scholarly journals Waves in Two Coaxial Elastic Cubically Nonlinear Shells with Structural Damping and Viscous Fluid Between Them

Symmetry ◽  
2020 ◽  
Vol 12 (3) ◽  
pp. 335
Author(s):  
Lev Mogilevich ◽  
Sergey Ivanov
Keyword(s):  
Exact Solution ◽  
Incompressible Fluid ◽  
Difference Scheme ◽  
Viscous Incompressible Fluid ◽  
Initial Conditions ◽  
Numerical Study ◽  
Fluid Motion ◽  
Structural Damping ◽  
The Difference ◽  
The Mathematical Model

This article investigates longitudinal deformation waves in physically nonlinear coaxial elastic shells containing a viscous incompressible fluid between them. The presence of a viscous incompressible fluid between the shells, as well as the influence of the inertia of the fluid motion on the amplitude and velocity of the wave, are taken into account. The mathematical model phenomenon is constructed by means of the method of two-scale asymptotic expansion. Structural damping in the shells and surrounding elastic media did not allow discovery of the exact solution of the problem of the deformation waves propagation. This leads to the need for numerical methods. A numerical study of the model constructed in the course of this work is carried out by using a difference scheme for the equation similar to the Crank–Nicholson scheme for the heat equation. In the absence of the structural damping and surrounding media influences, and under the similar initial conditions for both shells, the velocity and amplitude of the wave do not change. The result of the numerical experiment coincides with the exact solution, which is found in the case of the absence of the structural damping and surrounding media influences; therefore, the difference scheme is adequate to the generalized modified Korteweg–de Vries equations system. There is energy is transferred in the presence of the fluid, between the shells. The presence of inertia of the fluid motion leads to a decrease in the velocity of the deformation wave.

scholarly journals On the existence of an exact solution of the Navier-Stokes equation pertaining to certain vortex motion

1972 ◽  
Vol 13 (4) ◽  
pp. 456-460
Author(s):  
K. Kuen Tam
Keyword(s):  
Exact Solution ◽  
Incompressible Fluid ◽  
Stokes Equation ◽  
Viscous Incompressible Fluid ◽  
Fluid Motion ◽  
Vortex Motion ◽  
Polar Coordinates ◽  
Navier Stokes ◽  
Navier Stokes Equation ◽  
Cylindrical Polar Coordinates

In 1942, Burgers [1] observed that in cylindrical polar coordinates, the steady Navier-Stokes equation governing viscous incompressible fluid motion can be reduced to a set of ordinary differential equations if the velocity components vr, vo and vz are assumed to have a special form.

On the convergence of difference schemes for the generalized BBM–Burgers equation

2019 ◽  
Vol 26 (3) ◽  
pp. 341-349 ◽  
Author(s):  
Givi Berikelashvili ◽  
Manana Mirianashvili
Keyword(s):  
Exact Solution ◽  
Sobolev Space ◽  
Difference Scheme ◽  
Absolute Stability ◽  
Burgers Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Algebraic Equations ◽  
The Difference ◽  
Initial Boundary Value

Abstract A three-level finite difference scheme is studied for the initial-boundary value problem of the generalized Benjamin–Bona–Mahony–Burgers equation. The obtained algebraic equations are linear with respect to the values of the desired function for each new level. The unique solvability and absolute stability of the difference scheme are shown. It is proved that the scheme is convergent with the rate of order {k-1} when the exact solution belongs to the Sobolev space {W_{2}^{k}(Q)} , {1<k\leq 3} .

scholarly journals Stability of the Difference Schemes for the Equations of Weakly Compressible Liquid

2007 ◽  
Vol 7 (3) ◽  
pp. 208-220 ◽  
Author(s):  
P. Matus ◽  
O. Korolyova ◽  
M. Chuiko
Keyword(s):  
Initial Data ◽  
Difference Scheme ◽  
Initial Conditions ◽  
Compressible Liquid ◽  
Stability Conditions ◽  
Differential Problem ◽  
Weakly Compressible ◽  
Linear Problems ◽  
The Difference ◽  
The Stability

Abstract A priory estimates of the stability in the sense of the initial data of the difference scheme approximating weakly compressible liquid equations in the Riemann invariants have been obtained. These estimates have been proved without any assumptions about the properties of the solution of the differential problem and depend only on the behavior of the initial conditions. As distinct from linear problems, the obtained estimates of stability in the general case exist only for a finite instant of time t≤t_0. In particular, this is confirmed by the fact, that nonfulfilment of these stability conditions lead to the appearance of supersonic flows or domains with large gradients. The questions of uniqueness and convergence of the difference solution are considered also. The results of the computating experiment confirming the theoretical conclusions are given.

scholarly journals Rayleigh–Taylor instability at a tilted interface in laboratory experiments and numerical simulations

2003 ◽  
Vol 21 (3) ◽  
pp. 419-423 ◽  
Author(s):  
JOANNE M. HOLFORD ◽  
STUART B. DALZIEL ◽  
DAVID YOUNGS
Keyword(s):  
Numerical Simulations ◽  
Laboratory Experiments ◽  
Initial Conditions ◽  
Fluid Motion ◽  
Mixing Efficiency ◽  
Molecular Mixing ◽  
High Prandtl Number ◽  
Rayleigh Taylor Instability ◽  
The Difference ◽  
Rt Instability

This article investigates the molecular mixing caused by Rayleigh–Taylor (RT) instability of a gravitationally unstable density interface tilted at a small angle to the horizontal. The mixing is measured by the increase in background potential energy, and the mixing efficiency, or fraction of energy irreversibly lost to fluid motion doing work against gravity, is calculated. Laboratory experiments are carried out using saline and fresh water, and modeled with compressible numerical simulations, with a suitable choice of parameters and initial conditions. The experiments show that the high cumulative efficiency of mixing in RT instability at a horizontal interface is only slightly reduced by an interface tilt of up to 10°, despite the strong overturning that occurs. Instantaneous mixing efficiencies as high as 0.5–0.6 are measured, when RT instability is active, with lower values of about 0.35 during the subsequent overturning. The numerical simulations capture the most unstable scales and the overturning motion well, but generate more mixing than the experiments, with the instantaneous mixing efficiency remaining at 0.5 for most of the run. The difference may be due to restratification at small scales in the high Prandtl number experiments.

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