TABLE OF CONTENTS

I am grateful to him for allowing me to work with him as a research student and for every effort he put in to get me on track with my thesis. Khandker Farid Uddin Ahmed, Professor and Head, Department of Mathematics, Bangladesh University of Engineering and Technology, for his professional guidance and valuable suggestions. I would like to express my sincere thanks to all other respected teachers of this department for their valuable comments, inspiration, occasional guidance and all necessary help during M.Sc.

I would like to thank the staff of the Department of Mathematics, BUET, for their cooperation in this work. I would like to thank the Ministry of Science and Technology and the Government of the People's Republic of Bangladesh for providing NST grants during this work. Numerical results are presented for the two-dimensional flow in a wedge separated by an angle 2𝛼 and bounded by circular arcs at the inlet/outlet for the radial outflow of the fluid.

Also, the stability of the solutions is shown by the fork bifurcation and the relation 𝛼 − 𝑅𝑒 for two different types of input profiles. Comparisons with previously published results were performed and the results were found to be in excellent agreement.

• Overview of Viscous Flows in Sectors and Domains with Corners
• Jeffery –Hamel Flows
• Stability Analysis for Systems
• Basic Concepts of Bifurcation Analysis
• The Pitchfork Bifurcation
• Hopf Bifurcation
• Objectives of the Present Work
• Organization of the Thesis

In this regard, those points in the 𝛼 − 𝑅𝑒 plane that correspond to bifurcations of Jeffery-Hamel states may be of interest. 13] identified the limit ℬ2 as the point at which the Jeffery-Hamel solutions provide a good description of the flow in an expanding channel. 15] examined an experimental study of flow in a wedge angle of 0.28 and a radius ratio (𝜂) of approximately 29.

For values ​​of Reynolds number greater than 𝑅𝑒̂𝑐, which suggests some kind of instability of the symmetric Jeffery-Hamel solution. Tutty [16] interpreted the existence of the wave for Reynolds number less than 𝑅𝑒𝑐 the flow becomes unstable. Unstable: A critical point is unstable if at least one eigenvalue of the Jacobian matrix has positive or positive real parts.

Hyperbolic point: An equilibrium is hyperbolic if all eigenvalues ​​of the Jacobi matrix have non-zero real parts. Bifurcation analysis is a powerful method for studying the steady-state nonlinear dynamics of a system. Similarly, a pitchfork bifurcation is called subcritical if a non-trivial fixed point occurs at parameter values ​​lower than the bifurcation value.

The present study is a stability analysis of the Jeffery-Hamel solution and its relation to flow in a diverging channel.

Figure 1.2: The Flow regimes 𝐼𝐼 1 , 𝐼𝐼 2 , 𝐼𝑉 1 and 𝑉 1

• Numerical Solution Method of Computational FluidDynamics
• Mathematical Model
• Discretization Method
• Numerical Grid
• Finite Approximations
• Solution Technique
• Discretization Approaches
• Finite Element Method
• Mesh Generation
• Computational Procedure of Finite Element Formulation
• General
• Physical Configurations
• MathematicalFormulation
• Governing equations
• Dimensional Analysis
• NumericalAnalysis
• Mathematical Formulation for Finite Element Method
• Hermite-Pad𝑒 Approximant Solution
• Validation of numerical scheme

It is essentially a discrete representation of the geometric domain on which the problem is to be solved. Some of the available options are structural (regular) grid, block structured grid, unstructured grid, etc. The FEM is a numerical method that searches for an approximate solution of the distribution of field variables in the problem domain that is analytically difficult to obtain.

Development of the finite element model of the problem using its weighted-integral or weak form. The number of nodes in each element depends not only on the number of vertices in the element, but also on the type of element interpolation function. Then, the numerical solutions of the variables are obtained while the convergent criterion is satisfied.

Expanding the left side of the equation in ∑𝑑𝑖 = 0𝑃𝑁 [𝑖] (𝑥) 𝑈𝑖 (𝑥) = 𝑂 (𝑥𝑁) 𝑎𝑠 𝑥 → 0 powers of x and equated to the first N equations in the system equal to zero, a system of . In order to calculate the coefficients of Hermite-Pad𝑒́ polynomials, we required some form of normalizations, such as It is important to emphasize that the only input required for computing Hermite-Pad𝑒́ polynomials is the first N coefficient of the series 𝑈0,.

After non-dimensioning, the domain size will be parameterized by the radius ratio 𝜂. Some steps of the mathematical formulation for the above physical configurations are shown below. These assumptions imply that the velocity field is of the form 𝑽 = [𝑢𝑟, 𝑢𝜃] where 𝑢𝑟 is a function of 𝑟 and 𝜃.

The solution of equation (3.8) is particularly simple and inspires to form the non-dimensional solution of the problem. We solve the resulting dimensionless form of the Navier-Stokes equations (3.20) using the finite element method. The finite domain problem is raised by non-dimensionality in terms of constant radial volume flux of inlet Q and outlet radius 𝑅̂0.

Finite Element Library FEM Multiphysics solver with a MATLAB interface used to construct and solve the system of algebraic equations associated with the weak form of the Navier-Stokes equations in plane polar coordinates on a finite domain. Implicit in our finite element formulation is the so-called natural boundary condition that the flow is "pseudo-drag free" and that the entrance/exit of the finite domain is yes.

Figure 2.1: A typical two-dimensional Finite Element mesh (Reddy and Gartling [25])

• Effect of angle (𝛼)
• Effect of Reynolds number (Re)
• Effect of radius ratio (𝜂)
• The Bifurcation Structure in Finite Domain
• The Physical Relevance of Numerical results

At 𝛼 = 0.7, the maximum velocity for the divergent channel current is 2.307 and the maximum velocity for the Jeffery-Hamel current is 1.370. It can be seen from Figure 4.1 that the flow velocity profile of the diverging channel is larger than the Jeffery-Hamel flow velocity profile at a constant radial volume flux and at a fixed radius ratio 𝜂. As angle 𝛼 increases, the velocity profile decreases for both the Diverging Channel current and the Jeffery-Hamel current.

At angle 𝛼 and 0.7, there are no visible differences between the streamlines of the Divergence Channel current and the streamlines of the Jeffery-Hamel current. At angle 𝛼 = 1.39, the streamlines of the Diverging Channel current have changed slightly with the streamlines of the Jeffery-Hamel current. At 𝑅𝑒 = 40, the maximum velocity for the divergent channel flow is 3.456 and the maximum velocity for the Jeffery-Hamel flow is 2.398.

At 𝑅𝑒 = 100, the maximum velocity for the divergent channel flow is 3.459 and the maximum velocity for the Jeffery-Hamel flow is 2.414. At 𝑅𝑒 = 120, the maximum velocity for the divergent channel flow is 3.463 and the maximum velocity for the Jeffery-Hamel flow is 2.416. Figure 4.3 shows that the velocity profile of the divergent channel flow is larger than the Jeffery-Hamel flow velocity profile at a constant radial volume flux, a fixed angle 𝛼 = 0.4 and a fixed radius ratio 𝜂.

When the angle 𝑅𝑒 increases, the velocity profile also increases for both Diverging Channel flow and Jeffery-Hamel flow. 𝑅𝑒 = 40 and 80 there are no visible differences between streamlines of Divergence Channel flow and streamlines of Jeffery-Hamel flow. At Reynolds number 𝑅𝑒 = 100 and 120, the streamlines of Diverging Channel flow are slightly changed with streamlines of Jeffery-Hamel flow.

The radius ratio 𝜂 has a significant effect on the divergent channel flow velocity profile and the Jeffery-Hamel flow velocity profile over the physical domain. At 𝜂 = 100, the maximum velocity for divergent channel flow is 3.307 and the maximum velocity for Jeffery-Hamel flow is 2.414. At 𝜂 = 20, the maximum velocity for divergent channel flow is 0.661 and the maximum velocity for Jeffery-Hamel flow is 0.483.

When the radius ratio 𝜂 decreases, the velocity profile also decreases for both divergent channel flow and Jeffery–Hamel flow. At 𝜂 the Diverging Channel streamlines are slightly interchanged with the Jeffery-Hamel streamlines.

Figure 4.1: Effect of 𝛼 on velocity profile at 𝜂 = 100

Summary of the Major Outcome

In the finite domain, the Jeffery-Hamel flow has a sequence of pitchfork bifurcations in the limit of ​​𝑅𝑒 ≫ 1 and 𝛼 ≪ 1. The following can be put forward to the further works as follow-ups of the present research as. So this consideration can be extended to three-dimensional dimensional analysis to investigate the effects of different parameters.

7] Rosenhead, L., (1940), The Steady Two-dimensional Radial Flow of Viscous Fluid between Two Inclined Plane Walls, Proceedings of the Royal Society of London, vol. E., (1962), Laminêre vloei in simmetriese kanale met effens geboë mure, I: On the Jeffery-Hamel Solutions for Flow between Plane Walls, Proceeding of the Royal Society of London, vol. 11] Goldshitik, M., Hussain, F., en Shtern, V., (1991), Symmetry Breaking in Vortex-sources and Jeffery-Hamel Flows, Journal of Fluid Mechanics, vol.

15] Putkaradze, V., and Vorobieff, P., (2006), Instabilities, bifurcations and multiple solutions in expanding channel flows, Physical Review Letters, vol. R., (1996), Nonlinear development of flow in channels with nonparallel walls, Journal of Fluid Mechanics, vol. L., (2011), The Jeffery Hamel Similarity Solution and its Relation to Flow in a Diverging Channel, Journal of Fluid Mechanics, vol.