Weak Solution for 3D-Stochastic Third Grade Fluid Equations

Adilson Almeida,Fernanda Cipriano

doi:10.3390/w12113211

Abstract

This article studies the stochastic evolution of incompressible non-Newtonian fluids of differential type. More precisely, we consider the equations governing the dynamic of a third grade fluid filling a three-dimensional bounded domain O, perturbed by a multiplicative white noise. Taking the initial condition in the Sobolev space H2(O), and supplementing the equations with a Navier slip boundary condition, we establish the existence of a global weak stochastic solution with sample paths in L∞(0,T;H2(O)).

Highlights

The incompressible Newtonian fluids described by the Navier-Stokes equation are characterized by the Newton law of the viscosity, which corresponds to a linear relation between the shear stress and the rate-of-strain tensors
Some biological fluids and many fluids used in the industry and in the food processing do not satisfy such linear relation and are named non-Newtonian fluids
We studied the third grade fluid equations in the three-dimensional physical space, perturbed by a multiplicative white noise

Summary

Introduction

The incompressible Newtonian fluids described by the Navier-Stokes equation are characterized by the Newton law of the viscosity, which corresponds to a linear relation between the shear stress and the rate-of-strain tensors. Some biological fluids (as the blood) and many fluids used in the industry and in the food processing do not satisfy such linear relation and are named non-Newtonian fluids (see for instance [1,2,3,4,5]). The fluids of grade n constitute a special class of non-Newtonian fluids. For these fluids the stress tensor corresponds to a polynomial of degree n of the first n Rivlin-Ericksen kinematic tensors in [6].

Results

Discussion

Conclusion