Basic Fluid Dynamics

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# Basic Fluid Dynamics - PowerPoint PPT Presentation

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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### 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.

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 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

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