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

[Ludwik Kowalski <kowalskiL@MAIL.MONTCLAIR.EDU>]

Hi David:

Thanks for correcting my mistake, and for pushing the
analogy further than I did.

If an analogy is to be useful then comparing the d.o.p. to
pressure can lead in the wrong direction. My choices of
J/kg and kg/s corresponded to V=J/C and A=C/s. All
analogies have limitations but some are more limited
than others. And, as Donald Simanek often reminds us,
some analogies are not worth using.

Ludwik Kowalski

> 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