An Approximate Computational Method for the Fluid Stiction Problem of Two Separating Parallel Plates With Cavitation

2016 ◽  
Vol 138 (6) ◽  
Author(s):  
Rudolf Scheidl ◽  
Christoph Gradl
Keyword(s):  
Differential Equation ◽  
Approximate Calculation ◽  
Approximate Model ◽  
Computational Method ◽  
Limiting Factor ◽  
Solution Properties ◽  
Parallel Plates ◽  
Level Curves ◽  
Elliptic Domain ◽  
Pressure Boundary Conditions

Stiction forces exerted by a fluid in a thin, quickly widening gap to its boundaries can become a strongly limiting factor of the performance of technical devices, like compressor valves or hydraulic on–off valves. In design optimization, such forces need to be properly and efficiently modeled. Cavitation during parts of a stiction process plays a strong role and needs to be taken into account to achieve a meaningful model. The paper presents an approximate calculation method which uses qualitative solution properties of the non cavitating stiction problem, in particular of its level curves and gradient lines. In this method, the formation of the cavitation boundaries is approximated by an elliptic domain. The pressure distribution along its principle axis is described by a directly integrable differential equation, the evolutions of its boundaries is guided just by pressure boundary conditions when the cavitation zone expands and by a nonlinear differential equation when it shrinks. The results of this approximate model agree quite well with the solutions of a finite volume (FV) model for the fluid stiction problem with cavitation.

Magnetohydrodynamic Squeeze Film Characteristics Between Parallel Circular Plates Containing a Single Central Air Bubble in the Inertial Flow Regime

1999 ◽  
Vol 66 (4) ◽  
pp. 1021-1023 ◽  
Author(s):  
R. Usha ◽  
P. Vimala
Keyword(s):  
Magnetic Field ◽  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Bubble Radius ◽  
Fluid Film ◽  
Squeeze Film ◽  
Parallel Plates ◽  
Circular Plates ◽  
Air Bubble ◽  
Approximate Analytical Solutions

In this paper, the magnetic effects on the Newtonian squeeze film between two circular parallel plates, containing a single central air bubble of cylindrical shape are theoretically investigated. A uniform magnetic field is applied perpendicular to the circular plates, which are in sinusoidal relative motion, and fluid film inertia effects are included in the analysis. Assuming an ideal gas under isothermal condition for an air bubble, a nonlinear differential equation for the bubble radius is obtained by approximating the momentum equation governing the magnetohydrodynamic squeeze film by the mean value averaged across the film thickness. Approximate analytical solutions for the air bubble radius, pressure distribution, and squeeze film force are determined by a perturbation method for small amplitude of sinusoidal motion and are compared with the numerical solution obtained by solving the nonlinear differential equation. The combined effects of air bubble, fluid film inertia, and magnetic field on the squeeze film force are analyzed.

Fast diffusion with loss at infinity

1995 ◽  
Vol 36 (4) ◽  
pp. 438-459 ◽  
Author(s):  
J. R. Philip
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Fast Diffusion ◽  
Solution Properties ◽  
Positive Power ◽  
Negative Power ◽  
Exponential Decrease ◽  
Class 1 ◽  
Image Position ◽  
Instantaneous Source

AbstractWe study the equationHere s is not necessarily integral; m is initially unrestricted. Material-conserving instantaneous source solutions of A are reviewed as an entrée to material-losing solutions. Simple physical arguments show that solutions for a finite slug losing material at infinity at a finite nonzero rate can exist only for the following m-ranges: 0 < s < 2, −2s−1 < m ≤ −1; s > 2, −1 < m < −2s−1. The result for s = 1 was known previously. The case s = 2, m = −1, needs further investigation. Three different similarity schemes all lead to the same ordinary differential equation. For 0 < s < 2, parameter γ (0 < γ < ∞) in that equation discriminates between the three classes of solution: class 1 gives the concentration scale decreasing as a negative power of (1 + t/T); 2 gives exponential decrease; and 3 gives decrease as a positive power of (1 − t/T), the solution vanishing at t = T < ∞. Solutions for s = 1, are presented graphically. The variation of concentration and flux profiles with increasing γ is physically explicable in terms of increasing flux at infinity. An indefinitely large number of exact solutions are found for s = 1,γ = 1. These demonstrate the systematic variation of solution properties as m decreases from −1 toward −2 at fixed γ.

scholarly journals An Axisymmetric Squeezing Fluid Flow between the Two Infinite Parallel Plates in a Porous Medium Channel

2011 ◽  
Vol 2011 ◽  
pp. 1-10 ◽  
Author(s):  
S. Islam ◽  
Hamid Khan ◽  
Inayat Ali Shah ◽  
Gul Zaman
Keyword(s):  
Differential Equation ◽  
Porous Medium ◽  
Fluid Flow ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Parallel Plates ◽  
Transform Method ◽  
Navier Stokes Equations ◽  
Differential Transform ◽  
Fluid Parameters

The flow between two large parallel plates approaching each other symmetrically in a porous medium is studied. The Navier-Stokes equations have been transformed into an ordinary nonlinear differential equation using a transformationψ(r,z)=r2F(z). Solution to the problem is obtained by using differential transform method (DTM) by varying different Newtonian fluid parameters and permeability of the porous medium. Result for the stream function is presented. Validity of the solutions is confirmed by evaluating the residual in each case, and the proposed scheme gives excellent and reliable results. The influence of different parameters on the flow has been discussed and presented through graphs.

scholarly journals A novel computational method for solving nonlinear Volterra integro-differential equation

2020 ◽  
Vol 48 (1) ◽  
Author(s):  
Musa Cakir ◽  
◽  
Baransel Gunes ◽  
Hakki Duru ◽  
◽  
...  
Keyword(s):  
Differential Equation ◽  
Remainder Term ◽  
Integral Form ◽  
Computational Method ◽  
Weight Functions ◽  
Asymptotic Estimates ◽  
Numerical Examples ◽  
Theoretical Results ◽  
Integral Identities ◽  
Quasilinearization Technique

In this paper, we study quasilinear Volterra integro-differential equations (VIDEs). Asymptotic estimates are made for the solution of VIDE. Finite difference scheme, which is accomplished by the method of integral identities using interpolating quadrature rules with weight functions and remainder term in integral form, is presented for the VIDE. Error estimates are carried out according to the discrete maximum norm. It is given an effective quasilinearization technique for solving nonlinear VIDE. The theoretical results are performed on numerical examples.

Compensation of aberrations of deflected electron probe by means of dynamical focusing with stigmator

1978 ◽  
Vol 36 (1) ◽  
pp. 32-33
Author(s):  
Norio Baba ◽  
Yasuhiro Ito ◽  
Koichi Kanaya
Keyword(s):  
Field Strength ◽  
Electron Probe ◽  
Static Field ◽  
Limiting Factor ◽  
Two Dimensional ◽  
Parallel Plates ◽  
Axial Field ◽  
Electron Microscopes ◽  
Image Position ◽  
Scanning Electron Microscopes

An electron beam passing through a deflecting field is, in general, subject to aberrations such as distortion, astigmatism and coma in accordance with the deflecting angle. Accordingly, aberration defects of deflected beam are most serious limiting factor in performance of high resolution scanning electron microscopes.Based on the aberration formula(Wendt 1942, 1947 and Glaser 1956a), the distorted spot patterns of an electron probe deflected at two dimensional directions are calculated by using the conventional form of deflector field f (z) = [1+(z/a)2]-m. The important compensation conditions are induced by means of dynamical focusing with the stigmator.1] Field distribution and magnitude of deflectionThe magnitude of deflection by the electro-static field is given by,where E(z) is the axial field strength, Φ the accelerating potential, and za and zb are the co-ordinates of the entrance and specimen position, respectively. Now, assuming the function E(z)/Eo=[1+(z/a)2]-m (1) which corresponds to the axial field strength distributions of semi-infinite co-planar sheets(m=l/2), parallel cylinders or wires(m=l) and parallel plates with fringing effects(m=l∼5), deflection distances e and angles |e'|=α for various deflection geometry can be expressed analytically.

On Calculation of Means of SDE with Drift

2020 ◽  
Vol 23 (3) ◽  
pp. 299-305
Author(s):  
Anatoly Zherelo
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Numerical Method ◽  
Approximate Calculation

The numerical method of approximate calculation of mean of a stochastic differential equation with a drift was constructed. The proposed method is based on the so called weak method of approximations and does not require simulations of trajectories of the solutions.

On Thermodynamics of Gas-Turbine Cycles: Part 2—A Model for Expansion in Cooled Turbines

1986 ◽  
Vol 108 (1) ◽  
pp. 151-159 ◽  
Author(s):  
M. A. El-Masri
Keyword(s):  
Gas Turbine ◽  
Gas Turbines ◽  
Approximate Model ◽  
Operating Conditions ◽  
Limiting Factor ◽  
Inlet Temperature ◽  
Air Cooling ◽  
Closed Form Solutions ◽  
Key Factor ◽  
Expansion Path

While raising turbine inlet temperature improves the efficiency of the gas-turbine cycle, the increasing turbine-cooling losses become a limiting factor. Detailed prediction of those losses is a complex process, thought to be possible only for specific designs and operating conditions. A general, albeit approximate, model is presented to quantify those cooling losses for different types of cooling technologies. It is based upon representing the turbine as an expansion path with continuous, rather than discrete, work extraction. This enables closed-form solutions to be found for the states along the expansion path as well as turbine work output. The formulation shows the key factor in determining the cooling losses is the parameter scaling the ratio of heat to work fluxes loading the machine surfaces. Solutions are given for three cases: internal air-cooling, transpiration air cooling, and internal liquid cooling. The first and second cases represent lower and upper bounds respectively for the performance of film-cooled machines. Irreversibilities arising from flow-path friction, heat transfer, cooling air throttling, and mixing of coolant and mainstream are quantified and compared. Sample calculations for the performance of open and combined cycles with cooled turbines are presented. The dependence and sensitivity of the results to the various loss mechanisms and assumptions is shown. Results in this paper pertain to Brayton-cycle gas turbines with the three types of cooling mentioned. Reheat gas turbines are more sensitive to cooling losses due to the larger number of high-temperature stages. Those are considered in Part 3.

The Squeezing of a Fluid Between Two Plates

1976 ◽  
Vol 43 (4) ◽  
pp. 579-583 ◽  
Author(s):  
C.-Y. Wang
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Viscous Fluid ◽  
Stokes Equations ◽  
Nonlinear Ordinary Differential Equation ◽  
Similarity Solutions ◽  
Navier Stokes ◽  
Normal Velocity ◽  
Parallel Plates ◽  
Navier Stokes Equations

A viscous fluid lies between two parallel plates which are being squeezed or separated. If the normal velocity is proportional to (1 − αt)−1/2 the unsteady Navier-Stokes equations admit similarity solutions. The resulting nonlinear ordinary differential equation is governed by a parameter S which characterizes unsteadiness. Asymptotic solutions for small S and for large positive S are found which compare well with those obtained by numerical integration. It is found that the resistance is proportional to (1 − αt)−2 but is not necessarily opposite to the direction of motion.

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