Abstract
This work deals with the creeping flow of an incompressible viscous fluid through a membrane. It is assumed that the membrane is composed of nonhomogeneous porous cylindrical particles with radially varying permeability enclosing a cavity. The flow within the nonhomogeneous porous medium is governed by the Darcy equation. The flow inside the cavity and outside the nonhomogeneous porous region is governed by the Stokes equations. An analytical solution of the problem is obtained by using the cell model technique. Exact expressions for the drag force acting on the membrane and hydrodynamic permeability of the membrane are derived. The influence of radially varying permeability on flow parameters is considered. The effects of various parameters of the problem on hydrodynamic permeability of the membrane are discussed for four models. Some previous results for hydrodynamic permeability are verified as special limiting cases.
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References
J. Happel, “Viscous Flow Relative to Array of Cylinders,” AIChE J. 5(2), 174–177 (1959).
S. Kuwabara, “The Force Experienced by Randomly Distributed Parallel Circular Cylinders or Spheres in a Viscous Flow at Small Reynolds Number,” J. Phys. Soc. Jpn. 14, 527–532 (1959).
A. G. Kvashnin, “Cell Model of Suspension of Spherical Particles,” Fluid Dyn. 14(4), 598–602 (1980).
G. D. Mehta and T. F. Morse, “Flow Through Charged Membranes,” J. Chem. Phys. 63(5), 1878–1889 (1974).
D. D. Joseph and L. N. Tao, “The Effect of Permeability on the Slow Motion of a Porous Sphere in a Viscous Liquid,” Z. Angew. Math. Mech. 44(8/9), 361–364 (1964).
G. S. Beavers and D. D. Joseph, “Boundary Conditions at a Naturally Permeable Wall,” J. Fluid Mech. 30(1), 197–207 (1967).
G. P. Raja Sekhar and T. Amaranath, “Stokes Flow Inside a Porous Spherical Shell,” Z. Angew. Math. Mech. 51(3), 481–490 (2000).
S. Deo, “Stokes Flow Past a Swarm of Porous Circular Cylinders with Happel and Kuwabara Boundary Conditions,” Sadhana 29(3), 381–387 (2004).
A. N. Filippov, D. Y. Khanukaeva, S. I. Vasin, et al., “Liquid Flow Inside a Cylindrical Capillary with Walls Covered with a Porous Layer (Gel),” Colloid J. 75(2), 214–225 (2013).
M. P. Singh and J. L. Gupta, “The Flow of a Viscous Fluid Past an Inhomogeneous Porous Cylinder,” Z. Angew. Math. Mech. 51(1), 17–25 (1971).
I. V. Chernyshev, “The Stokes Problem for a Porous Particle with Radially Non-Uniform Porosity,” Fluid Dyn. 35(1), 147–152 (2000).
D. Palaniappan and K. Archana, “Two-Dimensional Creeping Flows with Permeable Cylinders,” Z. Angew. Math. Mech. 77(10), 791–796 (1997).
P. D. Noymer, L. R. Glicksman, and A. Devendran, “Drag on a Permeable Cylinder in Steady Flow at Moderate Reynolds Numbers,” Chem. Eng. Sci. 53(16), 2859–2869 (1998).
S. Deo and P. K. Yadav, “Stokes Flow Past a Swarm of Porous Nanocylindrical Particles Enclosing a Solid Core,” Int. J. Math. Math. Sci. 2008, 1–8 (2008).
S. Deo, P. K. Yadav, and A. Tiwari, “Slow Viscous Flow Through a Membrane Built up from Porous Cylindrical Particles with an Impermeable Core,” Appl. Math. Model. 34(5), 1329–1343 (2010).
S. I. Vasin and T. V. Kharitonova, “Flow of Liquid Around the Encapsulated Drop of Another Liquid,” Colloid J. 73(3), 297–302 (2011).
P. K. Yadav, “Slow Motion of a Porous Cylindrical Shell in a Concentric Cylindrical Cavity,” Meccanica 48(7), 1607–1622 (2013).
A. N. Filippov, S. I. Vasin, and V. M. Starov, “Mathematical Modeling of the Hydrodynamic Permeability of a Membrane Built up from Porous Particles with a Permeable Shell,” Colloids Surfaces, A: Physicochem. Eng. Aspects 282/283, 272–278 (2006).
P. K. Yadav, A. Tiwari, S. Deo, et al., “Hydrodynamic Permeability of Biporous Membrane,” Colloid J. 75(4), 473–482 (2013).
I. B. Stechkina, “Drag of Porous Cylinders in a Viscous Fluid at Low Reynolds Numbers,” Fluid Dyn. 14(6), 912–915 (1979).
M. Kohr, J. Prakash, G. P. Raja Sekhar, and W. L. Wendland, “Expansions at Small Reynolds Numbers for the Flow Past a Porous Circular Cylinder,” Appl. Anal. 88(7), 1093–1114 (2009).
M. S. Valipour, S. Rashidi, M. Bovand, and R. Masoodi, “Numerical Modeling of Flow around and through a Porous Cylinder with Diamond Cross Section,” Europ. J. Mech., B-Fluid 46, 74–81 (2014).
S. I. Vasin and A. N. Filippov, “Hydrodynamic Permeability of the Membrane as a System of Rigid Particles Covered with Porous Layer (Cell Model),” Kolloid. Zh. 66(3), 305–309 (2004).
A. G. Skirtach, A. A. Antipov, D. G. Shchukin, and G. B. Sukhorukov, “Remote Activation of Capsules Containing Ag Nanoparticles and IR Dye by Laser Light,” Langmuir 20(17), 6988–6992 (2004).
J. J. L. Higdon and M. Kojima, “On the Calculation of Stokes’ Flow Past Porous Particles,” Int. J. Multiphase Flow 7(6), 719–727 (1981).
J. L. Auriault, “On the Domain of Validity of Brinkman’s Equation,” Transport Porous Med. 79, 215–223 (2009).
S. Veerapaneni and M. R. Wiesner, “Hydrodynamics of Fractal Aggregates with Radially Varying Permeability,” J. Colloid Interface Sci. 177(1), 45–57 (1996).
A. Tiwari, P. K. Yadav, and P. Singh, “Stokes Flow Through Assemblage of Non-Homogeneous Porous Cylindrical Particles Using Cell Model Technique,” Nat. Acad. Sci. Lett. 41(1), 53–57 (2018).
S. Deo, A. N. Filippov, A. Tiwari, et al., “Hydrodynamic Permeability of Aggregates of Porous Particles with an Impermeable Core,” Adv. Colloid Interface Sci. 164, 21–27 (2011).
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Original Russian Text © P.K. Yadav, P. Singh, A. Tiwari, S. Deo.
Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 60, No. 5, pp. 41–52, September–October, 2019.
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Yadav, P.K., Singh, P., Tiwari, A. et al. Stokes Flow Through a Membrane Built up by Nonhomogeneous Porous Cylindrical Particles. J Appl Mech Tech Phy 60, 816–826 (2019). https://doi.org/10.1134/S0021894419050055
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DOI: https://doi.org/10.1134/S0021894419050055