Re: fluid analogs of battery, resistor, capacitor, inductor

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

I won't comment of the various fluid models put forth as analogs for
electrical devices such as batteries, inductors, and capacitors, but
I do wish to point out a couple of wrinkles regarding the fluidic
analog of a resistor prompted by Ludwik's comment:

> If the unit of the gravitational d.o.p. is J/kg, and the unit of I
> is kg/s, then the unit of r will be watt/kg^2.  How many
> W/kg^2 will a pipe have when its length and area are given?
> I stop here. But this question can be answered, I suppose, if
> the viscosity of water is given. A frictionless pipe and an ideal
> fluid with be an analogy for a superconducting  wire.

First of all, I think Ludwik accidentally flipped his powers of seconds
in his dimensional analysis for his analog of resistance.  Note that
(J/kg)/(kg/s) = J*s/kg^2 = m^2/(kg*s) not W/kg^2.

Second, if one solves for the flow resistance R (using Ludwik's above
assignment of potential difference and current) for a fluid of mass
density [rho] and shear viscosity [eta] moving through an
unobstructed pipe of cross section area A and length L at a sufficiently
slow speed (low Reynolds number limit) that the current is proportional
to the potential drop across the pipe, then one finds that Poiseuille's
(sp?) Law gives: R = K*[eta]*L/([rho]*A)^2 , where K is a fixed
dimensionless proportionality constant that depends on the cross
section shape of the pipe.  (For a circular cross section we have
K = 2*[pi] which is also the upper bound that K can have for any shape
cross section.)

If, instead of using Ludwik's identification, we assign the potential
difference to be the *pressure* drop along the pipe then one of the
factors of [rho] in the denominator of the expression for R drops out.
If we also assign the current to be the volume/(unit time) flowing rather
than the mass/(unit time) then the other factor of [rho] drops out as
well.

This geometric dependence of R (i.e. R being proportional to L/A^2) is
inconsistent with that of electrical resistance in a uniform conducting
wire which is instead proportional to L/A.  The reason for this
difference of behavior is that in the case of electrical resistance the
dissipation is a bulk effect caused by carrier scattering throughout the
interior of the conductor.  Whereas in the fluid case the friction is
caused by the fluid being dragged along the fixed bounding surface of
the pipe which sets up a velocity gradient in the fluid across the pipe's
cross section so that the flow is inhomogeneous with the current density
is greatest in the center of the pipe and tapering to zero at the pipe's
inner surface.  IF the pipe/fluid interfacial surface was perfectly
frictionless then the flow in the pipe would be dissipationless, since in
that case, all of the fluid would travel at the same velocity and there
would be no internal shearing of the fluid to dissipate energy.  In the
electrical conduction case the current density tends to be uniform
(assuming a homogeneous conducting medium and uniform cross section)
throughout the conductor's interior.

In order for the flow analogy between the fluid and electric currents to
be in a closer correspondence we should not assume that the pipe is
unobstructed.  Rather we should think of the pipe as being loosely filled
with a fixed permeable wadding or stuffing that creates friction with the
moving fluid throughout the pipe's interior.  This will tend to cause the
pipe's resistance to be proportional to L/A.  I think D'Arcy's law
provides a better conduction analog to Ohm's law than Poiseuille's law
does.  Instead of an open pipe, a fluid flowing through soil or a sponge
(or maybe petroleum flowing through permeable rock) is a better, albeit a
higher resistance, analog.

David Bowman
dbowman@georgetowncollege.edu