On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film

Hana Taklit Lahlah; Hamid Benseridi; Bahri Cherif; Mourad Dilmi; Salah Boulaaras; Rabab Alharbi; Hana Taklit Lahlah; Hamid Benseridi; Bahri Cherif; Mourad Dilmi; Salah Boulaaras; Rabab Alharbi

doi:10.3934/math.2023635

AIMS Mathematics

2023, Volume 8, Issue 6: 12637-12656. doi: 10.3934/math.2023635

Previous Article Next Article

Research article

On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film

1.
Applied Mathematics Laboratory, Department of Mathematics, Faculty of Sciences, University of Ferhat ABBAS- Sétif 1, 19000, Algeria
2.
Department of Mathematics, College of Science and Arts at ArRass, Qassim University, Qassim, Saudi Arabia

Received: 12 December 2022 Revised: 02 March 2023 Accepted: 06 March 2023 Published: 29 March 2023
MSC : 35R35, 76F10, 78M35, 35B40, 35J85

The goal of this paper is to examine the strong convergence of the velocity of a non-Newtonian incompressible fluid whose viscosity follows the power law with Coulomb friction. We assume that the fluid coefficients of the thin layer vary with respect to the thin layer parameter $\varepsilon$ . 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.

Keywords:

Citation: Hana Taklit Lahlah, Hamid Benseridi, Bahri Cherif, Mourad Dilmi, Salah Boulaaras, Rabab Alharbi. On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film[J]. AIMS Mathematics, 2023, 8(6): 12637-12656. doi: 10.3934/math.2023635

Related Papers:

[1]	P. Pirmohabbati, A. H. Refahi Sheikhani, H. Saberi Najafi, A. Abdolahzadeh Ziabari . Numerical solution of full fractional Duffing equations with Cubic-Quintic-Heptic nonlinearities. AIMS Mathematics, 2020, 5(2): 1621-1641. doi: 10.3934/math.2020110
[2]	SAIRA, Wenxiu Ma, Suliman Khan . An efficient numerical method for highly oscillatory logarithmic-algebraic singular integrals. AIMS Mathematics, 2025, 10(3): 4899-4914. doi: 10.3934/math.2025224
[3]	Kai Wang, Guicang Zhang . Curve construction based on quartic Bernstein-like basis. AIMS Mathematics, 2020, 5(5): 5344-5363. doi: 10.3934/math.2020343
[4]	Taher S. Hassan, Amir Abdel Menaem, Hasan Nihal Zaidi, Khalid Alenzi, Bassant M. El-Matary . Improved Kneser-type oscillation criterion for half-linear dynamic equations on time scales. AIMS Mathematics, 2024, 9(10): 29425-29438. doi: 10.3934/math.20241426
[5]	Dexin Meng . Wronskian-type determinant solutions of the nonlocal derivative nonlinear Schrödinger equation. AIMS Mathematics, 2025, 10(2): 2652-2667. doi: 10.3934/math.2025124
[6]	Samia BiBi, Md Yushalify Misro, Muhammad Abbas . Smooth path planning via cubic GHT-Bézier spiral curves based on shortest distance, bending energy and curvature variation energy. AIMS Mathematics, 2021, 6(8): 8625-8641. doi: 10.3934/math.2021501
[7]	Chunli Li, Wenchang Chu . Remarkable series concerning $\binom{3n}{n}$ and harmonic numbers in numerators. AIMS Mathematics, 2024, 9(7): 17234-17258. doi: 10.3934/math.2024837
[8]	Beatriz Campos, Alicia Cordero, Juan R. Torregrosa, Pura Vindel . Dynamical analysis of an iterative method with memory on a family of third-degree polynomials. AIMS Mathematics, 2022, 7(4): 6445-6466. doi: 10.3934/math.2022359
[9]	A. Palanisamy, J. Alzabut, V. Muthulakshmi, S. S. Santra, K. Nonlaopon . Oscillation results for a fractional partial differential system with damping and forcing terms. AIMS Mathematics, 2023, 8(2): 4261-4279. doi: 10.3934/math.2023212
[10]	Tongzhu Li, Ruiyang Lin . Classification of Möbius homogeneous curves in $\mathbb{R}^4$ . AIMS Mathematics, 2024, 9(8): 23027-23046. doi: 10.3934/math.20241119

Abstract

1. Introduction

We consider the following family of nonlinear oscillators

$\begin{equation} y_{zz}+k(y)y_{z}^{3}+h(y)y_{z}^{2}+f(y)y_{z}+g(y) = 0, \end{equation}$

(1.1)

where $k$ , $h$ , $f\neq0$ and $g\neq0$ are arbitrary sufficiently smooth functions. Particular members of (1.1) are used for the description of various processes in physics, mechanics and so on and they also appear as invariant reductions of nonlinear partial differential equations ^[1,2,3].

Integrability of (1.1) was studied in a number of works ^{[4,5,6,7,8,9,10,11,12,13,14,15,16]}. In particular, in ^[15] linearization of (1.1) via the following generalized nonlocal transformations

$\begin{equation} w = F(y), \quad d\zeta = (G_{1}(y)y_{z}+G_{2}(y))dz. \end{equation}$

(1.2)

was considered. However, equivalence problems with respect to transformations (1.2) for (1.1) and its integrable nonlinear subcases have not been studied previously. Therefore, in this work we deal with the equivalence problem for (1.1) and its integrable subcase from the Painlevé-Gambier classification. Namely, we construct an equivalence criterion for (1.1) and a non-canonical form of Ince Ⅶ equation ^[17,18]. As a result, we obtain two new integrable subfamilies of (1.1). What is more, we demonstrate that for any equation from (1.1) that satisfy one of these equivalence criteria one can construct an autonomous first integral in the parametric form. Notice that we use Ince Ⅶ equation because it is one of the simplest integrable members of (1.1) with known general solution and known classification of invariant curves.

Moreover, we show that transformations (1.2) preserve autonomous invariant curves for equations from (1.1). Since the considered non-canonical form of Ince Ⅶ equation admits two irreducible polynomial invariant curves, we obtain that any equation from (1.1), which is equivalent to it, also admits two invariant curves. These invariant curves can be used for constructing an integrating factor for equations from (1.1) that are equivalent to Ince Ⅶ equation. If this integrating factor is Darboux one, then the corresponding equation is Liouvillian integrable ^[19]. This demonstrates the connection between nonlocal equivalence approach and Darboux integrability theory and its generalizations, which has been recently discussed for a less general class of nonlocal transformations in ^[20,21,22].

The rest of this work is organized as follows. In the next Section we present an equivalence criterion for (1.1) and a non-canonical form of the Ince Ⅶ equation. In addition, we show how to construct an autonomous first integral for an equation from (1.1) satisfying this equivalence criterion. We also demonstrate that transformations (1.2) preserve autonomous invariant curves for (1.1). In Section 3 we provide two examples of integrable equations from (1.1) and construct their parametric first integrals, invariant curves and integrating factors. In the last Section we briefly discuss and summarize our results.

2. Main results

We begin with the equivalence criterion between (1.1) and a non-canonical form of the Ince Ⅶ equation, that is ^[17,18]

$\begin{equation} w_{\zeta\zeta}+3w_{\zeta}+\epsilon w^{3}+2w = 0. \end{equation}$

(2.1)

Here $\epsilon\neq0$ is an arbitrary parameter, which can be set, without loss of generality, to be equal to $\pm 1$ .

The general solution of (1.1) is

$\begin{equation} w = {\rm e}^{-(\zeta-\zeta_{0})}\rm{cn}\left\{\sqrt{\epsilon}({\rm e}^{-(\zeta-\zeta_{0})}-C_{1}),\frac{1}{\sqrt{2}}\right\}. \end{equation}$

(2.2)

Here $\zeta_{0}$ and $C_{1}$ are arbitrary constants and $\rm{cn}$ is the Jacobian elliptic cosine. Expression (2.2) will be used below for constructing autonomous parametric first integrals for members of (1.1).

The equivalence criterion between (1.1) and (2.1) can be formulated as follows:

Theorem 2.1. Equation (1.1) is equivalent to (2.1) if and only if either

$\begin{equation} \begin{gathered} (I)\quad 25515lgp^{2}q_{y}+2352980l^{10}+\left(3430q-6667920p^{3}\right) l^{5}\\ -14580qp^{3}-10q^{2}-76545lgqpp_{y} = 0, \end{gathered} \end{equation}$

(2.3)

$\begin{equation} \begin{gathered} (II)\quad 343l^{5}-972p^{3} = 0, \end{gathered} \end{equation}$

(2.4)

holds. Here

$\begin{equation} \begin{gathered} l = 9(fg_{y}-gf_{y}+fgh-3kg^{2})-2f^{3}, \quad p = gl_{y}-3lg_{y}+l(f^{2}-3gh),\\ q = 25515g_{y}lp^{2}-5103lgpp_{y}+686l^{5}-8505p^{2}\left( f^{2}-3gh \right) l+6561p^{3}. \end{gathered} \end{equation}$

(2.5)

The expression for $G_{2}$ in each case is either

$\begin{equation} (I) \quad G_{2} = \frac {126 l^{2}qp^{2}}{470596 l^{10}- \left( 1333584 p^{3}+1372 q \right) l^{5}+q^{2}}, \end{equation}$

(2.6)

$\begin{equation} (II) \quad G_{2}^{2} = -\frac{49l^{3}G_{2}+9p^{2}}{189pl}. \end{equation}$

(2.7)

In all cases the functions $F$ and $G_{1}$ are given by

$\begin{equation} F^{2} = \frac{l}{81\epsilon G_{2}^{3}}, \quad G_{1} = \frac{G_{2}(f-3G_{2})}{3g}. \end{equation}$

(2.8)

Proof. We begin with the necessary conditions. Substituting (1.2) into (2.1) we get

$\begin{equation} y_{zz}+k(y)y_{z}^{3}+h(y)y_{z}^{2}+f(y)y_{z}+g(y) = 0, \end{equation}$

(2.9)

where

$\begin{equation} \begin{gathered} k = \frac{FG_{1}^{3}(\epsilon F^{2}+2)+3G_{1}^{2}F_{y}+G_{1}F_{yy}-F_{y}G_{1,y}}{G_{2}F_{y}}, \\ h = \frac{G_{2}F_{yy}+(6G_{1}G_{2}-G_{2,y})F_{y}+3FG_{2}G_{1}^{2}(\epsilon F^{2}+2)}{G_{2}F_{y}},\\ f = \frac{3G_{2}(F_{y}+FG_{1}(\epsilon F^{2}+2))}{F_{y}}, \quad g = \frac{FG_{2}^{2}(\epsilon F^{2}+2)}{F_{y}}. \end{gathered} \end{equation}$

(2.10)

As a consequence, we obtain that (1.1) can be transformed into (2.1) if it is of the form (2.9) (or (1.1)).

Conversely, if the functions $F$ , $G_{1}$ and $G_{2}$ satisfy (2.10) for some values of $k$ , $h$ , $f$ and $g$ , then (1.1) can be mapped into (2.1) via (1.2). Thus, we see that the compatibility conditions for (2.10) as an overdertmined system of equations for $F$ , $G_{1}$ and $G_{2}$ result in the necessary and sufficient conditions for (1.1) to be equivalent to (2.1) via (1.2).

To obtain the compatibility conditions, we simplify system (2.10) as follows. Using the last two equations from (2.10) we find the expression for $G_{1}$ given in (2.8). Then, with the help of this relation, from (2.10) we find that

$\begin{equation} 81\epsilon F^{2}G_{2}^{3}-l = 0, \end{equation}$

(2.11)

and

$\begin{equation} \begin{gathered} 567lG_{2}^{3}+(243lgh-81lf^{2}-81gl_{y}+243lg_{y})G_{2}-7l^{2} = 0,\\ 243lgG_{2,y}+324lG_{2}^{3}-81gl_{y}G_{2}+2l^{2} = 0, \end{gathered} \end{equation}$

(2.12)

Here $l$ is given by (2.5).

As a result, we need to find compatibility conditions only for (2.12). In order to find the generic case of this compatibility conditions, we differentiate the first equation twice and find the expression for $G_{2}^{2}$ and condition (2.3). Differentiating the first equation from (2.12) for the third time, we obtain (2.6). Further differentiation does not lead to any new compatibility conditions. Particular case (2.4) can be treated in the similar way.

Finally, we remark that the cases of $l = 0$ , $p = 0$ and $q = 0$ result in the degeneration of transformations (1.2). This completes the proof.

As an immediate corollary of Theorem 2.1 we get

Corollary 2.1. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then an autonomous first integral of this equation can be presented in the parametric form as follows:

$\begin{equation} y = F^{-1}(w), \quad y_{z} = \frac{G_{2}w_{\zeta}}{F_{y}-G_{1}w_{\zeta}}\,. \end{equation}$

(2.13)

Here $w$ is the general solution of (2.1) given by (2.2). Notice also that, formally, (2.13) contains two arbitrary constants, namely $\zeta_{0}$ and $C_{1}$ . However, without loss of generality, one of them can be set equal to zero.

Now we demonstrate that transformations (1.2) preserve autonomous invariant curves for equations from (1.1).

First, we need to introduce the definition of an invariant curve for (1.1). We recall that Eq (1.1) can be transformed into an equivalent dynamical system

$\begin{equation} y_{z} = P, \quad u_{z} = Q, \quad P = u, \quad Q = -k u^{3}-h u^{2}-f u -g. \end{equation}$

(2.14)

A smooth function $H(y, u)$ is called an invariant curve of (2.14) (or, equivalently, of (1.1)), if it is a nontrivial solution of ^[19]

$\begin{equation} PH_{y}+QH_{u} = \lambda H, \end{equation}$

(2.15)

for some value of the function $\lambda$ , which is called the cofactor of $H$ .

Second, we need to introduce the equation that is equivalent to (1.1) via (1.2). Substituting (1.2) into (1.1) we get

$\begin{equation} w_{\zeta\zeta}+\tilde{k}w_{\zeta}^{3}+\tilde{h}w_{\zeta}^{2}+\tilde{f}w_{\zeta}+\tilde{g} = 0, \end{equation}$

(2.16)

where

$\begin{equation} \begin{gathered} \tilde{k} = \frac{kG_{2}^{3}-gG_{1}^{3}+(G_{1,y}-hG_{1})G_{2}^{2}+(fG_{1}-G_{2,y})G_{1}G_{2}}{F_{y}^{2}G_{2}^{2}},\\ \tilde{h} = \frac{(hF_{y}-F_{yy})G_{2}^{2}-(2fG_{1}-G_{2,y})G_{2}F_{y}+3gG_{1}^{2}F_{y}}{F_{y}^{2}G_{2}^{2}},\\ \tilde{f} = \frac{fG_{2}-3gG_{1}}{G_{2}^{2}}, \quad \tilde{g} = \frac{gF_{y}}{G_{2}^{2}}. \end{gathered} \end{equation}$

(2.17)

An invariant curve for (2.16) can be defined in the same way as that for (1.1). Notice that, further, we will denote $w_{\zeta}$ as $v$ .

Theorem 2.2. Suppose that either (1.1) possess an invariant curve $H(y, u)$ with the cofactor $\lambda(y, u)$ or (2.16) possess an invariant curve $\widetilde{H}(w, v)$ with the cofactor $\tilde{\lambda}(w, v)$ . Then, the other equation also has an invariant curve and the corresponding invariant curves and cofactors are connected via

$\begin{equation} \begin{gathered} H(y,u) = \widetilde{H}\left(F,\frac{F_{y}u}{G_{1}u+G_{2}}\right), \quad \lambda(y,u) = (G_{1}u+G_{2})\tilde{\lambda}\left(F,\frac{F_{y}u}{G_{1}u+G_{2}}\right). \end{gathered} \end{equation}$

(2.18)

Proof. Suppose that $\widetilde{H}(w, v)$ is an invariant curve for (2.16) with the cofactor $\widetilde{\lambda}(w, v)$ . Then it satisfies

$\begin{equation} v\widetilde{H}_{w}+(-\tilde{k} v^{3}-\tilde{h} v^{2}-\tilde{f} v -\tilde{g})\widetilde{H}_{v} = \widetilde{\lambda}\widetilde{H}. \end{equation}$

(2.19)

Substituting (1.2) into (2.19) we get

$\begin{equation} uH_{y}+(-k u^{3}-h u^{2}-f u-g)H = (G_{1}u+G_{2}) \tilde{\lambda}\left(F,\frac{F_{y}u}{G_{1}u+G_{2}}\right) H . \end{equation}$

(2.20)

This completes the proof.

As an immediate consequence of Theorem 2.2 we have that transformations (1.2) preserve autonomous first integrals admitted by members of (1.1), since they are invariant curves with zero cofactors.

Another corollary of Theorem 2.2 is that any equation from (1.1) that is connected to (2.1) admits two invariant curves that correspond to irreducible polynomial invariant curves of (2.1). This invariant curves of (2.1) and the corresponding cofactors are the following (see, ^[23] formulas (3.18) and (3.19) taking into account scaling transformations)

$\begin{equation} \widetilde{H} = \pm i \sqrt{-2\epsilon}(v+w)+w^{2}, \quad \tilde{\lambda} = \pm \sqrt{-2\epsilon}w-2. \end{equation}$

(2.21)

Therefore, we have that the following statement holds:

Corollary 2.2. If coefficients of an equation from (1.1) satisfy either (2.3) or (2.4), then is admits the following invariant curves with the corresponding cofactors

$\begin{equation} H = \pm i \sqrt{-2\epsilon}\left(\frac{F_{y}u}{G_{1}u+G_{2}}+F\right)+F^{2}, \quad \lambda = \left(G_{1}u+G_{2}\right)\left(\pm \sqrt{-2\epsilon}F-2\right). \end{equation}$

(2.22)

Let us remark that connections between (2.1) and non-autonomous variants of (1.1) can be considered via a non-autonomous generalization of transformations (1.2). However, one of two nonlocally related equations should be autonomous since otherwise nonlocal transformations do not map a differential equation into a differential equation ^[5].

In this Section we have obtained the equivalence criterion between (1.1) and (2.1), that defines two new completely integrable subfamilies of (1.1). We have also demonstrated that members of these subfamilies posses an autonomous parametric first integral and two autonomous invariant curves.

3. Examples

In this Section we provide two examples of integrable equations from (1.1) satisfying integrability conditions from Theorem 2.1.

Example 1. One can show that the coefficients of the following cubic oscillator

$\begin{equation} y_{zz}-\frac{12\epsilon \mu y}{(\epsilon \mu^{2}y^{4}+2)^{2}}y_{z}^{3}-6\mu y y_{z}+2\mu^{2}y^{3}(\epsilon\mu^{2}y^{4}+2) = 0, \end{equation}$

(3.1)

satisfy condition (2.3) from Theorem 2.1. Consequently, Eq (3.1) is completely integrable and its general solution can be obtained from (2.2) by inverting transformations (1.2). However, it is more convenient to use Corollary 2.1 and present the autonomous first integral of (3.1) in the parametric form as follows:

$\begin{equation} y = \pm\sqrt{\frac{w}{\mu}}, \quad y_{z} = \frac{w(\epsilon w^{2}+2) w_{\zeta}}{2w_{\zeta}+w(\epsilon w^{2}+2)}, \end{equation}$

(3.2)

where $w$ is given by (2.2), $\zeta$ is considered as a parameter and $\zeta_{0}$ , without loss of generality, can be set equal to zero. As a result, we see that (3.1) is integrable since it has an autonomous first integral.

Moreover, using Corollary 2.2 one can find invariant curves admitted by (3.1)

$\begin{equation} \begin{gathered} H_{1,2} = \frac{y^{4}\left[(\sqrt{2}\pm \sqrt{-\epsilon}\mu y^{2})^{2}(\sqrt{2}\mp\sqrt{-\epsilon}\mu y^{2})+2(\epsilon\mu y^{2}\mp\sqrt{-2\epsilon})u\right]}{2\mu^{2}y^{2}(\epsilon\mu^{2}y^{4}+2)-4u},\\ \lambda_{1,2} = \pm\frac{2(\mu y^{2}(\epsilon\mu^{2}y^{4}+2)-2u)(\sqrt{-2\epsilon}\mu y^{2}\mp 2)}{y(\epsilon\mu^{2}y^{4}+2)} \end{gathered} \end{equation}$

(3.3)

With the help of the standard technique of the Darboux integrability theory ^[19], it is easy to find the corresponding Darboux integrating factor of (3.1)

$\begin{equation} M = \frac{(\epsilon \mu^{2}y^{4}+2)^{\frac{9}{4}}}{(2\epsilon u^{2}+(\epsilon\mu^{2}y^{4}+2)^{2})^{\frac{3}{4}}(\mu y^{2}(\epsilon \mu^{2}y^{4}+2)-2u)^{\frac{3}{2}}}. \end{equation}$

(3.4)

Consequently, equation is (3.1) Liouvillian integrable.

Example 2. Consider the Liénard (1, 9) equation

$\begin{equation} y_{zz}+(b_{i}y^{i}){y_{z}}+a_{j}y^{j} = 0, \quad i = 0,\ldots 4, \quad j = 0,\ldots,9. \end{equation}$

(3.5)

Here summation over repeated indices is assumed. One can show that this equation is equivalent to (2.1) if it is of the form

$\begin{equation} y_{zz}-9(y+\mu)(y+3\mu)^{3}y_{z}+2y(2y+3\mu)(y+3\mu)^{7} = 0, \end{equation}$

(3.6)

where $\mu$ is an arbitrary constant.

With the help of Corollary 2.1 one can present the first integral of (3.6) in the parametric form as follows:

$\begin{equation} y = \frac{3\sqrt{-2\epsilon}\mu w}{2-\sqrt{-2\epsilon}w}, \quad y_{z} = \frac{7776 \sqrt{2} \epsilon \mu^{5} w w_{\zeta}}{(\sqrt{-2\epsilon}w-2)^{5}(2\sqrt{-\epsilon}w_{\zeta}+(\sqrt{2}\epsilon w +2\sqrt{-\epsilon})w)}, \end{equation}$

(3.7)

where $w$ is given by (2.2). Thus, one can see that (3.5) is completely integrable due to the existence of this parametric autonomous first integral.

Using Corollary 2.2 we find two invariant curves of (3.6):

$\begin{equation} H_{1} = \frac{y^{2}[(2y+3\mu)(y+3\mu)^{4}-2u)]}{(y+3\mu)^{2}[(y+3\mu)^{4}y-u]}, \quad \lambda_{1} = \frac{6\mu(u-y(y+3\mu)^{4})}{y(y+3\mu)}, \end{equation}$

(3.8)

and

$\begin{equation} H_{2} = \frac{y^{2}(y+3\mu)^{2}}{y(y+3\mu)^{4}-u}, \quad \lambda_{2} = \frac{2(2y+3\mu)(u-2y(y+3\mu)^{4})}{y(y+3\mu)}. \end{equation}$

(3.9)

The corresponding Darboux integrating factor is

$\begin{equation} M = \left[y(y+3\mu)^{4}-u\right]^{-\frac{3}{2}}\left[(2y+3\mu)(y+3\mu)^{4}-2u\right]^{-\frac{3}{4}}. \end{equation}$

(3.10)

As a consequence, we see that Eq (3.6) is Liouvillian integrable.

Therefore, we see that equations considered in Examples 1 and 2 are completely integrable from two points of view. First, they possess autonomous parametric first integrals. Second, they have Darboux integrating factors.

4. Conclusions and discussion

In this work we have considered the equivalence problem between family of Eqs (1.1) and its integrable member (2.1), with equivalence transformations given by generalized nonlocal transformations (1.2). We construct the corresponding equivalence criterion in the explicit form, which leads to two new integrable subfamilies of (1.1). We have demonstrated that one can explicitly construct a parametric autonomous first integral for each equation that is equivalent to (2.1) via (1.2). We have also shown that transformations (1.2) preserve autonomous invariant curves for (1.1). As a consequence, we have obtained that equations from the obtained integrable subfamilies posses two autonomous invariant curves, which corresponds to the irreducible polynomial invariant curves of (2.1). This fact demonstrate a connection between nonlocal equivalence approach and Darboux and Liouvillian integrability approach. We have illustrate our results by two examples of integrable equations from (1.1).

Acknowledgments

The author was partially supported by Russian Science Foundation grant 19-71-10003.

Conflict of interest

The author declares no conflict of interest in this paper.

References

[1]	M. Anguiano, F. J. Suárez-Grau, Nonlinear Reynolds equations for non-Newtonian thin-film fluid flows over a rough boundary, IMA J. Appl. Math., 84 (2019), 63–95. https://doi.org/10.1093/imamat/hxy052 doi: 10.1093/imamat/hxy052
[2]	G. Bayada, K. Lhalouani, Asymptotic and numerical analysis for unilateral contact problem with Coulomb's friction between an elastic body and a thin elastic soft layer, Asymptotic Anal., 25 (2001), 329–362.
[3]	H. Benseridi, Y. Letoufa, M. Dilmi, On the asymptotic behavior of an interface problem in a thin domain, Proc. Natl. Acad. Sci., India, Sect. A Phys. Sci., 90 (2020), 547–556. https://doi.org/10.1007/s40010-019-00598-4 doi: 10.1007/s40010-019-00598-4
[4]	M. Boukrouche, G. Łukaszewicz, Asymptotic analysis of solutions of a thin film lubrication problem with Coulomb fluid-solid interface law, Int. J. Eng. Sci., 41 (2003), 521–537. https://doi.org/10.1016/S0020-7225(02)00282-3 doi: 10.1016/S0020-7225(02)00282-3
[5]	M. Boukrouche, R. E. Mir, On a non-isothermal, non-Newtonian lubrication problem with Tresca law: existence and the behavior of weak solutions, Nonlinear Anal., 9 (2008), 674–692. https://doi.org/10.1016/j.nonrwa.2006.12.012 doi: 10.1016/j.nonrwa.2006.12.012
[6]	M. Boukrouche, R. E. Mir, Asymptotic analysis of non-Newtonian fluid in a thin domain with Tresca law, Nonlinear Anal., 59 (2004), 85–105. https://doi.org/10.1016/j.na.2004.07.003 doi: 10.1016/j.na.2004.07.003
[7]	L. Consiglieri, Stationary solutions for a Bingham flow with nonlocal friction, Chapman and Hall/CRC, 1 Ed., 1992.
[8]	M. Dilmi, H. Benseridi, A. Saadallah, Asymptotic analysis of a Bingham fluid in a thin domain with Fourier and Tresca boundary conditions, Adv. Appl. Math. Mech., 6 (2014), 797–810. https://doi.org/10.1017/S2070073300001466 doi: 10.1017/S2070073300001466
[9]	B. Q. Dong, Z. M. Chen, Asymptotic stability of non-Newtonian flows with large perturbation in $R_{2}$ , Appl. Math. Comput., 173 (2006), 243–250. https://doi.org/10.1016/j.amc.2005.04.002 doi: 10.1016/j.amc.2005.04.002
[10]	G. Duvaut, J. L. Lions, Les inéquations en mécanique et en physique, Dunod, Paris, 1972.
[11]	G. Duvaut, Equilibre d'un solide élastique avec contact unilatéral et frottement de Coulomb, C. R. Math. Acad. Sci. Paris, 290 (1980), 263–265.
[12]	I. Ekeland, R. Temam, Analyse convexe et problèmes variationnels, Dunod, Gauthier-Villars, Paris, 1974.
[13]	V. Girault, P. A. Raviart, Finite element approximation of the Navier-Stokes equations, Springer-Verlag, 1979. https://doi.org/10.1007/BFb0063447
[14]	B. Guo, G. Lin, Existence and uniqueness of stationary solutions of non-Newtonian viscous incompressible fluids, Commun. Nonlinear Sci. Numer. Simul., 4 (1999), 63–68. https://doi.org/10.1016/S1007-5704(99)90060-6 doi: 10.1016/S1007-5704(99)90060-6
[15]	W. H. Herschel, R. Bulkley, Konsistenzmessungen von Gummi-Benzollösungen, Kolloid-Z., 39 (1926), 291–300. https://doi.org/10.1007/BF01432034 doi: 10.1007/BF01432034
[16]	A. Massmeyer, E. Di Giuseppe, A. Davaille, T. Rolf, P. J. Tackley, Numerical simulation of thermal plumes in a Herschel-Bulkley fluid, J. Non-Newtonian Fluid Mech., 195 (2013), 32–45. https://doi.org/10.1016/j.jnnfm.2012.12.004 doi: 10.1016/j.jnnfm.2012.12.004
[17]	G. P. Matson, A. J. Hogg, Two-dimensional dam break flows of Herschel-Bulkley fluids: the approach to the arrested state, J. Non-Newtonian Fluid Mech., 142 (2007), 79–94. https://doi.org/10.1016/j.jnnfm.2006.05.003 doi: 10.1016/j.jnnfm.2006.05.003
[18]	C. Nouar, M. Lebouché, R. Devienne, C. Riou, Numerical analysis of the thermal convection for Herschel-Bulkley fluids, Int. J. Heat Fluid Flow, 16 (1995), 223–232. https://doi.org/10.1016/0142-727X(95)00010-N doi: 10.1016/0142-727X(95)00010-N
[19]	C. Nouar, C. Desaubry, H. Zenaidi, Numerical and experimental investigation of thermal convection for a thermodependent Herschel-Bulkley fluid in an annular duct with rotating inner cylinder, Eur. J. Mech.-B/Fluids, 17 (1998), 875–900. https://doi.org/10.1016/S0997-7546(99)80018-1 doi: 10.1016/S0997-7546(99)80018-1
[20]	S. Poyiadji, K. D. Housiadas, K. Kaouri, G. C. Georgiou, Asymptotic solutions of weakly compressible Newtonian Poiseuille flows with pressure-dependent viscosity, Eur. J. Mech.-B/Fluids, 49 (2015), 217–225. https://doi.org/10.1016/j.euromechflu.2014.09.002 doi: 10.1016/j.euromechflu.2014.09.002
[21]	Y. Qin, X. Liu, X. G. Yang, Global existence and exponential stability of solutions to the one-dimensional full non-Newtonian fluids, Nonlinear Anal., 13 (2012), 607–633. https://doi.org/10.1016/j.nonrwa.2011.07.053 doi: 10.1016/j.nonrwa.2011.07.053
[22]	A. Saadallah, H. Benseridi, M. Dilmi, Study of the non-isothermal coupled problem with mixed boundary conditions in a thin domain with friction law, J. Sib. Federal Univ.-Math. Phys., 11 (2018), 738–752.
[23]	A. Saadallah, H. Benseridi, M. Dilmi, Asymptotic convergence of a generalized non-Newtonian fluid with Tresca boundary conditions, Acta Math. Sci., 40 (2020), 700–712. https://doi.org/10.1007/s10473-020-0308-1 doi: 10.1007/s10473-020-0308-1
[24]	K. C. Sahu, P. Valluri, P. D. M. Spelt, O. K. Matar, Linear instability of pressure-driven channel flow of a Newtonian and a Herschel-Bulkley fluid, Phys. Fluids, 19 (2007), 122101. https://doi.org/10.1063/1.2814385 doi: 10.1063/1.2814385
[25]	F. J. Suárez-Grau, Asymptotic behavior of a non-Newtonian flow in a thin domain with Navier law on a rough boundary, Nonlinear Anal., 117 (2015), 99–123. https://doi.org/10.1016/j.na.2015.01.013 doi: 10.1016/j.na.2015.01.013
[26]	H. S. Tang, D. M. Kalyon, Estimation of the parameters of Herschel–Bulkley fluid under wall slip using a combination of capillary and squeeze flow viscometers, Rheol. Acta, 43 (2004), 80–88. https://doi.org/10.1007/s00397-003-0322-y doi: 10.1007/s00397-003-0322-y
[27]	C. Zhao, Y. Li, A note on the asymptotic smoothing effect of solutions to a non-Newtonian system in 2-D unbounded domains, Nonlinear Anal., 60 (2005), 475–483. https://doi.org/10.1016/j.na.2004.09.014 doi: 10.1016/j.na.2004.09.014

This article has been cited by:

1.	Dmitry I. Sinelshchikov, Linearizabiliy and Lax representations for cubic autonomous and non-autonomous nonlinear oscillators, 2023, 01672789, 133721, 10.1016/j.physd.2023.133721
2.	Jaume Giné, Xavier Santallusia, Integrability via algebraic changes of variables, 2024, 184, 09600779, 115026, 10.1016/j.chaos.2024.115026
3.	Meryem Belattar, Rachid Cheurfa, Ahmed Bendjeddou, Paulo Santana, A class of nonlinear oscillators with non-autonomous first integrals and algebraic limit cycles, 2023, 14173875, 1, 10.14232/ejqtde.2023.1.50
4.	Jaume Giné, Dmitry Sinelshchikov, Integrability of Oscillators and Transcendental Invariant Curves, 2025, 24, 1575-5460, 10.1007/s12346-024-01182-x

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)

通讯作者: 陈斌, bchen63@163.com

1.
沈阳化工大学材料科学与工程学院沈阳 110142

AIMS Mathematics

1.8 3.4

Metrics

Article views(1317) PDF downloads(78) Cited by(1)

Preview PDF

Download XML

Export Citation

Article outline

Show full outline

AIMS Mathematics

On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film

Related Papers:

Abstract

1. Introduction

2. Main results

3. Examples

4. Conclusions and discussion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Catalog

AIMS Mathematics

On the strong convergence of the solution of a generalized non-Newtonian fluid with Coulomb law in a thin film

Related Papers:

Abstract

1. Introduction

2. Main results

3. Examples

4. Conclusions and discussion

Acknowledgments

Conflict of interest

References

This article has been cited by:

Reader Comments

通讯作者: 陈斌, bchen63@163.com

Metrics

Other Articles By Authors

Related pages

Tools

Export File

Citation

Format

Content

Catalog