1. Pharos University ME 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 for Two-Dimensional Incompressible Flow
Two-Dimensional Flow
Stream Function y

5. Stream Function for Two-Dimensional Incompressible 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 by Is this flow irrotational?

9. Momentum Equation
Newtonian Fluid: Navier–Stokes Equations

10. Example exact solution Poiseuille 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 refer to 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 2
Cont. 3 6 5 7
Cont. 6

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

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
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 Elementary Potential Flow Solutions y f Uniform Stream U∞y U∞x
Source/Sink mq
mln(r)
Vortex
-Kln(r) Kq 26 26

27. CH006-29

28. CH006-31

29. CH006-32

30. CH006-34

31. CH006-35

32. CH006-36