Mass in per unit time = rAv =
For steady flow, mass out per unit time =
Rate of momentum in =
Rate of momentum out =
Rate of increase of momentum from AB to CD = = rAv dv à 1
Force due to p in the direction of motion = pA
Force due to p + dp opposing the direction of motion = (p + dp)(A + dA)
Force due to pside producing a component in the direction of motion = psidedA
Force due to mg producing a component opposing the direction of motion = mgcos(q)
Resultant force in the direction of motion = pA - (p + dp)(A + dA) + psidedA - mgcos(q) à 2
The value of pside will vary from p at AB to p + dp at CD, and can be taken as p + kdp where k is fraction.
Mass of fluid element ABCD = m = rg(A + 1/2 dA) ds
And ds = dz/cos(q); since cos(q) = dz/ds
Substituting in equn.2,
Resultant force in the direction of motion = pA - (p + dp)(A + dA) + p + kdp - rg(A + 1/2 dA) dz
= -Adp - dpdA + kdpdA - rgAdz - 1/2 dAdz
Neglecting products of small quantities,
Resultant force in the direction of motion = -Adp - rgAdz à 3
Applying Newton's second law, (i.e., equating equns.1 & 3)
rAv dv = -Adp - rgAdz
dividing by rAds,
or, in the limit as ds à 0,
This is known as Euler's equation, giving, in differential form
the relationship between p, v, r and elevation z, along a streamline for steady flow.
It can not be integrated until the relationship density and pressure is known.
For incompressible fluid, r is constant; therefore the Euler's equation is integrated to give the following:
which is nothing but the Bernoulli equation.
Last Modified on: 14-Sep-2014
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