Non-Newtonian Lubrication With the Convected Maxwell Model

1996 ◽  
Vol 118 (2) ◽  
pp. 344-348 ◽  
Author(s):  
J. A. Tichy
Keyword(s):  
Time Dependence ◽  
Fluid Model ◽  
Time Derivative ◽  
Maxwell Model ◽  
Deborah Number ◽  
Time Dependent ◽  
Nonlinear Coupling ◽  
Surface Slope ◽  
Regular Perturbation ◽  
Leading Term

In certain applications where the lubricant is subjected to rapidly changing conditions along its flowing path (such as an elastohydrodynamic contact), the time dependent nature of the lubricant may be significant. One of the simplest types of models to account for such fluid time dependence is the Maxwell model. The time derivative used in such a model must be written with respect to coordinates which translate and rotate with the fluid, or coordinates which deform with the fluid. Unfortunately, such derivatives greatly complicate problems and are rarely used, due to nonlinear coupling of stresses. An admissible formulation of the Maxwell viscoelastic fluid model using the convected derivative has been applied to lubrication flow. Using a regular perturbation in the Deborah number, with the conventional lubrication solution as the leading term, a solution can be obtained. Viscoelasticity may raise or lower pressure depending on combinations of surface slope and curvature.

Perturbation Solution of Non-Newtonian Lubrication With the Convected Maxwell Model

2005 ◽  
Vol 127 (2) ◽  
pp. 302-305 ◽  
Author(s):  
Rong Zhang ◽  
Xueming He ◽  
Simon X. Yang ◽  
Xinkai Li
Keyword(s):  
Viscoelastic Fluid ◽  
Reynolds Equation ◽  
Fluid Model ◽  
Maxwell Model ◽  
Deborah Number ◽  
Perturbation Solution ◽  
Two Dimensional ◽  
Sliding Velocity ◽  
Regular Perturbation ◽  
Leading Term

There are many studies on variations of the Maxwell model. Tichy (1996) discussed an admissible formulation of the Maxwell viscoelastic fluid model using a convected derivative and applied it to two-dimensional lubrication flow. Tichy obtained a solution using a regular perturbation in the Deborah number with the conventional lubrication solution as the leading term. This paper extends Tichy’s model by using a double regular perturbation to the convected Maxwell model. A correspondence solution can also be obtained. Our sliding velocity solution is different from Tichy’s solution; and a modified Reynolds equation is also different from that by Tichy.

Non-Newtonian Lubrication With the Second-Order Fluid

1998 ◽  
Vol 120 (3) ◽  
pp. 622-628 ◽  
Author(s):  
W. G. Sawyer ◽  
J. A. Tichy
Keyword(s):  
Film Flow ◽  
Maxwell Model ◽  
Companion Paper ◽  
Deborah Number ◽  
Second Order ◽  
Thin Film Flow ◽  
Regular Perturbation ◽  
Leading Term ◽  
Strain Tensors ◽  
Newtonian Behavior

In certain applications where the lubricant is subjected to rapidly changing conditions along its flowing path (such as an elastohydrodynamic contact), the inherently time dependent nature of the lubricant may be significant. The simplest type of model to correctly account for such time dependence is the second-order fluid, which is a systematic small departure from Newtonian behavior, involving higher order rate-of-rate-of strain tensors. As in a companion paper using the Maxwell model, the formalities of applying such a model to thin film flow are emphasized. Using a regular perturbation in the Deborah number, with the conventional lubrication solution as the leading term, a solution can be obtained. Viscoelasticity may raise or lower pressure depending on the nature of edge boundary conditions.

On unwinding yarn from a cylindrical package

1992 ◽  
Vol 436 (1898) ◽  
pp. 479-498 ◽  
Keyword(s):  
Boundary Condition ◽  
Time Dependence ◽  
Moving Boundary ◽  
Equations Of Motion ◽  
Time Derivative ◽  
Perturbation Expansion ◽  
Helix Angle ◽  
Zero Order ◽  
Regular Perturbation ◽  
Moving Boundary Condition

The over-end unwinding of yarn from a stationary helically wound cylindrical package is considered. The motion of the yarn between the unwind point (where it first starts to slip across the package surface before flying into the unwinding balloon) and the guide eye located on the package axis is analysed. The motion is periodic as the unwind point moves backwards and forwards along the length of the package surface. In 1958 D. G. Padfield argued that, provided the helix angle is small, the time derivative terms in the equations of motion can be neglected and the problem can be reduced to a stationary (relative to rotating axes) balloon problem subject to a modified boundary condition at the unwind point. The problem of yarn slipping across the package surface has also been investigated by D. G. Padfield and by H. V. Booth. In the present paper a regular perturbation expansion is used to provide a theoretical framework for Padfield’s ideas and to remove the time dependence from the zero order equations of motion. To this order of approximation the time dependence appears in the ‘moving’ boundary condition at the unwind point. A new derivation of this boundary condition is given and a set of continuity conditions between the yarn slipping on the package and the yarn in the balloon is used to splice the two solutions together so that the package can be unwound through a complete period of the unwinding cycle.

scholarly journals Renormalization Group Approach to Pandemics as a Time-Dependent SIR Model

2021 ◽  
Vol 8 ◽  
Author(s):  
Michele Della Morte ◽  
Francesco Sannino
Keyword(s):  
Renormalization Group ◽  
Time Dependence ◽  
Recovery Rate ◽  
Sir Model ◽  
Time Dependent ◽  
Cumulative Number ◽  
Group Approach ◽  
Renormalization Group Approach

We generalise the epidemic Renormalization Group framework while connecting it to a SIR model with time-dependent coefficients. We then confront the model with COVID-19 in Denmark, Germany, Italy and France and show that the approach works rather well in reproducing the data. We also show that a better understanding of the time dependence of the recovery rate would require extending the model to take into account the number of deaths whenever these are over 15% of the cumulative number of infected cases.

scholarly journals Perturbation Approach in the Dynamic Buckling of a Model Structure with a Cubic-quintic Nonlinearity Subjected to an Explicitly Time Dependent Slowly Varying Load

2019 ◽  
pp. 1-19
Author(s):  
A. M. Ette ◽  
I. U. Udo-Akpan ◽  
J. U. Chukwuchekwa ◽  
A. C. Osuji ◽  
M. F. Noah
Keyword(s):  
Perturbation Technique ◽  
Dynamic Buckling ◽  
Buckling Load ◽  
Time Dependent ◽  
Initial Time ◽  
Perturbation Approach ◽  
Elastic Model ◽  
Model Structure ◽  
Regular Perturbation ◽  
Long Run

This investigation is concerned with analytically determining the dynamic buckling load of an imperfect cubic-quintic nonlinear elastic model structure struck by an explicitly time-dependent but slowly varying load that is continuously decreasing in magnitude. A multi-timing regular perturbation technique in asymptotic procedures is utilized to analyze the problem. The result shows that the dynamic buckling load depends, among other things, on the first derivative of the load function evaluated at the initial time. In the long run, the dynamic buckling load is related to its static equivalent, and that relationship is independent of the imperfection parameter. Thus, once any of the two buckling loads is known, then the other can easily be evaluated using this relationship.

scholarly journals A MATHEMATICAL MODEL OF NONSTATIONARY MOTION OF A VISCOELASTIC FLUID IN ROLLER BEARINGS

2019 ◽  
Vol 81 (4) ◽  
pp. 501-512
Author(s):  
I.A. Zhurba Eremeeva ◽  
D. Scerrato ◽  
C. Cardillo ◽  
A. Tran
Keyword(s):  
Mathematical Model ◽  
Viscoelastic Fluid ◽  
Viscoelastic Properties ◽  
Fluid Model ◽  
Maxwell Fluid ◽  
Initial Boundary ◽  
Deborah Number ◽  
Nonstationary Motion ◽  
Friction Forces ◽  
Roller Bearings

Nowadays, the emergence of new lubricants requires an enhancement of the rheological models and methods used for solution of corresponding initial boundary-value problems. In particular, models that take into account viscoelastic properties are of great interest. In the present paper we consider the mathematical model of nonstationary motion of a viscoelastic fluid in roller bearings. We used the Maxwell fluid model for the modeling of fluid properties. The viscoelastic properties are exhibited by many lubricants that use polymer additives. In addition, viscoelastic properties can be essential at high fluid speeds. Also, viscoelastic properties can be significant in the case of thin gaps. Maxwell's model is one of the most common models of viscoelastic materials. It combines the relative simplicity of constitutive equations with the ability to describe a stress relaxation. In addition, viscoelastic fluids also allow us to describe some effects that are missing in the case of viscous fluid. An example it is worth to mention the Weissenberg effect and a number of others. In particular, such effects can be used to increase the efficiency of the film carrier in the sliding bearings. Here we introduced characteristic assumptions on the form of the flow, allowing to significantly simplify the solution of the problem. We consider so-called self-similar solutions, which allows us to get a solution in an analytical form. As a result these assumptions, the formulae for pressure and friction forces are derived. Their dependency on time and Deborah number is analyzed. The limiting values of the flow characteristics were obtained. The latter can be used for steady state of the flow regime. Differences from the case of Newtonian fluid are discussed. It is shown that viscoelastic properties are most evident at the initial stage of flow, when the effects of non-stationarity are most important.

scholarly journals The Diffraction of Transient Electro-Magnetic Waves by a Wedge

1958 ◽  
Vol 11 (2) ◽  
pp. 95-103 ◽  
Author(s):  
A. C. Butcher ◽  
J. S. Lowndes
Keyword(s):  
Wave Equation ◽  
Time Dependence ◽  
Half Plane ◽  
Line Source ◽  
Time Dependent ◽  
Dimensional Case ◽  
Two Dimensional ◽  
Electro Magnetic ◽  
Infinite Wedge ◽  
Magnetic Waves

Much of the work on the theory of diffraction by an infinite wedge has been for cases of harmonic time-dependence. Oberhettinger (1) obtained an expression for the Green's function of the wave equation in the two dimensional case of a line source of oscillating current parallel to the edge of a wedge with perfectly conducting walls. Solutions of the time-dependent wave equation have been obtained by Keller and Blank (2), Kay (3) and more recently by Turner (4) who considered the diffraction of a cylindrical pulse by a half plane.

scholarly journals DARBOUX CLASS OF COSMOLOGICAL FLUIDS WITH TIME-DEPENDENT ADIABATIC INDICES

2000 ◽  
Vol 15 (15) ◽  
pp. 979-990 ◽  
Author(s):  
H. C. ROSU
Keyword(s):  
Mathematical Method ◽  
Cosmological Constant ◽  
Time Dependence ◽  
Adiabatic Index ◽  
Time Dependent ◽  
Accelerating Universe ◽  
Constant Parameter ◽  
Darboux Transformations ◽  
Scale Factors ◽  
Experimental Findings

A one-parameter family of time-dependent adiabatic indices is introduced for any given type of cosmological fluid of constant adiabatic index by a mathematical method belonging to the class of Darboux transformations. The procedure works for zero cosmological constant at the price of introducing a new constant parameter related to the time dependence of the adiabatic index. These fluids can be the real cosmological fluids that are encountered at cosmological scales and they could be used as a simple and efficient explanation for the recent experimental findings regarding the present day accelerating universe. In addition, new types of cosmological scale factors, corresponding to these fluids, are presented.

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