Effect of Pressure-Dependent Viscosity on the Exiting Sheet Thickness in the Calendering of Newtonian Fluids

2012 ◽  
Author(s):  
Alfredo Hernández ◽  
Oscar E. Bautista ◽  
Eric G. Bautista ◽  
Jose C. Arcos
Keyword(s):  
Mathematical Problem ◽  
Sheet Thickness ◽  
Pressure Effects ◽  
Momentum Balance ◽  
Maximum Pressure ◽  
Pressure Fields ◽  
Pressure Dependent Viscosity ◽  
Pressure Independent ◽  
Roll Separating Force ◽  
Dependent Viscosity

A theoretical model was developed to describe the calendering process in Newtonian sheets of finite initial thickness taking into account that the viscosity of the fluid is a well-defined function of the pressure. We predict the influence of the pressure effects on the leave-off distance that is related to the exiting sheet thickness in the calendering process. The mass and momentum balance equations, which are based on lubrication theory, were nondimensionalized and solved for the velocity and pressure fields by using perturbation techniques, where the leave-off distance represents an eigenvalue of the mathematical problem. When the above variables were obtained, the dimensionless leave-off distance in the calendering process was determined. Moreover, quantities of engineering interest were calculated, including the maximum pressure, the roll-separating force and the power transmitted to the fluid by the rolls. The analytical results show that the inclusion of pressure-dependent viscosity effect increases about 3.37 percent the dimensionless exiting sheet thickness or 9.4 percent the leave-off distance in comparison with the case of pressure-independent viscosity.

Two Dimensional Modified Reynolds Equation Including Pressure Dependent Viscosity Effect

2012 ◽  
Author(s):  
Jung Gu Lee ◽  
Alan Palazzolo
Keyword(s):  
Reynolds Equation ◽  
Elastohydrodynamic Lubrication ◽  
Fluid Film ◽  
Maximum Pressure ◽  
Elastic Solids ◽  
Pressure Distributions ◽  
Pressure Dependent Viscosity ◽  
Modified Reynolds Equation ◽  
Dependent Viscosity ◽  
Modified Equation

The Reynolds equation plays an important role for predicting pressure distributions for fluid film bearing analysis, One of the assumptions on the Reynolds equation is that the viscosity is independent of pressure. This assumption is still valid for most fluid film bearing applications, in which the maximum pressure is less than 1 GPa. However, in elastohydrodynamic lubrication (EHL) where the lubricant is subjected to extremely high pressure, this assumption should be reconsidered. The 2D modified Reynolds equation is derived in this study including pressure-dependent viscosity, The solutions of 2D modified Reynolds equation is compared with that of the classical Reynolds equation for the ball bearing case (elastic solids). The pressure distribution obtained from modified equation is slightly higher pressures than the classical Reynolds equations.

Calendering Pseudoplastic and Viscoplastic Fluids With Slip at the Roll Surface

2005 ◽  
Vol 73 (2) ◽  
pp. 291-299 ◽  
Author(s):  
E. Mitsoulis ◽  
S. Sofou
Keyword(s):  
Yield Stress ◽  
Power Law ◽  
Sheet Thickness ◽  
Power Input ◽  
Maximum Pressure ◽  
Slip Coefficient ◽  
Bingham Number ◽  
Roll Separating Force ◽  
Power Law Index ◽  
Infinite Reservoir

The lubrication approximation theory (LAT) is used to provide numerical results for calendering a sheet from an infinite reservoir. The Herschel–Bulkley model of viscoplasticity is employed, which reduces with appropriate modifications to the Bingham, power-law, and Newtonian models. A dimensionless slip coefficient is introduced to account for the case of slip at the roll surfaces. The results give the final sheet thickness as a function of the dimensionless power-law index (in the case of pseudoplasticity), the Bingham number or the dimensionless yield stress calculated at the nip (in the case of viscoplasticity), and the dimensionless slip coefficient in both cases. Integrated quantities of engineering interest are also calculated. These include the maximum pressure, the roll-separating force, and the power input to the rolls. Decreasing the power-law index or increasing the dimensionless yield stress lead to excess sheet thickness over the thickness at the nip. All engineering quantities calculated in dimensionless form increase substantially with the departure from the Newtonian values. The presence of slip decreases pressure and the engineering quantities and increases the domain in all cases.

The oil film in a closing gap

1962 ◽  
Vol 266 (1326) ◽  
pp. 312-328 ◽  
Keyword(s):  
High Pressure ◽  
Film Thickness ◽  
Pressure Coefficient ◽  
Lubricant Film ◽  
Polished Surface ◽  
Maximum Pressure ◽  
Pressure Dependent Viscosity ◽  
Special Cases ◽  
Dry Contact ◽  
Dependent Viscosity

This paper presents a solution to the elasto-hydrodynamic problem of normal approach of two cylindrical bodies separated by a lubricating film. Analytic solutions are found for the special cases of constant viscosity and rigid material and also for pressure-dependent viscosity. The more general case accounting for elastic deformation of the bodies with constant or pressure dependent viscosity was solved by using an iterative numerical process with the help of an electronic computer. It is found that a very high pressure may be developed in the lubricant film at a finite separation of the cylinders. As the film thickness is further reduced, the value of the maximum pressure decreases and as the film thickness approaches zero, the pressure distribution converges to the Hertzian dry contact form. For a given load applied to the cylinders, the value of the maximum pressure reached depends to a large extent upon the value of the parameter oc E , i.e. the product of the pressure coefficient of viscosity and the equivalent Young’s modulus of the elastic system. Also, once the pressure has reached a sufficiently high value it becomes extremely sensitive to an increase in load; a small increase in load will produce a large increase in maximum pressure. A number of experiments were performed in order to check some of the theoretical predictions made. In these experiments a loaded steel ball was allowed to approach the polished surface of various materials whose surfaces were covered by a lubricant film, and the plastic deformations produced in the surface were then measured. These tests showed clearly the influence of the lubricant in that in every case the depth of the impressions with lubricant was significantly larger than the corresponding ones produced under Hertzian, dry contact impacts. The experimental results indicate a correlation between maximum pressure and the value of ol E and its sensitivity to increase in load at high pressure as predicted by the theory.

scholarly journals Influence of Viscous Dissipation on the Exiting Sheet Thickness in the Calendering of Newtonian Fluids

2020 ◽  
Vol 7 ◽  
Keyword(s):  
Mathematical Problem ◽  
Sheet Thickness ◽  
Newtonian Fluids ◽  
Temperature Fields ◽  
Balance Equations ◽  
Initial Thickness ◽  
Temperature Dependent ◽  
Temperature Dependent Viscosity ◽  
Dependent Viscosity ◽  
Velocity Pressure

In this work we treat theoretically the calendering process of Newtonian fluids with finite sheet initial thickness, taking into account that the viscosity of the fluid is a welldefined function of the temperature. We predict the influence of the temperature-dependent viscosity on the exiting sheet thickness in the calendering process. The mass, momentum and energy balance equations, based on the lubrication theory, were nondimensionalized and solved for the velocity, pressure and temperature fields by using perturbation and numerical techniques, where the exiting sheet thickness represents an eigenvalue of the mathematical problem. The numerical results show that the inclusion of temperature-dependent viscosity effect reduces about 20% the leave-off distance in comparison with the case of temperature-independent viscosity.

A depth-averaged -rheology for shallow granular free-surface flows

2014 ◽  
Vol 755 ◽  
pp. 503-534 ◽  
Author(s):  
J. M. N. T. Gray ◽  
A. N. Edwards
Keyword(s):  
Cutoff Frequency ◽  
Free Surface Flows ◽  
Uniform Flow ◽  
Momentum Balance ◽  
Pressure Dependent Viscosity ◽  
Rough Bed ◽  
Uniform Solution ◽  
Ill Posed ◽  
Dependent Viscosity ◽  
Bed Friction

AbstractThe $\mu (I)$-rheology is a nonlinear viscous law, with a strain-rate invariant and pressure-dependent viscosity, that has proved to be effective at modelling dry granular flows in the intermediate range of the inertial number, $I$. This paper shows how to incorporate the rheology into depth-averaged granular avalanche models. To leading order, the rheology generates an effective basal friction, which is equivalent to a rough bed friction law. A depth-averaged viscous-like term can be derived by integrating the in-plane deviatoric stress through the avalanche depth, using pressure and velocity profiles from a steady-uniform solution to the full $\mu (I)$-rheology. The resulting viscosity is proportional to the thickness to the three halves power, with a coefficient of proportionality that is angle dependent. When substituted into the depth-averaged momentum balance this term generates second-order derivatives of the depth-averaged velocity, which are multiplied by a small parameter. Its inclusion therefore represents a singular perturbation to the equations. It is shown that a granular front propagating down a rough inclined plane is completely unaffected by the rheology, but, discontinuities, which naturally develop in inviscid roll-wave solutions, are smoothed out. By comparison with existing experimental data, it is shown that the depth-averaged $\mu (I)$-rheology accurately predicts the growth rate of spatial instabilities to steady-uniform flow, as well as the dependence of the cutoff frequency on the Froude number and inclination angle. This provides strong evidence that, in the steady-uniform flow regime, the predicted decrease in the viscosity with increasing slope is correct. Outside the range of angles where steady-uniform flows develop, the viscosity becomes negative, which implies that the equations are ill-posed. This is a signature of the ill-posedness of the full $\mu (I)$-rheology at both high and low inertial numbers. The depth-averaged $\mu (I)$-rheology therefore cannot be used outside the valid range of angles without additional regularization.

Elastohydrodynamic Theory of Spherical Bodies in Normal Approach

1970 ◽  
Vol 92 (1) ◽  
pp. 145-153 ◽  
Author(s):  
H. Christensen
Keyword(s):  
Film Thickness ◽  
Elastic Deformation ◽  
Numerical Solutions ◽  
Maximum Pressure ◽  
Mineral Oils ◽  
Elasticity Equations ◽  
Metallic Contacts ◽  
Pressure Dependent Viscosity ◽  
Approach Velocity ◽  
Dependent Viscosity

The elastohydrodynamic problem of normal approach of two spherical bodies is studied and the lubrication and elasticity equations governing this type of motion are established. Numerical solutions to the general case accounting for elastic deformation of the bodies and pressure dependent viscosity are presented. It is found that for values of central film thickness that are not too small, the load and relative approach velocity is much more influenced by the increase of viscosity with pressure than by the effects of elastic distortion. Once the separation of the two surfaces becomes small enough, however, the effects of elastic deformation will profoundly influence all aspects of the motion. The transition film thickness HT at which this change takes place is sharply defined and for metallic contacts lubricated with mineral oils quite small, even compared to the surface roughness. Very high pressure—considerably in excess of the Hertzian maximum pressure corresponding to the load—can be generated by the normal approach motion. The maximum value of pressure is generated when film thickness reaches its transition value HT for the load in question. For loads sufficiently large to generate a high enough pressure in the oil film a small increase in load will cause a large increase in maximum pressure. Once the pressure has reached a high enough value it becomes extremely sensitive to a further increase in load.

Stochastic Reynolds equation for the combined effects of piezo-viscous dependency and squeeze-film lubrication of micropolar fluids on rough porous circular stepped plates

2021 ◽  
pp. 135065012110224
Author(s):  
Hanumagowda Bannihalli Naganagowda ◽  
Sreekala Cherkkarathandayan Karappan
Keyword(s):  
Carrying Capacity ◽  
Stochastic Approach ◽  
Squeeze Film ◽  
Closed Form Expression ◽  
Load Carrying Capacity ◽  
Micropolar Fluids ◽  
Pressure Dependent Viscosity ◽  
Load Carrying ◽  
Dependent Viscosity ◽  
Film Characteristics

The aim of this paper is to presents a theoretical analysis on squeeze-film characteristics of a rough porous circular stepped plate in the vicinity of pressure-dependent viscosity and lubrication by micropolar fluids. A closed-form expression for non-dimensional pressure, load, and squeezing time is derived based on Eringen’s theory, Darcy’s equation, and Christensen’s stochastic approach. Results indicate that the effects of pressure-dependent viscosity, surface roughness, and micropolar fluids play an important role in increasing the load-carrying capacity and squeezing time, whereas the presence of porous media decreases the load-carrying capacity and squeezing time of the rough porous circular stepped plates.

scholarly journals Analytical Solutions of Upper Convected Maxwell Fluid with Exponential Dependence of Viscosity under the Influence of Pressure

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 334
Author(s):  
Constantin Fetecau ◽  
Dumitru Vieru ◽  
Tehseen Abbas ◽  
Rahmat Ellahi
Keyword(s):  
Fluid Motion ◽  
Maxwell Fluid ◽  
Newtonian Fluids ◽  
Exponential Dependence ◽  
Physical Parameters ◽  
Shear Stresses ◽  
Parallel Plates ◽  
Pressure Dependent Viscosity ◽  
Laplace Transform Technique ◽  
Dependent Viscosity

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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