Figue 6 shows the adial velocity distibution fo a given time level fo diffeent non-dimensional axial positions, namely fo z 0, 0.3, 0.8.
Advances in Fluid Mechanics VIII 73 Figue 5: Distibution of the axial velocity in tems of adius fo diffeent time levels. Figue 6: Distibution of the adial velocity in tems of adius fo diffeent axial positions. The adial velocity deceases fom the inlet to the outlet. The distibution of the tangential velocity found to satisfy the system of equations of motion is not a function of the axial distance, z accoding to equation (5c). So at any axial position, the tangential position has a constant adial distibution at given time levels. Figue 7 pesents the axial velocity distibution fo a given time level fo diffeent axial positions z 0, 0.3, 0.8. The axial velocity shows an incease along the gap of the cylindes fom the inlet to the outlet.
74 Advances in Fluid Mechanics VIII Figue 7: Distibution of the axial velocity in tems of adius fo diffeent axial positions. 3.2 Unsteady swiling flow in a otated poous pipe An incompessible fluid of dynamic viscosity μ and density ρ is consideed within a otating pipe of length L. The inne pipe can otate along its axis with peipheal velocity R. At time level t t0 the fluid entes the pipe unifomly at z 0 and exits at z L o at non-dimensional axial distance z 1. The following bounday conditions ae satisfied: * * Fo 0 u Ri Fo Ro u Ro 0 u Ri uo u Ro 0 uz 0 u o Ro uz Ro 1 0 Fo the test case selected, the value of the axial velocity is zeo at the oute adius. The axial velocity u z, the adial velocity u and the tangential velocity u, can expessed in tems of the functions: u z J 0 b e bz e kt A (1 2 ) u J1 b e bz ekt B u C (7a) (7b) (7c) J 0 b and J1 b ae the Bessel functions of the Fist kind and A, B, C ae integation constants. whee
Advances in Fluid Mechanics VIII 75 The static pessue field is then calculated analytically as: p(, z,, t ) BJ 1 k J 0 e bz ekt J12 J 02 e 2bz e2 kt 1 e bz e kt 2 b 2A J1 e bz ekt AJ 0 e bz ekt AJ 0 2 e bz ekt 2 ABz Az b (8) The solution of the Navie Stokes equations defined by equations (7) and (8) does satisfy the continuity equation (1) and momentum equations (2) to (4). Figue 8 shows the adial velocity distibution fo thee diffeent time levels, namely fo t 0, 1, 2 along the adius. A decaying behaviou is obseved in this figue. The tangential velocity distibution found in equation (11c) is independent of the time vaiable t. The axial velocity has a decaying distibution towads the oute adius, figue 9. Figue 8: Distibution of the adial velocity in tems of adius fo diffeent time levels. Figue 9: Distibution of the axial velocity in tems of adius fo diffeent time levels.
76 Advances in Fluid Mechanics VIII Figue 10: Distibution of the adial velocity in tems of adius fo diffeent axial positions. Figue 11: Distibution of the axial velocity in tems of adius fo diffeent axial positions. Figue 10 shows the adial velocity distibution fo a given time level fo diffeent non-dimensional axial positions, namely fo z 0, 0.3, 0.8. The distibution of the tangential velocity found to satisfy the system of equations of motion is not a function of the axial distance, z. So at any axial position, the tangential position has a constant adial distibution at given time levels. Figue 11 pesents the axial velocity distibution fo a given time level fo diffeent axial positions z 0, 0.3, 0.8. The axial velocity shows an incease along the gap of the cylindes fom the inlet to the outlet.
Advances in Fluid Mechanics VIII 77 4 Conclusions In this aticle, an oiginal wok pesenting exact solutions of the Navie Stokes equations in the pesence of poous boundaies of axisymmetic otating geometies is poposed. Such flows have significant industial applications including filtation and paticle sepaation. Two cases wee examined. The fist one is the unsteady flow between two otating poous cylindes and the second one is the unsteady flow inside otating poous pipes. In both cases, the Bessel functions of the fist kind wee used to compute the axial and adial components of the flow velocities, while the tangential flow velocity was found to depend only on the adius. Fo both cases, the velocity and pessue fields wee found by means of analytical methods to satisfy the Navie Stokes equations fo lamina, incompessible unsteady flows. Fo the case of the unsteady flow inside two otating cylindes, it was found that the maximum of the axial velocity shifts towads the inne cylinde. The axial and adial velocity components ae independent of the ates of otation of cylindes. The tangential flow velocity having the fom of fee votex type of flow was found to satisfy the equations of motion. Fo the case of the swil flow inside otating pipes, it was found that the maximum of the axial velocity is at the cente of the pipe and decays towads the poous bounday at the maximum adius. Vaiations wee obseved also fo the adial velocity component which also has a maximum close to the cente of the pipe. The linea vaiation of the tangential velocity having the fom of foced votex type of flow along the adius was found to satisfy the equations of motion. Refeences [1] Tukyilmazoglu, M, Exact solutions fo the incompessible viscous fluid of a poous otating disk, Intenational Jounal of Non-Linea Mechanics, 44 (2009), pp. 352-357. [2] Batchelo, G.K., An Intoduction to Fluid Dynamics, Cambidge Univesity Pess, Cambidge, 1967. [3] Polyanin A.D., Exact solution to the Navie-Stokes equations with genealized sepaation of vaiables, Dokt. Phys. 46 (2001), pp. 726-731. [4] Sheman, R.S., Viscous Flow, McGaw-Hill Inc., New Yok, 1990. [5] Beman, A.S., Lamina flow in channels with poous walls, J. Appl. Phys., 24 (1953), pp. 1232-1235. [6] Jain, R.K. Metha, K.N., Lamina hydodynamic flow in a ectangula channel with poous walls, Poc. Nat. Inst. Sci. India, 28 (1962), pp. 846856. [7] Teil R.M., Lamina flow in a unifomly poous channel, The Aeonautical Quately 15, 1964, pp.299. [8] Gupta, M.C., Goyal, M.C., Viscous incompessible steady lamina flow between two poous coaxial otating cicula cylindes with diffeent
78 Advances in Fluid Mechanics VIII [9] [10] [11] [12] [13] pemeability, Indian Jounal of pue applied Mathematics, Vol. 3, No. 3, pp.402-425. Gumeov, N. A., Dueiswami, R., Modeling of paticle motion in viscous swil flow between two poous cylindes, ASME Pape FEDSM98-5110 in the Poceedings of 1998 ASME Fluids Engineeing Division Summe Meeting, June 21-25, 1998. Jawaneh, A.M., Vatistas, G.H., Ababneh, A., Analytical appoximate solution fo decaying lamina swiling flows within a naow annulus, Jodan Jounal of Mechanical and Industial Engineeing, 2 (2008), pp. 101-109. Edogan, M.E., Imak, C.E., On the flow in a unifomly poous pipe, Intenational Jounal on Non-Linea Mechanics, 43 (2008), pp. 292-301. Moussy, Y., Snide, A.D., Lamina flow ove pipes with injection and suction though the poous wall at low Reynolds numbe, Jounal of Membane Science, 327 (2009), pp. 104-107. N. Vlachakis, A. Fatsis, A. Panoutsopoulou, E. Kioussis M. Kouskouti V. Vlachakis, An exact solution of the Navie-Stokes equations fo swil flow models though poous pipes, Poceedings of the 6th Intenational Confeence on Advances in Fluid Mechanics, Wessex Institute of Technology, (2006), pp. 583-591.