Basic Fluid Dynamics

Momentum

• P = mv

Viscosity

• Resistance to flow; momentum diffusion

• Low viscosity: Air

• High viscosity: Honey

• Kinematic viscosity

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 = v L/• L is a characteristic length in the system• 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 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.

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

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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.)

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Separation

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

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

Laplace Law

Pin Pout

r

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

Young-Laplace Law

• With solid surfaces, in addition to the fluid1/fluid2 interface – where the interaction is given by the interfacial tension -- we have interfaces between each fluid and the surface S1and S2

• Often one of the fluids preferentially ‘wets’ the surface

• This phenomenon is captured by the contact angle

• cos = (S2 - S1

Young-Laplace Law

• Zero contact angle means perfect wetting;

• P = cos /r