Vorticity and Stress Tensor
DOI:
https://doi.org/10.26713/jims.v10i1-2.1061Keywords:
Navier-Stokes equation, Stress tensor, Vorticity, Coefficients of viscosity, Maxwell stress tensorAbstract
The assumptions of the Navier-Stokes equations are reconsidered. Vorticity is one of the three basic isotropic motions of a fluid element about a point. Taking into consideration the isotropic and symmetric nature of the associated tensor, it was assumed that vorticity could not contribute to the viscous stresses and hence was omitted from subsequent derivations of the Navier-Stokes equations. Deviating from this classical approach, the effect of vorticity is included in the derivation of the stress tensor.The equation hence derived contains additional terms involving vortex viscosity. Analogous to Maxwell stress tensor in electromagnetic fields, another stress tensor is defined in a vorticity field. By treating vortices as physical structures, it is possible to define pressure and the shearing stress that deform the volume element.Downloads
References
G.K. Batchelor., An Introduction to Fluid Dynamics, Cambridge University Press, 141 – 148 (2009), reprinted.
C.H. Berdahl and W.Z. Strang, Behavior of a vorticity influenced asymmetric stress tensor in fluid flow, Flight Dynamics Laboratory, Ohio, Final Report, 1 July 1985-29 May 1986.
H. Brenner, Beyond Navier-Stokes, International Journal of Engineering Science 54 (2012), 67 – 98.
H. Brenner, Navier-Stokes revisited, Physica A, Statistical Mechanics and its Applications 349 (2005), 60 – 132.
G. Buresti, A Note on Stokes' Hypothesis, Acta Mechanica 226 (2015), 3555 – 3559.
S. Chapman and T.G. Cowling, The Mathematical Theory of Non Uniform Gases, Cambridge University Press, 3rd edition (1970).
A.J. Chorin and J.E. Marsden, A Mathematical Introduction to Fluid Mechanics, Springer, 3rd edition, 19 – 20 (2000).
K.J. Cook, H. Fearn and P. Millonni, Fizeaus experiment and the Aharonov-Bohm effect, Am. J. Phys. 63 (8) (1995), 705 – 710.
G. Emanuel and B.M. Argrow, Linear dependence of the bulk viscosity on shock wave thickness, Phys. Fluids 6 (1994), 3203 – 3205.
V.C.A. Ferraro and C. Plumpton, An Introduction to Magneto Fluid Mechanics, Clarandon Press, Oxford (1966).
M. Gad-el-Hak, Questions in fluid mechanics: Stokes' hypothesis for a Newtonian, isotropic fluid, J. Fluid Eng. T. ASME 117 (1995), 3 – 5.
S.M. Karim and L. Rosenhead, The second coefficient of viscosity of liquids and gases, Reviews of Modern Physics 24 (2) (1952), 108 – 116.
B. Lautrup, Physics of Continuous Matter Exotic and Everyday Phenomena in the Macroscopic World, Institute of Physics Publishing Ltd., 112 – 120 (2005).
J.C. Maxwell, On Physical lines of force, Part I: The theory of molecular vortices applied to magnetic phenomena, Philosophical Magazine and Journal of Science 21 (139) (1861), 161 – 175.
K.R. Rajagopal, A new development and interpretation to the Navier-Stokes fluid which reveals why the Stokes assumption is inapt, Int. J. Non Linear Mech. 50 (2013), 141–151.
L. Rosenhead, A discussion on the first and second coefficients of viscosities of fluids, Proc. R. Soc. Lond. A. Mat. 226 (1954), 1 – 69.
A. Rutherford, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Prentice-Hall, Englewood Cliffs, New Jersey, 33 – 35 (2000).
K. Stierstadt and M. Liu, Maxwell's stress tensor and the forces in magnetic liquids, ZAMM – Journal of Applied Mathematics and Mechanics/Zeitschriftfür Angewandte Mathematik und Mechanik 95 (1) (2015), 4 – 37.
G.G. Stokes, On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids, Trans. Camb. Philos. Soc. 8 (1845), 287 – 319.
P.A. Thompson, Compressible-Fluid Dynamics, 20-21, McGraw-Hill, New York (1972).
C. Truesdell, On the viscosity of fluids according to the kinetic theory, Zeitschrift fur Physic 131 (1952), 273 – 289.
C. Truesdell, The mechanical foundations of elasticity and fluid dynamics, Journal of Rational Mechanics and Analysis 1 (1952), 228 – 231.
M.J. Vedan, M.B. Rajeshwari Devi and Susan Mathew P., Maxwell stress tensor in hydrodynamics, IOSR Journal of Mathematics 11 (1) (2015), Ver. V, 58 – 60.
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