dp1 dp2
--- + --- = 0 or p1 + p2 = P (2)
dx dx
where P is the total pressure of the mixture, and is everywhere
constant, consistently with the conditions of mechanical equilibrium.
Now dp1/dx is the pressure-gradient of the first gas, and is, by
Boyle's law, equal to k1 times the corresponding density-gradient.
Again [rho]1u1 is the mass of gas flowing across any section per unit
time, and k1[rho]1u1 or p1u1 can be regarded as representing the flux
of partial pressure produced by the motion of the gas. Since the total
pressure is everywhere constant, and the ends of the cylinder are
supposed fixed, the fluxes of partial pressure due to the two gases
are equal and opposite, so that
p1u1 + p2u2 = 0 or k1[rho]1u1 + k2[rho]2u2 = 0 (3).