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1.The model of shear flow of Johnson-SegalmanIn this paper,we focus on the Johnson-Segalman(JS) model andconsider it as a model of the shesr flow of polymeric fluid .By meansof the Johnson-Segalman ,we develop computation methods has solveJS model.The Johnson-Segalman model that describes the flow fluid betweentwo parallel plates(located at x=-1/2,x=1/2)can be written aswhere,the dimensionless parameters are α := ρh~2λ~2/μ, ε := ηλ/μ,where,η is the coefficient of Newtonian viscosity, μ is an elastic shearmodulus,λ isa relaxation rate, f is the pressure gradient, σ is the shear stress,Z is the principal normalstress difference and the velocitu field is υ ( 0,υ(x,t)).For later purpose,notice that the momentum equation may be writtenαv t ?Tx=f,where T = σ +εvx denoted the total shear stress.2.Clubic functionThe most popular RBF’ s are:x ? xj (Linear)x ? xj 2 +C2 (Multiquadric)( )2exp ? x ?xj (Gussian)x ? xj 2 log??? x?xj??? (Douch)3x ? xj (Culbic).In this article,Cubicfunction abowe is used equations(4) as a spatialapproximation for the unknow function T n (x)∑( )== N?jjnjTn xxx1( )λ φ (4)3.Numerical schemeA free mesh computational mer\thod is ysed to slmulate the system(5)which is equivilent with equations?????????=???? ?????=+??? ?????=++??? ?????ZTZZTTTZTtttxxεσσσε σσαεε σσ(1)(1)(5)This paper to solve equation (5) and numerical examples are given.To thebest of our knowledge, so far this approach in literature.Moreover,the resultsof numerical computation in this paper reproduce the spurt phenomenon in thechemical experiment done in [2].
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