Unsteady Hydromagnetic Couette Flow with Oscillating Pressure Gradient

The study investigates unsteady Couette flow under an oscillating pressure gradient and uniform suction and injection, utilizing the Galerkin finite element method. The research focuses on the effect of suction, Hartmann number, Reynolds number, amplitude of pressure gradient, and frequency of oscillation on fluid velocity. Governing equations for the fluid flow are analyzed, providing insight into this complex system.

• Hydromagnetics
• Fluid Dynamics
• Couette Flow
• Finite Element Method
• Oscillating Pressure Gradient

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1. UNSTEADY HYDROMAGNETIC COUETTE FLOW UNDER AN OSCILLATING PRESSURE GRADIENT AND UNIFORM SUCTION AND INJECTION Jennilee Veronique1, Sreedhara Rao Gunakala2*, Victor M. Job3 Author 1, 2, 3: Department of Mathematics and Statistics, Faculty of Science and Technology, The University of the West Indies, Trinidad IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

2. INTRODUCTION The study of unsteady Couette flow under varying different physical effects has been studied extensively in the last few decades. Hydromagnetics, also known as magnetohydrodynamics (MHD), focuses on the dynamics of magnetic fields in electrically conducting fluids such as in liquid metals and plasma. Suction or injection on the boundary contributes significantly in the field of aerodynamics and space sciences. Its main purpose is to influence the flow of fluid in the channel. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

3. INTRODUCTION The finite element method is a powerful and effective numerical method which is applied to real world problems. In applying this method, the domain given is viewed as a collection of subdomains and over each subdomain, the governing equation is approximated using any of the traditional variational methods. In the present study, the Galerkin finite element method is used to investigate unsteady hydromagnetic Couette flow under a sinusoidally-oscillating pressure gradient and uniform suction and injection. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

4. OBJECTIVES To investigate the unsteady hydromagnetic Couette flow under an oscillating pressure gradient and uniform suction and injection. To examine the effect of - Suction (S) - Hartmann number (Ha) - Reynolds number (Re) - Amplitude of the pressure gradient (k) Frequency of oscillation (?) - on the velocity of the fluid. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

5. DISCRIPTION OF THE PROBLEM Figure 1- Schematic Diagram of the Physical System IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

6. DISCRIPTION OF THE PROBLEM Governing equations for the fluid flow: ??+ ??2? ??? ??+ ??0 ?? ??= ?? 2? ??2 ??0 The initial condition is: ? = 0 at ? = 0 The boundary conditions are: ? = 0 at ? = ? = ?0 at ? = IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

7. DISCRIPTION OF THE PROBLEM The governing equations , the initial conditions and the boundary conditions are transformed in dimensionless form by the non-dimensional variables defined as follows: ? , ? = ? , ? = ? ?0, ? = ? ??0 2, ? = ? = ??0 ?0 ?0 is the suction parameter, ?? = ?0 ? ? is the where ? = ? ?0 ? is the Reynolds number. Hartmann number and ?? = IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

8. DISCRIPTION OF THE PROBLEM The non-dimensional equation ?2? ??2 ??2 ?? ??+ ??? 1 ??= ? cos(??) + ??? ?? Where the non-dimensional pressure gradient is ?? ??= ?cos ?? with dimensionless amplitude ? and oscillation frequency (Strouhal number) ? of the pressure gradient. The dimensionless initial and boundary conditions are ? = 0 ?? ? = 0 ? = 0 ?? ? = 1 ? = 1 ?? ? = 1 IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

9. N NUMERICAL SOLUTION OF THE PROBLEM The equation is solved numerically using Galerkin s finite element method under the initial and boundary conditions to determine the velocity distributions for different values of the parameters ??,??,?,? ??? ?. According to the Galerkin s finite element method, a spatial semi- discretization of the problem was performed such that the associated trial and test function spaces are equal, and the finite element approximation is represented using Lagrange basis functions. The Crank-Nicholson scheme was then applied to obtain a system of linear equations, which is solved using the MATLAB. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

10. RESULTS Figure 2- Time development of the velocity ? IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

11. RESULTS Figure 3- Effect of varying Hartmann number (Ha) and Suction (S) on the time variation of ? IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

12. RESULTS Figure 4-Effect of varying Reynolds number (Re) and amplitude of the pressure gradient (k) on the time variation of ? IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

13. RESULTS Figure 5-Effect of varying amplitude of the pressure gradient (k) on the time variation of ? IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

14. RESULTS Figure 6-Effect of varying of frequency of oscillation (?) on the time variation of ? IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

15. CONCLUSION It was revealed that the suction parameter (S), Hartmann number (Ha) and Reynolds number (Re) has a marked effect on velocity. In the case of ? and various values of the frequency of oscillation (?), it was found that velocity increases as the value of ? increases. In addition, as the frequency of oscillation increases, the velocity decreases. IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

16. REFERENCES Attia, Hazem Ali. Unsteady MHD Couette flow with heat transfer in the presence of uniform suction and injection. Mechanics and Mechanical Engineering 12, no. 2 (2008): 165 170. Dorch, Soren Bertil F. Magnetohydrodynamics. Scholarpedia 2, no. 4 (2007): 2295. C. Johnson. Numerical Solution of Partial Differential Equations by the Finite Element Method. Dover Publications: New York. Uwanta, IJ, and MM Hamza. Effect of suction/injection on unsteady hydromagnetic convective flow of reactive viscous fluid between vertical porous plates with thermal diffusion. International scholarly research notices 2014 (2014). IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago

17. QUESTIONS? THANK YOU! IConETech-2020, Faculty of Engineering, The UWI, St. Augustine, Trinidad and Tobago