## Bernoulli's Equation

From the Equation of Contunity,

A _{1}v_{1}= A_{2}v_{2}we know that the fluid must be moving slower at position 1 where the cross section A

_{1}is larger and it must be moving faster at position 2 where cross section A_{2}is smaller. That is, the fluid mustaccelerateas is moves from position 1 to position 2. That means thepressureon the fluid at position 1 must be greater than thepressureat position 2 in order to provide a net force to cause this acceleration. This is an example ofBernoulli's Principlethatthepressureexerted by a moving fluid isgreaterwhere the speed of the fluid issmallerand the pressure issmallerwhere the speed of the fluid isgreater.Now consider fluid that flows -- along a stream tube -- with a change in cross sectional area

anda change in height. Work must be done on the fluid to change itskinetic energyand itspotential energy.At position 1, the force on the shaded portion of the fluid is

F _{1}= P_{1}A_{1}Likewise, at position 2,

F _{2}= P_{2}A_{2}The

workdone at the two positions isW _{1}= F_{1}l_{1}= P_{1}A_{1}l_{1}and

W _{2}= - F_{2}l_{2}= - P_{2}A_{2}l_{2}Gravity also does work,

W _{grav}= m g y_{1}- m g y_{2}= - m g (y_{2}- y_{1})where

m = _{1}A_{1}l_{1}=_{2}A_{2}l_{2}so that

W _{net}= W_{1}+ W_{2}+ W_{grav}We

knowthat the net work on anything equals thechange in kinetic energy,W _{net}= KE = (^{1}/_{2}) m v_{2}^{2}- (^{1}/_{2}) m v_{1}^{2}W

_{1}+ W_{2}+ W_{grav}= (^{1}/_{2}) m v_{2}^{2}- (^{1}/_{2}) m v_{1}^{2}P

_{1}A_{1}l_{1}- P_{2}A_{2}l_{2}- m g (y_{2}- y_{1}) = (^{1}/_{2}) m v_{2}^{2}- (^{1}/_{2}) m v_{1}^{2}(

^{1}/_{2}) m v_{1}^{2}+ P_{1}A_{1}l_{1}+ m g y_{1}= (^{1}/_{2}) m v_{2}^{2}+ P_{2}A_{2}l_{2}+ m g y_{2}(

^{1}/_{2})_{1}A_{1}l_{1}v_{1}^{2}+ P_{1}A_{1}l_{1}+_{1}A_{1}l_{1}g y_{1}== (

^{1}/_{2})_{2}A_{2}l_{2}v_{2}^{2}+ P_{2}A_{2}l_{2}+_{2}A_{2}l_{2}g y_{2}Recall that

A _{1}l_{1}= A_{2}l_{2}= V(

^{1}/_{2})_{1}V v_{1}^{2}+ P_{1}V +_{1}V g y_{1}= (^{1}/_{2})_{2}V v_{2}^{2}+ P_{2}V+_{2}V g y_{2}(

^{1}/_{2})_{1}v_{1}^{2}+ P_{1}+_{1}g y_{1}= (^{1}/_{2})_{2}v_{2}^{2}+ P_{2}+_{2}g y_{2}This means

( ^{1}/_{2}) v^{2}+ P + g y = constantor

( ^{1}/_{2})_{1}v_{1}^{2}+ P_{1}+_{1}g y_{1}= (^{1}/_{2})_{2}v_{2}^{2}+ P_{2}+_{2}g y_{2}If the vertical height y does not change, this means

( ^{1}/_{2}) v^{2}+ P = constantor

( ^{1}/_{2})_{1}v_{1}^{2}+ P_{1}= (^{1}/_{2})_{2}v_{2}^{2}+ P_{2}

Venturi tube or venturi flow meter:Click

herefor anotherExample.

(c) 2005, Doug Davis; all rights reserved

Equation of ContinuityApplications of Bernoulli's EquationReturn to ToC, Fluids