Page:On Faraday's Lines of Force.pdf/10

164 medium through which the ﬂuid is supposed to ﬂow. This resistance depends on the nature of the medium, and will in general depend on the direction in which the ﬂuid moves, as well as on its velocity. For the present we may restrict ourselves to the case of a uniform medium, whose resistance is the same in all directions. The law which we assume is as follows.

Any portion of the ﬂuid moving through the resisting medium is directly opposed by a retarding force proportional to its velocity.

If the velocity be represented by $$v$$, then the resistance will be a force equal to $$kv$$ acting on unit of volume of the ﬂuid in a direction contrary to that of motion. In order, therefore, that the velocity may be kept up, there must be a greater pressure behind any portion of the ﬂuid than there is in front of it, so that the difference of pressures may neutralise the effect of the resistance. Conceive a cubical unit of ﬂuid (which we may make as small as we please, by (5)), and let it move in a direction perpendicular to two of its faces. Then the resistance will be $$kv$$, and therefore the difference of pressures on the ﬁrst and second faces is $$kv$$, so that the pressure diminishes in the direction of motion at the rate of $$kv$$ for every unit of length measured along the line of motion; so that if we measure a length equal to $$h$$ units, the difference of pressure at its extremities will be $$kvh$$.

(11) Since the pressure is supposed to vary continuously in the ﬂuid, all the points at which the pressure is equal to a given pressure $$p$$ will lie on a certain surface which we may call the surface (p) of equal pressure. If a series of these surfaces be constructed in the ﬂuid corresponding to the pressures 0, 1, 2, 3 &c., then the number of the surface will indicate the pressure belonging to it, and the surface may be referred to as the surface 0, 1, 2 or 3. The unit of pressure is that pressure which is produced by unit of force acting on unit of surface. In order therefore to diminish the unit of pressure as in (5) we must diminish the unit of force in the same proportion.

(12) It is easy to see that these surfaces of equal pressure must be perpendicular to the lines of ﬂuid motion; for if the ﬂuid were to move in any other direction, there would be a resistance to its motion which could not be balanced by any difference of pressures. (We must remember that the ﬂuid here considered has no inertia or mass, and that its properties are those only which are formally assigned to it, so that the resistances and pressures are the only things