Article
Full entry |
PDF
(0.5 MB)
Feedback
PDF
(0.5 MB)
Feedback
Keywords:
pressure dependent viscosity; implicit constitutive theory; Poiseuille flow
pressure dependent viscosity; implicit constitutive theory; Poiseuille flow
Summary:
Stokes recognized that the viscosity of a fluid can depend on the normal stress and that in certain flows such as flows in a pipe or in channels under normal conditions, this dependence can be neglected. However, there are many other flows, which have technological significance, where the dependence of the viscosity on the pressure cannot be neglected. Numerous experimental studies have unequivocally shown that the viscosity depends on the pressure, and that this dependence can be quite strong, depending on the flow conditions. However, there have been few analytical studies that address the flows of such fluids despite their relevance to technological applications such as elastohydrodynamics. Here, we study the flow of such fluids in a pipe under sufficiently high pressures wherein the viscosity depends on the pressure, and establish an explicit exact solution for the problem. Unlike the classical Navier-Stokes solution, we find the solutions can exhibit a structure that varies all the way from a plug-like flow to a sharp profile that is essentially two intersecting lines (like a rotated V). We also show that unlike in the case of a Navier-Stokes fluid, the pressure depends both on the radial and the axial coordinates of the pipe, logarithmically in the radial coordinate and exponentially in the axial coordinate. Exact solutions such as those established in this paper serve a dual purpose, not only do they offer solutions that are transparent and provide the solution to a specific but simple boundary value problems, but they can be used also to test complex numerical schemes used to study technologically significant problems.
References:
[1] G. G. Stokes: On the theories of the internal friction of fluids in motion and of the equilibrium motion of elastic solids. Trans. Cambridge Phil. Soc. 8 (1845), 287–305.
[2] P. W. Bridgman: The Physics of High Pressure. The MacMillan Company, New York, 1931.
[3] K. R. Rajagopal, A. S. Wineman: On constitutive equations for branching of response with selectivity. Intl. J. Non-Linear Mech. 15 (1980), 83–91. DOI 10.1016/0020-7462(80)90002-5 | MR 0580724
[4] W. G. Cutler, R. H. McMickle, W. Webb, and R. W. Schiessler: Study of the compressions of several high molecular weight hydrocarbons. J. Chem. Phys. 29 (1958), 727–740.
[5] E. M. Griest, W. Webb, and R. W. Schiessler: Effect of pressure on viscosity of high hydrocarbons and their mixtures. J. Chem. Phys. 29 (1958), 711–720.
[6] K. L. Johnson, R. Cameron: Shear behaviour of elastohydrodynamic oil films at high rolling contact pressures. Prof. Instn. Mech. Engrs. 182 (1967), 307–319. DOI 10.1243/PIME_PROC_1967_182_029_02
[7] K. L. Johnson, J. L. Tevaarwerk: Shear behaviour of elastohydrodynamic oil films. Proc. R. Soc. Lond. Ser. A 356 (1977), 215–236.
[8] K. L. Johnson, J. A. Greenwood: Thermal analysis of an Eyring fluid in elastohydrodynamic traction. Wear 61 (1980), 355–374.
[9] S. Bair, W. O. Winer: The high pressure high shear stress rheology of fluid liquid lubricants. J. Tribology 114 (1992), 1–13.
[10] A. Z. Szeri: Fluid Film Lubrication: Theory and Design. Cambridge University Press, Cambridge, 1998. Zbl 1001.76001
[11] J. Hron, J. Málek, and K. R. Rajagopal: Simple flows of fluids with pressure dependent viscosities. Proc. R. Soc. Lond., Ser. A 457 (2001), 1603–1622. DOI 10.1098/rspa.2000.0723
[12] J. Málek, J. Nečas and K. R. Rajagopal: Global analysis of the flows of fluid with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165 (2002), 243–267. DOI 10.1007/s00205-002-0219-4 | MR 1941479
[13] E. L. Ince: Ordinary Differential Equations. Dover Publications, New York, 1944. MR 0010757 | Zbl 0063.02971
[14] N. W. McLachlan: Bessel Functions for Engineers. Clarendon Press, Oxford, 1955.
[15] A. Malewsky, D. Yen: Strongly chaotic non-Newtonian mantle convection in the Earth’s mantle. Geophys. Astrophys. Fluid Dynamics 65 (1996), 149–171.
Search
Partner of
