DARCY-FORCHHEIMER NANO LIQUID FLOW BASED ON CONVECTIVE CONDITIONS pptx

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1. THE UNIVERSITY OF AZAD JAMMU AND KASHMIR MUZAFFARABAD Thesis Defense for Master of Philosophy in Mathematics Title : DARCY-FORCHHEIMER NANO LIQUID FLOW BASED ON CONVECTIVE CONDITIONS . Presented by: Asma Aeman M.Phil . Scholar, DEPARTMENT OF MATHEMATICS University of Azad Jammu & Kashmir Muzaffarabad Supervised by: Dr. Muhammad Zaheer Kiyani

2. OUTLINE:

3. INTRODUCTION The Darcy-Forchheimer law is named after Henry Darcy and Paul Forchheimer. Darcy's law describes fluid flow through porous media based on pressure gradient and permeability. Forchheimer extended Darcy's work to include non-linear and inertial effects. The Darcy- Forchheimer law is a comprehensive model for fluid flow in complex porous media systems. It is a valuable tool in hydrogeology and petroleum engineering for predicting fluid behavior in porous media.

4. Why we study Darcy- Forchheimer law? The Darcy- Forchheimer law is a fundamental concept in fluid dynamics, particularly in the study of porous media and fluid flow through packed beds. We study the Darcy- Forchheimer law for several reasons: Porous media applications : Understanding fluid flow through porous media is crucial in various fields, such as: Groundwater flow and contaminant support Oil and gas reservoir engineering Chemical engineering (e.g., filtration, catalysis) Biological systems (e.g., blood flow, tissue engineering) Pressure drop prediction: The Dracy-Forchheimer law helps predict pressure drops in porous media,

5. essential for designing and optimizing systems. 3) Permeability and inertial effects : The Darcy- Forchheimer law accounts for both permeability(Darcy’s law) and inertial effects( Forchheimer’s extension), providing a comprehensive understanding of fluid flow through porous media. 4) Foundation for advanced models: The Darcy- Forchheimer law serves as a foundation for more complex models, such as the Brinkman- Forchheimer and Navier -Stokes equations. 5) Flow regime characterization : The law distinguishes between laminar and turbulent flow regimes, crucial for understanding and modeling fluid behavior.

6. APPLICATIONS Biomedical Engineering Groundwater Flow Oil Reservoirs Filtration Systems Heat Exchangers Research and Optimization

7. Use of Darcy Forchheimer law in Biomedical Engineering

8. OBJECTIVES Develop a mathematical model for modified Darcy Nano liquid flow with convective conditions. Find numerical algorithm to solve the equations. Analyze the solutions of velocity, temperature and concentration profiles .

9. Basic Definitions & Basic Laws Time Dependent Nano Liquid Flow Capturing Magneto Hydrodynamics & Radiation Effects Numerical Study Of Modified Darcy Fluid Flow With Effects Of Heat Generation & Chemical Reaction LAYOUT OF THESIS

10. Homotopy Analysis Method: The Homotopy Analysis Method (HAM) is a mathematical technique used to solve nonlinear differential equations and systems. It was developed by Ji-Huan in the 1990s. Keller Box Method: A numerical technique designed for solving boundary value problems using discretization and finite difference methods . NDSolve : A powerful function in Mathematica that provides numerical solutions for a wide range of differential equations, utilizing various numerical methods.

11. ASSUMPTIONS OF THE PROBLEM Laminar flow Viscous flow Incompressible flow Two dimensional flow Unsteady flow

12. GEOMETRY OF THE PROBLEM

13. Governing equations under the boundary layer approximation are (1) u F , ( 2 ) (3) ) (4)   PROBLEM FORMULATION

14. , , (5) u , T as y Here u and v are velocity components in x and y directions respectively, T is the temperature and C is the concentration.   BOUNDARY CONDITIONS

15. 𝜂 = y (𝜂) , v =- (6) ,   Here we used the following similarity transformations SIMILARITY TRANSFORMATIONS

16. RESULTING EQUATIONS Equations (2-4) are transformed as , (7) r 𝜃 = 0 , (8) (9)  

17. With corresponding dimensionless boundary conditions :

18. , , = , , , , , (11) , , ,   DIMENSIONLESS PARAMETERS

19. PHYSICAL QUANTITIES The dimensionless forms of Skin friction coefficient, local Nusselt number and local Sherwood number are        

20. To solve Eqs . (7-9), the implicit finite difference scheme (Keller, J. B , 1978) is used. The first step is to convert the Eqs. (7-9) to first order by introducing new variables u , v, t , and w (where f′=u , u ′=v , θ ′=t and φ′=w) and thus Eqs. (7-9) becomes ( 15) ( 16) ( 17)   NUMERICAL SOLUTION

21. Continue.. Centralizing via ( ), we get  

22. Continue.. Using centering difference approximations about ( ) for Eqs. (15-17 ), we have = ( 22) ( 23) ( 24)  

23. BOUNDARY CONDITIONS = S , = 1, = 0 = 1, = ( 25) = 1, = 0 The above nonlinear algebraic equations are linearized using Newton's linearization technique. We then have a linear system which is solved using the block elimination method, as discussed by Keller .  

24. Table 1: Comparison for R when R=0.1, Pr =1.2, n=1.6, =0.5= , A=0.3, =0.01, S=1.1, =0.01= , Le=1.3   Keller Box NDSolve 0.0 0.4 0.6 2.34262 2.34475 0.2 2.43761 2.43225 0.4 2.51944 2.51636 0.2 2.37564 2.37437 0.4 2.43761 2.43225 0.6 2.48181 2.48816 0.3 2.34929 2.3446 0.5 2.40986 2.40357 0.7 2.46492 2.46044 Keller Box NDSolve 0.0 0.4 0.6 2.34262 2.34475 0.2 2.43761 2.43225 0.4 2.51944 2.51636 0.2 2.37564 2.37437 0.4 2.43761 2.43225 0.6 2.48181 2.48816 0.3 2.34929 2.3446 0.5 2.40986 2.40357 0.7 2.46492 2.46044

25. Table 2: Comparison for R when Pr =1.2, Fr=0.2, n=1.4, =0.6, =0.5, A=0.3, =0.01, Le=1.3, S=1.1, =0.01= , M=0.4   (0) Keller Box NDSolve 0.1 0.5 1.6 0.0128282 0.0128286 0.3 0.01581995 0.0158206 0.5 0.0188011 0.0188033 0.7 0.0188009 0.0188021 0.9 0.01880069 0.0188019 1.1 0.01880048 0.0188017 1.4 0.01806084 0.0180621 1.6 0.01880048 0.0188017 1.8 0.019512 0.0195131 Keller Box NDSolve 0.1 0.5 1.6 0.0128282 0.0128286 0.3 0.01581995 0.0158206 0.5 0.0188011 0.0188033 0.7 0.0188009 0.0188021 0.9 0.01880069 0.0188019 1.1 0.01880048 0.0188017 1.4 0.01806084 0.0180621 1.6 0.01880048 0.0188017 1.8 0.019512 0.0195131

26. Table 3:Comparison for R when R=0.1, Pr =1.2, Fr=0.2, =0.6, =0.5, A=0.3, = 0.01, S=1.1, =0.01= , M=0.4   (0) (0) Keller Box NDSolve 1.0 0.5 1.6 0.01123191 0.011233 1.5 0.01128751 0.0112877 2.0 0.01131523 0.0113154 0.7 0.01132716 0.0113273 0.9 0.01133379 0.0113339 1.1 0.01133801 0.0113381 1.4 0.0108929 0.0108929 1.6 0.01133801 0.0113381 1.8 0.01176628 0.0117664 Keller Box NDSolve 1.0 0.5 1.6 0.01123191 0.011233 1.5 0.01128751 0.0112877 2.0 0.01131523 0.0113154 0.7 0.01132716 0.0113273 0.9 0.01133379 0.0113339 1.1 0.01133801 0.0113381 1.4 0.0108929 0.0108929 1.6 0.01133801 0.0113381 1.8 0.01176628 0.0117664

27. Now, we discuss the graphical results of different parameters on velocity, temperature fields and concentration. The system of nonlinear ODE's with boundary conditions solved numerically by Keller box method. The results obtained show the effects of the dimensionless parameters, namely magnetic field parameter(M), porosity parameter , suction parameter(S), Forchheimer number(Fr), Brownian movement parameter , Thermophoresis element( ), Prandtl number(Pr), Radiation parameter(R), Heat source/sink variable( ), Lewis number(Le), Chemical reaction parameter( ), Nusselt number and Sherwood number.   DISCUSSION

28. GRAPHICAL RESULTS Fig 1: Impact of on   Fig 2: Impact of on  

29. Fig 3: Impact of on   Fig 4: Impact of Fr on   GRAPHICAL RESULTS

30. F ig 5: Impact of on 𝜃   Fig 6: Impact of on 𝜃   GRAPHICAL RESULTS

31. F ig 8: Impact of M on   Fig. 7. Impact of Pr on 𝜃(𝜂) Fig. 8. Impact of R on 𝜃(𝜂) GRAPHICAL RESULTS

32. Fig 9 : Impact of S on θ   Fig 10: Impact of on θ   GRAPHICAL RESULTS

33. Fig 11: Impact of Le on 𝜙   Fig 12: Impact of on 𝜙   GRAPHICAL RESULTS

34. Fig 13:Impact of on 𝜙   GRAPHICAL RESULTS

35. Fig 14: Nusselt number vs. R & Pr Fig 15: Sherwood number vs. Le &   GRAPHICAL RESULTS

36. CONCLUSIONS

37. THANK YOU