Regularity Criterion for a Two Dimensional Carreau Fluid Flow

Carreau 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 u10,u20∈H1Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( u_{10},u_{20}\right) \in H^{1}\left( \Omega \right) $$\end{document} where Ω=0,L×\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Omega =\left[ 0,L\right] \times $$\end{document}0,∞.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left[ 0,\infty \right) .$$\end{document} Secondly, the analysis focuses on a regularity criteria when u10,u20∈L4Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( u_{10},u_{20}\right) \in L^{4}\left( \Omega \right) $$\end{document} and, lastly, similar results are obtained for u10,u20∈H2Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( u_{10},u_{20}\right) \in H^{2}\left( \Omega \right) $$\end{document} while the fluid velocity vertical component, u2(x,y)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$u_2 (x,y)$$\end{document}, is such that ∂u2∂x∈L4Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) $$\end{document} and ∂∇u2∂y,Δu2∈L2Ω\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\left( \frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) $$\end{document}.


Introduction
Most of the fluid conceptions used to model flows in typical media like air, water or oil, are based on a Newtonian description. In an important number of cases, the supposition of Newtonian behavior does not seem to be accurate enough, leading to expand the theory to introduce additional rheological properties that end in non-Newtonian descriptions. As a set of examples that preclude its ubiquity, the non-Newtonian flow is experienced in industries of mines, where sludges and muds are frequently dealt, in lubrication and biomedical flows.
Currently, there is not a wide literature focused on developing the regularity criteria for equations obtained upon a Carreau fluid. Nonetheless, there exists an extensive literature developing the regularity criteria for Navier-Stokes equations (see [18][19][20][21][22][23]). Motivated by such facts, our intention in this study is to develop global regularity criteria for a magnetohydrodynamic (MHD) flow of Carreau fluid equations. The fluid is considered to flow between two stationary plates. Flow between the plates is induced by an initial velocity profile and a stationary linear law at y = 0 to the x-axis velocity component. In addition, a uniform magnetic field is applied in the transverse direction to the flow.
The paper layout is as follows: Firstly, the Carreau fluid model is discussed based on the general theory of fluid dynamics of non-Newtonian flows. In addition, some preliminary required results are introduced. Afterwards the theorems on existence of regular solutions are presented. The paper introduces progressively, in different sections, each of the required proofs for each of the regularity Theorems together with the supporting information required.

Model Proposal
We consider the two dimensional incompressible fluid flow equations of a Carreau fluid. The fluid is electrically conducting in the presence of an applied magnetic field B 0 . The MHD flow is governed by the following set of equations: where is the velocity field, t the time, is the fluid density, is the Cauchy stress tensor, is the current density and = + is the magnetic field for a Carreau fluid, which is given by with where p is the pressure field, the identity tensor, 0 the zero-shear-rate viscosity, ∞ the infinite-shear-rate viscosity, a material time constant, and n express the power law index (since it describes the slope of 0 − ∞ 0 + ∞ in the power law region). The shear rate ⋅ is defined by Here is the second invariant strain rate tensor and 1 is given by Note that (⋅) T denotes the transpose of a matrix. From (2.1) to (2.6), the governing equations in the absence of pressure gradient are given by where = 0 is the kinematic viscosity and is related with the electrical charges distribution.
Note that the subject boundary conditions are together with Where U is a constant. In addition, the following initial conditions hold (2.7)

3
such that u 10 (x, y), u 20 (x, y) corresponds to the vector of initial velocities. Note that the domain is given as In addition, consider that the shear stress is zero at y = 0. As a consequence, the following holds leading to u 1 y = 0 at y = 0.

Previous Results
Consider the well-known norm ‖⋅‖ L p in the Lebesgue functional space L p (Ω) together with the usual Sobolev order m functional space defined by with the norm In addition, the following lemma is also needed (refer to Lemma 1 in [23])

Proof of Theorem 3.1
Firstly, the following proposition is required to be shown Integrating by parts After using Eq. (2.1), the following reads Integrating again we get I 1 = 0 . Introducing the value of I 1 into Eq. (4.1) and after Young's inequality, the following applies The Grownwall's inequality yields where ∼ C 1 depends on M and T. ◻ In addition, the following Proposition is also required.
Expanding the term on the right side and neglecting the higher powers of Γ 4 where Integrating I 2 , where (2.1) has been employed. Now, integrating again we get I 2 = 0. Introducing I 2 in Eq. (4.2) and applying Young's inequality, the following holds

Proof of Theorem 3.2
To this end, the following Proposition is required where ∼ C 3 depends on M and T. ◻ Note that the Theorem 3.2 is shown as per the Proposition 5.1 introduced and proved.

Proof of Theorem 3.3
The Theorem is shown based on the coming Propositions.
Expanding the term on the right side in the last equation and neglectng the higher powers of Γ 4 ,

Aplying interation by parts which implies that
where Now, the intention is to further develop the integral I 4 , to this end From Eq. (2.1), the following holds Integrating again where Eq. (2.1) has been used, therefore I 4 becomes x dxdy.
Introducing the assessed integral I 4 into Eq. (6.1) and after using Proposition 3 and Proof Apply the monotone operator Δ to Eq. (2.7) and take inner product with Δu 1 . After integration by parts, the following reads

3
Expanding the terms on the right side and neglecting the higher powers of Γ 4 ,

After Integration
where Now for I 5 (2.13) Δu 1 dxdy, Applying integration by parts and continuity

Then
In order to solve the above six terms, the Lemma 1 is employed so that (6.2) (6.4) (6.5) (6.6) (6.7) (6.9) where ∼ C 5 depends on suitable constants and T. ◻ Finally, the Theorem 3.3 is shown by using Proposition 6.1 and Proposition 6.2.

Conclusion
The proposed Theorems 3.1, 3. Funding Not applicable.

Data Availability
The authors declare that data can be made available upon request.

Conflict of Interest
The authors have no competing interests to declare that are relevant to the content of this article.

Consent for Publication Authors declare the consent for manuscript publication
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