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Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure. MATHEMATICS 2021. [DOI: 10.3390/math9040334] [Citation(s) in RCA: 7] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
Some unsteady motions of incompressible upper-convected Maxwell (UCM) fluids with exponential dependence of viscosity on the pressure are analytically studied. The fluid motion between two infinite horizontal parallel plates is generated by the lower plate, which applies time-dependent shear stresses to the fluid. Exact expressions, in terms of standard Bessel functions, are established both for the dimensionless velocity fields and the corresponding non-trivial shear stresses using the Laplace transform technique and suitable changes of the unknown function and the spatial variable in the transform domain. They represent the first exact solutions for unsteady motions of non-Newtonian fluids with pressure-dependent viscosity. The similar solutions corresponding to the flow of the same fluids due to an exponential shear stress on the boundary as well as the solutions of ordinary UCM fluids performing the same motions are obtained as limiting cases of present results. Furthermore, known solutions for unsteady motions of the incompressible Newtonian fluids with/without pressure-dependent viscosity induced by oscillatory or constant shear stresses on the boundary are also obtained as limiting cases. Finally, the influence of physical parameters on the fluid motion is graphically illustrated and discussed. It is found that fluids with pressure-dependent viscosity flow are slower when compared to ordinary fluids.
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Abstract
The development of EHL theory from its tentative beginnings is outlined, with an account of how Ertel explained its relation to Hertz contact theory. The problems caused by the failure of the early numerical analysts to understand that the film thickness depends on only two variables are emphasised, and answers of the form H = F ( P , S ) given. Early methods of measuring the film thickness are described, but these became archaic with the development of optical EHL. The behaviour of surface roughness as it passes through the high pressure region and suffers elastic deformation is described, and the implication for the traditional Λ -ratio noted. In contrast, the understanding of traction is far from satisfactory. The oil in the high pressure region must become non-Newtonian: the early explanation that the viscosity reduction is the effect of temperature proved inadequate. There must be some form of shear thinning (perhaps according to the Eyring theory), but also a limiting shear stress under which the lubricant shears as an elastic solid. It seems that detailed, and difficult, measurements of the high pressure, high shear-rate behaviour of individual oils are needed before traction curves can be predicted.
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Daniels BK. Non-Newtonian Thermo-Viscoelastic EHD Traction from Combined Slip and Spin. ACTA ACUST UNITED AC 2008. [DOI: 10.1080/05698198008982955] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/22/2022]
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Jang J, Khonsari M, Bair S. On the elastohydrodynamic analysis of shear-thinning fluids. Proc Math Phys Eng Sci 2007. [DOI: 10.1098/rspa.2007.0062] [Citation(s) in RCA: 52] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Realistic prediction of the characteristics of the elastohydrodynamic lubrication (EHL) contact requires consideration of the appropriate constitutive equation for the lubricant. In many applications, the lubricant exhibits a shear-thinning behaviour which significantly affects the film thickness. In this paper, we present a generalized formulation that can efficiently treat shear-thinning fluids with provision for compressibility in the EHL line contact. Specifically, the Carreau model and the sinh-law model are investigated. An extensive set of numerical solutions and comparison with experiments reveal that the Carreau equation properly captures the film thickness behaviour under both rolling and sliding conditions.
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Affiliation(s)
- J.Y Jang
- Department of Mechanical Engineering, Louisiana State UniversityBaton Rouge, LA 70803, USA
| | - M.M Khonsari
- Department of Mechanical Engineering, Louisiana State UniversityBaton Rouge, LA 70803, USA
| | - S Bair
- Center for High-Pressure Rheology, Georgia Institute of TechnologyAtlanta, GA 30332, USA
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Mathematical Properties of the Solutions to the Equations Governing the Flow of Fluids with Pressure and Shear Rate Dependent Viscosities* *The contribution of J. Málek to this work is a part of the research projects MSM 0021620839 and LC06052 financed by MSMT. J. Málek thanks also the Czech Science Foundation, the project GACR 201/06/0321, for its support. K.R. Rajagopal thanks the National Science Foundation for its support. HANDBOOK OF MATHEMATICAL FLUID DYNAMICS 2007. [DOI: 10.1016/s1874-5792(07)80011-5] [Citation(s) in RCA: 17] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/10/2023]
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Franta M, Málek J, Rajagopal KR. On steady flows of fluids with pressure– and shear–dependent viscosities. Proc Math Phys Eng Sci 2005. [DOI: 10.1098/rspa.2004.1360] [Citation(s) in RCA: 43] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
There are many technologically important problems such as elastohydrodynamics
which involve the flows of a fluid over a wide range of pressures. While the density
of the fluid remains essentially constant during these flows whereby the fluid can be
approximated as being incompressible, the viscosity varies significantly by several
orders of magnitude. It is also possible that the viscosity of such fluids depends on
the shear rate. Here we consider the flows of a class of incompressible fluids with
viscosity that depends on the pressure and shear rate. We establish the existence of
weak solutions for the steady flows of such fluids subjected to homogeneous Dirichlet
boundary conditions and to specific body forces that are not necessarily assumed to
be small. A novel aspect of the study is the manner in which we treat the pressure
that allows us to establish its compactness, as well as that of the velocity gradient.
The method draws upon the physics of the problem, namely that the notion of
incompressibility is an idealization that is attained by letting the compressibility of
the fluid to tend to zero.
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Affiliation(s)
- M. Franta
- Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
| | - J. Málek
- Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
| | - K. R. Rajagopal
- Department of Mechanical Engineering, Texas A&M University, College Station, TX 77843, USA
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Rajagopal K, Szeri A. On an inconsistency in the derivation of the equations of elastohydrodynamic lubrication. Proc Math Phys Eng Sci 2003. [DOI: 10.1098/rspa.2003.1145] [Citation(s) in RCA: 55] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- K.R Rajagopal
- Department of Mechanical Engineering, Texas A&M University, 3123 College Station, TX 77843‐3123, USA
| | - A.Z Szeri
- Department of Mechanical Engineering, University of Delaware, 108 Spencer Lab, Newark, DE 19716, USA
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Hron J, Málek J, Rajagopal KR. Simple flows of fluids with pressure–dependent viscosities. Proc Math Phys Eng Sci 2001. [DOI: 10.1098/rspa.2000.0723] [Citation(s) in RCA: 110] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Affiliation(s)
- J. Hron
- Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
| | - J. Málek
- Mathematical Institute of Charles University, Sokolovská 83, 186 75 Prague 8, Czech Republic
| | - K. R. Rajagopal
- Department of Mechanical Engineering Texas A&M University, College Station, TX 77843, USA
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Fang N, Chang L, Johnston G, Webster M, Jackson A. An experimental/theoretical approach to modeling the viscous behavior of liquid lubricants under EHL conditions. TRIBOLOGY SERIES 2001. [DOI: 10.1016/s0167-8922(01)80157-7] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 01/27/2023]
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Abstract
A simple constitutive equation is proposed for the isothermal shear of lubricant films in rolling/sliding contacts. The model may be described as nonlinear Maxwell, since it comprises nonlinear viscous flow superimposed on linear elastic strain. The nonlinear viscous function can take any convenient form. It has been found that an Eyring ‘sinh law’ fits the measurements on five different fluids, although the higher viscosity fluids at high pressure are well described by the elastic/perfectly plastic equations of Prandtl-Reuss. The proposed equation covers the complete range of isothermal behaviour: linear and nonlinear viscous, linear viscoelastic, nonlinear viscoelastic and elastic/plastic under any strain history. Experiments in support of the equations are described. The nonlinear Maxwell constitutive equation is expressed in terms of three independent fluid parameters: the shear modulus
G
, the zero-rate viscosity η and a reference stress τ
0
. The variations of these parameters with pressure and temperature, deduced from the experiments, are found to be in broad agreement with the Eyring theory of fluid flow.
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Abstract
A now method of measuring refractive index in a lubricant point contact is described which allows the density, pressure and viscosity in the fluid to be determined. The viscosity measurements are absolute, only the elastic and optical constants of the glass are needed. Due to the very high pressures developed (1 GPa) in the 0.3 mm diameter contact very large pressure gradients are produced, hence viscosities four orders higher than conventional high-pressure viscometers normally reach, can be measured. Four lubricants tested all showed that the pressure viscosity coefficient dropped sharply above 10
3
Pa s (10
4
poise); the limit of the normal viscometer. One fluid - a polyphenylether - apparently vitrified at high pressures to a limiting viscosity of 10
6
Pa s. Ball bounce is shown to limit the range of fluids that can be tested with this apparatus.
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Kyotani T, Yoshitake H, Ito T, Tamai Y. Correlation Between Flow Properties and Traction of Lubricating Oils. ACTA ACUST UNITED AC 1986. [DOI: 10.1080/05698198608981665] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Carmichael GDT, Davies PB. Factors Which Affect the Transient Behaviour of Preloaded Ball Bearing Assemblies. ACTA ACUST UNITED AC 1972. [DOI: 10.1080/05698197208981397] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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Kauzlarich JJ, Greenwood JA. Elastohydrodynamic Lubrication With Herschel-Bulkley Model Greases. ACTA ACUST UNITED AC 1972. [DOI: 10.1080/05698197208981427] [Citation(s) in RCA: 48] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/23/2022]
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