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Abstract.To estimate the solutionof the coupled first-order hyperbolic partial differential equations，we use both the boundary-layermethod and numeric analysis to study the Cauchy fluid equations andP-T/T stress equation. On the macroscopic scale the free surface elements generateflow singularity and stress uncertainty by excessive tensile stretch. A numerical super-convergence semi-discrete finite element scheme is used to solve the time dependent equations. The coupled nonlinear solutions are estimated by boundary-layer approximation. Its numericalsuper convergence is proposed with the a priori and a posteriori error estimates.
Keywords：Non-Newtonian fluid； Semi-discrete finite element method； Super convergence； Boundary-layer solution
INTRODUCTION
Insidethe micro-scale materials Non-Newtonian fluid mechanics is anothercomplex subject developed in recent decades to observe the fluid-structure interactions. It originated from polymer processing and involves many experimental fields. It is justan interdisciplinary subject， including mechanics， modern mathematics， chemical and engineering sciences， especially material science. Also， it is an important branch of modern fluid mechanics and is an important part of modern theology for virtual test. With further study for experts in the difficult and expensive chemical industry， oil， water conservancy， bioengineering， light industry and food material science， more realization of the computing accuracy impacted on the testing results become important issue in the complex testing conditions which is virtually designed by the CAD/CAE tools. The specific mathematical model is also widely used and promotedvery rapidly to solve the practical aspect of the problems.
In this paper， fluid-solid coupling fractional differential equations： Cauchy equation （fluid equation） and P-T＼T equation （solid equation） are used to describe the rheological process on non-Newtonian complex contact surface. The P-T＼Tequation adds exponential impact item to Maxwell equation which allowsus to study the boundary-layer near discontinues interface to pin-point stress uncertainty and singularity. Rheological problems in cellular porous structural material are dealt within detail in [1， 2] which describe the background of the deformable materials [5， 6].
1. MACROSCOPIC EQUATION
1.1 Fluid--Solid Coupling Equations
In the paper the contact surface （Hermit boundary conditions） and non-contact surface are taken as fluid composed of free surface element. In nature it is usually singular by excessive tensile stretch in the boundary-layer and randomly by the point force uncertainty. On theone hand， using standard variables of material science， the resistance of extensional and simple shear rate is analyzed by the characteristics ofnon-Newtonian P-T＼Tequation （1）[4， 8]. Coupled with Cauchy conservation equation （2）[3， 7]， the elastic-plastic material deformation （shear thinning） caused by distribution changes in the macro-stress field τ can be calculated. Here we review Cauchy equation
Keywords：Non-Newtonian fluid； Semi-discrete finite element method； Super convergence； Boundary-layer solution
INTRODUCTION
Insidethe micro-scale materials Non-Newtonian fluid mechanics is anothercomplex subject developed in recent decades to observe the fluid-structure interactions. It originated from polymer processing and involves many experimental fields. It is justan interdisciplinary subject， including mechanics， modern mathematics， chemical and engineering sciences， especially material science. Also， it is an important branch of modern fluid mechanics and is an important part of modern theology for virtual test. With further study for experts in the difficult and expensive chemical industry， oil， water conservancy， bioengineering， light industry and food material science， more realization of the computing accuracy impacted on the testing results become important issue in the complex testing conditions which is virtually designed by the CAD/CAE tools. The specific mathematical model is also widely used and promotedvery rapidly to solve the practical aspect of the problems.
In this paper， fluid-solid coupling fractional differential equations： Cauchy equation （fluid equation） and P-T＼T equation （solid equation） are used to describe the rheological process on non-Newtonian complex contact surface. The P-T＼Tequation adds exponential impact item to Maxwell equation which allowsus to study the boundary-layer near discontinues interface to pin-point stress uncertainty and singularity. Rheological problems in cellular porous structural material are dealt within detail in [1， 2] which describe the background of the deformable materials [5， 6].
1. MACROSCOPIC EQUATION
1.1 Fluid--Solid Coupling Equations
In the paper the contact surface （Hermit boundary conditions） and non-contact surface are taken as fluid composed of free surface element. In nature it is usually singular by excessive tensile stretch in the boundary-layer and randomly by the point force uncertainty. On theone hand， using standard variables of material science， the resistance of extensional and simple shear rate is analyzed by the characteristics ofnon-Newtonian P-T＼Tequation （1）[4， 8]. Coupled with Cauchy conservation equation （2）[3， 7]， the elastic-plastic material deformation （shear thinning） caused by distribution changes in the macro-stress field τ can be calculated. Here we review Cauchy equation