Pharos University ME 253 Fluid Mechanics 2

1. Pharos UniversityME 253 Fluid Mechanics 2 Revision for Mid-Term Exam Dr. A. Shibl

2. Streamlines • A Streamline is a curve that is everywhere tangent to the instantaneous local velocity vector. • Consider an arc length • must be parallel to the local velocity vector • Geometric arguments results in the equation for a streamline

3. Kinematics of Fluid Flow

4. Stream Function forTwo-DimensionalIncompressible Flow • Two-Dimensional Flow • Stream Function y

5. Stream Function forTwo-DimensionalIncompressible Flow • Cylindrical Coordinates • Stream Function y(r,q)

6. Is this a possible flow field

7. Given the y-component Find the X- Component of the velocity,

8. Determine the vorticity of flow field described byIs this flow irrotational?

9. Momentum Equation • Newtonian Fluid: Navier–Stokes Equations

10. Example exact solutionPoiseuille Flow

11. Example exact solution Fully Developed Couette Flow • For the given geometry and BC’s, calculate the velocity and pressure fields, and estimate the shear force per unit area acting on the bottom plate • Step 1: Geometry, dimensions, and properties

12. Fully Developed Couette Flow • Step 2: Assumptions and BC’s • Assumptions • Plates are infinite in x and z • Flow is steady, /t = 0 • Parallel flow, V=0 • Incompressible, Newtonian, laminar, constant properties • No pressure gradient • 2D, W=0, /z = 0 • Gravity acts in the -z direction, • Boundary conditions • Bottom plate (y=0) : u=0, v=0, w=0 • Top plate (y=h) : u=V, v=0, w=0

13. Fully Developed Couette Flow Note: these numbers referto the assumptions on the previous slide • Step 3: Simplify 3 6 Continuity This means the flow is “fully developed”or not changing in the direction of flow X-momentum 7 5 6 6 3 Cont. 2 Cont.

14. Fully Developed Couette Flow • Step 3: Simplify, cont. Y-momentum 7 3 2,3 3,6 3 3 3 3 Z-momentum 2,6 6 6 6 6 6 6 7

15. Fully Developed Couette Flow • Step 4: Integrate X-momentum integrate integrate Z-momentum integrate

16. Fully Developed Couette Flow • Step 5: Apply BC’s • y=0, u=0=C1(0) + C2C2 = 0 • y=h, u=V=C1hC1 = V/h • This gives • For pressure, no explicit BC, therefore C3 can remain an arbitrary constant (recall only P appears in NSE). • Let p = p0 at z = 0 (C3 renamed p0) Hydrostatic pressure Pressure acts independently of flow

17. Fully Developed Couette Flow • Step 6: Verify solution by back-substituting into differential equations • Given the solution (u,v,w)=(Vy/h, 0, 0) • Continuity is satisfied 0 + 0 + 0 = 0 • X-momentum is satisfied

18. Fully Developed Couette Flow • Finally, calculate shear force on bottom plate Shear force per unit area acting on the wall Note that w is equal and opposite to the shear stress acting on the fluid yx(Newton’s third law).

19. Momentum Equation • Special Case: Euler’s Equation

20. Inviscid Flow for Steady incompressible • For steady incompressible flow, the equation reduces to where  = constant. • Integrate from a reference at  along any streamline =C :

21. Two-Dimensional Potential Flows • Therefore, there exists a stream function such that in the Cartesian coordinate and in the cylindrical coordinate

22. Potential Flow

23. Two-Dimensional Potential Flows • The potential function and the stream function are conjugate pair of an analytical function in complex variable analysis. • The constant potential line and the constant streamline are orthogonal, i.e., and to imply that .

24. Stream and Potential Functions • If a stream function exists for the velocity field u = a(x2 -- y2) & v = - 2axy & w = 0 Find it, plot it, and interpret it. • If a velocity potential exists for this velocity field. Find it, and plot it.

26. Summary 26 Elementary Potential Flow Solutions

27. CH006-29

28. CH006-31

29. CH006-32

30. CH006-34

31. CH006-35

32. CH006-36