Transcript Basic Fluid Dynamics - Florida International University

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:
G 2
2
u ( x) =
(a  x )
2m
• G can be gravitational acceleration
times density or (linear) pressure
gradient (Pin – Pout)/L
u(x)
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
Re = 41
Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford
University Press, Oxford. 519 pp.
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:
1
1 
DP = g   
 R1 R2 
• R2 can sometimes be infinite
• But for full- or semi-spherical meniscii –
drops, bubbles, and capillary tubes – the
two radii are the same and
2
DP = g  
R
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 = cosq 1/Rsize:
R
Rsize
q
0
30
60
90
R/Rsize
1
1.15
2

Young-Laplace Law
• The contact angle affects the radius of the
meniscus as 1/R = cosq 1/Rsize, so we end up
with


1
1 

DP = g cos q

 Rsize Rsize 
1
2 

• If the two Rsizes are equal (as in a capillary tube),
we get
 1 

DP = 2g cos q 
 Rsize 
• If one Rsize is infinity (as in a slit), then
 1 

DP = g cos q 
 Rsize 