Nonlinear Phenomena of Fluid Flow in a Bioinspired Two-Dimensional Geometric Symmetric Channel with Sudden Expansion and Contraction

Abstract

Inspired by the airway for phonation, fluid flow in an idealized model within a sudden expansion and contraction channel with a geometrically symmetric structure is investigated, and the nonlinear behaviors of the flow therein are explored via numerical simulations. Numerical simulation results show that, as the Reynolds number (Re = U₀H/ν) increases, the numerical solution undergoes a pitchfork bifurcation, an inverse pitchfork bifurcation and a Hopf bifurcation. There are symmetric solutions, asymmetric solutions and oscillatory solutions for flows. When the sudden expansion ratio (Er) = 6.00, aspect ratio (Ar) = 1.78 and Re ≤ Rec1 (≈185), the numerical solution is unique, symmetric and stable. When Rec1 < Re ≤ Rec2 (≈213), two stable asymmetric solutions and one symmetric unstable solution are reached. When Rec2 < Re ≤ Rec3 (≈355), the number of numerical solution returns one, which is stable and symmetric. When Re > Rec3, the numerical solution is oscillatory. With increasing Re, the numerical solution develops from periodic and multiple periodic solutions to chaos. The critical Reynolds numbers (Rec1, Rec2 and Rec3) and the maximum return velocity, at which reflux occurs in the channel, change significantly under conditions with different geometry. In this paper, the variation rules of Rec1, Rec2 and Rec3 are investigated, as well as the maximum return velocity with the sudden expansion ratio Er and the aspect ratio Ar.

1. Introduction

Titze et al. [1] found that the process of human phonation is related to the flow of air in the larynx. Computational fluid dynamics (CFD) is an effective tool for investigating the flow characteristics of airflow through the larynx, which are important for the prevention and treatment of voice disorders and diseases [2,3]. A clinical study has demonstrated the differences in vocal tract geometry between males and females [4]. McCollum et al. [5] numerically simulated the airflow in nine channel models with different geometry and gave the vibrations of vocal tracts under the action of airflow, thus indicating the effect of changes in geometry on vocalization. Šidlof [6] has also explored the vocalization process of the human vocal tract through numerical simulation.

It is notable that the geometries of the channels considered in the literature [5,6] are symmetric, while the numerical results of the flow fields are asymmetric at some specific value ranges of the characteristic parameter Reynolds number (Re). In addition, all the boundary conditions of the given problem are independent of time; however, the numerical results of the flow field are oscillatory and time dependent when Re exceeds a certain critical value (Rec). This exhibits a nonlinear phenomenon of flow, where the numerical solution bifurcates.

This paper aims to investigate nonlinear phenomena in the physical modeling of the fluid flow in a sudden expansion and contraction channel. In this paper, to simplify the problem and focus the study on the nonlinear behavior of fluid flow, the biological larynx is simplified to an idealized geometric model: a two-dimensional symmetric sudden expansion and contraction channel. Numerical simulation was carried out to study the fluid flow behavior, especially the nonlinear behavior in a sudden expansion and contraction channel, as well as the effect of geometry on these characteristics to support the prevention and treatment of voice disorders and diseases.

Researchers from different engineering backgrounds have explored the nonlinear phenomena of fluid flow within various geometrical structures and the bifurcation of their solutions [7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23]. Except in the study of Erturk and Allahviranloo [7], the physical models studied in this literature [8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23] share one common feature: some kind of symmetry in geometrical structure. However, the flow in these models may be asymmetric and steady-state (i.e., static bifurcation has occurred). Furthermore, another feature shared by these physical models is that the boundary conditions are time-independent. However, the flow therein may be variable and oscillate over time (i.e., dynamic bifurcation has occurred).

Erturk and Allahviranloo [7] studied forced flow within a semi-elliptical cavity. Although this is not a symmetric structure, a bifurcation of the numerical solution still occurs, and the solution is not unique. The nonlinear properties of the flow in the sudden expansion channel have been the most studied [8,9,10,11,12,13,14,15,16]. When Re increases to a specific value of Rec, a stable asymmetric solution is observed and the flow starts to be asymmetric. Different geometrical models and different sudden expansion ratios (Er) have values that correspond to Rec. Drikakis [8] and Revuelta [9] found that Rec decreases with increasing Er. The numerical calculations used by Sobey and Drazin [10] and Fearn et al. [11], as well as the linear stability analysis of Weihs [12], show that symmetry breaks as a result of supercritical pitchfork bifurcation as a solution to the Navier-Stokes equations. Battaglia et al. [13] pointed out the coexistence of the two stable solutions under the above value of Rec. Sobey and Drazin [10] found that there was another unstable solution. Stable bifurcations of three-dimensional flows with sudden expansion through plane symmetry have been studied in the literature [14], and it has been shown that two stable asymmetric solutions and one unstable symmetric solution coexist in it. Guevel et al. [15] investigated the fluid flow inside a three-dimensional sudden expansion channel by varying the geometrical parameters and derived several major Recs of bifurcation. Moallemi and Brinkerhoff [16] directly simulated the two-dimensional sudden expansion flow condition, observed the bifurcation phenomena of the flow and determined the value of Rec for each Er.

Working in a biomedical research context similar to that of this paper, Saha [17] numerically modeled blood flow in the human heart by assuming blood to be a Newtonian fluid. The bifurcation of blood flow behavior, nonlinear behavior and bifurcation of solutions within the planar contraction-expansion channel are analyzed at different Re and contraction ratios. Laskos et al. [18] simulated the nonlinear behavior of flow in a three-dimensional rectangular cross-sectional channel with a surface-mounted rib. Wang et al. [19] investigated the nonlinear behavior of flow and heat transfer in a protruding channel with a circular obstacle. In addition to the geometrical structure, they also investigated coupled temperature distribution and heat transfer. The nonlinear properties of flow and mass transfer in symmetric circular and semicircular shallow chambers have been studied in the literature [20] using a model coupling concentration distribution and mass transfer. The nonlinear behavior of natural convection in a horizontal cavity has been studied [21]; the model studied here still has a symmetric structure, but natural convective heat and mass transfer are coupled and the flow is driven by buoyancy due to the density and concentration changes in a gravitational field.

In the studies of Mizushima et al. [22,23], the geometries used are similar to those in the present study but are different in terms of aspect ratio and sudden expansion ratio. These two papers used the vorticity–stream function method and the finite element method. In contrast, this paper uses the SIMPLE method, based on the finite volume of the primitive variables numerical method, to deal with the coupling of velocity and pressure. The boundary conditions of the vorticity–stream function method are approximate; the result is not in one-to-one correspondence with the numerical results calculated by the primitive variable method in this study and the nonlinear behavior of the resulting flow and velocity fields may not be the same. This study shows the numerical solution of the primitive variable method, and it may be a good topic for future research if further study is carried out to see if the nonlinear behaviors of the two methods agree well.

In this paper, the SIMPLE method based on primitive variables is used to explore the nonlinear behavior of the numerical solution of the flow in a sudden expansion and contraction channel, such as the occurrence of pitchfork bifurcation, inverse pitchfork bifurcation and Hopf bifurcation. The critical Reynolds numbers Rec1, Rec2 and Rec3 for the pitchfork bifurcation, inverse pitchfork bifurcation and Hopf bifurcation are quantitatively given. Moreover, the effects of geometrical factors, such as sudden expansion ratio Er and aspect ratio Ar, on the critical Reynolds numbers Rec1, Rec2 and Rec3 at the bifurcation point, where the bifurcation occurs, and on the maximum return velocity are also investigated. This is a new conclusion, obtained from the numerical simulations carried out in this paper while changing the geometry. The results derived from this paper can be used as a reference for biomedical engineering and other related areas.

2. Materials and Methods

2.1. Physical Models

Figure 1 is a schematic diagram of the sudden expansion and contraction channel. The channel consists of a narrow channel at both ends and a wide channel between the two ends, and the medium flows through the channel in the process of sudden expansion and contraction of the flow. As shown in Figure 1, the channel is completely symmetrical about the horizontal central axis (x-axis). The channel inlet section inlet velocity is uniform for U_m, and the inlet and exit section heights (h) are the same. The middle section height is H and the length of the channel is L₀. The intersection of the sudden expansion cross-section and the axis of symmetry is at point O. A Cartesian coordinate system is established with point O as the origin and the flow direction is in the x-axis positive direction. The point with horizontal distance H from point O is P (x, y) = (H, 0). The velocity U₀ at point O, which is the maximum velocity throughout the channel and taken as the reference velocity for the non-dimensional disposal of velocity, is dimensionless, and the dimensionless instantaneous velocity V(t) in the y-direction at P is used as a representative quantity for nonlinear characteristics.

It is worth noting that the lengths L₁ and L₂ of the inlet and outlet sections of the channel are taken to be 60 h in order to ensure that the flow velocity at the inlet and outlet reaches a fully developed state.

2.2. Numerical Method

Define the following dimensionless parameters:

$$U=\frac{u}{U_0},V=\frac{v}{U_0},P=\frac{P}{\rho U_0^2},X=\frac{x}{H},Y=\frac{y}{H},T=\frac{U_{0}t}{H}.$$

Here $u$ and $v$, are the velocity components in the horizontal direction x and vertical direction y, respectively, $U_0$ is the velocity at point O, P is the pressure and $\upsilon$ and $\rho$ are the kinematic viscosity and density, respectively.

The dimensionless eigenvalue Reynolds number Re is defined as follows:

$$Re=\frac{U_{0}H}{\upsilon }$$

Dimensionless geometric quantities are defined as follows:

$$Er=\frac{H}{h}$$

$$Ar=\frac{Lo}{H}$$

Assuming that the fluid is incompressible, the dimensionless control differential equations for the forced flow in the sudden expansion and contraction channel shown in Figure 1 are as follows:

$$\frac{\partial U}{\partial T}+U\frac{\partial U}{\partial X}+V\frac{\partial U}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{1}{Re}(\frac{{\partial }^{2}U}{\partial X^2}+\frac{{\partial }^{2}U}{\partial Y^2}),$$

(1)

$$\frac{\partial V}{\partial \mathrm{T}}+U\frac{\partial V}{\partial X}+V\frac{\partial V}{\partial Y}=-\frac{\partial P}{\partial X}+\frac{1}{Re}(\frac{{\partial }^{2}V}{\partial X^2}+\frac{{\partial }^{2}V}{\partial Y^2}),$$

(2)

$$\frac{\partial U}{\partial X}+\frac{\partial V}{\partial Y}=0.$$

(3)

The corresponding dimensionless boundary conditions are

$$\mathrm{Inlet} U=\frac{2}{3};V=0,$$

(4)

$$Outlet \frac{\partial U}{\partial X}=0; \frac{\partial V}{\partial X}=0,$$

(5)

$$Wall U=V=0,$$

(6)

Initial conditions are

$$T=0, U=V=0, P=0$$

(7)

All computational tasks in this paper are performed using the commercial computational software Ansys Fluent, Canonsburg, PS, USA (ANAYS 2020 R2) [24]. The momentum Equations (1) and (2) based on the primitive variables are discretized using the finite-volume method and the QUICK scheme, the unsteady state is discretized using the second-order implicit difference scheme and the coupling of pressure and velocity is performed using the SIMPLE method [25]. The convergence index for the iterative computation is that the iterative residuals become less than 1 × 10⁻⁵.

2.3. Grid Validation and Time Step Setting

In order to ensure computational accuracy and computational efficiency, a two-dimensional unstructured mesh is used to divide the computational region. Considering the local flow details, the meshes near the wall and the sudden expansion and sudden contraction cross-sections are refined. The mesh division is shown in Figure 2.

For Er = 6, Ar = 1.78, the sudden expansion and contraction channel, with different grid totals, are used to compare the velocities at point O and point P for grid independence verification. Based on the results of the grid independence verification, 60,000 grid totals are selected. The results are shown in Figure 3.

When the Re range is less than Rec3 (numerical solution of the critical Reynolds number of the occurrence of oscillation), the flow is calculated to be in a steady state and time-independent. At this time, as the value of the infinitesimal time step does not affect the convergence of the calculation results, the value of the time step was set as a fixed ΔT = 1 × 10⁻². When the Re is greater than the Rec3, the flow is time-varying and oscillating, and the ΔT was set in the range of 1 × 10⁻³~1 × 10⁻⁴; the larger the Re, the smaller the value of the infinitesimal time step.

3. Results

3.1. Numerical Solutions with Re and Bifurcation of the Solution

The computational results indicate that as Re increases, the obtained numerical solution exhibits bifurcation. Based on their main characteristics, the solutions can be divided into four regions: one unique symmetric solution, two asymmetric solutions (pitchfork bifurcation) and one symmetric solution (unstable), a high-order Re unique symmetric solution (inverse pitchfork bifurcation), and an oscillatory solution (Hopf bifurcation).

The distribution of the velocity component U in the x-direction for several different cross-sections at Re = 85 and Re = 185 is shown in Figure 4. In Figure 4a, it can be seen that the cross-section velocity distribution curve at Re = 85 is perfectly symmetric about the symmetry axis (y = 0) and there is a return flow of the fluid (u/U_o < 0).

As can be seen in Figure 4b, the given cross-sectional velocity distribution curve is asymmetric about the axis of symmetry (y = 0), and there is also a return flow of the fluid (u/U₀ < 0). It should be noted that whether Re = 85 or Re = 185, the numerical solution, after going through the initial time-depended phase, no longer varies with time, i.e., it enters the steady state flow. When Re = 85, the numerical solution is unique, symmetric and steady-state; but when Re = 185, the numerical solution is not unique and there are at least three solutions. Figure 4b shows one of these three solutions, which is steady-state and asymmetric. One other solution is symmetric, but it is unstable and numerical calculations cannot obtain it. It is easy to prove that, for the mathematical models given in this paper (Equations (1)–(7)), symmetric solutions exist. However, a symmetric solution can be obtained by numerical calculation only if it is stable, such as the velocity distribution at Re = 85 given in Figure 4a. At Re = 185, there is a third solution, which is steady-state, asymmetric and antisymmetric with respect to the axis of symmetry (y = 0) in the one given earlier (Figure 4b). This solution can be obtained by numerical calculations.

Figure 5 shows a more typical flow diagram for several different Re values when numerical simulations are obtained. As can be seen from Figure 5a,b, the numerical results given for Re = 85 and Re = 173 are perfectly symmetric and steady-state and do not vary with time. As can be seen in Figure 5c,d, at Re = 185, two different solutions can be obtained, both steady-state and asymmetric, but anti-symmetric about the axis of symmetry (y = 0). As mentioned earlier, at Re = 185, there are at least three steady-state numerical solutions, two of which are asymmetric (antisymmetric to each other) and are called the pair of solutions, while one is symmetric. However, the pair of solutions is unstable and cannot be obtained by numerical calculations. As can be seen in Figure 5e,f, when Re = 248 and Re = 345, the numerical results given are completely symmetric and steady-state and do not vary with time.

In summary, the numerical results show that, for the physical and mathematical models given in this paper, there are three critical Reynolds numbers (Rec, Rec1, Rec2 and Rec3) for the numerical solution to bifurcate. When Re ≤ Rec1 (≈185), there is only one symmetric numerical solution. When Rec1 < Re ≤ Rec2 (≈213), there are two asymmetric numerical solutions and one unstable symmetric solution—a pitchfork bifurcation occurs. When Rec2 < Re ≤ Rec3 (≈355), the numerical solution returns to the unique symmetric solution and an inverse pitchfork bifurcation occurs. When Re > Rec3, the numerical solution is an oscillatory solution and a Hopf bifurcation occurs.

Regarding the case when Re > Rec3, only the nonlinear phenomenon that the solution is oscillating is pointed out here; it will be further discussed later.

The critical Rec at which bifurcation occurs is related to the geometric structure and will be discussed in the following two sections.

3.2. Influence of Geometry on Critical Reynolds Number Rec

As mentioned above, the critical Reynolds number Rec, at which bifurcation occurs, is related to the geometry. The effects of changes in the sudden expansion ratio Er and the aspect ratio Ar on the critical Reynolds number Rec are discussed below.

3.2.1. Effect of Er on Critical Reynolds Number Rec

The changes in the critical Reynolds number, Rec, are discussed when Ar = 2.15 and 4.0 ≤ Er ≤ 6.3.

The variation of the dimensionless velocity component V with Re in the y-direction at observation point P (X, Y) = (1, 0) is given in Figure 6 for Ar = 2.15 and 4.0 ≤ Er ≤ 6.3. From the variation of V with Re given in Figure 6, it is possible to give the critical Reynolds numbers Rec1 and Rec2 at the bifurcation point, where pitchfork bifurcation and inverse pitchfork bifurcation occur.

As shown in Figure 6a, for Er = 4.0, 4.5 and 5.0, the critical Rec1 ≈ 157.21, 152.24 and 145.87, respectively, which decreases with the increase of Er. As shown in Figure 6b, for Er = 5.2, 6.0 and 6.3, the critical Rec1 ≈ 141.94, 149.12 and 154.15, respectively, which increases with the increase of Er. Rec1 takes the minimum value at Er = 5.2. The results given above show that the critical Reynolds number Rec1 at the bifurcation point first decreases and then increases with the increase of Er, and Rec1 takes the minimum value when Er = 5.2, as shown in Figure 7a.

Figure 6 also shows the variation of critical Rec2 with Er for the occurrence of antitone fork bifurcation. In Figure 6a, for Er = 4.0, 4.5, and 5.0, the critical Rec2 ≈ 278.33, 304.29, and 328.45, respectively; in Figure 6b, for Er = 5.2, 6.0, and 6.3, the critical Rec2 ≈ 336.89, 370.24, and 383.06, respectively. Rec2 increases with the increase of Er, as shown in Figure 7b.

3.2.2. Effect of Ar on Critical Reynolds Number Rec

The variation of the dimensionless velocity component V in the y-direction with Re at observation point P(X, Y) = (1, 0) is given in Figure 8 for Er = 6.0 and 1.78 ≤ Ar ≤ 2.15.

As shown in Figure 8, the critical Rec1 for the occurrence of tuning fork bifurcation is about 181.5 when Ar = 1.78, while the critical Rec1 for the occurrence of pitchfork bifurcation decreases to 147.32 when Ar = 2.15. The variation of the critical Reynolds number, Rec1, with respect to Ar is given in Figure 9a. From Figure 9a, it can be seen that Rec1 decreases with the increase in Ar.

Figure 8 also shows the variation of critical Rec2 with Ar for the occurrence of inverse pitchfork bifurcation. As shown in Figure 8, the critical Rec2 for the occurrence of inverse pitchfork bifurcation is about 212.58 when Ar = 1.78, whereas the critical Rec2 for the occurrence of inverse pitchfork bifurcation increases to 369.35 when Ar = 2.15. The variation in the critical Reynolds number Rec2 with Ar is given in Figure 9b. From Figure 9b, it can be seen that Rec2 increases with the increase in Ar.

It should be noted that the above Rec1 and Rec2 values given in this paper are obtained by numerical approximation, and therefore, strictly speaking, they are approximate values.

3.3. Influence of Geometry on Maximum Return Velocity

In this paper, the main flow direction of the channel is in the x-direction of the coordinate axis. Due to the influence of this geometrical structure of sudden expansion and contraction, as shown in Figure 5, all the flow line diagrams given have reflux vortices, i.e., there is a flow in the direction opposite to the direction of the x-axis, so that the velocity in the recirculation zone is less than zero. The so-called maximum return velocity is the maximum value of velocity that is opposite to the x-axis direction. The absolute value of the ratio of the maximum return velocity to the characteristic velocity (Uo) is defined as the dimensionless maximum return velocity Ur(max). The effect of geometric dimensions on Ur(max) is discussed below.

3.3.1. Effect of Er on Ur(max)

For a fixed Ar = 2.15, the general trend is to increase Ur(max) with increasing Re. As shown in Figure 10a, when 4.0 ≤ Er ≤ 5.0 and Re < Rec2, Ur(max) increases with increasing Re, and Ur(max) also increases with increasing Er. When Re = Rec2, Ur(max) shows one drop, then Ur(max) increases with Re. When 5.2 ≤ Er ≤ 6.3, as shown in Figure 10b, Ur(max) increases with Re but decreases with Er. At Re = Rec2, Ur(max) also falls.

3.3.2. Effect of Ar on Ur(max)

As shown in Figure 11, for a fixed Er = 6.00, the overall trend of Ur(max) increases with Re, with a dip in the middle at Re = Rec2. When Re < Rec2 and 1.78 ≤ Ar ≤ 1.90, the maximum reflux velocity Ur(max) increases with increasing Re and increasing Ar. At Re = Rec2, there is a drop in Ur(max) and then, as Re increases, Ur(max) increases with increasing Re but decreases with increasing Ar. In the range 2.00 ≤ Ar ≤ 6.3, as shown in Figure 12b, the increase of Ar does not have a significant effect on Ur(max), and as Re increases, the maximum reflux velocity Ur(max) increases.

3.4. Oscillatory Solution

As mentioned earlier, the Hopf bifurcation occurs when Re > Rec3 and the numerical solution is an oscillatory solution.

The velocity vs. time series for point P (X, Y) = (1, 0) at Re = 370 is shown in Figure 12. As seen in Figure 12a, the velocity v of the observed point P (X, Y) = (1, 0) oscillates periodically with time. The velocity phase diagram of observation point P (X, Y) = (1, 0) given in Figure 12b can be seen as a closed circular trajectory, indicating that the numerical solution oscillates periodically with time.

Figure 13 shows the streamline diagrams of the flow in the sudden expansion and contraction channel simulated at Re = 370 at several time points during a cycle, where C is the dimensionless time of one cycle. As seen in Figure 13, the main flow regularly undergoes upward bias to downward bias and returns to the original position to complete the cycle. New vortices are regularly generated inside the upper and lower reflux zones by growing, shrinking and shedding. It can be clearly seen from Figure 13 that the shedding of the vortex-like flow pattern is antisymmetric and that it is a process of one against the other. This periodic flow is also called the “flapping” mode.

The velocity v time series and velocity phase diagrams for point P (X, Y) = (1, 0) at Re = 850 and Re = 1200 are given in Figure 14 and Figure 15, respectively.

The velocity v at observation point P (X, Y) = (1, 0) oscillates periodically with time when Re is increased to 850. The phase diagram of the velocity of observation point P (X, Y) = (1, 0) given in Figure 14b is seen as a closed loop with trajectory lines, indicating that the numerical solution oscillates (quasiperiodically) with time.

When Re increases to 1200, the velocity v of observation point P (X, Y) = (1, 0) oscillates with time and becomes irregular. The velocity phase diagram of observation point P (X, Y) = (1, 0), given in Figure 15b, is seen to be a cluttered and disordered trajectory, indicating that flow enters the chaos state.

4. Conclusions

The following results were obtained by numerical simulation of the fluid flow in the sudden expansion and contraction channel.

1. When Ar = 1.78 and Er = 6.00, the numerical solution undergoes pitchfork bifurcation, inverse pitchfork bifurcation, and Hopf bifurcation as the Re number increases. According to the classification of flow characteristics, the numerical solutions are divided into four regions: when Re ≤ Rec1 (≈185), there is only one stable symmetric solution; when Rec1 < Re ≤ Rec2 (≈213), there are two stable asymmetric numerical solutions and one symmetric unstable solution; when Rec2 < Re ≤ Rec3 (≈355), the numerical solution returns to a single stable symmetric solution, and an inverse pitchfork bifurcation occurs; when Re > Rec3, the numerical solution is oscillatory, and a Hopf bifurcation occurs.

2. When Ar = 2.15, 4.0 ≤ Er ≤ 6.3, with the increase of the sudden expansion ratio Er, the critical Reynolds number Rec1 for the occurrence of the pitchfork bifurcation point first decreases and then increases; the critical Reynolds number Rec2 for the occurrence of the inverse pitchfork bifurcation point shows a monotonic increase with increasing Er. When Er = 6.00 and 1.78 ≤ Ar ≤ 2.15, Rec1 decreases monotonically and Rec2 increases with the increasing Ar.

3. Maximum return velocity Ur(max) increases with increasing Re when Re ≤ Rec2; at Re = Rec2, Ur(max) suddenly falls. When Rec2 < Re≤ Rec3, Ur(max) increases monotonically with increasing Re. When Ar = 2.15 and 4.0 ≤ Er ≤ 6.3, Ur(max) first decreases to a minimum value with increasing Er and then increases. When Er = 6.0 and 1.78 ≤ Ar ≤ 2.15, for Ar ≤ 1.9, Ur(max) increases with increasing Re when Re ≤ Rec2 and decreases with increasing Re when Rec2 < Re ≤ Rec3; for Ar > 1.9, the change of Ar has little effect on Ur (max).

4. When Ar = 1.78, Er = 6.00 and Re > Rec3 (≈355), a Hopf bifurcation occurs; as Re increases, the numerical solution appears as a periodic solution, a multiplicative periodic solution, and a chaotic solution.