Ricci curvature of semi-slant warped product submanifolds in generalized complex space forms

1.   Introduction

Non-Newtonian fluids are defined as fluids with an extra-tension tensor that cannot be expressed as a linear, isotropic function of the components of the strain rate tensor.One of the goals of asymptotic analysis is to obtain and describe a two-dimensional problem from a three-dimensional problem, passing to the limit on the thickness of the domain assumed to be already thin. In this context, several previous studies have been conducted to deal with this problem.

The first study we mention is what the authors have done in ^[4], where they mainly examine the existence and behavior of weak solutions for a lubrication problem with Tresca law. In another study, the authors in ^[1] gave the nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary. Suárez-Grau in ^[25] studied the asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary. The convergence stability of the solutions for the non-Newtonian fluid motion with large perturbation in $\mathbb{R}^{2}$ has been given in ^[9]. In ^[20], the authors presented an extension of the results related to the solutions of weakly compressible fluids with pressure-dependent viscosity. In contrast, the existence and uniqueness of stationary solutions of non-Newtonian viscous incompressible fluids were obtained in ^[11]. Other contexts and problems be found in the monographs such as in ^[21,27], and in the literature quoted within.

The Herschel-Bulkley fluid is a generalized model of a non-Newtonian fluid. The name is related to Winslow Herschel and Ronald Bulkley ^[15], and it was first mentioned, in 1926, where the relationship between the stress tensor $\sigma^{\varepsilon}$ and the symmetric deformation velocity $d(u^{\varepsilon})$ is given by:

where, $d(u^{\varepsilon}) = \dfrac{1}{2}(\nabla u^{\varepsilon}+\left(\nabla u^{\varepsilon}\right) ^{T})$, $u^{\varepsilon}$ is the velocity field, $\mu > 0$ is the viscosity constant, $\pi ^{\varepsilon }$ is the pressure, $\delta^{\varepsilon}\geq0$ is the yield stress, $1 < r\leq2$ is the power law exponent of the material and $\delta_{ij}$ is the Kronecker symbol.

In this paper, we will adopt the constitutive law by considering that a Herschel-Bulkley incompressible fluid whose viscosity will follow the power law with a liquid-solid friction condition of Coulomb in three-dimensional domain $\mathbb{Q}^{\varepsilon}\subset$ $\mathbb{R}^{3}$.

The Herschel-Bulkley fluid has been studied intensively by mathematicians, physicists, and engineers as intensively as the Navier-Stokes. For example, we mention the studies carried out in the fields of metal fluxes, plastic solids and some polymers. The literature concerning this topic is extensive; see e.g. ^[24,26] and many others references. More recently, the authors in ^[17], have studied the two-dimensional slow flow of non-Newtonian fluids of the Herschel-Bulkley type an inclined plane. In the context of the Bingham fluid, $r = 2$, the authors in ^[8,22] proved the asymptotic convergence of this fluid in the isothermal and non-isothermal case with non linear friction law. In the case $\delta ^{\varepsilon} = 0$, with the particular conditions of Tresca, this problem has been studied by ^[5,6] respectively in both non-isothermal and isothermal study cases. Benseridi et al. in ^[3] studied theasymptotic analysis of a contact between two general Bingham fluids, however Saadallah et al. in ^[23] studied the analog of the problem presented in this work but in thevery particular case where the velocity on the surface $\Gamma_b$ is null with the friction of the Tresca type. We can also mention others studies where authors gave the numerical solutions of the Herschel-Bulkley fluid but in other particular cases (see ^{[14,16,18,19]}).

In this study, the objective is to make an extension of our previous works ^[8,22,23] and to improve the result obtained in ^[5,6].

The novelty of our study can be summarized in following two major points. First, we take into account a generalized model of a non-Newtonian fluid ($1 < r\leq2$ and $\delta^{\varepsilon}\neq0$). Second, we choose the Coulomb friction with the velocity of the lower surface $\Gamma_b$ different to zero, since all previously mentioned works were restricted only to the particular friction of Tresca.

From our side, this choice will cause different difficulties in other parts of the study, especially with regard to Lemma 5.1, Theorems 4.2–4.4 and the uniqueness theorem.

Accordingly, this work makes the following new contributions by finding solutions to these problems:

The first contribution consists of finding the solution for the first difficulty coming from the fact that the integral on $\Gamma_b$ has no clear meaning. In our study, we will replace the normal stress by some regularization as in ^[10]. The second contribution consists of dealing with the problem of choosing the test functions. In fact, we cannot choose the test functions as it was done in ^[5,6], their work does not contain the yield stress $\delta ^{\varepsilon}$.

This remaining of our paper is organized as follows: Section 2 will summarize the description of the problem and the basic equations. Moreover, we introduce some notations and preliminaries that will be used in other sections. Section 3 will be reserved to the proof of the related weak formulation. We will also discussing the problem in transpose form. The corresponding main convergence results will be stated in Theorems $(4.j)$, $j = 1$ to $5$ of Section 4. The mathematical proofs will be presented in Section 5.

2.   Description of the problem and basic equations

We start by introducing some notations used in the paper. Motivated by lubrication problems, we consider:

the domain of the flow, where $\Gamma_b$ is a non-empty bounded domain of $\mathbb{R} ^{2}$ with a Lipschitz continuous boundary, $h\left(.\right)$ is a Lipschitz continuous function defined on $\Gamma_b$ such that $0 < h_{\star }\leq h(y')\leq h^{\star}$, for all ($y', 0$) in $\Gamma_b$ and $\varepsilon$ is a small parameter that will tend to zero.

We decompose the boundary of $\mathbb{Q}^{\varepsilon}$ as $\Gamma^{\varepsilon } = \overline{\Gamma }_{u}^{\varepsilon}\cup \overline{\Gamma}_{l}^{\varepsilon }\cup \overline{\Gamma}_b$ with

where ${\Gamma}_b$ is the bottom of the domain, $\Gamma _{u}^{\varepsilon }$ is the upper surface and $\Gamma _{l}^{\varepsilon }$ the lateral part of $\Gamma ^{\varepsilon }$. Let $u^{\varepsilon }(y):\mathbb{Q} ^{\varepsilon }\rightarrow \mathbb{R}^{3}$ be the velocity and $\pi ^{\varepsilon }\left(y\right) :\mathbb{Q} ^{\varepsilon }\rightarrow \mathbb{R}$ the pressure of the fluid. We denote by $\eta = (\eta_{1}, \eta_{2}, \eta_{3})$ the unit outward normal to the boundary $\Gamma ^{\varepsilon }$, and we define the normal and tangential velocities of $u^{\varepsilon }$ on $\Gamma^{\varepsilon }$ as:

Similarly, for a regular tensor field $\sigma ^{\varepsilon }$, we denote by $\sigma _{\eta}^{\varepsilon }$ and $\sigma _{\tau }^{\varepsilon }$ the normal and tangential components of $\sigma ^{\varepsilon }$ given by

Let $\mathbb{S}$ be denotes the set of all symmetric $3\times 3$ matrices and for $\eta, \zeta \in \mathbb{S}$, we define the scalar product and the corresponding norm by

The boundary-value problem describing the stationary flow for generalized non-Newtonian and incompressible fluid is described by:

Problem$\mathcal{P}^{\varepsilon}$. Find the pressure $\pi ^{\varepsilon }:\mathbb{Q}^{\varepsilon}\rightarrow \mathbb{R}$ and a velocity field $u^{\varepsilon}:\mathbb{Q}^{\varepsilon}\rightarrow \mathbb{R} ^{3 }\ $ such that

where, $f^{\varepsilon} = (f_{i}^{\varepsilon})_{1\leq i\leq3}$ is the body forces, $s$ is the velocity of the bottom boundary $\Gamma_b$. Furthermore, the Eq $(2.1)$ represents the law of conservation of momentum. Relation $(2.2)$ gives the law of behavior of the Herschel-Bulkley fluid. The formula $(2.3)$ represents the incompressibility equation. Equations $(2.4)$ and $(2.5)$ represent the velocity on $\Gamma_{u} ^{\varepsilon}$ and $\Gamma_{l}^{\varepsilon}$ respectively. On the other hand, Eq $(2.6)$ justified the no-flux through on $\Gamma_b$. However, assuming that the friction is sufficiently large, the tangential velocity is unknown and satisfies the Coulomb boundary condition $(2.7)$ on the part $\Gamma_b$, with $k^{\varepsilon }$ is the friction coefficient. This law introduced by ^[2] is one of the most spread laws in mathematics and it is more realistic than the law of Tresca.

Suppose that the function $g = (g_{i})_{1\leq i\leq3}$ is in $\left(W^{1-1/r, r}(\Gamma^{\varepsilon })\right) ^{3}$, the space of traces of functions from $\left(W^{1, r}(\mathbb{Q}^{\varepsilon})\right) ^{3}$ on $\Gamma^{\varepsilon}$ which will define in the next section. Due to ${ \int_{\Gamma^{\varepsilon}}} g.nd\sigma = 0$ that there exists a function $G^{\varepsilon}$ (^[10]):

Also, we suppose that $g_{3} = 0$ on $\Gamma^{\varepsilon}$ and $g = s$ on $\Gamma_b$.

3.   Study of the weak solution

3.1.   Functional framework and regularization of $\sigma_{\eta }^{\varepsilon}$

Before starting this study, we need to introduce the functional framework and the functional spaces that we use in the rest of this work: Let $L^{r}(\mathbb{Q}^{\varepsilon})$ represents the Lebesgue space for the norm $\Vert.\Vert_{L^{r}(\mathbb{Q}^{\varepsilon})}$ and $W^{1, r} (\mathbb{Q}^{\varepsilon})$ are the standard Sobolev spaces given by

for $1 < r < \infty$, and $W_{0}^{1, r}(\mathbb{Q}^{\varepsilon})$ is the closure of $\mathcal{D}(\mathbb{Q}^{\varepsilon})$ in $W^{1, r}(\mathbb{Q} ^{\varepsilon})$. We denoted by $W^{-1, q}(\mathbb{Q}^{\varepsilon})$ the dual space of $W_{0}^{1, r}(\mathbb{Q}^{\varepsilon})$, where $r^{-1}+q^{-1} = 1$.

Moreover, we need the following functional spaces

Assume that the problem $(\mathcal{P}^{\varepsilon})$ admits a solution denoted by $(u^{\varepsilon}, \pi ^{\varepsilon })$, with sufficient regularity. Multiplying $(2.1)$ by $(v-u^{\varepsilon})\in E^{\varepsilon}$ and then using Green's formula, along with the boundary conditions (2.4)–(2.7), we obtain:

Problem$\mathcal{P}_{\mathcal{K}}^{\varepsilon}$. We are looking for the velocity $u^{\varepsilon}\in E_{div}^{\varepsilon}$ and $\pi ^{\varepsilon }\in E_{0}^{q}(\mathbb{Q}^{\varepsilon})$, which verify:

where

The integral $\widetilde{j}(u^{\varepsilon}, v)$ has no meaning for $u^{\varepsilon}\in E^{\varepsilon}$. Indeed, $\sigma_{\eta}^{\varepsilon}$ is defined by duality as an element of $W^{-\frac{1}{2}, r}(\Gamma_b)$ and $|\sigma_{\eta}^{\varepsilon}|$ is not well defined on $\Gamma_b$. So following ^[10], we replace $\sigma_{\eta}^{\varepsilon}$ by some regularization $R(\sigma_{\eta}^{\varepsilon})$, where $R$ is a regularization operator from $W^{-\frac{1}{2}, r}(\Gamma_b)$ into $L^{r}(\Gamma_b)$ can be obtained by convolution with a positive regular function and defined by

$\phi$ is a given positive function of class $C^{\infty}$ with compact support in $\Gamma_b$ and $W^{-\frac{1}{2}, r}(\Gamma_b)$ is the dual space to $W_{00}^{\frac{1}{2}, r}(\Gamma_b) = \{v_{|_{\Gamma_b}}:v\in W^{1, r}(\mathbb{Q} ^{\varepsilon}), v = 0\, \ $on$\, \ \Gamma_{u}^{\varepsilon}\cup \Gamma _{l}^{\varepsilon}\}$.

After the regularization, we get the new problem:

Problem$\mathcal{P}_{\mathcal{K}}^{\varepsilon, r}$. Find $\left(u^{\varepsilon}, \pi ^{\varepsilon }\right) \in E_{div}^{\varepsilon}$ $\times E_{0}^{q}(\mathbb{Q}^{\varepsilon})$, provided it verifies the problem:

where

Remark 3.1. If $v \in E_{div}^{\varepsilon}$ the inequality $(3.7)$ becomes

Theorem 3.1. For $f^{\varepsilon}\in L^{q}(\mathbb{Q}^{\varepsilon})^{3}$ and $k^{\varepsilon} > 0$ in $L^{\infty }(\Gamma_b)$; then the problem $\mathcal{P}_{\mathcal{K} }^{\varepsilon, r}$ admits a unique pair $(u^{\varepsilon }, \pi ^{\varepsilon })\in E_{div}^{\varepsilon}\times E_{0}^{q}(\mathbb{Q} ^{\varepsilon})$ verifying $(3.7)$. Moreover, for a small value of the friction threshold $k^{\varepsilon}$, this solution becomes unique.

Proof. To show the existence and uniqueness result of (3.7), we define the following intermediate problem:

where, $Y$ defined from $L^{r}(\Gamma_b)$ into $L^{r}(\Gamma_b)$ as: $Y\rightarrow -k^{\varepsilon}R(\sigma_{\eta}^{\varepsilon})$ and

By the analog of the techniques used in ^[16], it is easy to see that $F(u^{\varepsilon}, v-u^{\varepsilon})$ is bounded coercive hemicontinuous and strictly monotone.

$Y+\delta_{E_{\operatorname{div}}^{\varepsilon}}$ is a proper, convex and continuous function on $L^{r}(\Gamma_b)$, then by Tichovo's fixed point theorem (as in ^[7]), we ensure the existence of a unique $u^{\varepsilon}$ $\in E_{\operatorname{div}}^{\varepsilon}$ verifying the variational inequality $(3.9)$. The existence of the pressure $\pi ^{\varepsilon }\in E_{0}^{q}(\mathbb{Q} ^{\varepsilon})$ such that $(u^{\varepsilon}, \pi ^{\varepsilon })$ satisfy (3.7) is found in ^[12].

3.2.   Problem in transpose form

In this subsection, we use the dilatation in the variable $y_3$ given by $y_{3} = z{\varepsilon }$, then our problem will be defined on a domain $\mathbb{Q}$ does not depend on $\varepsilon$ given by:

and its boundary $\Gamma = \overline{\Gamma}_u\cup\overline{\Gamma}_l\cup\overline{\Gamma}_b$.

After this change of scale following the third component, it is normal to give the new functions and the new data defined on the new fixed domain $\mathbb{Q}$:

with all the new notations given in $(3.11)$ and $(3.12)$ do not depend on $\varepsilon$.

Also, we denote by:

where the condition $(3.13)$ is given by

Finally, the Banach space $\Theta _{z}$ and its linear subspace $\widetilde\Theta _{z}$ are denoted by:

with the norm of $\Theta _{z}$ is given as follows:

By introducing all these new notations into the variational inequality $(3.7)$, and then multiplying all the terms deduced by $\varepsilon^{r-1}$ after this scaling, then the problem $\mathcal{P}_{\mathcal{K}}^{\varepsilon, r}$ takes the following form:

Problem $\mathcal{P}_{\mathcal{K}}$. Find $\left(\hat {u}^{\varepsilon}, \hat\pi ^{\varepsilon }\right) \in E_{div}$ $\times E_{0} ^{q}(\mathbb{Q})$, such that

where

We introduce some results found in ^[4] which we will need to use in the rest of this work.

4.   Main convergence results

The convergence results of $\left(\hat{u}^{\varepsilon}, \widehat{\pi}^{\varepsilon }\right)$ towards $(u^{\star}, \pi^{\star})$ as well as the limit problem independently of the parameter $\varepsilon$ will be given in the next of this subsection.

Theorem 4.1. Assume that the assumptions of Theorem 3.1 hold, there exist $\pi^{\star}\in E_{0}^{q}(\mathbb{Q})$ and $u^{\star} = \left(u_{1}^{\star}, u_{2}^{\star}\right) \in \widetilde\Theta _{z}$ satisfy the following convergences:

Theorem 4.2. With the same assumptions as Theorem 4.1, the pair $\left(u^{\star}, \pi^{\star}\right)$ satisfies:

Theorem 4.3. Suppose that the assumptions of the previous theorem hold, and if $\left \vert \dfrac{\partial u^{\star}}{\partial z}\right \vert \neq0$, the solution $(u^{\star}, \pi^{\star})$ satisfies

Theorem 4.4. Suppose that the assumptions of Theorem 4.2 hold, then $\tau^{\star}$, $s^{\star}$ satisfy the inequality:

and the limit form of Coulomb law:

Also, the solution $(u^{\star}, \pi^{\star})$ satisfies the weak generalized form:

where

Theorem 4.5. For $\hat{f}\in L^{q}(\mathbb{Q})^{3}$ and $\hat{k} > 0$ in $L^{\infty }(\Gamma_b)$; there exists $\overline{k} > 0$ sufficiently small such that for $\Vert \widehat{k} \Vert_{L^{\infty}(\Gamma_b)}\leq \overline{k}$, the solution $\left(u^{\star}, \pi^{\star}\right)$ of the limiting problem $\left(4.8\right)$ is unique in $\widetilde\Theta _{z}\times \left(E_{0}^{q}\left(\Gamma_b \right) \cap W^{1, q}\left(\Gamma_b \right) \right) ^{2}$.

5.   Proofs of the main results

Proof of Theorem 4.1. Before starting the proof of this theorem, we need the following estimates which can be considered as the key that allows us to make a passage to the limit when $\varepsilon$ tends to zero.

Lemma 5.1. Assume that $f^{\varepsilon }\in L^{q}\left(\mathbb{Q} ^{\varepsilon }\right) ^{3}$ and let $\left(u^{\varepsilon }, \pi ^{\varepsilon }\right) \in E^{\varepsilon }_{\operatorname{div} }\times E_{0}^{q}\left(\mathbb{Q}^{\varepsilon } \right)$ be a solution of $\mathcal{P}_{\mathcal{K} }^{\varepsilon, r}$, where the friction coefficient $k^{\varepsilon } > 0$ in $L^{\infty }\left(\Gamma_b \right)$. Then there exists a constant $C$ independent of $\varepsilon$ such that

Proof of Lemma 5.1. Choosing $v =$ $G^{\varepsilon}$ in $(3.8)$ and using the fact that $G^{\varepsilon} = s$ on $\Gamma_b$, we find

By applying Korn's inequality, we ensure the existence of a constant $C_{K} > 0$ that does not depend on $\varepsilon$ with:

Now we apply Hölder's inequality and then Young's, the increase of the first term of $(5.4)$ is given by

By $(3.15)$, the inequality $(5.6)$ becomes

We apply $(3.16)$ and $(3.17)$, we obtain the analogue of $(5.7)$

Now, from $(5.4)$–$(5.9)$, we obtain

We multiply $(5.10)$ by $\varepsilon^{r-1}$ then using the fact that

and

for $i = 1, 2$, we deduce $(5.1)$ with

For get the estimate $\left(5.2\right)$, we choose in $(3.14),$ $\hat{v} = \hat{u}^{\varepsilon}+\phi$, with $\phi \in W_{0}^{1, r}\left(\mathbb{Q} \right) ^{3}$, we find

then

as

we obtain

As

By Hölder's inequality, we get

We apply the results of $(5.1)$, we have:

The same, we choose in $(3.14)$: $\hat{v} = \hat {u}^{\varepsilon}-\phi, \phi \in W_{0}^{1, r}\left(\mathbb{Q} \right) ^{3}$, we obtain

From $(5.12)$ and $(5.13)$, we deduce

Choosing $\phi = \left(\phi_{1}, 0, 0\right)$ then $\phi = \left(0, \phi_{2}, 0\right)$, in $(5.14)$, we find

Then $(5.2)$ follows for $i = 1, 2$.

For get $(5.3)$, we take in the inequality $(5.14)$, $\phi = \left(0, 0, \phi_{3}\right),$ we find

Which completes the proof of Lemma 5.1.

Now, the convergence $(4.1)$–$(4.6)$ of Theorem 4.1 are a direct result of inequalities $(5.1)$–$(5.3)$. Indeed, by $(5.1)$, $\exists C > 0$ not related to $\varepsilon$, and verifying

It is clear that $(4.1)$ deduces directly from $(5.15)$ and the using of the Poincaré's inequality in the fixed domain $\mathbb{Q}$. Also $(4.2)$–$(4.4)$ follows from $(5.1)$. The obtaining of $(4.5)$ is done as in ^[6]. Finally, it is easy $(4.6)$ follows from $(5.2)$ and $(5.3)$.

In order to proceed to the proof of strong convergence $(4.7)$ of Theorem 4.2, it suffices to demonstrate the strong convergence of the integral term defined on $\Gamma_b$.

Lemma 5.2. Let $R$ is a regularization operator from $W^{-\frac{1}{2}, r}(\Gamma_b)$ into $L^{r}(\Gamma_b)$, then the choice of $R$ ensures the existence of a subsequence of $R(\widehat{\sigma}_{\eta}^{\varepsilon}\left(\hat{u}^{\varepsilon }, \widehat{\pi}^{\varepsilon}\right))$ strongly converges to $R\left(-\pi^{\star}\right)$ in $L^{r}\left(\Gamma_b \right)$.

Proof of Lemma 5.2. From the equilibrium Eq $(2.1)$, we have

with $f^{\varepsilon}\in \left(L^{q}\left(\mathbb{Q} \right) \right) ^{3}$. By the results of Theorem 4.1, we deduce that $\left(\hat {u}^{\varepsilon}, \widehat{\pi}^{\varepsilon}\right)$ are bounded in $\widetilde\Theta _{z}$ $\times E_{0}^{q}(\mathbb{Q})$, then $\widehat {\sigma}^{\varepsilon}$ is bounded in

which shows that there exists a subsequence converging weakly towards $\sigma^{\star}$. Now, we show that $\widehat{\sigma}_{\eta} (\widehat{u}^{\varepsilon}, \widehat{\pi}^{\varepsilon})$ converges weakly to $\left(-\pi^{\star}\right)$ in $W^{\frac{-1}{2}, r}(\Gamma_b)$.

Indeed, as $\sigma_{\eta}^{\varepsilon} = \sigma_{ij}^{\varepsilon}\eta_{i}\eta_{j},$ $1\leq i, j\leq3$, we have

Since $\widehat{\sigma}^{\varepsilon}$ is bounded in $H_{div} (\mathbb{Q})$, then there exists a subsequence converging weakly towards $\sigma^{\star}$ in $H_{div}(\mathbb{Q})$. Using the fact that the trace operator is continuous from $H_{div}(\mathbb{Q})$ into $W^{\frac{-1}{2}, r}(\Gamma_b)$, we therefore obtain the weakly convergence of $\widehat {\sigma}_{\eta}(\widehat{u}^{\varepsilon}, \widehat{\pi}^{\varepsilon})$ to $\widehat{\sigma}_{\eta}(u^{\star}, \pi^{\star})$ in $W^{\frac{-1}{2}, r}(\Gamma_b)$. We apply now the results of Theorem 4.1 in the formula of $\widehat{\sigma}_{\eta}(\widehat{u}^{\varepsilon}, \widehat{\pi}^{\varepsilon})$, we obtain the desired result.

For the rest of proof, using the same techniques as in ^[2,13], we get the result.

Proof of Theorem 4.2. For $u^{\varepsilon}$ the solution on $(3.8)$, we obtain for $v \in E_{div}^{\varepsilon }$

Using the inequality as (^[24])

and by using the Korn's inequality, we find

We multiply the last formula by $\varepsilon^{r-1}$, as well as the convergence of Theorem 4.1, we get in the fixed domain $\mathbb{Q}$

We pose, $\overline{u}^{\varepsilon} = (\hat{u}_{1}^{\varepsilon}, \hat{u}_{2}^{\varepsilon}),$ $u^{\star} = (u_{1}^{\star}, u_{2}^{\star}),$ $\overline{v} = (\widehat{v}_{1}, \widehat{v}_{2})$, so $\overline{v}\in \widetilde{\Xi}(E)$ and

Consequently,

for $\varepsilon < \varepsilon(\eth)$, where $\eth > 0$ is arbitrary.

Therefore, $\exists \overline{v}\in \widetilde{\Xi}(E):\overline{v}\rightarrow u^{\star}$ in $\widetilde\Theta _{z}$, which gives

Now, since $\lim \inf \widehat{j}\left(\overline{u}^{\varepsilon }\right) \geq \widehat{j}\left(u^{\star}\right)$, we deduce: $\overline{u}^{\varepsilon}\rightharpoonup u^{\star}$ in $\widetilde\Theta _{z}$. Furthermore, $\widehat{j}\left(\overline{u}^{\varepsilon }, \overline{u}^{\varepsilon}\right) \rightarrow \widehat{j}(\overline {u}^{\varepsilon}, u^{\star})$ for $\varepsilon \rightarrow0$, which gives the convergence $(4.7)$.

If $r = 2$, we follow the same techniques but $(5.16)$ we will be replaced by

For the proof of the inequality $(4.8)$, we introduce in $(3.14)$ the condition of incompressibility of the fluid ($\operatorname{div}\left(\hat{u}^{\varepsilon}\right)$ $= 0$ in $\mathbb{Q}$), then by the application of Minty's Lemma, we deduce:

We apply the convergence of Theorem 4.1, Lemma 5.2 and the fact $\widehat{j}$ is convex and lower semi-continuous, we obtain

From [5,Lemma 5.1$]$, $\pi^{\star}$ independent of $z$, then applying Minty's lemma for the second time, we deduce $(4.8)$.

Proof of Theorem 4.3. Choosing $\hat{v}$ in $(4.8)$ (as in ^[3]) by: $\hat{v}_{i} = u_{i}^{\star }+\phi_{i},$ $i = 1, 2$, with $\phi_{i}\in W_{0}^{1, r}(\mathbb{Q})$, we find

By using Green's formula, and choosing in the first step $\phi_{1}$ $= 0$ and $\phi_{2}$ $\in W_{0}^{1, r}(\mathbb{Q})$ then reversing this choice in the second step, we find $(4.9)$.

Now, for the prove of $(4.10)$, we cannot choose the test function as in ^[5,6], since their works do not contain the term $\hat{\delta}\frac{\sqrt{2}}{2}\int_{\mathbb{Q}}\partial \hat{v}/\partial z$ $dy'dz$. For this, we use the following techniques. Firstly, we choose $\hat{v}$ in $(4.8)$ by $v = u^{\star}+\lambda \phi$ then $v = u^{\star}-\lambda \phi$, $\phi \in W_{0}^{1, r}\left(\mathbb{Q} \right) ^{2}$, we obtain

Secondly, dividing $\left(5.18\right) \;$and $\left(5.19\right)$ by $\lambda$ and the passage to the limit when $\lambda$ tends to zero, we find

So the last two formulas, we give:

By the Green's formula, we get $(4.10)$.

Proof of Theorem 4.4. We take in $(4.8)$, $\hat{v} _{i} = u_{i}^{\star}+\lambda \phi_{i}$ for $i = 1, 2$, where $\phi_{i}\in W_{\Gamma_{u}\cup \Gamma_{l}}^{1, r}\left(\mathbb{Q} \right)$ and

then

Dividing the last inequality by $\lambda$ and the passage to the limit when $\lambda$ tends to zero, we find

Finally, using the Green formula in $(5.23)$ and from $(4.10)$, we find

This last formula holds for any $\phi \in D(\Gamma_b)^{2}$, but given the density of $D(\Gamma_b)$ in $L^{r}(\Gamma_b)$, we find the desired result $(4.11)$. For the proof of $(4.12)$, we follow the same techniques as in ^[4].

To establish $\left(4.13\right)$, we integrate twice $(4.10)$ from $0$ to $z$, we get

Substituting $z$ by $h$ in $(4.24)$, we get

We integrate $(5.24)$ from $0$ to $z$, it comes:

From $(5.25),$ we deduce

By $(5.26)$ and $(5.27)$, we deduce $\left(4.13\right)$.

Proof of Theorem 4.5. Suppose that the boundary value problem $(4.8)$ admits two solutions which we denote by $\left(u^{\star, 1}, \pi^{\star, 1}\right)$ and $\left(u^{\star, 2}, \pi^{\star, 2}\right)$. Taking $\hat{v} = u^{\star, 2}$ and $\hat{v} = u^{\star, 1}$ respectively, as test function in $\left(4.8\right)$ then by summing two inequalities, we get

We apply $(5.16)$ and $(5.17)$, we obtain

By the inequality $(3.16)$, then we apply the Hölder inequality on the second term of $(5.29)$, we have

whence

Using the fact that $R$ is a linear continuous operator $W^{-\frac{1}{2}, r}(\Gamma_b)$ into $L^{r}(\Gamma_b)$, there exists a constant $C_{1}$ depending on $R$, such that

Combining $(5.30)$ and $(5.31)$ we deduce that if $\Vert \widehat{k} \Vert_{L^{\infty}(\Gamma_b)}\leq \overline{k}$ for sufficiently small $\overline{k}$, then we have

Using Poincaré's inequality, we get

The uniqueness of the $\pi ^{\ast }$ in the $E_{0}^{q}\left(\Gamma_b \right)$ follows from $(4.13)$, in fact we take first in the Reynolds equation $(4.13)$ the pressure value $\pi ^{\ast } = \pi^{\star, 1}$ then $\pi ^{\ast } = \pi^{\star, 2}$ respectively, at the end by subtracting the equations obtained, it becomes:

Choosing $v = \pi^{\star, 1}-\pi^{\star, 2}$, and by Poincaré's inequality, we find

This ends the proof of the Theorem 4.5.

6.   Conclusions

The aim of this study is to examine the strong convergence of the velocity of a non-Newtonian incompressible fluid whose viscosity follows the power law with Coulomb friction, where we give in a first step the description of the problem and basic equations. Then, we present the functional framework. The following paragraph is reserved for the main convergence results. Finally, we give the detail of the proofs of these results. In the future work we will extend and develop our work to new space.

Acknowledgements

Researchers would like to thank the Deanship of Scientific Research, Qassim University for funding publication of this project.

Conflict of interest

The authors declares that they have no conflicts of interest.