Regularity criteria for a two dimensional Erying-Powell fluid flowing in a MHD porous medium

1.   Introduction

A non-Newtonian description of a fluid may be required to model complex processes in physics and engineering. The constitutive equations, particularly the momentum, of such a non-Newtonian fluid give rise to mathematical concerns due to the additional rheological parameters in the constitutive relations. Unlike in the classical Newtonian viscous fluids, there is not a single constitutive equation for non-Newtonian fluids, which can describe in a general basis, their rheological properties. As a consequence, the non-Newtonian fluids are considered under particular descriptions supported by the observations and/or by theoretical arguments. One of this, that has attracted much attention in the last decades, is known as Eyring-Powell fluid. The diffusion operator associated to this kind of fluids emerges from the application of the kinetic theory of liquids, instead of particular experimental principles about viscosity (refer to ^[1,2] and the studies cited therein).

Few recent attempts, focused on the solutions of Eyring-Powell fluid flow equations, can be mentioned through the studies from ^{[1,2,3,4,5,6,7,8,9,10]}. It should be noted that the cited references can be understood as representative of the state of the art. Therein, the reader can find remarkable applications of the mentioned fluid together with dedicated solutions obtained under analytical and numerical procedures. Nonetheless, the solutions are provided with no additional explorations about their regularity in the appropriate general mathematical spaces. On the contrary, such regularity principles are currently available in the literature for Navier-Stokes equations (see the references ^{[11,12,13,14,15,16]} for some analysis on the matter).

We focus our interest now on the discussions about some additional and remarkable studies involving analysis in fluid flows. These discussions may be regarded as additional justifications of the model to come. From an analytical perspective, we shall mention that the authors in ^[17] introduced a characterization of perfect fluid spacetimes based on Ricci solitons, gradient A-Einstein solitons, gradient Ricci solitons and gradient Schouten solitons. In ^[18], the authors obtained regularity results in $L^p$ for the highest order derivative in an elliptic set of equations with coefficients in the space of Sarason vanishing mean oscillation (VMO). Some additional regularity conditions in fluids are given in ^[19], where the authors studied a blow-up condition for the set of solutions to a MHD fluid involving the Lebesgue space $L^3$. In ^[20], the authors study the heat dissipation of a stretching sheet under a MHD mixed convective flow of Eyring-Powell fluid. The solutions are obtained based a shooting method with a Fourth-order Runge-Kutta approach. In addition, in ^[21], the authors provide a comparative study between a nanofluid and water and the exploration of solutions complying with the oscillatory pattern of the boundary layer equations via a Perturbation Technique. In ^[22], a theoretical analysis, about the physical characteristics of an unsteady nanofluid flow, is provided based on a Parametric Continuation Method (PCM) validated with the Matlab numerical routine bvp4c. Another interesting analysis, involving the application of non-Newtonian fluids, is given in ^[23]; wherein the authors studied a mucus fluid transportation with the Laplace transform technique and the use of programming. In ^[24], the authors analyze the problem of a two dimensional laminar flow by converting the set of PDE into nonlinear and higher-order ODEs. The new system of ODEs is treated with a fourth-order Runge-Kutta method combined with shooting approach.

Supported by the fact that none of the mentioned references provides a general frame dealing with regularity of solutions, the purpose of the presented study is to develop the global regularity conditions for the solutions of a two-dimensional flow of Eyring-Powell fluid type arising in Magnetohydrodynamic (MHD) of porous materials. To make our analysis tractable, it should be noted that the induced magnetic field is considered to be negligible for small magnetic Reynolds numbers. This means that the electromagnetic interaction between charges is assumed to be small in comparison with the external applied magnetic field that induces the charged particles to move.

1.1.   Model description

Mathematically, we consider a two-dimensional Eyring-Powell fluid with boundary layers and with, at least, a preliminary generalized solution belonging to the bounded mean oscillation ($BMO$) space. The existence of such a kind of solution is not a hard hypothesis, on the contrary, the references ^{[2,3,4,5,6,7,8,9,10]} provide different forms of solutions constructed by analytical and numerical means. Hence and given the existing vast literature dealing with solutions, it seems to us that the existence of such solutions is not a big issue at this stage, but the regularity and boundedness of them is.

Note that the fluid is considered to be incompressible and flowing through a porous medium. In this case, the flow velocity vector $\mathbf{V}$, the continuity and dynamical equations for the unsteady two-dimensional Eyring-Powell fluid flow with nano-boundary conditions (see ^[25]) are given by (note that we develop the analysis for the velocity component $u$. Similarly, it can be done for the other velocity component $v$)

The related boundary conditions are expressed as follows

Note that at $y = 0$, the velocity component $u$ is requested to satisfy an stretching condition with the constant $U$. This is introduced to ensure that the fluid flow complies with the continuity equation. Indeed any variation of the velocity component $v$ shall comply with $\frac{\partial v}{\partial y} = -U$. Typically, this last value can be regarded as arbitrarily small. In the region where the fluid flow is fully developed (outside the stretching area), the stretching condition is considered as negligible. Assume that such a region is defined in the $y-$interval, $(y_m, \infty),$ where $y_m > 0$. Then we can assume that the kinematic Dirichlet condition in the nanoboundary applies as follows

In addition, the initial conditions is given by

It should be noted that $u$ and $v$ are the first and second velocity components respectively, while $U$ is a constant. Recall that $u_{0}(x, y)$ is the initial velocity profile, $\nu$ is the kinematic viscosity of a fluid, defined as a ratio of dynamic viscosity $\mu$ and fluid density $\rho$, and $\beta$, $d$, $\phi$ and $k$ refer to some fluid parameters that shall be obtained in accordance with the physical phenomenon to model (see a dedicated example in ^[25]).

2.   Summary of results

Our results are stated as follows:

Theorem 2.1. Assume $u_{0}\in H^{1}(\Omega),$ with $\left(u, \frac{\partial u}{ \partial y}, \frac{\partial ^{2}u}{\partial y^{2}}\right) \in L^{2}(0, T, BMO)$, then the Eq $(1.2)$ has solutions, $u(x, y, t)$, complying with a regularity condition in $[0, T]$. Note that here $\Omega = [y_m, L]\times \lbrack 0, \infty).$

Theorem 2.2. Suppose that $u_{0}\in L^{P}(\Omega),$ with $\left(u^{\frac{P-2}{2}}\frac{ \partial u}{\partial y}, \frac{\partial u}{\partial y}\right) \in L^{2}(0, T, BMO),$ then the Eq $(1.2)$ has solutions, $u(x, y, t)$, complying with a regularity condition in $[0, T]$. Note that here $\Omega = [y_m, L]\times \lbrack 0, \infty).$

Theorem 2.3. Assuming $u_{0}\in H^{2}(\Omega),$ together with $\left(\frac{\partial u}{\partial y}, \frac{ \partial ^{2}u}{\partial y^{2}}, \frac{\partial \nabla u}{\partial y}, \Delta u, \frac{\partial \Delta u}{\partial y}\right) \in L^{2}(0, T, BMO)$ and $\left(\Delta v. \, \frac{\partial \nabla v}{\partial y}\right)\in L^{2}(\Omega)$. In addition, assume that $\frac{\partial^{2}u}{\partial x^2}$ and $\frac{\partial^{2}u}{\partial y^2}$ are of the same sign. Then the Eq $(1.2)$ has solutions, $u(x, y, t)$, complying with a regularity condition in $[0, T]$. Note that here $\Omega = [y_m, L]\times \lbrack 0, \infty).$

3.   Preliminaries

In this section, we introduce some notations and collect some preliminary results that will be used in the coming analysis. We introduce the well known functional space $L_{p}\left(\Omega \right)$ with the norm $\left \Vert \cdot \right \Vert _{L^{p}}$ :

The usual Sobolev space of order $m$ is defined by

with the norm

In addition, we make use of the homogeneous space of bounded mean oscillation whose norm is defined as (see ^[26])

Lemma 3.1. (See ^[27]) Let us consider $1 < b < a < \infty$; then,

Lemma 3.2. The following anisotropic Sobolev inequality holds:

Let $f, g, h\in C_{c}^{\infty }\left(\mathbb{R}^{3}\right)$

Note that this last inequality is employed in $\mathbb{R}^2$ in this analysis.

4.   Proof of Theorem 2.1

For proving Theorem 2.1, the following Propositions are firstly shown

Proposition 1. (General bound and regularity in the first spatial derivative in the flow direction) Given the initial data $u_{0} \in L^2 (\Omega)$, then any solution of Eq $\left(1.2\right)$ satisfies

where $\eta$ depends on a suitable constant to be introduced, namely $C_{5}$. In addition, the bounding constant $\psi$ depends on another constant $C_{6}$ and the existence time $T$.

Proof. Multiplying the Eq $(1.2)$ with $u$ and applying integration by parts, we have

where

For $I_{1},$ we have

Applying integration by parts

where we used Eq $\left(1.1\right).$ After integration, we get $I_{1} = 0.$

For $I_{2},$ we consider the Young inequality, so that:

Introducing the values of $I_{1}$ and $I_{2}$ in Eq $\left(4.1\right)$ we get

where we used Lemma 3.1. Since $\left(\frac{\partial u}{\partial y}, u\right) \in L^{2}(0, T, BMO)$

which implies that

where $v+\frac{1}{\rho \beta d} > \frac{C_{2}}{6\rho \beta d^{3}}+\frac{ C_{4}\phi }{12k\rho \beta d^{3}}$. Therefore

The Gronwall inequality yields to

where $\eta$ depends on $C_{5}$ and $\psi$ depends on $C_{6}$ and $T$.

Proposition 2. (Regularity in the first and second spatial derivatives in the flow direction) Assume that the initial condition satisfies $\Vert \frac{\partial u_{0}}{\partial y} \Vert_{L^2} < \infty$ in $\Omega$ and that any solution of Eq $\left(1.2\right)$ complies with $\left(\frac{\partial u}{\partial y}, \frac{\partial ^{2}u }{\partial y^{2}}\right) \in L^{2}(0, T, BMO)$; then,

where $\eta$ depends on the constant $C_{10}$ (to be introduced) and $\phi$ depends on another constant $C_{11}$ and the time $T,$ such that in the interval $(0, T)$ the solutions are considered to exist.

Proof. Multiplying Eq $(1.2)$ with $-\frac{\partial ^{2}u}{\partial y^{2}}$ and then solving by standard integration, we have

where

and

Applying Lemma 3.1 on Eq $\left(4.2\right),$ we have

Since $\frac{\partial u}{\partial y}\in L^{2}(0, T, BMO),$ the following holds

For $I_{3},$ using integration by parts, we have

From Eq $(1.1)$, the following holds

Integrating the second term on the right hand side with regards to $x$

Applying Young inequality on $I_{4},$

where we used Lemma 2.1$.$ Since $\left(\frac{\partial u}{\partial y}, \frac{\partial ^{2}u}{\partial y^{2}}\right) \in L^{2}(0, T, BMO)$; then,

Introducing the values of $I_{3\text{ }}$and $I_{4}$ in Eq $(4.3)$:

Choosing $v+\frac{1}{\rho \beta d} > \frac{C_{9}}{4\rho \beta d^{3}}$; then,

and for simplicity we can choose the constant value in the right hand side of expression $(4.4)$:

Then, the Eq $\left(4.4\right)$ becomes

Applying the Gronwall inequality,

where $\eta$ depends on $C_{10}$ and $\psi$ depends on $C_{11}$ and $T.$

Finally, the Theorem 2.1 proof is completed by using Propositions 1 and 2, as both propositions provide evidences on the global bound of solutions and regularity of the fluid in the flowing direction in $\Omega \times (0, T]$.

5.   Proof of Theorem 2.2

The Theorem 2.2 provides a general bound and regularity in the first spatial derivative in the flow direction within the space $L^p (\Omega)$, $p > 2$, and locally in the interval $(0, T]$.

To prove such a theorem, we start by multiplying the Eq $(1.2)$ with $\left \vert u\right \vert ^{P-2}u$. After integration, the following holds:

where we used Lemma 3.1. In addition

and

Applying integration by parts on $I_{5},$ we have

By using Eq $(1.1)$, we can write

Integrating again we obtain $I_{5} = 0.$ Applyling integration by parts on $I_{6}$, we have

where we used Holder inequality and Lemma 3.1$.$ Since $\left(u^{\frac{p-2}{ 2}}\frac{\partial u}{\partial y}, \frac{\partial u}{\partial y}\right) \in L^{2}\left(0, T, BMO\right)$ and after using Proposition 2, we get

Introducing $I_{5}$ and $I_{6}$ in Eq $\left(5.1\right),$ we get

which implies that

As $p > 2$ and $v+\frac{1}{\rho \beta d} > C_{14}$; then, $C_{16} = \left[P(P-1)\left(v+\frac{1}{\rho \beta d}\right) -C_{14} \right] > 0$ and $C_{17} = P\left[C_{15}-\frac{\phi }{k}\left(v+\frac{1}{\rho \beta d}\right) \right]$. Consequently, the Eq $\left(5.2\right)$ becomes

The Gronwall inequality yields to

where $\eta$ depend on $C_{16}$ and $\psi$ depend on $C_{17}$ and $T.$

6.   Proof of Theorem 2.3

Firstly, the following Propositions are required.

Proposition 3. (General bound and regularity is the spatial gradient) Assume that the initial condition satisfies $\Vert \nabla u_{0} \Vert _{L^{2}}^{2} < \infty$ in $\Omega$ and for any time in $(0, T]$. Consider that $u$ is a solution to Eq $\left(1.2\right)$ with $\left(\frac{ \partial u}{\partial y}, \nabla u, \frac{\partial }{\partial y}\left(\nabla u\right) \right) \in L^{2}\left(0, T, BMO\right)$; then, $u\left(x, y, t\right)$ satisfies

where $\eta$ depends on other constants, namely $v, \rho, \beta, d, C_{18}$ and $\in$ and the bounding constant $\psi$ depends on $C, \phi, k, v, \rho, \beta, d, C_{19}$ and the time $T,$ such that in the interval $(0, T)$ the solutions are considered to exist.

Proof. Taking the inner product of Eq $(1.2)$ with $\Delta u$ and upon integration

where

Applying integration by parts on $I_{7},$

From Eq $(1.1),$ we have

Integrating the second term on the right hand side, we have

Integrating the third term on the right hand side, the following holds

where we used Eq $(1.1)$. Therefore $I_{7}$ becomes

Applying Young inequality, we have

Integrating $I_{8}$ by parts, we obtain

From Young inequality, we obtain

Since $\frac{\partial \nabla u}{\partial y}\in L^{2}(0, T, BMO)$ and also applying Proposition 1, we can write

Applying integration by parts on $I_{9},$ we have

Integrating again, we have

where we used Holder inequality, Young inequality and Lemma 3.1.

Since $(\nabla u, \frac{\partial ^{2}u}{\partial y^{2}})\in L^{2}(0, T, BM0)$ the following holds

Introducing the values of $I_{7}, I_{8}$ and $I_{9}$ in Eq $\left(6.1\right),$ we get

since $\frac{\partial v}{\partial x}\in L^{2}\left(0, T\right),$ and after using Propositions 1 and 2

Applying Gronwall inequality, we get

where $v+\frac{1 }{\rho \beta d} > \frac{C_{18}}{4\rho \beta d^{3}} > \frac{\in }{2},$ and $\eta$ depends on $v, \rho, \beta, d, C_{18}$ and $\in$ and $\psi$ depends on $C, \phi, k, v, \rho, \beta, d, C_{19}$ and the time $T,$ such that in the interval $(0, T)$ the solutions are considered to exist.

Proposition 4. (General bound and regularity is the spatial laplacian derivatives) Assume that the initial condition satisfies $\left \Vert \Delta u_{0}\right \Vert _{L^{2}}^{2} < \infty$ in $\Omega$. Then, the solution of Eq $\left(1.2\right)$ with $\left(\frac{\partial u}{\partial y}, \frac{\partial ^{2}u}{\partial y^{2}}, \frac{ \partial }{\partial y}\left(\nabla u\right), \nabla u, \Delta u, \frac{ \partial }{\partial y}\left(\Delta u\right) \right) \in L^{2}\left(0, T, BMO\right)$ and $\left(\Delta v. \, \frac{\partial \nabla v}{\partial y}\right)\in L^{2}\left(0, T\right)$ satisfies

where $\eta$, $\kappa$ and $\zeta$ depend on suitable constants to be introduced, and the bounding $\psi$ depends on another constant and the time $T,$ such that in the interval $(0, T)$ the solutions are considered to exist.

Proof. Applying the $\Delta$ operator to Eq $(1.2)$ and testing with $\Delta u$

where

For $I_{10, }$ we have

Applying integration by parts and the continuity relation we obtain

Therefore $I_{10}$ becomes

In order to solve the above named six terms, we use the inequality in Lemma 3.2,

For $k_{2},$ we have

After applying Eq $\left(1.1\right)$ on $k_{3},$we obtain

For $k_{4},$ we have

For $k_{5},$ we have

For $k_{6},$ we have

Introducing the values of $k_{1}, k_{2}, k_{3}, k_{4}, k_{5}$ and $k_{6}$ in Eq $\left(6.3\right) \ $, considering the Propositions 2 and 3, and letting $\left(\Delta v. \, \frac{\partial \nabla v}{\partial y}\right)\in L^{2}\left(0, T\right)$

For $I_{12},$we can write

which implies that

where

In order to solve above pointed terms, we use the Young inequality on $k_{7}$

where we used Lemma 3.1. Since $\frac{\partial \nabla u}{\partial y} \in L^{2}\left(O, T, BMO\right)$ :

Applying Young inequality on $k_{8}$, we get

where we used Lemma 3.1. Since$\left(\frac{\partial \left(\Delta u\right) }{\partial y}, \frac{\partial u}{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$ it holds that

Integrating $k_{9},$we have

Integrating again, we obtain

Using Holder Inequality, we obtain

where we used Young inequality and Lemma 3.1. Since $\left(\frac{\partial \left(\nabla u\right) }{\partial y}, \frac{\partial u}{\partial y}, \frac{ \partial \Delta u}{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$, we can write

For $k_{10},$ we have

Applying Young inequality, we get

where we used Lemma 3.1, since $\left(\frac{\partial u}{\partial y}, \frac{ \partial \left(\Delta u\right) }{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$ :

Introducing the values of $k_{7}, k_{8}, k_{9},$and $k_{10}$ in Eq $\left(6.4\right),$ we get

For $I_{12},$ we have

For $k_{1}^{\prime },$ applying Young inequality, we get

where we used Lemma 3.1$.$ Since $\left(\Delta u, \nabla u\right) \in L^{2}\left(O, T, BMO\right)$:

Integration $k_{2}^{\prime },$ we have

Integrating again, we have

Applying Young inequality, we get

where we used Lemma 3.1$.$ Since $\left(\Delta u, \nabla u\right) \in L^{2}\left(O, T, BMO\right)$:

For $k_{3}^{\prime },$ applying integration by parts

Applying Young inequality

where we used Lemma 3.1$,$ since $\left(u, \Delta u, \frac{\partial \Delta u}{ \partial y}, \frac{\partial \Delta u}{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$:

For $k_{4}^{\prime },$ we have

Applying Holder Inequality, we get

Applying Young inequality, we obtain

where we used Lemma 3.1. Since $\left(u, \nabla u, \Delta u, \frac{\partial \Delta u}{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$:

Integrating $k_{5}^{\prime },$ we obtain

By integrating $k_{6}^{\prime },$ we have

where we used Lemma 3.1. Since $\left(u, \nabla u, \Delta u, \frac{\partial \Delta u}{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$:

For $k_{7}^{\prime },$ we have

By Holder Inequality, we get

Applying Young inequality, we obtain

where we used $\left(1.1\right)$. since $\left(u, , \frac{\partial u}{ \partial y}, \Delta u, \frac{\partial \Delta u}{\partial y}\right) \in L^{2}\left(O, T, BMO\right)$ :

Introducing the values of $k_{1}^{\prime }k_{2}^{\prime }, k_{3}^{\prime }, k_{4}^{\prime }, k_{5}^{\prime }, k_{6}^{\prime },$ and $k_{7}^{\prime }$ and from Propositions 1–3, the Eq $\left(5.5\right)$ becomes

Utilizing the values of $I_{10}, I_{11},$ and $I_{12}$ in Eq $\left(6.1\right),$ and after using the Propositions 1–3

Taking

Then the Eq $\left(6.6\right)$ becomes

The Gronwall inequality yields to

where $\eta$ depends on $C_{50},$ $\kappa$ depends on $C_{51},$ $\zeta$ depends on $C_{52}$ and $\psi$ depends on $C_{53}$ and $T.$

Eventually, the Theorem 2.3 is proved by using Propositions 3 and 4, as both propositions provide evidenced on the global bound of spatial gradients and laplacian derivatives for solutions in $\Omega \times (0, T]$.

7.   Conclusions

The proofs of the proposed Theorems have been provided together with the different supporting propositions required. The Theorems have permitted to show the regularity criteria for different functional spaces conditions. The assessed bounds and regularity results have been applied to an Eyring-Powell fluid in $\Omega = [0, L]\times \lbrack y_m, \infty)$, where $y_m > 0$ is the minimum value of $y$ for which the kinematic Dirichlet conditions of the flow holds, and for $T \in (0, \infty)$, where $T$ refers to the maximal time for existence of solutions.

Acknowledgments

The authors would like to express their gratitude to the anonymous reviewers and the supporting organizations.

Conflict of interest

On behalf of all authors, the corresponding author states that there is no conflict of interest.