On the integro-differential general solution for the unsteady micropolar Stokes flow of a conducting ferrofluid
Author:
Panayiotis Vafeas
Journal:
Quart. Appl. Math. 76 (2018), 19-37
MSC (2010):
Primary 35C05, 35C10, 35C15, 35J25, 76W05, 76D07, 65N99
DOI:
https://doi.org/10.1090/qam/1467
Published electronically:
March 31, 2017
MathSciNet review:
3733092
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Additional Information
Abstract: The three–dimensional (3–D) unsteady creeping motion, corresponding to Stokes flow, of a non–conductive colloidal suspension of ferromagnetic particles, which are embedded within an otherwise electrically conducting, viscous and incompressible, carrier liquid, is considered in this contribution. This group of micropolar conducting ferrofluids comprises a novel class of engineering materials that respond in the presence of a general externally applied magnetic field, which is arbitrarily orientated in the three–dimensional domain of practical interest. Therein, an induced magnetic field of minor importance is created, while the effective viscosity of the fluid is increasing and an additional magnetic pressure appears. In order to be compatible with the principles of both ferrohydrodynamics and magnetohydrodynamics, we readily include the magnetization and the electrical conductivity of the magnetic fluid, respectively into the governing partial differential equations of the particular physical system. Employing the potential representation theory, we fabricate a new integro–differential general solution for the situation under investigation, which provides the time–dependent velocity and total pressure fields in a 3–D spaced closed form and in terms of easy–to–find potentials, via a semi–analytical shape. This generalized representation is proved to be complete, whilst it is valid for any non–axisymmetric geometry. We demonstrate the applicability of our analytical approach, by introducing a basic degenerate case of the aforementioned method to simulate the time–dependent creeping flow of a micropolar fluid with conductive properties inside a circular duct.
References
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- B. S. Padmavathi, G. P. Raja Sekhar, and T. Amaranath, A note on complete general solutions of Stokes equations, Quart. J. Mech. Appl. Math. 51 (1998), no. 3, 383–388. MR 1639020, DOI https://doi.org/10.1093/qjmam/51.3.383
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- Panayiotis Vafeas, Polycarpos K. Papadopoulos, and Pavlos M. Hatzikonstantinou, Analytical integro-differential representation of flow fields for the micropolar Stokes flow of a conducting ferrofluid, IMA J. Appl. Math. 80 (2015), no. 3, 839–864. MR 3394306, DOI https://doi.org/10.1093/imamat/hxu016
- A. Venkatlaxmi, B. S. Padmavathi, and T. Amaranath, Unsteady Stokes equations: some complete general solutions, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 2, 203–213. MR 2062400, DOI https://doi.org/10.1007/BF02829854
- D. Palaniappan, On some general solutions of transient Stokes and Brinkman equations, Journal of Theoretical and Applied Mechanics 52 (2014), 405–415.
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Volumes I and II, McGraw–Hill, New York, 1953.
- E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
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References
- B. Berkovski and V. Bashtovoy, Magnetic Fluids and Applications Handbook, Begell House, New York, 1996.
- V. G. Bashotovoy, B. M. Berkovski and A. N. Vislovich, Introduction to Thermomechanics of Magnetic Fluids, Hemisphere Publishing Corporation, New York, 1988.
- R.E. Rosensweig, Ferrohydrodynamics, Dover Publications, New York, 1997.
- J. Happel and H. Brenner, Low Reynolds Number Hydrodynamics, Prentice Hall: Englewood Cliffs, NJ, 1965 and Martinus Nijholl Publishers: Dordrecht, 1986.
- P. M. Hatzikonstantinou and P. Vafeas, A general theoretical model for the magnetohydrodynamic flow of micropolar magnetic fluids. Application to Stokes flow, Math. Methods Appl. Sci. 33 (2010), no. 2, 233–248. MR 2597203, DOI https://doi.org/10.1002/mma.1170
- P. Chadwick and E. A. Trowbridge, Elastic wave fields generated by scalar wave functions, Proc. Cambridge Philos. Soc. 63 (1967), 1177–1187. MR 0218047
- Xin Sheng Xu and Min Zhong Wang, General complete solutions of the equations of spatial and axisymmetric Stokes flow, Quart. J. Mech. Appl. Math. 44 (1991), no. 4, 537–548. MR 1144982, DOI https://doi.org/10.1093/qjmam/44.4.537
- D. Palaniappan, S. D. Nigam, T. Amaranath, and R. Usha, Lamb’s solution of Stokes’s equations: a sphere theorem, Quart. J. Mech. Appl. Math. 45 (1992), no. 1, 47–56. MR 1154762, DOI https://doi.org/10.1093/qjmam/45.1.47
- B. S. Padmavathi, G. P. Raja Sekhar, and T. Amaranath, A note on complete general solutions of Stokes equations, Quart. J. Mech. Appl. Math. 51 (1998), no. 3, 383–388. MR 1639020, DOI https://doi.org/10.1093/qjmam/51.3.383
- G. D. McBain, Convection in a horizontally heated sphere, J. Fluid Mech. 438 (2001), 1–10. MR 1849873, DOI https://doi.org/10.1017/S0022112001003913
- B. Sri Padmavati and T. Amaranath, A note on decomposition of solenoidal fields, Appl. Math. Lett. 15 (2002), no. 7, 803–805. MR 1920978, DOI https://doi.org/10.1016/S0893-9659%2802%2900045-9
- Panayiotis Vafeas, Polycarpos K. Papadopoulos, and Pavlos M. Hatzikonstantinou, Analytical integro-differential representation of flow fields for the micropolar Stokes flow of a conducting ferrofluid, IMA J. Appl. Math. 80 (2015), no. 3, 839–864. MR 3394306, DOI https://doi.org/10.1093/imamat/hxu016
- A. Venkatlaxmi, B. S. Padmavathi, and T. Amaranath, Unsteady Stokes equations: some complete general solutions, Proc. Indian Acad. Sci. Math. Sci. 114 (2004), no. 2, 203–213. MR 2062400, DOI https://doi.org/10.1007/BF02829854
- D. Palaniappan, On some general solutions of transient Stokes and Brinkman equations, Journal of Theoretical and Applied Mechanics 52 (2014), 405–415.
- P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Volumes I and II, McGraw–Hill, New York, 1953.
- E. W. Hobson, The theory of spherical and ellipsoidal harmonics, Chelsea Publishing Company, New York, 1955. MR 0064922
- L. M. Delves and J. Walsh, Numerical Solutions of Integral Equations, Clarendon, Oxford, 1974.
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Additional Information
Panayiotis Vafeas
Affiliation:
Department of Chemical Engineering, University of Patras, 265 04 Patras, Greece
MR Author ID:
684750
Email:
vafeas@chemeng.upatras.gr
Keywords:
Differential and integral solutions,
magnetohydrodynamics,
ferrohydrodynamics,
unsteady micropolar Stokes flow
Received by editor(s):
February 11, 2017
Published electronically:
March 31, 2017
Article copyright:
© Copyright 2017
Brown University