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Non-Newtonian flow at lowest order,the role of the Reiner–Rivlin stress
Institution:1. Department of Mathematics, Quaid-I-Azam University, 45320, Islamabad 44000, Pakistan;2. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Department of Mathematics, Faculty of Science, King Abdulaziz University, 80203, Jeddah 21589, Saudi Arabia;1. Department of Mathematics, B V Raju Institute of Technology, Narsapur, Medak 502313, Telangana, India;2. Fluid Mechanics, Propulsion and Nanosystems, Aeronautical and Mechanical Engineering, Newton Bldg., University of Salford, Manchester M54WT, UK;3. Materials and Corrosion, Petroleum and Gas Engineering, Newton Bldg., University of Salford, Manchester M54WT, UK;1. COMSATS Institute of Information Technology, Abbottabad, Pakistan;2. Department of Mathematics, Faculty of Sciences, HITEC University Taxila Cantt, Pakistan
Abstract:In 1963 Giesekus H. Giesekus, Die simultane Translations- und Rotationsbewegung einer Kugel in einer elastikoviskosen Flüssigkeit, Rheol. Acta 3 (1962) 59–71] showed that a Stokes velocity field also satisfies the equilibrium equation for the flow of a restricted form of the second order fluid. The same result was found by Tanner R.I. Tanner, Plane creeping flows of incompressible second order fluids, Phys. Fluids 9 (1966) 1246–1247] in 1966 in the context of plane flow for which the restrictions on the second order fluid are not relevant. Tanner R.I. Tanner, Some extended Giesekus-type theorems for non-Newtonian fluids, Rheol. Acta 28 (1989) 449–452] later showed that the velocity field for the inertialess, plane flow of the generalized Newtonian fluid is also the velocity field for the flow of a special form of the Criminale–Ericksen–Filbey (CEF) stress system W.O. Criminale Jr., J.L. Ericksen, G.L. Filbey Jr., Steady flow of non-Newtonian fluids, Arch. Rat. Mech. 1 (1958) 410–417]. In this paper it will be shown that the results of Giesekus and Tanner are special cases of a more general theorem in which the velocity field, in any dimension, of the equilibrium Reiner–Rivlin problem also satisfies the corresponding problem for the materially steady stress system (a generalization of the CEF system) provided the coefficients of the Reiner–Rivlin stress M. Reiner, A mathematical theory of dilatancy, Am. J. Math. 67 (1945) 350–362; R.S. Rivlin, The hydrodynamics of non-Newtonian fluids, Proc. R. Soc. Lond. 193 (1948) 260–281] are derivable from a strain-rate potential. As with the Giesekus–Tanner theorems the new theorem holds generally for velocity boundary conditions, but in some cases, such as the free jet, stress boundary conditions can be imposed.
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