Global Existence of Bounded Solutions for Eyring–Powell Flow in a Semi-Inﬁnite Rectangular Conduct

: The purpose of the present study is to obtain regularity results and existence topics regarding an Eyring–Powell ﬂuid. The geometry under study is given by a semi-inﬁnite conduct with a rectangular cross section of dimensions L × H . Starting from the initial velocity proﬁles ( u 01 , u 02 ) in xy -planes, the ﬂuid ﬂows along the z -axis subjected to a constant magnetic ﬁeld and Dirichlet boundary conditions. The global existence is shown in different cases. First, the initial conditions are considered to be squared-integrable; this is the Lebesgue space ( u 01 , u 02 ) ∈ L 2 ( Ω ) , Ω = [ 0, L ] × [ 0, H ] × ( 0, ∞ ) . Afterward, the results are extended for ( u 01 , u 02 ) ∈ L p ( Ω ) , p > 2. Lastly, the existence criteria are obtained when ( u 01 , u 02 ) ∈ H 1 ( Ω ) . A physical interpretation of the obtained bounds is provided, showing the rheological effects of shear thinningand shear thickening in Eyring–Powell ﬂuids.


Introduction
An Eyring-Powell fluid is a sub-class of a non-Newtonian fluid of interest in applied sciences. To cite some examples, we remark the applications in manufacturing engineering [1,2] and biological technology [3,4].
The Eyring-Powell model has been of interest for the description of magnetohydrodynamics (MHD). As representative of previous studies, Akbar et al. [5] carried out the analysis of solutions in a two-dimensional MHD fluid. Hina [6] considered an Eyring-Powell fluid for MHD purposes to study heat-transfer processes. Afterward, Bhatti et al. [7] proposed an analysis for a stretching surface under MHD physical principles. Similarly, other references can be cited describing analyses of Eyring-Powell fluids, combining analytical and numerical approaches, from purely mathematical principles to applications in different physical scenarios [8][9][10][11][12][13][14][15][16][17][18][19].
It shall be noted that there exists much literature dealing with the existence criteria of solutions when a fluid is formulated with the classical Newtonian viscosity involved in the Navier-Stokes equations; see the remarkable studies [20][21][22][23][24][25][26][27][28][29]. Nonetheless, the specific rheological properties of a fluid may lead to the exploration of other kinds of viscosity formulations. One of these formulations, based on the kinetic theory of liquids, led to the mentioned Eyring-Powell fluid. To the best of our knowledge, there is not a wide range of literature dealing with the existence and regularity of solutions in Eyring-Powell fluids in three-dimensional geometry. Consequently, our main objective is to introduce such an analysis under the most general conditions. Considering some recent achievements related to the application of advanced analytical tools to non-Newtonian fluids, we can highlight the recent work of Bilal et al. [30], where the (G /G 2 )−expansion method was employed to obtain exact wave solutions to a Dullin-Gottwald-Holm system. In addition, the solutions to a Korteweg-deVries-Zakharov-Kuznetsov equation were explored in Ref. [31]. Based on a modified extended direct algebraic method, these authors found solutions in the form of solitary, shock, singular, shock singular, solitary shock and double singular solitons. In the present manuscript, we are concerned with the regularity and existence of solutions rather than with the specific forms of such solutions. This, however, establishes a basis for future research topics.
The paper layout is as follows. First, we introduce the framework of our study and describe the Eyring-Powell fluid model. Secondly, a set of three theorems is given so that their proofs permit to draw a conclusion on the regularity and existence of solutions to the proposed Eyring-Powell formulation. The introduced theorems are supported by a number of lemmas that are provided for the sake of clarity and by some propositions that are proved. The involved assessments follow a process that can be introduced sequentially as follows: -Formulation of the involved equations in integral form. -Derivation of a temporal differential equation in terms of spatial distributions in L p (p ≥ 2); the involved integrals are assessed, typically by parts. -Introduction of hypotheses in a space of bounded mean oscillations that assure a bounded solution, and obtain the bounding constants. -Application of the Gronwall theorem for a bound in the temporal law, and under spatial distributions in L p (p ≥ 2).

Model Formulation
We consider a flow of an electrically conducting Eyring-Powell fluid. The selection of this type of fluid is justified based on the following ideas. Firstly, the rheological properties of an Eyring-Powell fluid are derived based on the kinetic theory of liquids, instead of empirical or quasi-experimental principles; this can be the case of a Darcy-Forchheimer or a power-law fluid. Deducing a rheological law from a well-known theory makes the Eyring-Powell fluid interesting for purely mathematical assessments such that the analytical concepts rely on theoretical and well-proven physical aspects. Secondly, the Eyring-Powell rheological properties can be understood as an expansion of a typical linear fluid rheology. Then, the scope of our analysis contains some mathematical ideas that can be applied for the study of simpler rheological laws; this naturally extends to Newtonian fluids described by the classical continuity and momentum Navier-Stokes equations.
The Cartesian coordinates (x, y, z), with the corresponding velocity components V = (u 1 , u 2 , u 3 ), are chosen such that the origin is located in the plane sheet at z = 0. The fluid occupies the region z > 0, and flows from the sheet z = 0 to z → ∞.
The conservation of mass and momentum are described in a general basis as where dV/dt refers to the total derivative of the velocity field, ρ is the fluid density, B is the applied longitudinal (along the z-axis) magnetic field of magnitude B 0 driving the flow, J is the current charges density, and τ refers to the Cauchy stress tensor, which is given by where p is the pressure field in the fluid, I is the identity tensor and τ ij is the stress tensor typical in Eyring-Powell fluid models.
Based on the kinetic theory of liquids [32], a formulation to such stress tensor is where µ is the dynamic viscosity, and β and γ are two characteristic constants related to the fluid spatial behavior and its characteristic relaxation frequency, respectively [33]. By considering Note that we may consider higher-order terms, denoted by '. . . ' in the expression above, when approximating the sinh −1 function, or even other forms of rheological behavior; see the work of Oke [34] for additional insights.
We assume a boundary layer is developed and analyze the velocity profiles in each xy-plane for which the following Dirichlet boundary conditions apply: Based on the exposed arguments and taking L, γ −1 and ρ as characteristic values for length, time and density, the governing equations written in dimensionless parameters read [32] where Γ = H/L refers to the cross-sectional aspect ratio, Re = ργL 2 /µ is the Reynolds number, M = 1/(γβµ) characterizes the rheological behavior of the fluid, and B = σB 2 0 /(ργ) > 0 is the dimensionless effective magnetic field inducing the flow. The kinematic Dirichlet boundary conditions now read u 1 = u 2 = 0 at x = 0, 1 and y = 0, Γ. Note that the own magnetic field generated by the charges motion is assumed to be negligible.

Previous Definitions and Results
Consider the Lebesgue norm · L p to define the functional space L p (Ω). In addition, the usual Sobolev functional space of order m is considered as As it will be specified later, we will establish the regularity criteria if ∂u 1 /∂z 2 BMO , ∂u 2 /∂z 2 BMO , ∇u 1 2 BMO , and ∇u 2 2 BMO are sufficiently small. Note that BMO denotes the homogeneous space of 'bounded mean oscillations' associated with the norm For further details on BMO spaces, we refer the reader to Ref. [35].
In addition, we recall the following two lemmas.
For the proof of Lemma 1, we refer the reader to Ref. [35].

Statement of Results
The main results obtained in this analysis are stated as follows.
In addition, assume that ∂u 1 /∂z 2 BMO and ∂u 2 /∂z 2 BMO are sufficiently small, then system  The proposed theorems are shown in the coming sections.

Proof of Theorem 1
The first intention is to show that the two-dimensional velocity profiles (u 1 , u 2 ) are globally bounded when the fluid is flowing through the z-axis. This means that for any level in the z-axis, the fluid flow exhibits a regular behavior. The following proposition is required to support the proof of Theorem 1.
where C 3 and C 4 are suitable constants related with the set of parameters involved in Equations (3)-(5).

Proof. Multiplying Equation (3) by u 1 and integrating
which implies that where Note that we used Equation (1). Developing further the integration on I 1 : Since u 1 = 0 at x = 0, 1, then I 1 = 0 and Equation (7) becomes where we used Lemma 1. Now, provided that ∂u 1 /∂z 2 BMO is sufficiently small, we can choose and therefore, the above equation becomes which implies that Similarly, multiplying Equation (4) by u 2 and integrating again, we obtain Adding Equations (8) and (9): In most realistic cases, the rheological parameter is small |M| 1, and one can apply the Gronwall inequality to obtain sup 0≤t≤T u 1 2 in [0, T] × Ω, where C 3 , C 4 depend on Re, and M and C 2 should be upper bounded by C 2 < 6(1 + M)/M.
From a physical point of view, this upper bound only applies to shear-thinning fluids with M > 0. This can be understood as a bound for the viscosity reduction that ensures an exponential or sub-exponential decrease in (u 1 , u 2 ) as z → ∞ so that the associated integrals remain finite. For shear-thickening fluids with M < 0, in contrast, the increase in viscosity with the applied shear ensures such exponential (or sub-exponential) decay and, in practice, removes any condition on C 2 . For additional insights about the rheological properties of Eyring-Powell fluids, the reader is referred to Ref. [34].
Note that the Theorem 1 is proved simply using the bound properties shown in Proposition 1.

dzdy.
As u 1 = 0 at x = 0, 1, then I 2 = 0 and Equation (10) simplifies to Since by initial assumption u p−2 2 1 ∂u 1 /∂z 2 BMO is sufficiently small, we can take The application of the Gronwall inequality yields where C 8 and C 9 refer to the Gronwall constants, compiling C 6 and C 7 , that depend on the dimensionless parameters of the problem. Again, C 6 should obey the same upper bound derived for C 2 .
Proceeding similarly, multiplying the Equation (4) by |u 2 | p−2 u 2 , p > 2, we obtain with the same bounds for the involved constants.
We recall the previous discussion about these bounds in the shear-thinning and shearthickening cases.

Proof of Theorem 3
Before showing the Theorem 3, the following proposition is required.
where C 14 is a suitable constant related to the dimensionless parameters involved in the set of Equations (3)-(5) and the BMO bound hypothesis.
Proof. Take the inner product in Equation (3) with ∆u 1 and integrate with regards to the spatial variables to obtain where Using Equation (1), we have where we used Lemma 1.
Then, we arrive to where C 13 = C 12 − B > 0. Similarly, multiplying Equation (4) by ∇u 2 and after integration by parts, we have Adding Equations (11) and (12), Finally, the Gronwall inequality yields where C 14 depends on the dimensionless parameters of the problem.
Theorem 3 is shown by making use of the results obtained in Proposition 1 and Proposition 2.
Compared to the previous results, the unique bound required here is C 12 > B, which can be understood as a bound for the applied magnetic field.

Conclusions
In this paper, we developed the global existence of regular solutions for an Eyring-Powell fluid flowing along a semi-infinite conduct with a rectangular cross-section of dimensions [0, 1] × [0, Γ], subjected to a constant longitudinal magnetic field of (dimensionless) magnitude B. The initial velocity profiles (u 0 1 , u 0 2 ) were given in xy-planes along the z-axis, and the flow developed in the region z > 0. The following results were provided. Firstly, for (u 0 1 , u 0 2 ) ∈ L 2 (Ω), Ω = [0, 1] × [0, Γ] × (0, ∞), a regular global solution was shown to hold. A similar existence result was proved in the case (u 0 1 , u 0 2 ) ∈ L p (Ω), p > 2. Finally, we obtained similar existence criteria for (u 0 1 , u 0 2 ) ∈ H 1 (Ω). The proposed results can be of practical use to support the resolution of the Eyring-Powell fluid with numerical means. Prior to starting any numerical assessment, the regularity of the solutions can be interpreted based on the results outlined in this work. As a future research topic related to the proposed Eyring-Powell fluid, one can consider the possibility of understanding the behavior of the solutions together with their increasing or decreasing rate. A remarkable question to explore is related to the existence of an exponential profile for a special class of solutions known as traveling waves. The fact of having an exponential behavior leads to state the regularity of the solutions, and shall be compliant with the obtained results.