| Chronology | Current Month | Current Thread | Current Date |
| [Year List] [Month List (current year)] | [Date Index] [Thread Index] | [Thread Prev] [Thread Next] | [Date Prev] [Date Next] |
The steady flow of a *sufficiently* viscous liquid through a pipe is
governed by Poiseuille's Law. The steady flow of an electric current
through an electric conductor is governed by Ohm's Law. They are
*not* very analogous when looked at in some detail.
Regarding the analogies mentioned by John Cockman:current,
There are two analogies that I like to use when I introduce this subject:
Voltage is like a water tower high above the ground -- the water is
and a resistor is like a narrow section of pipe. It is easy to visualize
the pipes in series and parallel.
For R = rho(L/A), a resistor is like a crowded hallway. The longer and
narrower the hall, the greater the resistance. Rho can be the number of
people per length of hallway.
The water-in-a-pipe analogy is *not* particularly accurate in terms
of modeling the behavior of electric current in an ohmic resistive
medium. The problem is not with the resistance being proportional to
the path length, but with it being inversely proportional to the
cross section area of the flowing current.
In its usual form the flow of a liquid through a pipes does *not*
have the same mathematical description of for the fluid flow as that
of DC electric currents in electrical conductors (unless the pipe
happens to be stuffed with a very fine-pore porous wadding). The
main conceptual discrepancy is related to the fact that for steady
electric current in a conductor the electric field is *locally*
proportional to the current density. This is the essense of Ohm's
law. It is a local interior *bulk* effect. It has as a consequence
that the resistance of a current carrying conductor is inversely
proportional to the cross section area of the channel (for a given
length of the conductor) and the resistance of a given length of
uniform wire does not depend on the cross sectional shape of the
wire. IOW, the DC resistance of a meter length of square cross
sectional Cu wire of 1 mm^2 area is the same as the DC resistance of
a meter length of circular cross sectional Cu wire whose cross
section area is also 1 mm^2.
*But* in the hydraulic analogy involving the steady flow of a viscous
liquid through a pipe the dissipation is *not* quite a local bulk
effect in that the local pressure gradient is not proportional to the
current density. In the pipe the source of drag is the mismatch
between the velocity of the walls of the pipe and the velocity of the
fluid in the center of the pipe. In this case the velocity field is
non-uniform over the cross section of the pipe with the fluid moving
the fastest in the center, and slowest near the pipe walls. The
entire fluid's flow is only impeded by the stickiness of the fluid
touching the *surface* walls and the resulting relative viscous
shearing action of the various parts of the fluid at different
distances from the wall. (Here we are assuming that the Reynolds
number for the flow is sufficiently low enough so that we don't need
to consider turbulence effects, eddy viscosity, and the like.) In
fact, if the *surface* walls did not exert any frictional forces on
the fluid film in contact with them, then the entire steady state
flow of the fluid in the pipe would be a *dissipationless*
'supercurrent' where all the fluid flowed uniformly throughout the
pipe at one velocity with no loss or pressure drop. This different
kind of drag (relative to the case of electric current) causes the
flow resistance of a given length of pipe to be inversely
proportional to the *square* of the pipe's cross sectional area,
*and* the resistance depends strongly on the actual shape of the
cross section (a circular cross section gives the smallest flow
resistance of all shapes of a given cross section area and overall
length).
The steady flow of a *sufficiently* viscous liquid through a pipe is
governed by Poiseuille's Law. The steady flow of an electric current
through an electric conductor is governed by Ohm's Law. They are
*not* very analogous when looked at in some detail. *But*, if we
change our analogy from that of flow through a pipe to that of fluid
flow through a fine porous medium, such as water through a sponge or
filter, or ground water or petroleum through rocks, then the analogy
with electric current is much better. In this later case the fluid
flow is governed by D'Arcy's Law where the pressure gradient *is*
locally (here 'locally' means not quite *so* fine scale that the
medium's manifest inhomogeneity is apparent) proportional to the
current density, and this kind of flow really *is* mathematically
analogous to DC electric current flow through a conducting medium
(at least from the point of view of the macroscopic equations
describing the steady state flow).
David Bowman
David_Bowman@georgetowncollege.edu