Accounting for Quadratic and Cubic Invariants in Continuum Mechanics–An Overview

The differential equations of continuum mechanics are the basis of an uncountable variety of phenomena and technological processes in fluid-dynamics and related fields.These equations contain derivatives of the first order with respect to time.The derivation of the equations of continuum mechanics uses the limit transitions of the tendency of the volume increment and the time increment to zero.Derivatives are used to derive the wave equation.The differential wave equation is second order in time.Therefore,increments of volume and increments of time in continuum mechanics should be considered as small but finite quantities for problems of wave formation.This is important for calculating the generation of sound waves and water hammer waves.Therefore,the Euler continuity equation with finite time increments is of interest.The finiteness of the time increment makes it possible to take into account the quadratic and cubic invariants of the strain rate tensor.This is a new branch in hydrodynamics.Quadratic and cubic invariants will be used in differential wave equations of the second and third order in time.

An estimation of the velocity profile for the laminar-turbulent transition in the plane jet on the basis of the theory of stochastic equations and equivalence of measures

The theory of stochastic equations and the theory of equivalence of measures previously applied to flows in the boundary layer and in the pipe are considered to calculate the velocity profile of the flat jet.This theory previously made it possible to determine the critical Reynolds number and the critical point for the flow of the plane jet.Here based on these results the analytical dependence for the index of the velocity profile is derived.Velocity profiles are calculated for a laminar-turbulent transition in the jet.This formula reliably reflects an increase of the energy transferred from a deterministic state to a random one with an increase of the index of the velocity profile.Results show satisfactory agreement with the known experimental data for the velocity profile of the flat jet.Using obtained results it is possible to determine the location of technical devices for laminarization of the flow in the jet.This is important both for reducing friction in the flow around aerodynamic vehicles and for maintaining the jet profile if it is necessary to ensure the stability of the flow characteristics.Also the obtained relations can be useful for researching of the processes in combustion chambers,in the case of welding and in other technical devices.