Analysis of a Bifurcation and Stability of Equilibrium Points for Jeffrey Fluid Flow through a Non-Uniform Channel

Abstract

This study aims to analyze how the parameter flow rate and amplitude of walling waves affect the peristaltic flow of Jeffrey’s fluid through an irregular channel. The movement of the fluid is described by a set of non-linear partial differential equations that consider the influential parameters. These equations are transformed into non-dimensional forms with appropriate boundary conditions. The study also utilizes dynamic systems theory to analyze the effects of the parameters on the streamline and to investigate the position of critical points and their local and global bifurcation of flow. The research presents numerical and analytical methods to illustrate the impact of flow rate and amplitude changes on fluid transport. It identifies three types of streamline patterns that occur: backwards, trapping, and augmented flow resulting from changes in the value of flow rate parameters.

1. Introduction

Peristalsis is regarded as one of the main mechanisms of fluid movement in physiological systems such as human and mammalian. Peristalsis occurs due to the expansion and contraction of sinusoidal waves across the walls of a channel. These mechanisms are important in physiology and industry: for instance, the flow of blood in arterioles, movement of urine from the kidney to the bladder, venules and capillaries, roller and finger pumps, waste management pumps in nuclear industry, hoses, and domestic pumps. Also, the technique of this transport is used in devices like heart–lung machines to pump blood.

The main researcher on this subject is Latham [1], who introduced an exhaustive analysis of peristaltic transport in the context of mechanical pumping. The mathematical formulation between motion of fluid and the amplitude ratio of the peristaltic transport was done by [2]. A further extension by Shapiro et al. [3] studied peristaltic movement under the conditions of large wavelengths at tiny Reynolds numbers.

Raju and Devanathan [4] were the first who investigated the peristaltic phenomenon for non-Newtonian fluids in a tube by considering blood as a model. Many studies discussed different features, such as the velocity and amplitude of the fluid, pumping characteristics, and the conditions for which trapping happens. Trapping refers to the situation when a trapped bolus of fluid is transported in the flow, with the wave speed, under particular values of the parameters [5,6,7].

Subsequently, many academics have presented studies about the mechanism of peristalsis due to the influence analysis of peristaltic flows in engineering and biomedicine. Researchers get different results about peristaltic flows in different types of fluids with varied types of flow channels [8,9,10,11,12,13]. In addition, the double-lid-driven rectangular cavity, which consists of two moving lids and two solid walls, has been extensively studied by various authors [14,15,16]. Recently, many researchers have examined several hybrid nanofluid flows with varying parameters, including thermal radiation [17,18,19,20].

In recent years, interest has begun in studying the bifurcations of stagnation points. Moreover, a bifurcation in a flow refers to the changes in streamline topologies that occurs in the flow due to the variation in involved parameters. In addition, bifurcation allows inquiry about the state of fluid flow in flow channels [21,22,23]. We review some of the research in this regard where Jménez-Lozano and Sen were interested in analyzing the peristaltic transport of the Newtonian fluid in a symmetric channel and tube. In addition, they concentrated on the theory of dynamical systems to analyze the streamline topologies and their bifurcations [24,25,26].

Asghar and Ali presented two separate studies [27,28]. They analyzed the topology of streamlines and bifurcations of the compressible peristaltic flow through asymmetric channels. In addition, they investigated the nature and stability of equilibrium points and highlighted different streamlined topologies by employing the theory of the dynamical system. Subsequently, Ullah et al. [29] analyzed the stability status of critical points and streamline topologies of different flows and investigated their bifurcations for peristaltic flow of a power-law fluid in an axisymmetric channel. Ali et al. [9] considered a two-dimensional model to study a bifurcation analysis for the peristaltic movement of power-law fluid through an asymmetric channel.

Ullah and Ali [30] analyzed the bifurcation of stagnation points for a peristaltic transport of micropolar fluids through a symmetric channel with slip condition. This research aims to investigate the bifurcation analysis and global dynamics of a biomedical drug delivery model that has been designed as an application for the peristaltic flow of fluids through a two-dimensional channel. Hosham and Sellami provided an effective study related to the same phenomena in a biomedical drug delivery model. In addition, they developed an application for the peristaltic flow of thermal fluids based on bifurcation theory [31].

Inspired by previous work and others that we have not listed, we present a mathematical model whose purpose is to study and analyze the effect of flow rate and amplitude of wall waves on the peristaltic flow of a Jeffrey fluid. Furthermore, we analyze the streamlined topologies and flow of local and global bifurcations in a 2D asymmetric channel. In the literature, such an analysis of the peristaltic transport of Jeffrey fluid through an asymmetric conduit has not been performed. The research is divided into seven sections: the first is a simple overview of previous work, and the second is the creation of a mathematical model based on the shape of the channel and the type of fluid. The third part includes the solution method using non-dimensional parameters; calculating the flow rate is in the fourth part of the research. The fifth part includes solving the problem by determining the formula for functions (velocity, stream function, shear stress, and pressure gradient). The main part of the work is located in the sixth section through the use of dynamic systems theory to explore the position of critical points and their ramifications. The last section includes the conclusions.

2. Mathematical Formulations

A peristaltic flow of Jeffrey fluid is considered laminar and flows in a two-dimensional channel with an irregular solid wall. In a cartesian coordinate system $(\overline{X},\overline{Y})$, the channel walls are shown in Figure 1, which has the upper and lower cosine wave equations $H_l(\overline{x},\overline{t})$ and $-H_l(\overline{x},\overline{t})$, respectively, and is given by

$$H_l(\overline{x},\overline{t})=l_1-d{cos}^2\left({\displaystyle \frac{\pi }{\sigma }}(\overline{X}-\alpha \overline{t})\right).$$

(1)

In the above equation, $l_1$ is the average radius of the tube, d is the amplitude of a peristaltic wave, $\sigma$ is a wavelength, $\alpha$ is a wave propagation speed, and $\overline{t}$ is time.

The equations of motion (conservation of mass and momentum) are

$$\begin{array}{ccc}\hfill \nabla .\overline{U}& =& 0(\mathrm{continuity}\mathrm{equation}),\hfill \\ \hfill \rho (\overline{U}.\nabla )\overline{U}& =& -\nabla \overline{P}+div\overline{S}\left(\mathrm{momentum}\mathrm{equation}\right),\hfill \end{array}$$

(2)

where $U\equiv (U_1(X,Y,t),U_2(X,Y,t),0)$ is a velocity field, $\rho$ is a density, $\overline{P}$ is a pressure, and $\overline{S}$ is an extra stress tensor. The constitutive equation of the extra stress tensor for a Jeffrey fluid is given by

$$\overline{S}={\displaystyle \frac{\kappa }{1+{\lambda }_1}}(\overline{\dot{\gamma }}+{\lambda }_2\overline{\ddot{\gamma }}),$$

(3)

where $\kappa$ is a fluid viscosity, ${\lambda }_1$ is the ratio of relaxation to retardation times, ${\lambda }_2$ is the retardation time, and $\overline{\dot{\gamma }}$ is the shear rate (tensor form).

Peristaltic motion is a normally unsteady phenomenon and has the test form $(\overline{X},\overline{Y}).$ It is possible to transfer it to the steady state by reference assumption of wave frame $(\overline{x},\overline{y})$ (by converting constant formulas) with the relations

$$\begin{array}{ccc}\hfill \overline{X}& =& \overline{x}+\alpha \overline{t},\hfill \\ \hfill \overline{Y}& =& \overline{y},\hfill \\ \hfill {\overline{U}}_1& =& {\overline{u}}_1+\alpha ,\hfill \\ \hfill {\overline{U}}_2& =& {\overline{u}}_2,\hfill \\ \hfill \overline{P}& =& \overline{p},\hfill \end{array}$$

(4)

where $({\overline{u}}_1,{\overline{u}}_2)$ and $({\overline{U}}_1,{\overline{U}}_2)$ are velocity elements of the moving and stationary structures, respectively. As a sequence for these transformations, the equations of continuity and momentum (2) become

$${\displaystyle \frac{\partial ({\overline{u}}_1+\alpha )}{\partial \overline{x}}}+{\displaystyle \frac{\partial {\overline{u}}_2}{\partial \overline{y}}}=0,$$

(5)

$$\begin{array}{ccc}\hfill \rho ({\overline{u}}_1+\alpha ){\displaystyle \frac{\partial ({\overline{u}}_1+\alpha )}{\partial \overline{x}}}+\rho {\overline{u}}_2{\displaystyle \frac{\partial {\overline{u}}_2}{\partial \overline{y}}}-{\displaystyle \frac{\partial {\overline{S}}_{\overline{x}x}}{\partial \overline{x}}}-{\displaystyle \frac{\partial {\overline{S}}_{\overline{x}y}}{\partial \overline{y}}}+{\displaystyle \frac{\partial \overline{p}}{\partial \overline{x}}}& =& 0,\hfill \\ \hfill \rho ({\overline{u}}_1+\alpha ){\displaystyle \frac{\partial {\overline{u}}_2}{\partial \overline{x}}}+\rho {\overline{u}}_2{\displaystyle \frac{\partial {\overline{u}}_2}{\partial \overline{y}}}-{\displaystyle \frac{\partial {\overline{S}}_{\overline{x}y}}{\partial \overline{x}}}-{\displaystyle \frac{\partial {\overline{S}}_{\overline{y}y}}{\partial \overline{y}}}+{\displaystyle \frac{\partial \overline{p}}{\partial \overline{y}}}& =& 0.\hfill \end{array}$$

(6)

The boundary conditions for the dimensionless no-slip (frame of reference) are as follows:

$$\begin{array}{ccc}\hfill {\overline{u}}_1\left(\overline{y}\right)& =& -\alpha at\overline{y}=\mp H_l\left(\overline{x}\right)=\mp (l_1-\overline{d}{cos}^2\left({\displaystyle \frac{\pi }{\sigma }}\left(\overline{x}\right)\right)),\hfill \\ \hfill {\overline{u}}_2\left(\overline{y}\right)& =& 0at\overline{y}=H_l\left(\overline{x}\right)=l_1-\overline{d}{cos}^2\left({\displaystyle \frac{\pi }{\sigma }}\left(\overline{x}\right)\right).\hfill \end{array}$$

(7)

The stress component are:

$$\begin{array}{ccc}\hfill {\overline{S}}_{\overline{x}\overline{x}}& =& \left({\displaystyle \frac{2\kappa }{1+{\lambda }_1}}\right)(1+{\lambda }_2[({\overline{u}}_1+\alpha ){\displaystyle \frac{\partial }{\partial \overline{x}}}+{\overline{u}}_2{\displaystyle \frac{\partial }{\partial \overline{y}}}])\left({\displaystyle \frac{\partial {\overline{u}}_1}{\partial \overline{x}}}\right),\hfill \\ \hfill {\overline{S}}_{\overline{y}\overline{y}}& =& \left({\displaystyle \frac{2\kappa }{1+{\lambda }_1}}\right)(1+{\lambda }_2[({\overline{u}}_1+\alpha ){\displaystyle \frac{\partial }{\partial \overline{x}}}+{\overline{u}}_2{\displaystyle \frac{\partial }{\partial \overline{y}}}])\left({\displaystyle \frac{\partial {\overline{u}}_2}{\partial \overline{y}}}\right),\hfill \\ \hfill {\overline{S}}_{\overline{x}\overline{y}}& =& {\overline{S}}_{\overline{y}\overline{x}}=\left({\displaystyle \frac{\kappa }{1+{\lambda }_1}}\right)(1+{\lambda }_2[({\overline{u}}_1+\alpha ){\displaystyle \frac{\partial }{\partial \overline{x}}}+{\overline{u}}_2{\displaystyle \frac{\partial }{\partial \overline{y}}}])({\displaystyle \frac{\partial {\overline{u}}_1}{\partial \overline{y}}}+{\displaystyle \frac{\partial {\overline{u}}_2}{\partial \overline{x}}}).\hfill \end{array}$$

(8)

The corresponding stream functions are ${\overline{u}}_1={\displaystyle \frac{\partial \overline{\mathsf{\Psi }}}{\partial y}}$ and ${\overline{u}}_2=-{\displaystyle \frac{\partial \overline{\mathsf{\Psi }}}{\partial x}}$.

3. Solution Method

In order to simplify the governing equations of the problem, we introduce the following dimensionless transformations as follows:

$$\begin{array}{ccc}\hfill x& =& {\displaystyle \frac{\overline{x}}{\sigma }},y={\displaystyle \frac{\overline{y}}{l_1}},\zeta ={\displaystyle \frac{l_1}{\sigma }},u_1={\displaystyle \frac{{\overline{u}}_1}{\alpha }},u_2={\displaystyle \frac{\sigma {\overline{u}}_2}{\alpha l_1}},p={\displaystyle \frac{l_1^2\overline{p}}{\kappa \sigma \alpha }},t={\displaystyle \frac{\alpha \overline{t}}{\sigma }},\mathsf{\Psi }={\displaystyle \frac{\overline{\mathsf{\Psi }}}{l_1\alpha }},d={\displaystyle \frac{\overline{d}}{l_1}},\hfill \\ \hfill h& =& {\displaystyle \frac{H_l}{l_1}},R_e={\displaystyle \frac{\rho \alpha l_1}{\kappa }},Q_1={\displaystyle \frac{{\overline{Q}}_1}{l_1\alpha }},q_1={\displaystyle \frac{{\overline{q}}_1}{l_1\alpha }},S_{xx}={\displaystyle \frac{\sigma {\overline{S}}_{\overline{x}\overline{x}}}{\kappa \alpha }},S_{xy}={\displaystyle \frac{l_1{\overline{S}}_{\overline{x}\overline{y}}}{\kappa \alpha }},S_{yy}={\displaystyle \frac{\sigma {\overline{S}}_{\overline{y}\overline{y}}}{\kappa \alpha }},\hfill \end{array}$$

(9)

where $Re$ is a “Reynolds number”, d is an amplitude ratio, $\zeta$ is a dimensionless wave number, and $\mathsf{\Psi }$ is a “stream function”. Substituting Equation (9) into Equations (5)–(8) under the long-wavelength assumption $\zeta \ll 1$ become the governing equations after simplification:

$${\displaystyle \frac{\partial u_1}{\partial x}}+{\displaystyle \frac{\partial u_2}{\partial y}}=0$$

(10)

$${\displaystyle \frac{\partial p}{\partial x}}={\displaystyle \frac{\partial S_{xy}}{\partial y}},\mathrm{and}{\displaystyle \frac{\partial p}{\partial y}}=0$$

(11)

$$S_{xy}=\left({\displaystyle \frac{1}{1+{\lambda }_1}}\right){\displaystyle \frac{\partial u_1}{\partial y}}$$

(12)

The boundary conditions become:

$$\begin{array}{ccc}\hfill u_1\left(y\right)& =& u_1(-y)=-1\mathrm{at}y=h=1-d{cos}^2\left(\pi x\right),\hfill \\ \hfill u_2\left(y\right)& =& 0\mathrm{at}y=h=1-d{cos}^2\left(\pi x\right).\hfill \end{array}$$

(13)

4. Rate of Volume Flow

In fixed coordinate system, the instantaneous volume of flow rate is given by:

$${\widehat{\mathbf{Q}}}_1={\int }_{-H}^H{\overline{U}}_1(\overline{X},\overline{Y},\overline{t})d\overline{Y}.$$

(14)

Substituting Equation (4) into (13) and then integrating gets:

$${\widehat{\mathbf{Q}}}_1={\overline{q}}_1+2\alpha H\mathrm{where}{\overline{q}}_1={\int }_{-H}^H{\overline{u}}_1\left(\overline{x,\overline{y}}\right)d\overline{y}$$

The time-mean flow over a period $T={\displaystyle \frac{\sigma }{\alpha }}$ is defined as

$${\overline{Q}}_1={\displaystyle \frac{1}{T}}{\int }_0^T{\widehat{\mathbf{Q}}}_{1}d\overline{t}={\displaystyle \frac{1}{T}}{\int }_0^T({\overline{q}}_1+2\alpha H)d\overline{t}={\overline{q}}_1+2\alpha (l_1-{\displaystyle \frac{\overline{d}}{2}}).$$

(15)

Using Equation (9) yields ${\mathbf{Q}}_1=q_1+2(1-{\displaystyle \frac{d}{2}})$, so that we have $q_1={\mathbf{Q}}_1-2(1-{\displaystyle \frac{d}{2}}),$ where $q_1$ is the dimensionless volume flow rate in the wave frame defined by

$$q_1={\int }_{-h}^h{u}_1(x,y)dy={\int }_{-h}^h\left({\displaystyle \frac{\partial \mathsf{\Psi }}{\partial y}}\right)dy=\mathsf{\Psi }\left(h\left(x\right)\right)-\mathsf{\Psi }(-h\left(x\right)).$$

(16)

With boundary conditions: ${\displaystyle \frac{\partial \mathsf{\Psi }}{\partial y}}{{|}_{y=h}={\displaystyle \frac{\partial \mathsf{\Psi }}{\partial y}}|}_{y=-h}=-1,\mathsf{\Psi }\left(h\right)={\displaystyle \frac{1}{2}}{Q}_1\mathsf{\Psi }(-h)=-{\displaystyle \frac{1}{2}}{Q}_1.$

5. Solution of the Problem

Substituting Equation (12) into Equation (11) and using the boundary conditions (13) and volume flow rate $q_1$, we get the solution to the velocity equation and, therefore, the formula for the velocity function is

$$u_1={\displaystyle \frac{(1+{\lambda }_1)}{2}}{\displaystyle \frac{\partial p}{\partial x}}(y^2-h^2)-1.$$

(17)

The corresponding stream function is $\mathsf{\Psi }={\displaystyle \frac{(1+{\lambda }_1)}{2}}{\displaystyle \frac{\partial p}{\partial x}}({\displaystyle \frac{1}{3}}{y}^3-h^{2}y)-y$. The shear stress $S_{xy}=(1+{\lambda }_1){\displaystyle \frac{\partial p}{\partial x}}y,$ where the evaluate pressure gradient is ${\displaystyle \frac{\partial p}{\partial x}}={\displaystyle \frac{-3}{2(1+{\lambda }_1)h^3}}({\mathbf{Q}}_{\mathbf{1}}+d-2+2h).$

The above solution of the streamline function with specific values of the volume flow rate $Q_1=0.9,1.3$, and $1.7$ and amplitude $d=0.5$ can be analyzed in three different scenarios of the flow field (seen in Figure 2). First, backward flow occurs when the entire flow moves with the opposite direction of the wave motion (in panel $A_1$). Then, panel $A_2$ shows trapping phenomena that happen when boluses composed of fluid particles appear in the flow motion. It can be described as a closed path of streamlines. Finally, remarkable change appears when the trapped boluses are under the wave crests and between them there exist some flow transports through the centerline in the wave direction; see panel $A_3$.

6. The Nonlinear System and Its Bifurcation

The dynamical system theory is useful to analyze qualitative behaviors of the flow of fluid. From Equations (10) and (17), we get the following coupled nonlinear differential equations system for the original problem with two parameters, $Q_1$ and d:

$$\begin{array}{ccc}\hfill u_1& =& -(3/8)((2Q_1+d-4d{(cos\left(\pi x\right))}^2)(-{(1-d{(cos\left(\pi x\right))}^2)}^2\hfill \\ & +& {\left(y\right)}^2\left)\right)/\left({(1-d{(cos\left(\pi x\right))}^2)}^3\right)-1=f(x,y),\hfill \\ \hfill u_2& =& -(9/4)((2Q_1+d-4d{(cos\left(\pi x\right))}^2)(-{(1-d{(cos\left(\pi x\right))}^2)}^{2}y\hfill \\ & +& (1/3){\left(y\right)}^3)dcos\left(\pi x\right)sin\left(\pi x\right)\pi )/\left({(1-d{(cos\left(\pi x\right))}^2)}^4\right)\hfill \\ & +& (3dcos\left(\pi x\right)sin\left(\pi x\right)\pi (-{(1-d{(cos\left(\pi x\right))}^2)}^{2}y\hfill \\ & +& (1/3){\left(y\right)}^3\left)\right)/\left({(1-d{(cos\left(\pi x\right))}^2)}^3\right)-(3/2)\left(\right(2Q_1+d\hfill \\ & -& 4d{(cos\left(\pi x\right))}^2)dcos\left(\pi x\right)sin\left(\pi x\right)\pi y)/\left({(1-d{(cos\left(\pi x\right))}^2)}^2\right)=g(x,y).\hfill \end{array}$$

(18)

The significant step in studying the above system is finding equilibrium points in the flow of fluid by putting

$$f(x,y)=g(x,y)=0.$$

The system (18) can be solved numerically by using Maple software and getting the low point $(x_i,y_i)$. It is noted that the nature and stability of the critical points $(x_i,y_i)$ in the flow field of the channel wall are changed due to varying values of the two parameters $Q_1$ and d and have a remarkable effect on the behavior of streamline (see Figure 2).

• $(x_{1,2},y_{1,2})=\left({\displaystyle \frac{arccos(\pm \sqrt{\frac{-6Q_1-3d+8}{4d}})}{\pi }},0\right)$,
• $(x_{3,4},y_{3,4})=\left({\displaystyle \frac{1}{2}},\pm \sqrt{-{\displaystyle \frac{-6Q_1-3d+8}{6Q_1+3d}}}\right)$,
• $(x_{5,6},y_{5,6})=\left(1,\pm \sqrt{-{\displaystyle \frac{-6Q_1-d+8}{6Q_1-9d}}}\right)(1-d)$,
• $(x_{7,8},y_{7,8})=\left(0,\pm \sqrt{-{\displaystyle \frac{-6Q_1-d+8}{6Q_1-9d}}}\right)(1-d)$.

The Hartman–Grobman theorem has been adopted as an effective method for studying local behavior and stability in a neighborhood of the hyperbolic equilibrium point. It is called a nondegenerate point because it has a Jacobian matrix with trace and determinant not equal to zero. The technique of this theorem finds local linearization close to equilibrium points to investigate the stability status and nature of these equilibrium points. Therefore, the Jacobian matrix A at a critical point $(x,y)$ is a matrix containing the first-order partial derivatives of functions. It is required with its eigenvalues to analyze the stability of the system.

$${A|}_{\{x,y\}}=\left(\begin{array}{cc}{f}_x& f_y\\ g_x& g_y\end{array}\right),$$

where

$$\begin{array}{ccc}\hfill f_x={\displaystyle \frac{\partial f}{\partial x}}& =& {\displaystyle \frac{9}{4}}{\displaystyle \frac{B_1(-{\left(B_2\right)}^2+y^2)dcos\left(\pi x\right)sin\left(\pi x\right)\pi }{{\left(B_2\right)}^4}}\hfill \\ & -& {\displaystyle \frac{3dcos\left(\pi x\right)sin\left(\pi x\right)\pi (-{\left(B_2\right)}^2+y^2)}{{\left(B_2\right)}^3}}\hfill \\ & +& {\displaystyle \frac{3}{2}}{\displaystyle \frac{B_{1}dcos\left(\pi x\right)sin\left(\pi x\right)\pi }{{\left(B_2\right)}^2}},\hfill \\ \hfill f_y={\displaystyle \frac{\partial f}{y}}& =& -{\displaystyle \frac{3}{4}}{\displaystyle \frac{B_{1}y}{B_2^3}},\hfill \\ \hfill g_x={\displaystyle \frac{\partial g}{\partial x}}& =& {\displaystyle \frac{18B_1{d}^{2}cos{\left(\pi x\right)}^{2}sin{\left(\pi x\right)}^2{\pi }^2(-{\left(B_2\right)}^{2}y+\frac{1}{3}{y}^3)}{{\left(B_2\right)}^5}}\hfill \\ & -& {\displaystyle \frac{36d^{2}cos{\left(\pi x\right)}^{2}sin{\left(\pi x\right)}^2{\pi }^2(-{\left(B_2\right)}^{2}y+\frac{1}{3}{y}^3)}{{\left(B_2\right)}^4}}\hfill \\ & +& {\displaystyle \frac{9}{4}}{\displaystyle \frac{B_{1}dsin{\left(\pi x\right)}^2{\pi }^2(-{\left(B_2\right)}^{2}y+\frac{1}{3}{y}^3)}{{\left(B_2\right)}^4}}\hfill \\ & -& {\displaystyle \frac{9}{4}}{\displaystyle \frac{B_{1}dcos{\left(\pi x\right)}^2{\pi }^2(-{\left(B_2\right)}^{2}y+\frac{1}{3}{y}^3)}{{\left(B_2\right)}^4}}\hfill \\ & +& {\displaystyle \frac{15B_1{d}^{2}cos{\left(\pi x\right)}^{2}sin{\left(\pi x\right)}^2{\pi }^{2}y}{{\left(B_2\right)}^3}}\hfill \\ & -& {\displaystyle \frac{3dsin{\left(\pi x\right)}^2{\pi }^2(-{\left(B_2\right)}^{2}y+\frac{1}{3}{y}^3)}{{\left(B_2\right)}^3}}\hfill \\ & +& {\displaystyle \frac{3dcos{\left(\pi x\right)}^2{\pi }^2(-{\left(B_2\right)}^{2}y+\frac{1}{3}{y}^3)}{{\left(B_2\right)}^3}}\hfill \\ & -& {\displaystyle \frac{24d^{2}cos{\left(\pi x\right)}^{2}sin{\left(\pi x\right)}^2{\pi }^{2}y}{{\left(B_2\right)}^2}}\hfill \\ & +& {\displaystyle \frac{3}{2}}{\displaystyle \frac{B_{1}dsin{\left(\pi x\right)}^2{\pi }^{2}y}{{\left(B_2\right)}^2}}\hfill \\ & -& {\displaystyle \frac{3}{2}}{\displaystyle \frac{B_{1}dcos{\left(\pi x\right)}^2{\pi }^{2}y}{{\left(B_2\right)}^2}},\hfill \end{array}$$

$$\begin{array}{ccc}\hfill g_y={\displaystyle \frac{\partial g}{\partial y}}& =& -{\displaystyle \frac{9}{4}}{\displaystyle \frac{B_1(-{\left(B_2\right)}^2+y^2)dcos\left(\pi x\right)sin\left(\pi x\right)\pi }{{\left(B_2\right)}^4}}\hfill \\ & +& {\displaystyle \frac{3dcos\left(\pi x\right)sin\left(\pi x\right)\pi (-{\left(B_2\right)}^2+y^2)}{{\left(B_2\right)}^3}}\hfill \\ & -& {\displaystyle \frac{3}{2}}{\displaystyle \frac{B_{1}dcos\left(\pi x\right)sin\left(\pi x\right)\pi }{{\left(B_2\right)}^2}},\hfill \end{array}$$

(19)

where $B_1=2Q_1+d-4dcos{\left(\pi x\right)}^2$ and $B_2=1-dcos{\left(\pi x\right)}^2.$

From the above matrix, we can get the quadratic function ${\mu }^2+\tau \mu +\delta =0$, such that ${\tau |}_{\{x,y\}}=f_x+g_y$ and ${\delta |}_{\{x,y\}}=dat\left(A\right)$. The solution of this equation is two eigenvalues ${\mu }_1$ and ${\mu }_2$. The fixed point $p^{\ast }$ is called hyperbolic (nondegenerate) if the real part of the eigenvalues ${\mu }_1$ and ${\mu }_2$ of the matrix A are $nonzero.$ A hyperbolic fixed point can be classified according to its eigenvalues: a sink, source, or center. It is a stable sink or node if it has eigenvalues with a negative real parts, $Re({\mu }_{1,2}<0)$. In addition, it is called an unstable sours or node point if the real parts of all eigenvalues are positive, $Re({\mu }_{1,2}>0)$. Finally, it is called a saddle if it is neither a sink nor a source; this would be the case if the real part of eigenvalues satisfy $Re\left({\mu }_1\right)>0$ and $Re\left({\mu }_2\right)<0$, or vice versa. A non-hyperbolic critical point is said to have a center that has eigenvalues that are only a conjugate imaginary part.

In this section, we describe the changes in the flow patterns for the stream function that are given in Equation (18). These changes are shown in Figure 2 and Figure 3 with labels $A_i,B_i(i=1,2,3,4)$ at two constant amplitude ratios of the flow channel $d=0.5,0.7$, respectively, with different values for the rate of flow $Q_1\in (0.5,1.9)$. Also for system (18), a local bifurcation diagram is plotted in $(Q_1,u_2)$ planes with different fluid behavior indices according to change values of d; it is displayed in Figure 4A–C. Moreover, a parameter space shows the bifurcation parameter is numerically estimated for a regular range of d and $Q_1$, shown in Figure 4E.

The important qualitative observation is in Figure 2, where the value of $d=0.5.$ In panel $A_1,$ there are saddle points for the flow pattern at $Q_1<1.083$, which represent heteroclinic connections between them. This case shows backward flow and corresponds to the blue branch in the bifurcation diagram (Figure 4A).

After the transition from A₁ to A₂ where $Q1=1.089$ and crosses the pink bifurcation point $Q_1\simeq 1.083$ (Figure 4A), a small trapping zone will appear, and the distance between two centers under the same wave crest is small. With an increased value of the flow rate $Q_1=1.3$, shown in Figure 2A₃, the trapping region will expand and center points will gradually move away from each other.

Moreover, it is observed that there are black unstable saddle points $(x_{3,4},y_{3,4})$ on the centerline and red center points $(x_{1,2},y_{1,2}).$ The nondegenerate saddle points are plotted as a single black line, and red center points represent two red branches of lines in Figure 4A at the light yellow area. There are more consequences when the value of the flow rate crosses the pink bifurcation. Every two saddle points come together to create a heteroclinic connection that provides a global bifurcation picture of the dynamics. Due to the existence of saddle connections and center points under the wave crest, the streamlines create an enclosed flow zone, which is characterized as a “trapping” bolus phenomenon. The trapping zone appears near the upper and lower centerlines of the channel.

This phenomenon is continuous with growing value of the flow rate until the second cyan bifurcation point $Q_1\simeq 1.417$ emerges, where the saddle–node bifurcation will occur. With more explanation, a saddle point splits into two black saddle points $(x_{5,6},y_{5,6})$ and $(x_{7,8},y_{7,8})$ under the wave trough. They shift away from $u_1-axis$ and create a saddle connection, which is known as a heteroclinic connection that happens between two different saddle points. These changes are depicted as double black curves and indicate unstable critical points in the bifurcation diagram Figure 4A. The new value of the bifurcation parameter $Q_1$ causes the appearance of augmented flow phenomena. In other words, when the trapped bolus splits and there exists some flow going forward between them, the two remaining eddies are below the wave crests and inside the heteroclinic orbit.

With more clarification, we can divide the bifurcation diagram Figure 4A into three regions: the first one is light blue I, where backward flow occurs; the next region is light yellow, $III$, where the trapping zone is seen; and finally, at light green $IV$, there is the appearance of augmented flow phenomena. The Roman numbers refer to the number of equilibrium points. These regions and the pink and cyan bifurcation points are also determined in the parameter space $(Q_1,d),$; see Figure 4E. This figure displays the relationship between effective parameters in the streamline.

The topological appearance in Figure 2 and Figure 3 has a similar scenario with different values of the amplitude $d=0.5, 0.7,$ respectively. However, there is a core difference between them. As the fluid amplitude d increases approximately $\Delta d=0.2,$ the distance between two bifurcation points (pink at $Q_1\simeq 1.45$ and cyan at $Q_1\simeq 0.9833$) also increases. The effect of that change is that the trapping zone is lengthened with respect to the flow rate, such that $\Delta Q_{1_{\left(0.7\right)}}-\Delta Q_{1_{\left(0.5\right)}}=0.1327,$; see Figure 4A, B (light yellow region) and Figure 4E (solid and dashed line).

Parameter space provides techniques for analyzing the stability and qualitative change of the system when parameters are changed; see Figure 4E. The information gained from this space allows us to predict and understand the behavior of the streamline of the system. For instance, by observation, when the amplitude is $d=0.9$, in Figure 4C, it can be seen that the pink/cyan bifurcation points for $Q1\simeq 0.8333, 1.483$, respectively, both of them lay on the branches of saddle–node bifurcation (black dash-circle lines) in the light yellow (parameter plane). The findings from these notes tell us that there exists a trapping zone that expands more than in previous states, without the need to plot more streamline patterns. Moreover, it can be predicted what will happen in a complicated behavior of flow in the future.

7. Conclusions

This paper analyzes the flow of Jeffrey’s fluid through a wave channel that simulates blood flow through arterioles. Dynamic system theory is used to analyze the streamlined topologies. In addition, it explores the stability of equilibrium points along with their ramifications for peristaltic fluid transport. The critical points are classified by the eigenvalues extracted from the Jacobian matrix of the linearization system. There are two types of equilibrium points: centers and unstable saddles. The center shows the eddying motion in the flow, whereas the point where separatrices divide the flow corresponds to the saddle. The local bifurcations of the critical point are presented to display effective changes in amplitude and the flow rate parameter on fluid transport. Three types of streamline patterns occur: backward, trapping, and augmented flow in the channel through movement due to a change in the value of the rate parameter. These phenomena are plotted in streamline patterns. The changes in the behavior of fluid through the channel correspond to the bifurcation phenomenon where non-hyperbolic degenerate points change their behavior to form the heteroclinic connection between saddles. The stability and nature of critical points and local topological changes are explained graphically through bifurcation diagrams and parameter space, which are found numerically and analytically.

Furthermore, bifurcation figures were utilized to identify the trapping region. This can help in avoiding trapping behavior in various phenomena, such as mechanical instruments that operate based on the peristaltic principle, like dialysis machines and infusion pumps. In medical applications like drug delivery, it can enhance the effectiveness of treatments by controlling the trapping of particles within biological fluids.