On the Variational Principle and Lagrange Equations in Studies of Gasdynamic Lubrication

1964 ◽  
Vol 31 (1) ◽  
pp. 43-46 ◽  
Author(s):  
L. N. Tao
Keyword(s):  
Variational Formulation ◽  
Lagrangian Density ◽  
Reynolds Equation ◽  
Hydrodynamic Lubrication ◽  
Minimum Energy ◽  
Classical Dynamics ◽  
Lagrange Equations ◽  
Rayleigh Principle ◽  
Special Case ◽  
Minimum Energy Dissipation

The paper is concerned with the variational formulation in studies of gasdynamic lubrication. It is shown that Reynolds’ equation of lubrication is equivalent to a set of Lagrange equations similar to those in classical dynamics. The Lagrangian and the dissipation-production are defined. Furthermore, based on the Hamiltonian principle for the field of a continuum, the Lagrangian density and the dissipation-production density are established. This formulation includes the incompressible problem, which is obtainable from the Helmholtz-Rayleigh principle of minimum energy-dissipation, as a special case. Hence a unification of the variational methods for both gasdynamic and hydrodynamic lubrication is accomplished.

An Integral Variational Equation for Transport Processes in a Moving Fluid

1989 ◽  
Vol 56 (1) ◽  
pp. 208-210 ◽  
Author(s):  
E. S. Geskin
Keyword(s):  
Energy Dissipation ◽  
Lagrangian Density ◽  
Variational Equation ◽  
Boundary Energy ◽  
Minimum Energy ◽  
Transport Processes ◽  
Lagrange Equations ◽  
Flow Through ◽  
Moving Fluid ◽  
Minimum Energy Dissipation

An integral variational equation can adequately describe heat, mass, and momentum transfer in a moving chemically reactive fluid. The Euler-Lagrange equations corresponding to the suggested variational equation are identical to the equations of entropy, momentum, angular momentum, and mass balance. The constructed Lagrangian density relates energy change in the system to the work and energy dissipation of the system. For steady-state processes, the Lagrangian density includes convective energy flow through the system boundary, energy dissipation in the system, and work of the system. The proposed variational equation is equivalent to the expansion of the principle of minimum energy dissipation.

scholarly journals On the Consistent Formulation and Approximation of EHL Theory

1989 ◽  
Vol 111 (1) ◽  
pp. 108-113 ◽  
Author(s):  
R. Verstappen ◽  
E. van Groesen
Keyword(s):  
Differential Equations ◽  
Boundary Conditions ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Hydrodynamic Lubrication ◽  
Physical Principle ◽  
Creeping Flow ◽  
Total Power ◽  
Finite Element Approximations ◽  
Consistent Formulation

In this paper a quasi-variational formulation for the creeping flow of a lubricant between deformable surfaces is derived on the basis of a principle of virtual power. This physical principle states that the state of the lubricated system is such that any change of the deformation or of the flow increases the total power. The principle correctly produces the governing differential equations and boundary conditions. Performing the usual approximations that are based on the scaling of the problem directly into the power functional (and not into the differential expressions), an approximate variational formulation is obtained that contains only quantities for and at the deformed surface. In deriving the differential equations and boundary conditions from this approximate functional it is noted that the usual formulation is obtained provided the sliding speed of the surfaces vanishes at the cavitation boundary. When restricted to hydrodynamic lubrication, a true variational principle results; the functional correctly produces the Reynolds equation, but differs from the functional that is commonly derived solely from this equation. The quasi-variational formulation that is derived for EHL gives an alternative way to formulate finite element approximations. Furthermore it specifies the relevance of the contribution of centrifugal forces on the deformation.

scholarly journals Mathematical analysis of laboratory microbial experiments demonstrating deterministic chaotic dynamics

2021 ◽  
Author(s):  
Fred Molz ◽  
Boris Faybishenko
Keyword(s):  
Dilution Rate ◽  
Mathematical Analysis ◽  
Chaotic Dynamics ◽  
Minimum Energy ◽  
Living Systems ◽  
Microbial Populations ◽  
Classical Dynamics ◽  
Monod Kinetics ◽  
Model Simulations ◽  
Minimum Energy Dissipation

AbstractPresented is a system of four ordinary differential equations and a mathematical analysis of microbiological experiments in a four-component chemostat—nutrient n, rods r, cocci c, and predators p. The analysis is consistent with the conclusion that previous experiments produced features of deterministic chaotic and classical dynamics depending on dilution rate. The surrogate model incorporates as much experimental detail as possible, but necessarily contains unmeasured parameters. The objective is to understand better the differences between model simulations and experimental results in complex microbial populations. The key methodology for simulation of chaotic dynamics, consistent with the measured dilution rate and microbial volume averages, was to cause the preference of p for r vs. c to vary with the r and c concentrations, to make r more competitive for nutrient than c, and to recycle some dying p biomass, leading to a modified version of the Monod kinetics model. Our mathematical model demonstrated that the occurrence of chaotic dynamics requires a predator, p, preference for r versus c to increase significantly with increases in r and c populations. Also included is a discussion of several generalizations of the existing model and a possible involvement of the minimum energy dissipation principle. This principle appears fundamental to thermodynamic systems including living systems. Several new experiments are suggested.

scholarly journals Second-order integrable Lagrangians and WDVV equations

2021 ◽  
Vol 111 (2) ◽  
Author(s):  
E. V. Ferapontov ◽  
M. V. Pavlov ◽  
Lingling Xue
Keyword(s):  
Lagrangian Density ◽  
Theta Functions ◽  
Second Order ◽  
Elementary Functions ◽  
Lagrange Equations ◽  
Wdvv Equations ◽  
Integrability Conditions ◽  
Jacobi Theta Functions

AbstractWe investigate the integrability of Euler–Lagrange equations associated with 2D second-order Lagrangians of the form $$\begin{aligned} \int f(u_{xx},u_{xy},u_{yy})\ \mathrm{d}x\mathrm{d}y. \end{aligned}$$ ∫ f ( u xx , u xy , u yy ) d x d y . By deriving integrability conditions for the Lagrangian density f, examples of integrable Lagrangians expressible via elementary functions, Jacobi theta functions and dilogarithms are constructed. A link of second-order integrable Lagrangians to WDVV equations is established. Generalisations to 3D second-order integrable Lagrangians are also discussed.

scholarly journals Which Is the Motion State of a Droplet on an Inclined Hydrophilic Rough Surface in Gravity: Pinned or Sliding?

2021 ◽  
Vol 11 (9) ◽  
pp. 3734
Author(s):  
Jian Dong ◽  
Youhai Guo ◽  
Long Jiao ◽  
Chao Si ◽  
Yinbo Bian ◽  
...  
Keyword(s):  
Rough Surface ◽  
Contact Angle Hysteresis ◽  
Minimum Energy ◽  
Contact Angles ◽  
Microfluidic Chips ◽  
Motion State ◽  
Receding Contact ◽  
Novel Design ◽  
Minimum Energy Dissipation ◽  
Good Agreement

The motion state of a droplet on an inclined, hydrophilic rough surface in gravity, pinned or sliding, is governed by the balance between the driving and the pinned forces. It can be judged by the droplet’s shape on the inclined hydrophilic rough surface and the droplet’s contact angle hysteresis. In this paper, we used the minimum energy theory, the minimum energy dissipation theory, and the nonlinear numerical optimization algorithm to establish Models 1–3 to calculate out the advancing/receding contact angles (θa/θr), the initial front/rear contact angles (θ1−0/θ2−0) and the dynamic front/rear contact angles (θ1−*/θ2−*) for a droplet on a rough surface. Also, we predicted the motion state of the droplet on an inclined hydrophilic rough surface in gravity by comparing θ1−0(θ2−0) and θ1−*(θ2−*) with θa(θr). Experiments were done to verify the predictions. They showed that the predictions were in good agreement with the experimental results. These models are promising as novel design approaches of hydrophilic functional rough surfaces, which are frequently applied to manipulate droplets in microfluidic chips.

Computational Evolution of Optimal Iceform Shapes Over a Flat Plate

1991 ◽  
Author(s):  
Ronald S. LaFleur
Keyword(s):  
Energy Dissipation ◽  
Flat Plate ◽  
Thermal Field ◽  
Fluid Dynamic ◽  
Minimum Energy ◽  
Thermal Parameter ◽  
Thermal Constraints ◽  
Ice Water ◽  
Nonlinear Geometric ◽  
Minimum Energy Dissipation

Abstract This paper presents the computational evolution of minimum energy dissipation iceform contours. The ice/water interface is shaped according to fluid dynamic and heat transfer characteristics of the flow field near the interface. A Couette iceform design model is used to approximate flow and thermal field behavior near the interface. The theory used to calculate the interface shape is based on a wedge model of the ice contour over a cold flat plate. The steady state ice profile is calculated when Reynolds number and the thermal parameter are selected. The generation function, designation function and energy dissipation are related to the nonlinear geometric development. An optimal preprocess criterion is prescribed as zero evolution length. The result is optimal geometries that are adapted to the flow and thermal constraints.

scholarly journals Analysis for Flow of an Incompressible Brinkman-Type Fluid in Thin Medium with Friction

2021 ◽  
Vol 2021 ◽  
pp. 1-8
Author(s):  
Soumia Manaa ◽  
Salah Boulaaras ◽  
Hamid Benseridi ◽  
Mourad Dilmi ◽  
Sultan Alodhaibi
Keyword(s):  
Fluid Flow ◽  
Variational Formulation ◽  
Reynolds Equation ◽  
Three Dimensional ◽  
Stationary Regime ◽  
Limit Problem ◽  
Brinkman Equation ◽  
Asymptotic Convergence ◽  
Thin Domain ◽  
Uniqueness Of The Solution

In this paper, we consider the Brinkman equation in the three-dimensional thin domain ℚ ε ⊂ ℝ 3 . The purpose of this paper is to evaluate the asymptotic convergence of a fluid flow in a stationary regime. Firstly, we expose the variational formulation of the posed problem. Then, we presented the problem in transpose form and prove different inequalities for the solution u ε , p ε independently of the parameter ε . Finally, these estimates allow us to have the limit problem and the Reynolds equation and establish the uniqueness of the solution.

On the effects of two-dimensional Reynolds roughness in hydrodynamic lubrication

1978 ◽  
Vol 364 (1716) ◽  
pp. 89-106 ◽  
Keyword(s):  
Surface Roughness ◽  
Numerical Computation ◽  
Autocorrelation Function ◽  
Reynolds Equation ◽  
Hydrodynamic Lubrication ◽  
The Other ◽  
Roughness Height ◽  
Two Dimensional ◽  
Roughness Effects ◽  
Dimensional Surface

The hydrodynamic lubrication of rough surfaces is analysed with the Reynolds equation, whose application requires the roughness spacing to be large, and the roughness height to be small, compared with the thick­ness of the fluid film. The general two-dimensional surface roughness is considered, and results applicable to any roughness structure are obtained. It is revealed analytically that two types of term contribute to roughness effects: one depends on the shape of the autocorrelation function and the other does not. The former contribution was neglected by previous workers. The numerical computation of an example shows that these two contributions are comparable in magnitude.

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