Ad hoc closed form solutions of the two-dimensional non-linear steady small perturbation equation in fluid mechanics

1995 ◽  
Vol 30 (4) ◽  
pp. 597-608 ◽  
Author(s):  
D.E. Panayotounakos ◽  
M. Markakis
Keyword(s):  
Fluid Mechanics ◽  
Closed Form ◽  
Small Perturbation ◽  
Ad Hoc ◽  
Perturbation Equation ◽  
Two Dimensional ◽  
Closed Form Solutions ◽  
Non Linear

Approximate Analysis of Penetration of a Cylindrical Roller Into a Thin Elastic Coating

1990 ◽  
Vol 112 (3) ◽  
pp. 460-468 ◽  
Author(s):  
Tsung-Ju Gwo ◽  
Thomas J. Lardner
Keyword(s):  
Approximate Solution ◽  
Closed Form ◽  
Approximate Analytical Solution ◽  
Experimental Studies ◽  
Preliminary Design ◽  
Two Dimensional ◽  
Rigid Substrate ◽  
Closed Form Solutions ◽  
Finite Element Calculations ◽  
Cylindrical Roller

An approximate analytical solution to the problem of two-dimensional indentation of a frictionless cylinder into a thin elastic coating bonded to a rigid substrate has been obtained using the approach introduced by Matthewson for axisymmetric indentation. We show by comparing the results of the approximate solution to the exact solutions and to finite element calculations that the approximate solution is accurate for a/h> 2. The advantage of this approach is that the results are expressed in closed form and the accuracy of the approximate solution improves with increasing values of a/h. For a/h>2, for a given load, the theory overestimates the value of a/h compared to the exact solution by less than 10 percent. In many experimental studies and in preliminary design, it is convenient to have closed-form solutions exhibiting the dependence of the parameters.

On the Approximate Solution of Viscous-Flow Problems

1963 ◽  
Vol 30 (2) ◽  
pp. 263-268 ◽  
Author(s):  
J. A. Schetz
Keyword(s):  
Approximate Solution ◽  
Exact Solutions ◽  
Viscous Flow ◽  
Closed Form ◽  
New Method ◽  
Two Dimensional ◽  
General Technique ◽  
Closed Form Solutions ◽  
Flow Problems

The need for a general technique for the approximate solution of viscous-flow problems is discussed. Existing methods are considered and a new method is presented which results in simple closed-form solutions. The accuracy of the method is demonstrated by comparisons with the results of known exact solutions, and finally the general technique is employed to determine a new solution for the fully expanded two-dimensional laminar nozzle problem.

Faraday instability threshold in large-aspect-ratio containers

2002 ◽  
Vol 467 ◽  
pp. 307-330 ◽  
Author(s):  
FRANCISCO J. MANCEBO ◽  
JOSÉ M. VEGA
Keyword(s):  
Aspect Ratio ◽  
Numerical Solution ◽  
Closed Form ◽  
Ad Hoc ◽  
Linear Problem ◽  
Experimental Results ◽  
Large Aspect Ratio ◽  
Faraday Waves ◽  
Closed Form Solutions ◽  
Faraday Instability

We consider the Floquet linear problem giving the threshold acceleration for the appearance of Faraday waves in large-aspect-ratio containers, without further restrictions on the values of the parameters. We classify all distinguished limits for varying values of the various parameters and simplify the exact problem in each limit. The resulting simplified problems either admit closed-form solutions or are solved numerically by the well-known method introduced by Kumar & Tuckerman (1994). Some comparisons are made with (a) the numerical solution of the original exact problem, (b) some ad hoc approximations in the literature, and (c) some experimental results.

Two-dimensional planar plumes: non-Boussinesq effects

2014 ◽  
Vol 750 ◽  
pp. 245-258 ◽  
Author(s):  
T. S. van den Bremer ◽  
G. R. Hunt
Keyword(s):  
Closed Form ◽  
Uniform Density ◽  
Two Dimensional ◽  
Boussinesq Model ◽  
Axisymmetric Case ◽  
Closed Form Solutions ◽  
Plume Model ◽  
Entrainment Velocity ◽  
Planar Area ◽  
Pure Plume

AbstractIn an accompanying paper (van den Bremer & Hunt, J. Fluid Mech., vol. 750, 2014, pp. 210–244) closed-form solutions, describing the behaviour of two-dimensional planar turbulent rising plumes from horizontal planar area and line sources in unconfined quiescent environments of uniform density, that are universally applicable to Boussinesq and non-Boussinesq plumes, are proposed. This universality relies on an entrainment velocity unmodified by non-Boussinesq effects, an assumption that is derived in the literature based on similarity arguments and is, in fact, in contradiction with the axisymmetric case, in which entrainment is modified by non-Boussinesq effects. Exploring these solutions, we show that a non-Boussinesq plume model predicts exactly the same behaviour with height for a pure plume as would a Boussinesq model, whereas the effects on forced and lazy plumes are opposing. Non-intuitively, the non-Boussinesq model predicts larger fluxes of volume and mass for lazy plumes, but smaller fluxes for forced plumes at any given height compared to the Boussinesq model. This raises significant questions regarding the validity of the unmodified entrainment model for planar non-Boussinesq plumes based on similarity arguments and calls for detailed experiments to resolve this debate.

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