1. Basic Fluid Dynamics

2. Momentum
p = mv

3. Viscosity Resistance to flow; momentum diffusion Low viscosity: Air
High viscosity: Honey

4. Viscosity
Dynamic viscosity m
Kinematic viscosity n [L2T-1]

5. Shear stress
Dynamic viscosity m
Shear stress t = m u/y

6. Reynolds Number
The Reynolds Number (Re) is a non-dimensional number that reflects the balance between viscous and inertial forces and hence relates to flow instability (i.e., the onset of turbulence)
Re = u L/n
L is a characteristic length in the system
n is kinematic viscosity
Dominance of viscous force leads to laminar flow (low velocity, high viscosity, confined fluid)
Dominance of inertial force leads to turbulent flow (high velocity, low viscosity, unconfined fluid)

7. Poiseuille Flow
In a slit or pipe, the velocities at the walls are 0 (no-slip boundaries) and the velocity reaches its maximum in the middle
The velocity profile in a slit is parabolic and given by:
u(x)
G can be gravitational acceleration times density or (linear) pressure gradient (Pin – Pout)/L
x = 0
x = a

8. Poiseuille Flow
S.GOKALTUN
Florida International University

9. Entry Length Effects
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.

10. Re << 1 (Stokes Flow)
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.

11. Separation

12. Eddies and Cylinder Wakes
Re = 30
Re = 40
Re = 47
Re = 55
Re = 67
Re = 100
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.
Re = 41

13. Eddies and Cylinder Wakes
S.Gokaltun
Florida International University
Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g= , L=300 lu, D=100 lu)

14. Eddies and Cylinder Wakes
S.Gokaltun
Florida International University
Streamlines for flow around a circular cylinder at 40 ≤ Re ≤ 50.(g=0.0001, L=300 lu, D=100 lu) (Photograph by Sadatoshi Taneda. Taneda 1956a, J. Phys. Soc. Jpn., 11, )

15. Laplace Law
There is a pressure difference between the inside and outside of bubbles and drops
The pressure is always higher on the inside of a bubble or drop (concave side) – just as in a balloon
The pressure difference depends on the radius of curvature and the surface tension for the fluid pair of interest: DP = g/r in 2D

16. Laplace Law In 3D, we have to account for two primary radii:
R2 can sometimes be infinite
But for full- or semi-spherical meniscii – drops, bubbles, and capillary tubes – the two radii are the same and

17. 2D Laplace Law DP = g/r → g = DP/r,
which is linear in 1/r (a.k.a. curvature) r
Pin
Pout

18. Young-Laplace Law
When solid surfaces are involved, in addition to the fluid1/fluid2 interface – where the interaction is given by the surface tension -- we have interfaces between both fluids and the surface
Often one of the fluids preferentially ‘wets’ the surface
This phenomenon is captured by the contact angle
Zero contact angle means perfect wetting
In 2D: DP = g cos q/r

19. Young-Laplace Law
The contact angle affects the radius of the meniscus as 1/R = cos 1/Rsize: R
Rsize  30 60 90
R/Rsize 1
1.15 2 

20. Young-Laplace Law
The contact angle affects the radius of the meniscus as 1/R = cos 1/Rsize, so we end up with
If the two Rsizes are equal (as in a capillary tube), we get
If one Rsize is infinity (as in a slit), then