Bernoullis equation 1738

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Bernoullis equation (1738)
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Bernoullis equation
z
Same speed as if we dropped the water!
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What about tubes and viscosity?
Newtonian Fluid
U
Shear stress is proportional to strain rate
Predicting frictional losses in tubes stumped
scientists for 100-150 years!
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Leonhard Euler

• Came up with Eulers equations, 1752.
• Modern presentation of the derivation and
  equations.
• Equations were not solvable at the time.
• First time to have a field built on equations
  rather than general principles expressed in
  words.

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Claude-Louis Navier

• Problem of fluid resistance was unknown at the
  time Euler equations had little practicality.
• Proposed Navier-Stokes equation in 1822.
• Boundary conditions were unknown Navier used
  the wrong conditions to match bad experimental
  data.
• useless equation for pipe flow.
• Attacked by Cauchy, Saint-Venant, Poisson and
  others.

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George Gabriel Stokes

• Studied fluid friction in 1840s
• Re-derived, clarified NS equations.
• Experiments on damped pendulum in fluid.
• Derived drag laws, got the right boundary
  conditions.

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Osborne Reynolds

• Established criteria for laminar-turbulent
  transition, 1883.
• Credited with non-dimensional numbers but not
  really

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Why use dimensionless data?
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Dimensionless representation
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Moody chart f(Re)
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Modified Bernoullis equation(Flow in tubes)
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Tank to tank (laminar)
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Flow prop. to effort!
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Tank to ground (laminar)
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Tank to ground (laminar)
General case
Short tube limit (Bernoulli)
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Long tube limit
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What happens here?
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Properties of water

• Density 1000 kg/cubic meter
• Dynamic viscosity 0.001 Pascal seconds
• Tube diameter 1/8 inch 3.175 mm