Regularity Criterion for a Two Dimensional Carreau Fluid Flow

Abstract

AbstractCarreau fluids are a source of research from both theoretical and applied approaches. They have been considered to model different non-newtonian phenomena such as blood flow, plasma and viscoeslastic materials. The purpose of this study is to develop the global regularity criteria for a Carreau fluid in two dimensions flowing in a strip. Firstly, a regularity criteria is shown for the initial set $$\left( u_{10},u_{20}\right) \in H^{1}\left( \Omega \right) $$ u 10 , u 20 ∈ H 1 Ω where $$\Omega =\left[ 0,L\right] \times $$ Ω = 0 , L × $$\left[ 0,\infty \right) .$$ 0 , ∞ . Secondly, the analysis focuses on a regularity criteria when $$\left( u_{10},u_{20}\right) \in L^{4}\left( \Omega \right) $$ u 10 , u 20 ∈ L 4 Ω and, lastly, similar results are obtained for $$\left( u_{10},u_{20}\right) \in H^{2}\left( \Omega \right) $$ u 10 , u 20 ∈ H 2 Ω while the fluid velocity vertical component, $$u_2 (x,y)$$ u 2 ( x , y ) , is such that $$\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) $$ ∂ u 2 ∂ x ∈ L 4 Ω and $$\left( \frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) $$ ∂ ∇ u 2 ∂ y , Δ u 2 ∈ L 2 Ω .