Perturbation Analysis Of An Inter Gas To Slow Acceleration

Acoustic waves in the cylinder can cause oscillations in thermodynamic variables and gas velocity,thereby affecting the efficiency.So the aerodynamic process in the piston-cylinder is crucial to internal combustion engines.This process can be described by Euler’s equations.For many nonlinear problems,we can not find an exact solution to the equation,so we sometimes use an approximate solution instead,such as numerical solutions and analytical solutions.In this paper,we use the multiscale method to get the asymptotic solution.This paper studies the initial boundary value problem of the slow piston acceleration motion that occurs only on the piston time scale in the absence of an initial disturbance:（?）Where τ=εt,U_P（τ）is the dimensionless piston velocity.Based on the first term of the approximate solution given by Wang and Kassoy,we use multi-scale method by transforming the time and get the second term of the approximate solution.To get the error estimate of the approximate solution,we establish the second-order equation system of the remainder.Then we solve it by using the energy integration method and the Gronwall inequality.The following conclusions are proved:Theorem:Suppose（1）U_P（τ）has a third-order continuous derivative,U_P（0）=0;（2）There is （?）>0,M>0,such that 1/ρ≤（?）≤ρ;（3）ρ and v have second-order continuous derivatives at t>0,1<s<1,and the first-order derivatives are bounded at t≥0,1≤ s≤ 1;（4）v（x,t;ε）=v₀+εv₁+εR_v,ρ（x,t;ε）=ρ₀+ε²ρ₂+εR_ρ,e where （?） then there is T>0 and positive real numbers ε₀,M₀（T,ε₀）,so that when 0<εt<T,0<ε<ε₀,（?）and（?）We use numerical methods to obtain the solution then contrast it with asymptotic solution,which proves the rationality of the solution.