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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 [L 2 T -1 ]. Shear stress. Dynamic viscosity m Shear stress t = m  u /  y. Reynolds Number.

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momentum
Momentum
  • p = mv
viscosity
Viscosity
  • Resistance to flow; momentum diffusion
  • Low viscosity: Air
  • High viscosity: Honey
viscosity4
Viscosity
  • Dynamic viscosity m
  • Kinematic viscosity n [L2T-1]
shear stress
Shear stress
  • Dynamic viscosity m
  • Shear stress t = m u/y
reynolds number
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
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 flow8
Poiseuille Flow

S.GOKALTUN

Florida International University

entry length effects
Entry Length Effects

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

re 1 stokes flow
Re << 1 (Stokes Flow)

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

eddies and cylinder wakes

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

eddies and cylinder wakes13
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 wakes14
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
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 law16
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
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
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 law19
Young-Laplace Law
  • The contact angle affects the radius of the meniscus as 1/R = cos 1/Rsize:

R

Rsize

young laplace law20
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