scholarly journals Analysis of an Exact Solution of Problem of the Evaporative Convection (Review). Part II. Three-dimensional Flows

2018 ◽  
Vol 11 (3) ◽  
pp. 342-355 ◽  
Author(s):  
Bekezhanova Victoria B. ◽  
◽  
Goncharova Olga N. ◽  
Shefer Ilia A. ◽  
◽  
...  
Keyword(s):  
Exact Solution ◽  
Three Dimensional ◽  
Evaporative Convection

The generalized Burgers and Zabolotskaya-Khokhlov equations

1994 ◽  
Vol 447 (1930) ◽  
pp. 253-270 ◽  
Keyword(s):  
Exact Solution ◽  
Plane Waves ◽  
Three Dimensional ◽  
Two Dimensions ◽  
Compressive Wave ◽  
Similarity Reduction ◽  
Weakly Nonlinear ◽  
Cylindrically Symmetric ◽  
Generalized Burgers Equation ◽  
Long Time

A point transformation between forms of the generalized Burgers equation (g b e) first given by Cates (1989) is investigated. Applications include generalizations of Scott’s (1981) classification of long-time behaviour for compressive wave solutions of the GBE and the equivalence of the exponential and cylindrical forms of the GBE, yielding an exact solution for the exponential GBE. Applications to nonlinear diffractive acoustics are considered by using a similarity reduction of the dissipative Zabolotskaya-Khokhlov (dzk) equation (describing the evolution of nearly plane waves in a weakly nonlinear medium with allowance for transverse variation effects) onto the GBE. The result is that waves from parabolic sources may be described by the cylindrical GBE in the case of two dimensions, and by the spherical GBE in the three-dimensional, cylindrically symmetric case. Furthermore, results on the formation of shocks and caustics in the context of the ZK equation are presented, along with an exact solution to the DZK equation. Exact solutions with caustic singularities are studied, along with a possible mechanism for their control. Finally, results on the evolution of a shock approaching a caustic are given through the identification of a series of parameter regimes dependent on the diffusivity.

SOME PERIODIC AND BLOW-UP SOLUTIONS FOR LANDAU–LIFSHITZ EQUATION

2011 ◽  
Vol 26 (32) ◽  
pp. 2437-2452 ◽  
Author(s):  
PENGHONG ZHONG ◽  
SHU WANG ◽  
SHENGTAO CHEN
Keyword(s):  
Exact Solution ◽  
Periodic Solution ◽  
Exact Solutions ◽  
Finite Time ◽  
Blow Up ◽  
Three Dimensional ◽  
Ferromagnetic Materials ◽  
Lifshitz Equation ◽  
Dimensional Spacetime ◽  
Vortex Solution

In this paper, we construct the exact solution of two- or three-dimensional spacetime Landau–Lifshitz equation raised in the ferromagnetic materials. Under suitable transformations, some exact solutions are obtained in the radially symmetric coordinates and nonsymmetric coordinates. The type of solutions cover the finite time blow-up solution, vortex solution and periodic solution. In the end, we sketch some solutions and their spatial curvatures.

On metastable approximations in co-operative assemblies

1956 ◽  
Vol 235 (1201) ◽  
pp. 247-259 ◽  
Keyword(s):  
Exact Solution ◽  
Specific Heat ◽  
Curie Point ◽  
Lattice Structure ◽  
Three Dimensional ◽  
Entropy Change ◽  
Successive Approximations ◽  
Series Expansions ◽  
Temperature Expansion ◽  
Heat Curve

The exact solution of the three-dimensional Ising model of a ferromagnetic presents diffi­culties of a very fundamental nature. It therefore seems that the most reliable information on the behaviour of the model is provided by exact series expansions of the partition func­tion at low and high temperatures. However, the usual low -temperature expansion fails to converge in the neighbourhood of the critical point. By rearranging the terms of the series on the basis of physical considerations, it is possible to obtain a systematic set of successive approximations, each approximation taking exact account of clusters of a given size or less (metastable approximations). By extrapolation accurate estimates can be derived of the Curie point and critical values of the energy and entropy. It is found that there is a marked difference in behaviour between two- and three-dimensional lattices, a far larger proportion of the entropy change taking place in the temperature region below the Curie point in the latter case. The corre­sponding specific heat curves are therefore much closer to those observed experimentally. Finally, a brief discussion is given of the dependence of the specific heat curve on lattice structure.

Flow Due to Eccentric Rotating a Porous Disk and a Fluid at Infinity

1976 ◽  
Vol 43 (2) ◽  
pp. 203-204 ◽  
Author(s):  
M. Emin Erdogan
Keyword(s):  
Exact Solution ◽  
Velocity Distribution ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Navier Stokes ◽  
Porous Disk ◽  
Navier Stokes Equations ◽  
Uniform Suction ◽  
Asymptotic Profile

An exact solution of the steady three-dimensional Navier-Stokes equations is obtained for the case of flow due to noncoaxially rotations of a porous disk and a fluid at infinity. It is shown that for uniform suction or uniform blowing at the disk an asymptotic profile exists for the velocity distribution.

An Exact Solution for the Natural Frequencies of Flexible Beams Undergoing Overall Motions

2003 ◽  
Vol 9 (11) ◽  
pp. 1221-1229 ◽  
Author(s):  
Ali H Nayfeh ◽  
S.A. Emam ◽  
Sergio Preidikman ◽  
D.T. Mook
Keyword(s):  
Exact Solution ◽  
Natural Frequencies ◽  
Three Dimensional ◽  
Beam Theory ◽  
Free Vibrations ◽  
Mode Shapes ◽  
Flexible Beam ◽  
Bernoulli Beam ◽  
Flexible Beams ◽  
Euler Bernoulli Beam

We investigate the free vibrations of a flexible beam undergoing an overall two-dimensional motion. The beam is modeled using the Euler-Bernoulli beam theory. An exact solution for the natural frequencies and corresponding mode shapes of the beam is obtained. The model can be extended to beams undergoing three-dimensional motions.

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