Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 1
2014Feb18
Lecture Notes (#8) 2014Feb 13, Hydrodynamics
A. The Continuity Equation
Total Mass is conserved
1. Volume Flux (volume flow through a surface)
The inclusion of the cosine of the angle of incidence is known as “Lambert’s Law” (usually it is
in the context of flux of light through a surface).
Board #10: Assuming no sources or sinks of fluids, the total flux into a closed surface
must be zero (i.e. as much flows in one side must flow out the other side)
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 2
2014Feb18
Board #11: If there is a source of flux contained inside of the closed surface, then
there can be a net flux out of the surface. Example: consider a water tank inside of the
region. The flow out of the region (e.g. a hose) must be due to the loss of water in the
tank. Hence the total mass flux out of the surface must be due to loss in time of mass
density inside of the surface. This is a form of “Gauss’s law”. Below it is expressed in
integral as well as differential form.
Board #12: B. Bernoulli’s Law
Bernoulli’s law contains 3 laws in one: Pascal’s law of depth, Torricelli’s Law and the
Venturi effect.
1. Torricelli’s Law: [A student of Galileo] The flow speed out of the bottom of a tank is
the same as the speed of a mass dropping the height from top of fluid to exit point.
Note then the result is independent of the density of the fluid (analogous to Galileo’s
law that all bodies fall at same rate independent of their mass).
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 3
2014Feb18
Board #13: 2. The Bernoulli Effect (Venturi Effect)
If a fluid speeds up, there is a corresponding drop in pressure. For example, if there is
a narrowing of a tube, by the continuity equation we know the speed must increase.
The pressure will be lower in the narrow tube (Venturi effect).
The rationalization made on the left side of the board is that pressure is due to
collisions of molecules with the wall. If the fluid is moving fast, then there is more
distance between successive collisions with the wall, hence lower pressure. We will
discuss this “kinetic theory” of pressure in more detail in the section on
thermodynamics.
On the right side, we showed that combining the Bernoulli effect with with Pascal’s law
we can derive Torricelli’s law.
Board #14: 3. The Bernoulli Equation
Combine Pascal’s law, Torricelli’s law and the Venturi effect into one law. The sum of
the pressure, kinetic energy (density) and potential energy (density) of a fluid at any
point in the fluid will sum to the same constant.
Note that point 1 and point 2 need not be connected by a streamline!
--This is where we ended in lecture 8.--
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 4
2014Feb18
Lecture Notes (#9) 2014Feb 18, Hydrodynamics Continued
[Note the first 2 boards were devoted to a Hydrostatics problem]
We can derive Bernoulli’s law by first looking at Newton’s 2nd law per unit volume of the
fluid. Force is time change of momentum. Force density is the time change of
momentum density of the fluid. This has two contributions, acceleration of the fluid,
and the change in density.
Board #4:
We consider two different types of forces on the fluids. The “body force” is that gravity
acts on every part of the fluid. The “contact” force is the pressure on the body due to
contact with the rest of the fluid. If the pressure is bigger on one side than the other
than there is a “pressure gradient”, and hence a net force on the body.
Putting it together, we get a differential equation of motion for the fluid. If either gravity
or a body force is present, it can “accelerate” the fluid.
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 5
2014Feb18
Board #5:
Details [Reference Feynman 40-4]: The time derivative of the momentum should be

v , which
with respect to a particular piece of the fluid. This is not quite the same as
t
would be the time derivative with respect to a fixed point of space because the fluid is
moving. One needs to use the “Convective derivative”. Hence the correct form of the
equation of motion would be:
  
v   v  v   P  g
t
Board #6: Review of the Work-Energy Theorem of Mechanics
Derivation of Bernoulli’s law goes in parallel with the derivation of the “Work-Energy”
theorem of mechanics [that the change in Kinetic energy is due to applied work on the
body]. At the bottom of board 5 we state the theorem, and below we show a formal
derivation using the Fundamental Theorem of Calculus.
We then generalize the derivation for fluids. The left side again involves the
fundamental theorem of calculus. The right side is the work, which has two terms to
integrate, the body force (gravity) and the surface force (pressure).
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 6
2014Feb18
Board 7: Gravitational Work
The gravity force work: If we assume that gravity is conservative, then it can be derived from a

potential: g   , where  is the gravitational potential. If we are near the surface of the
earth, the potential is just:   mgh . The work (per volume) done on the fluid is (minus) the
change in potential energy (per volume).
Board 8: Pressure Work
The work (per volume) due to the pressure gradient is quite simple because of the
fundamental theorem of calculus. The work (per volume) along a path is independent
of the path, it just depends upon the pressure at the endpoints.
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 7
2014Feb18
Board 9: Result: we derive the Bernoulli Equation
Put it all together, then the change in the fluid’s kinetic energy (per volume) between two points
is equal to the work done (per volume) connecting these two points, along ANY path you
choose, i.e. its path independent (it need NOT be an actual path along which the fluid flows!).
Rearranged you get the that the sum of three terms (pressure, kinetic energy density
and potential energy density) at any point in the fluid is the same constant as at any
other point in the fluid.
Board 10: Bernoulli’s equation is 3 in one
We now show that Bernoulli’s equation can be reduced to three simpler laws.
(1) Pascal’s law of depth is derived if you say that velocity at the two points is zero (note that it
might be non-zero between the points!).
(2) Torricelli’s law is derived if you choose two points which have the same pressure (although
the pressure might be different between the points!).
Dr. W. Pezzaglia
Las Positas College
Physics 8C, Spring 2014
Lecture #8 & 9: Hydrodynamics
Page 8
2014Feb18
Board 11: (3) The Venturi effect is derived if you choose two points in the fluid for which the
height is the same. It would be more correct to say two points for which the potential energy
density is the same (n.b. if you have two fluids with different densities in contact).
A very practical application is the “Venturi tube”, which is the basis of airspeed measurement in
airplanes and fluid flow meters in a pipe. A constriction in the tube makes the fluid flow faster
(due to Continuity Equation). The faster speed means a lower pressure (i.e. the “Venturi
effect”, also known as “Bernoulli effect”). If the fluid is a gas, then the difference in pressure
can be measured by a u-shaped Mercury Manometer (an application of Pascal’s law of depth,
the difference between the heights gives the pressure difference). If the fluid is a liquid (such
as drinking water) you may not want Mercury to mix with it. Instead use two vertical open-end
tubes on top. The difference in the heights of the columns of water supported will again give
Pressure difference (by Pascal’s law). Putting it all together, these equations give you the
ability to measure the incoming velocity!
C. Viscosity: [Addenda] The subject of viscosity (velocity friction) was not covered in the book. A light
treatment was given in the powerpoint slides, and of course we covered this topic in Lab. However, I will
state that the equation of motion for a fluid with viscosity  is approximately given by [Reference Feynman,
equation (41.15)],
  
v   v  v   P  g   2v
t
Derivation of Poiseuille’s Law: Assume constant flow of a level pipe, then: P  
people have shown that the velocity profile in a cylindrical pipe of radius “R” is
given:
2
v . Elsewhere
 r2 
v(r )  v0 1  2  where by integration over the area you can show
 R 
the average velocity is just half the velocity in the center v0. Applying
Laplace’s operator in cylindrical coordinates you find the force density due to viscosity is a constant:
 1   v  
4v
8v
f visc   2v   
 r     20   2 where the average speed “v” can be related to the flow
R
R
 r r  r  
2
rate “Q” of the pipe: Q  vR . The pressure gradient is the change in pressure over the length “L”, so
8LQ
one derives Poiseuille’s law: P 
R 4
-end of hydrodynamics-