Influence of an Inlet Oil Meniscus Geometry on Parameters of a Point Elastohydrodynamic Contact

1997 ◽  
Vol 119 (1) ◽  
pp. 112-125
Author(s):  
I. I. Kudish ◽  
M. Ya. Panovko
Keyword(s):  
Free Boundary ◽  
Boundary Conditions ◽  
Reynolds Equation ◽  
Contact Region ◽  
Numerical Results ◽  
Contact Boundary ◽  
Frictional Stress ◽  
Stress Distributions ◽  
Pressure Gap ◽  
Subsurface Stress

In the present paper an isothermal elastohydrodynamic problem for a lightly loaded lubricated point contact of elastic bodies is formulated and studied numerically. Mathematical formulation of the problem is based on a steady nonlinear system of integrodifferential equations: Reynolds’ equation, equations of elasticity, boundary conditions for pressure, and the equilibrium condition. Nonlinearity of the problem is caused by nonlinearity of the Reynolds equation and the boundary conditions for pressure describing a free boundary. The inlet contact boundary is considered to be known and located close to the center of the contact. In order to determine the location of the free boundary (exit boundary of the contact region) the problem is formulated as a problem of complementarity by Kostreva (1984a) and Oh (1984). The dimensionless system of equations and inequalities for the elastohydrodynamic lubrication (EHL) problem is solved using the Newton-Raphson method. The inlet boundary of a contact region is considered to have a complex irregular shape (the inlet oil meniscus has some deep notches) which is taken into account while deriving the finite-difference equations. The effect of the shape and location of the inlet oil meniscus on the lubrication film thickness, pressure, gap, sliding and rolling frictional and subsurface stress distributions are considered. Some numerical results are presented for pressure, gap, frictional and subsurface stress distributions in EHL contact. These numerical results show that variations in the inlet meniscus shape and location may cause significant qualitative and quantitative changes in distributions of parameters of a lubricated contact. For instance, the maximum values of pressure may change by 20 percent, and for rolling frictional stress by 100 percent and even more.

A numerical method for subglacial cavitation posed as a variational inequality

2021 ◽  
Author(s):  
Gonzalo Gonzalez de Diego ◽  
Patrick Farrell ◽  
Ian Hewitt
Keyword(s):  
Variational Inequality ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Numerical Method ◽  
Stokes Equations ◽  
Water Pressure ◽  
High Water ◽  
Contact Boundary ◽  
Novel Approach ◽  
Discrete Contact

<p>Subglacial cavitation is a phenomenon that occurs at the base of an ice sheet or a glacier where the ice detaches from the bedrock at high water pressures. The process is recognised as an essential mechanism in glacial sliding. A mathematical description of subglacial cavitation involves a free boundary equation and a Stokes equation with contact boundary conditions. These contact boundary conditions model the process of detachment from the bed at each instant in time. <br><br>In this talk we show that the problem can be written as a variational inequality and present a novel approach to solving the equations with finite element methods that exploit the structure of the variational inequality. In particular, we present a formulation involving Lagrange multipliers, which allows us to solve the discrete contact conditions exactly. Thanks to this latter property, the Stokes equations can be solved together with the free boundary equations in a robust and stable manner. A similar method should also prove useful for improving grounding-line calculations.<br><br>With this numerical method, we compute a friction law <span>(the relation between sliding velocity and shear stress) for ice flowing</span> over a periodic bed.  <span>We recover existing results for the case when the cavities are in a steady state for a given effective pressure.</span> We extend these results to consider time-varying cavitation driven by changes in subglacial water pressure.</p>

scholarly journals Asymptotic Behavior of Solutions to Free Boundary Problem with Tresca Boundary Conditions

2021 ◽  
Vol 2021 ◽  
pp. 1-9
Author(s):  
Abdelkader Saadallah ◽  
Nadhir Chougui ◽  
Fares Yazid ◽  
Mohamed Abdalla ◽  
Bahri Belkacem Cherif ◽  
...  
Keyword(s):  
Asymptotic Behavior ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Boundary Problem ◽  
Free Boundary Problem ◽  
Reynolds Equation ◽  
Limit Problem ◽  
Asymptotic Behavior Of Solutions ◽  
Thin Domain

In this paper, we study the asymptotic behavior of an incompressible Herschel-Bulkley fluid in a thin domain with Tresca boundary conditions. We study the limit when the ε tends to zero, we prove the convergence of the unknowns which are the velocity and the pressure of the fluid, and we obtain the limit problem and the specific Reynolds equation.

Non-axisymmetric magnetohydrodynamic shear layers in a rotating spherical shell

2000 ◽  
Vol 408 ◽  
pp. 239-274 ◽  
Author(s):  
ANDREW M. SOWARD ◽  
RAINER HOLLERBACH
Keyword(s):  
Magnetic Field ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Spherical Shell ◽  
Shear Layers ◽  
Numerical Results ◽  
Free Boundary Conditions ◽  
Viscous Forces ◽  
Inner Sphere ◽  
Stress Free Boundary

Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric magnetic field. Slow steady non-axisymmetric motion is driven by a prescribed non-axisymmetric body force; both rigid and stress-free boundary conditions are considered. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E limit analytically for values of the Elsasser number Λ less than order unity and they are compared with new numerical results. The analytic study focuses on the nature of the various shear layers on the equatorial tangent cylinder attached to the inner sphere. Though the ageostrophic layers correspond to those previously isolated by Kleeorin et al. (1997) for axisymmetric flows, the quasi-geostrophic layers have a new structure resulting from the asymmetry of the motion.In the absence of magnetic field, the inviscid limit exhibits a strong shear singularity on the tangent cylinder only removeable by the addition of viscous forces. With the inclusion of magnetic field, large viscous forces remain whose strength [Zscr ] was measured indirectly by Hollerbach (1994b). For magnetic fields with dipole parity, cf. Kleeorin et al. (1997), [Zscr ] increases throughout the range Λ [Lt ] 1; whereas, for quadrupole parity, cf. Hollerbach (1994b), [Zscr ] only increases for Λ [Lt ] E1/5.The essential difference between the dipole and quadrupole fields is the magnitude of their radial components in the neighbourhood of the equator of the inner sphere. Its finite value for the quadrupole parity causes the internal shear layer – the Hartmann–Stewartson layer stump – to collapse and merge with the equatorial Ekman layer when Λ = O(E1/5). Subsequently the layer becomes an equatorial Hartmann layer, which thins and spreads polewards about the inner sphere surface as Λ increases over the range E1/5 [Lt ] Λ [Lt ] 1. Its structure for the stress-free boundary conditions employed in Hollerbach's (1994b) model is determined through matching with a new magnetogeostrophic solution and the results show that the viscous shear measured by [Zscr ] decreases with increasing Λ. Since [Zscr ] depends sensitively on the detailed boundary layer structure, it provides a sharp diagnostic of new numerical results for Hollerbach's model; the realized [Zscr ]-values compare favourably with the asymptotic theory presented.

scholarly journals Local Well-Posedness for Free Boundary Problem of Viscous Incompressible Magnetohydrodynamics

Mathematics ◽  
2021 ◽  
Vol 9 (5) ◽  
pp. 461
Author(s):  
Kenta Oishi ◽  
Yoshihiro Shibata
Keyword(s):  
Free Surface ◽  
Free Boundary ◽  
Boundary Conditions ◽  
Stokes Equations ◽  
Mhd Flow ◽  
Transmission Conditions ◽  
Well Posedness ◽  
Anisotropic Space ◽  
Free Boundary Conditions ◽  
Incompressible Magnetohydrodynamics

In this paper, we consider the motion of incompressible magnetohydrodynamics (MHD) with resistivity in a domain bounded by a free surface. An electromagnetic field generated by some currents in an external domain keeps an MHD flow in a bounded domain. On the free surface, free boundary conditions for MHD flow and transmission conditions for electromagnetic fields are imposed. We proved the local well-posedness in the general setting of domains from a mathematical point of view. The solutions are obtained in an anisotropic space Hp1((0,T),Hq1)∩Lp((0,T),Hq3) for the velocity field and in an anisotropic space Hp1((0,T),Lq)∩Lp((0,T),Hq2) for the magnetic fields with 2<p<∞, N<q<∞ and 2/p+N/q<1. To prove our main result, we used the Lp-Lq maximal regularity theorem for the Stokes equations with free boundary conditions and for the magnetic field equations with transmission conditions, which have been obtained by Frolova and the second author.

The Bending, Buckling, and Flexural Vibration of Simply Supported Polygonal Plates by Point-Matching

1961 ◽  
Vol 28 (2) ◽  
pp. 288-291 ◽  
Author(s):  
H. D. Conway
Keyword(s):  
Hydrostatic Pressure ◽  
Boundary Conditions ◽  
Flexural Vibration ◽  
Frequency Equation ◽  
Numerical Results ◽  
Special Technique ◽  
Point Matching ◽  
Buckling Loads ◽  
Simply Supported ◽  
Flexural Vibrations

The bending by uniform lateral loading, buckling by two-dimensional hydrostatic pressure, and the flexural vibrations of simply supported polygonal plates are investigated. The method of meeting the boundary conditions at discrete points, together with the Marcus membrane analog [1], is found to be very advantageous. Numerical examples include the calculation of the deflections and moments, and buckling loads of triangular square, and hexagonal plates. A special technique is then given, whereby the boundary conditions are exactly satisfied along one edge, and an example of the buckling of an isosceles, right-angled triangle plate is analyzed. Finally, the frequency equation for the flexural vibrations of simply supported polygonal plates is shown to be the same as that for buckling under hydrostatic pressure, and numerical results can be written by analogy. All numerical results agree well with the exact solutions, where the latter are known.

Measurement of the Interfacial Stresses in Rolling Using the Elastic Deformation of the Roll

1987 ◽  
Vol 109 (4) ◽  
pp. 362-369 ◽  
Author(s):  
D. J. Meierhofer ◽  
K. A. Stelson
Keyword(s):  
Elastic Deformation ◽  
Normal Pressure ◽  
New Method ◽  
Frictional Stress ◽  
Stress Distributions ◽  
Strain Gages ◽  
Experimental Rolling ◽  
Mechanical Isolation ◽  
Roll Gap ◽  
Future Work

A new method to measure the frictional stresses and normal pressure in the roll gap during cold rolling, and experimental verification of this new method, are presented. The method overcomes many of the shortcomings of pin-type sensors. The elastic deformation of the roll itself is measured with strain gages, and is used to calculate the stresses between the sheet and the roll. Since no modification of the roll is necessary, the deformation process is undisturbed by the measurement. Mechanical isolation of the sensor is unnecessary. The mathematical procedure used to calculate the normal pressure and frictional stresses from the measured strains explicitly acknowledges that these strains are the result of the entire distribution of pressures and shears in the roll gap. An experimental rolling mill was constructed to verify the proposed method. Lead was rolled, and the resulting pressure and frictional stress distributions in the roll gap were measured. Several features of these distributions are in agreement with measurements made by various investigators using other techniques, thereby confirming the usefulness of the new method. Future work is proposed to increase the accuracy with which the roll gap stresses may be measured.

NEW MODELS IN MICROPOLAR FLUID AND THEIR APPLICATION TO LUBRICATION

2005 ◽  
Vol 15 (03) ◽  
pp. 343-374 ◽  
Author(s):  
GUY BAYADA ◽  
NADIA BENHABOUCHA ◽  
MICHÈLE CHAMBAT
Keyword(s):  
Boundary Condition ◽  
Friction Coefficient ◽  
Boundary Conditions ◽  
Existence And Uniqueness ◽  
Micropolar Fluid ◽  
Reynolds Equation ◽  
Slip Boundary Condition ◽  
Slip Boundary ◽  
New Models ◽  
Uniqueness Of The Solution

A thin micropolar fluid with new boundary conditions at the fluid-solid interface, linking the velocity and the microrotation by introducing a so-called "boundary viscosity" is presented. The existence and uniqueness of the solution is proved and, by way of asymptotic analysis, a generalized micropolar Reynolds equation is derived. Numerical results show the influence of the new boundary conditions for the load and the friction coefficient. Comparisons are made with other works retaining a no slip boundary condition.

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