Basic Fluid Dynamics

Momentum p = mv

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

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

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

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)

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

Poiseuille Flow S.GOKALTUN Florida International University

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

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

Separation

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

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)

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.)

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

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

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

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

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 

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