Basic Fluid Dynamics

Momentum

• p = mv

Viscosity

• Resistance to flow; momentum diffusion

• Low viscosity: Air

• High viscosity: Honey

Viscosity

• Dynamic viscosity

• Kinematic viscosity [L2T-1]

Shear stress

• Dynamic viscosity

• Shear stress 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/• L is a characteristic length in the system• 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:

)(2

)( 22 xaG

xu

x = 0 x = a

u(x)

• G can be gravitational acceleration times density or (linear) pressure gradient (Pin – Pout)/L

Poiseuille Flow

S.GOKALTUNFlorida 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 = 41Tritton, D.J. Physical Fluid Dynamics, 2nd Ed. Oxford University Press, Oxford. 519 pp.

Eddies and Cylinder WakesS

.Go

kaltu

n

Flo

rida

Inte

rna

tion

al U

nive

rsity

Streamlines for flow around a circular cylinder at 9 ≤ Re ≤ 10.(g=0.00001, L=300 lu, D=100 lu)

Eddies and Cylinder Wakes

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.)

S.G

oka

ltun

Flo

rida

Inte

rna

tion

al U

nive

rsity

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: P = /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

21

11

RRP

R

P2

2D Laplace Law

Pin Pout

r

P = /r → = P/r,which is linear in 1/r (a.k.a. curvature)

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: P = cos /r

Young-Laplace Law• The contact angle affects the radius of the

meniscus as 1/R = cos 1/Rsize:

Rsize

R

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 = 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

21

11cos

sizesize RRP

sizeRP

1cos2

sizeRP

1cos