Abstract
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 \(\left( u_{10},u_{20}\right) \in H^{1}\left( \Omega \right) \) where \(\Omega =\left[ 0,L\right] \times \) \(\left[ 0,\infty \right) .\) Secondly, the analysis focuses on a regularity criteria when \(\left( u_{10},u_{20}\right) \in L^{4}\left( \Omega \right) \) and, lastly, similar results are obtained for \(\left( u_{10},u_{20}\right) \in H^{2}\left( \Omega \right) \) while the fluid velocity vertical component, \(u_2 (x,y)\), is such that \(\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) \) and \(\left( \frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) \).
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1 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.
The non-Newtonian fluids are characterized in different types depending on their rheological characteristics. One of such types is known as Carreau fluid (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17] for few relevant studies). More particularly, an interesting research on Carreau fluids is given in [4] where the fluid is analyzed over a shrinking surface in presence of an infinite shear rate viscosity. In addition, the study of Carreau flows in spheres are widely analyzed in [5, 6].
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.
2 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 \({\mathbf {V}}\) is the velocity field, t the time, \(\rho \) is the fluid density, \({\tau }\) is the Cauchy stress tensor, \({\mathbf {J}}\) is the current density and \(\mathbf {B=B_{0}+B_{1}}\) is the magnetic field for a Carreau fluid, which is given by
with
where p is the pressure field, \({\mathbf {I}}\) the identity tensor, \(\eta _{0}\) the zero-shear-rate viscosity, \(\eta _{\infty }\) the infinite-shear-rate viscosity, \(\varvec{\Gamma }\) a material time constant, and n express the power law index (since it describes the slope of \(\frac{\eta _{0}-\eta _{\infty }}{\eta _{0}+\eta _{\infty }}\) in the power law region). The shear rate \(\overset{{\cdot }}{\gamma }\) is defined by
Here \(\varvec{\Pi }\) is the second invariant strain rate tensor and \(\mathbf { A}_{1}\) is given by
Note that \(\left( \cdot \right) ^{T}\) 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 \(\nu =\frac{\eta _{0}}{\rho }\) is the kinematic viscosity and \(\sigma \) 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
such that \(\left( u_{10}\left( x,y\right) ,u_{20}\left( x,y\right) \right) \) corresponds to the vector of initial velocities. Note that the domain is given as \(\Omega = \lbrace (x,y) \in [0,L] \times [0, \infty ) \rbrace \).
In addition, consider that the shear stress is zero at \(y=0.\) As a consequence, the following holds
leading to \(\frac{ \partial u_{1}}{\partial y}=0\) at \(y=0.\)
3 Preliminaries and Statement of Results
3.1 Previous Results
Consider the well-known norm \(\left\| \cdot \right\| _{L^{p}}\) in the Lebesgue functional space \(L_{p}\left( \Omega \right) \) 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])
Lemma 1
The following anisotropic Sobolev inequality holds
3.2 Statement of Results
The main results are stated as follows:
Theorem 3.1
Assuming \(\left( u_{10},u_{20}\right) \in H^{1}(\Omega )\), where \(\left| \nabla u_{0}\right| =\left| \frac{\partial u_{0}}{\partial y}\right| ,\) the equation (2.7) has solutions on \(\left( 0,T\right] \) in the defined strip \(\Omega =\left[ 0,L\right] \times \) \(\left[ 0,\infty \right) .\)
Theorem 3.2
Assuming \(\left( u_{10},u_{20}\right) \in L^{4}(\Omega )\), then the system (2.7) has solutions on \(\left( 0,T\right] \) in the defined strip \(\Omega = \left[ 0,L\right] \times \) \(\left[ 0,\infty \right) .\)
Theorem 3.3
Assuming \(\left( u_{10},u_{20}\right) \in H^{2}(\Omega )\), then the system (2.7) has solutions on \(\left( 0,T\right] \) when \(\frac{\partial u_{2} }{\partial x}\in L^{4}\left( \Omega \right) \) and \(\left( \frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) \), in the defined strip \(\Omega =\left[ 0,L\right] \times \) \( \left[ 0,\infty \right) .\)
4 Proof of Theorem 3.1
Firstly, the following proposition is required to be shown
Proposition 4.1
Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). Neglecting the higher powers of \(\Gamma ^{4}\), then \(u_{1}(x,y,t)\) satisfies
where \(\overset{\sim }{C}_{1}\) depends on a suitable constant M (to be defined in the proof) and T.
Proof
Multiplying the Eq. (2.7) by \(u_{1}\), operating and neglecting higher power of \(\Gamma ^{4},\) the following holds
After using integration by parts
where
Integrating by parts
After using Eq. (2.1), the following reads
Integrating again we get \(\left( I_{1}=0\right) \). Introducing the value of \( I_{1}\) into Eq. (4.1) and after Young’s inequality, the following applies
The Grownwall’s inequality yields
where \(\overset{\sim }{C}_{1}\) depends on M and T. \(\square \)
In addition, the following Proposition is also required.
Proposition 4.2
Assume a solution \(u_{1}\) to set of the equation and conditions (2.7)–(2.11) departing from \(u_{10}\). Then, neglecting higher powers of \(\Gamma ^{4}\), \(\frac{\partial u_{1}}{\partial y}\) satisfies
where \(\overset{\sim }{\text { }C}_{2}\) depends on M and T.
Proof
Multiplying (2.7) by \(-\frac{\partial ^{2}u_{1}}{\partial y^{2}}\) and integrating by parts:
Expanding the term on the right side and neglecting the higher powers of \(\Gamma ^{4}\)
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
which implies that
Now, applying Gronwall’s inequality
where \(\overset{\sim }{\text { }C}_{2}\) depends on M and T. \(\square \)
Finally, note that the Theorem 3.1 is proved by using Propositions 4.1 and 4.2.
5 Proof of Theorem 3.2
To this end, the following Proposition is required
Proposition 5.1
Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). Neglecting the higher powers of \(\Gamma ^{4}\), then \(u_{1}(x,y,t)\) satisfies
where \(\overset{\sim }{C}_{3}\) depends on M and T.
Proof
Multiplying the Eq. (2.7) by \(u_{1}^{3}\), operating, neglecting higher power of \(\Gamma ^{4},\) and after using integration by parts
Afer using integration by parts
where
Applying integration by parts on the second term in right side and making use of Eq. (2.1)
Integrating again we get \(\left( I_{3}=0\right) \). Introducing the value of \( I_{3}\) in Eq. (5.1) and after Young’s inequality
The Grownwall’s inequality yields
where \(\overset{\sim }{C}_{3}\) depends on M and T. \(\square \)
Note that the Theorem 3.2 is shown as per the Proposition 5.1 introduced and proved.
6 Proof of Theorem 3.3
The Theorem is shown based on the coming Propositions.
Proposition 6.1
Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). In addition, assume the existence of \(\nabla u_{10}\) in \(L^{2}\left( \Omega \right) \) and that \(\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) \). Neglecting the higher power of \(\Gamma ^{4}\) then
where \(\overset{\sim }{\text { }C}_{4}\) depends on M and T.
Proof
Considering the inner product in Eq. (2.7) with \(\Delta u_{1}\) and integrating
Expanding the term on the right side in the last equation and neglectng the higher powers of \(\Gamma ^{4},\)
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
Introducing the assessed integral \(I_{4}\) into Eq. (6.1) and after using Proposition 3 and \(\frac{\partial u_{2}}{\partial x}\in L^{4}\left( \Omega \right) \)
From Grownwall’s inequality
where \(\overset{\sim }{\text { }C}_{4}\) depends on M and T. \(\square \)
Proposition 6.2
Assume \(u_{1}\) is a solution departing from \(u_{10}\) to the set of the equation and conditions (2.7) to (2.11). assume that in the strip \(\Omega \), there exists \(\left( \Delta u_{10},\frac{\partial \nabla u_{2}}{\partial y},\Delta u_{2}\right) \in L^{2}\left( \Omega \right) .\) Neglecting the higher power of \(\Gamma ^{4}\), then
where \(\overset{\sim }{\text { }C}_{5}\) depends on constants and T.
Proof
Apply the monotone operator \(\Delta \) to Eq. (2.7) and take inner product with \(\Delta u_{1}\). After integration by parts, the following reads
Expanding the terms on the right side and neglecting the higher powers of \(\Gamma ^{4},\)
After Integration
where
Now for \(I_{5}\)
Applying integration by parts and continuity
Then
In order to solve the above six terms, the Lemma 1 is employed so that
Integrating \(I_{6}\), we have
where we made use of the Young’s inequality. Similarly to solve \(I_{7}\)
Combining Eq. (2.13) to Eq. (2.22), After using Proposition 1, Proposition 2, Proposition 3 and \(\Delta u_{2},\frac{\partial \nabla u_{2}}{\partial y}\in L^{2}\left( \Omega \right) ,\) we get
where \(C_{1}\) is suitable constant. Applying Grownwall’s inequality again
where \(\overset{\sim }{\text { }C}_{5}\) depends on suitable constants and T. \(\square \)
Finally, the Theorem 3.3 is shown by using Proposition 6.1 and Proposition 6.2.
7 Conclusion
The proposed Theorems 3.1, 3.2 and 3.3 have been shown in the different supporting propositions. Such Theorems lead to confirm on the existence of regular solutions departing from an initial data generalized under the defined functional spaces. The shown solutions are applicable to the two dimensional Carreau fluid on the defined strip \(\Omega = [0,L] \times [0, \infty )\).
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José L. Díaz (JL), Saeed Rahman (SR), M. Khan (MK) and Guang-Zhong Yin (GZ). JL, SR and MK conceived the study and the overall manuscript design. JL, SR and MK carried out the analytical studies. SR has performed the specific analytical assessment with the revision of JL. SR, GZ and JL have made a revision of the manuscript. GZ has made the final edition and overall supervision. All authors read and approved the final manuscript.
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Díaz Palencia, J.L., Rahman, S., Khan, M. et al. Regularity Criterion for a Two Dimensional Carreau Fluid Flow. J Nonlinear Math Phys 29, 731–749 (2022). https://doi.org/10.1007/s44198-022-00057-6
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DOI: https://doi.org/10.1007/s44198-022-00057-6