Basic Fluid Dynamics

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. Re = 30 Re = 40 Re = 47 Re = 55 Re = 67 Re = 100 Eddies and Cylinder Wakes 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=0.00001, 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, 302-307.)

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

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