Abstract
In the considered problem, we presented a model for the flow of immiscible fluids (non-Newtonian and Newtonian) through the cylindrical pipe. Cylindrical pipe is constituted by two porous cylindrical shell enclosing a cylindrical cavity. Non-Newtonian (micropolar) fluid is flowing through the middle porous cylindrical shell, and other immiscible Newtonian fluids are flowing through the cavity and outer porous cylindrical shell. The flow of fluid through the outer porous cylindrical shell and cavity is governed by well-known Brinkman and Stoke’s equation, respectively. However, the flow of micropolar fluid through the middle porous cylindrical shell is governed by the field equation given by Eringen (Springer, Berlin, 2001). An analytical solution of the problem has been obtained by using the justified boundary conditions. The effects of various non-dimensional parameters such as permeability parameters, micropolar parameter, and viscosity ratio on the linear flow velocity, microrotational flow velocity and flow rate are examined graphically. The results are validated with the help of previously established result.
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References
H.P.G. Darcy, Les Fontaines publiques de la ville de Dijon. Exposition et application des principes ‘a suivre et des formules ‘a employer dans les questions de distribution d’eau, etc. V. Dalamont (1856)
H. Brinkman, Flow Turbul. Combust. 1, 27 (1949a)
T. Zlatanovski, Q. J. Mech. Appl. Math. 52, 111–126 (1999)
N. Rudraiah, I. Shivakumaran, D. Palaniappan, D. Chandrashekar, Adv. Fluid Mech. 2, 253–256 (2004)
H. Ramkissoon, Acta Mech. 123, 227–233 (1997)
D. Palaniappan, Zeitschrift fur angewandte Mathematik und Physik ZAMP 45, 832–838 (1994)
P.K. Yadav, A. Tiwari, S. Deo, A. Filippov, S. Vasin, Acta Mech. 215, 193–209 (2010)
P.K. Yadav, S. Deo, Meccanica 47, 1499–1516 (2012)
P.K. Yadav, A. Tiwari, S. Deo, M.K. Yadav, A. Filippov, S. Vasin, E. Sherysheva, Colloid J. 75, 473–482 (2013)
P.K. Yadav, A. Tiwari, P. Singh, Acta Mech. 229, 1869–1892 (2018)
G.S. Beavers, D.D. Joseph, J. Fluid Mech. 30, 197–207 (1967)
S. Liu, J.H. Masliyah, Handbook of Porous Media (CRC Press, Boca Raton, 2005), pp. 99–160
P.K. Yadav, Meccanica 48, 1607–1622 (2013)
T. Ariman, M. Turk, N. Sylvester, Int. J. Eng. Sci. 11, 905–930 (1973)
A.C. Eringen, Int. J. Eng. Sci. 2, 205–217 (1964)
A.C. Eringen, J. Math. Mech. 16(1), 1–18 (1966)
A.C. Eringen, Microcontinuum field theories: II Fluent media (Springer, 2001)
T. Ariman, A.S. Cakmak, Rheol. Acta 7, 236–242 (1968)
T. Ariman, M. Turk, N. Sylvester, Int. J. Eng. Sci. 12, 273–293 (1974a)
S. Jaiswal, P.K. Yadav, Phys. Fluids 31, 071901 (2019)
J. Prakash, P. Sinha, Int. J. Eng. Sci. 13, 217–232 (1975)
M.A. Turk, N.D. Sylvester, T. Ariman, Trans. Soci. Rheol. 17, 1–21 (1973)
G. Bugliarello, J. Sevilla, Biorheology 7, 85–107 (1970)
T. Ariman, M. Turk, N. Sylvester, J. Appl. Mech. 41, 1–7 (1974b)
P. Chaturani, V. Upadhya, Biorheology 16, 419–428 (1979)
G. Lukaszewicz, Micropolar Fluids: Theory and Applications (Springer, Berlin, 1999)
B.D. Sharma, P.K. Yadav, Transp. Porous Media 120, 239–254 (2017)
D.Y. Khanukaeva, A. Filippov, P. Yadav, A. Tiwari, Eur. J. Mech.-B Fluids 76, 73–80 (2019)
D.Y. Khanukaeva, A. Filippov, P. Yadav, A. Tiwari, J. Mol. Liq. 294, 111558(1–8) (2019)
A.J. Chamkha, Mech. Res. Commun. 21(3), 281–288 (1994)
A.J. Chamkha, Int. J. Numer. Methods Heat Fluid Flow 11(5), 430–448 (2001)
J.C. Umavathi, A.J. Chamkha, A. Mateen, A. Al-Mudhaf, Nonlinear Anal. Model. Control 14(3), 397–415 (2009)
S. Parvin, R. Nasrin, M.A. Alim, N.F. Hossain, A.J. Chamkha, Int. J. Heat Mass Transf. 55, 5268–5274 (2012)
R. Mohebbi, M. Izadi, A.J. Chamkha, Phys. Fluids 29, 122009(1–13) (2017)
J. Raza, F.M. Oudina, A.J. Chamkha, Multidiscip. Model. Mater. Struct. 15(4), 737–757 (2019)
D. Toghraie, R. Mashayekhi, H. Arasteh, S. Sheykhi, M. Niknejadi, A.J. Chamkha, Int. J. Numer. Methods Heat Fluid Flow 30(4), 1795–1814 (2020)
J.C. Umavathi, J.P. Kumar, A.J. Chamkha, I. Pop, Transp. Porous Med. 61, 315–335 (2005)
R. Shail, Int. J. Eng. Sci. 11, 1103–1108 (1973)
A.J. Chamkha, J. Fluids Eng. 122, 117–124 (1999)
J.C. Umavathi, A.J. Chamkha, A. Mateen, A. Al-Mudhaf, Heat Mass Transf. 42, 81 (2005)
J.P. Kumar, J.C. Umavathi, A.J. Chamkha, I. Pop, Appl. Math. Mod. 34, 1175–1186 (2010)
A.J. Chamkha, T. Grosan, I. Pop, Int. Commun. Heat Mass Transf. 29(8), 1119–1127 (2002)
M. Keimanesh, M.M. Rashidi, A.J. Chamkha, R. Jafari, Comput. Math. Appl. 62, 2871–2891 (2011)
A.J. Chamkha, M. Molana, A. Rahnama, F. Ghadami, Powder Technol. 332, 287–322 (2018)
J.R. Murthy, J. Srinivas, Int. J. Heat Mass Transf. 65, 254–264 (2013)
A. Siddiqui, Q. Azim, M. Rana, Nonlinear Sci. Lett. A 1, 67–76 (2010)
M. Devakar, N.C. Ramgopal, J. Porous Med. 18, 549–558 (2015)
I.A. Ansari, S. Deo, Natl. Acad. Sci. Lett. 40, 211–214 (2017)
J.C. Umavathi, A.J. Chamkha, K.S.R. Sridhar, Transp. Porous Med. 85, 157–169 (2010)
F. Selimefendigil, H.F. Öztop, A.J. Chamkha, Int. J. Numer. Methods Heat Fluid Flow 30(4), 1755–1772 (2020)
P.K. Yadav, S. Jaiswal, T. Asim, R. Mishra, Eur. Phy. J. Plus 133, 247 (2018)
P. Yadav, S. Jaiswal, B. Sharma, Appl. Math. Mech. 39, 993–1006 (2018)
P.K. Yadav, S. Jaiswal, Can. J. Phys. 96, 1016–1028 (2018)
A.J. Chamkha, Int. J. Eng. Sci. 33(3), 437–446 (1995)
M. Devakar, A. Raje, Eur. Phy. J. Plus 133, 180 (2018b)
M. Devakar, A. Raje, J. Braz. Soc. Mech. Sci. Eng. 40, 325 (2018)
A.J. Chamkha, Int. J. Heat Mass Transf. 45, 2509–2525 (2002)
A.J. Chamkha, Numer. Heat Transf. Part A Appl. Int. J. Comput. Methodol. 32(6), 653–675 (1997)
A.J. Chamkha, Int. J. Heat Fluid Flow 21, 740–746 (2000)
A. Raje, M. Devakar, Numerical Heat Transfer and Fluid Flow (Springer, Berlin, 2019), pp. 55–64
S. Jaiswal, P.K. Yadav, Arab. J. Sci. Eng. 45, 921–934 (2020)
D.A. Nield, A. Bejan, Convection in Porous Media, 3rd edn. (Springer, New York, 2006)
B. Straughan, Stability and Wave Motion in Porous Media (Springer, New York, 2008)
J. Happel, H. Brenner, Low Reynolds number hydrodynamics: with special applications to particulate media (Springer, Netherlands, 1981)
J.A. Ochoa-Tapia, S. Whitaker, Int. J. Heat Mass Transf. 38(14), 2647–2655 (1995)
J.A. Ochoa-Tapia, S. Whitaker, Int. J. Heat Mass Transf. 38(14), 2635–2646 (1995)
F.A. Coutelieris, J.M.P.Q. Delgado, Transport Processes in Porous Media (Springer, Berlin, 2012)
Acknowledgements
Pramod Kumar Yadav is thankful to SERB, New Delhi for supporting this research work under the research Grant SR/FTP/MS-47/2012.
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Yadav, P.K., Verma, A.K. Analysis of immiscible Newtonian and non-Newtonian micropolar fluid flow through porous cylindrical pipe enclosing a cavity. Eur. Phys. J. Plus 135, 645 (2020). https://doi.org/10.1140/epjp/s13360-020-00672-6
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DOI: https://doi.org/10.1140/epjp/s13360-020-00672-6