Ideal fluids streamlines do not meet or cross velocity vector is tangent to streamline A streamline A volume of fluid follows a tube of flow bounded by streamlines Flow obeys continuity equation volume flow rate Q=Av is constant along flow tube A1V,=A2v2 follows from mass conservation if flow is compressible. Physics 121: Lecture 21, Pg 6
Physics 121: Lecture 21, Pg 6 Flow obeys continuity equation volume flow rate Q = A·v is constant along flow tube. follows from mass conservation if flow is incompressible. A1 A2 v1 v2 streamline A1v1 = A2v2 Ideal Fluids streamlines do not meet or cross velocity vector is tangent to streamline volume of fluid follows a tube of flow bounded by streamlines
Steady Flow of Ideal Fluids (actually laminar flow of real fluid) Physics 121: Lecture 21, Pg 7
Physics 121: Lecture 21, Pg 7 Steady Flow of Ideal Fluids (actually laminar flow of real fluid)
Lecture 21 Act 1 Continuity A housing contractor saves v1/2 some money by reducing the size of a pipe from1” diameter to 1/2 diameter at some point in your house 1)Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1 diameter pipe how fast is the water going in the 1/2 pipe? a)2 V1 b)4 V1 c)1/2v1c)1/4v1 Physics 121: Lecture 21, Pg 8
Physics 121: Lecture 21, Pg 8 1) Assuming the water moving in the pipe is an ideal fluid, relative to its speed in the 1” diameter pipe, how fast is the water going in the 1/2” pipe? Lecture 21 Act 1 Continuity A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. v1 v1/2 a) 2 v1 b) 4 v1 c) 1/2 v1 c) 1/4 v1
Conservation of Energy for Ideal fluid Recall the standard work-energy relation W=AK Apply the principle to a section of flowing fluid with volume 8V and mass Sm= psV(here W is work done on fluid W=W ravity Pressure y2 gravity=-om g(y2-y1) pavg(y2-y1) SV lese=p1y-P242∞2 (p1-p2)8v W=Ak=1,mvi,=pov(v2-v) Bernoulli Equation p,+pVi+pgy,=p2+2pv2+pgy2 Physics 121: Lecture 21, Pg 9
Physics 121: Lecture 21, Pg 9 Recall the standard work-energy relation Apply the principle to a section of flowing fluid with volume dV and mass dm = dV (here W is work done on fluid) ( p p ) V W p A x p A x 1 2 pressure 1 1 1 2 2 2 d d d = − = − Vg( y y ) W m g( y y ) 2 1 gravity 2 1 = − − = − − d d W K mv mv V(v v ) 2 1 2 2 2 2 1 2 1 2 1 2 2 1 = = d − d = d − 2 2 2 2 1 1 2 2 2 1 1 Bernoulli Equation p1 + v + gy = p + v + gy y 1 y 2 v 1 v 2 p 1 p 2 dV W =Wgravity +Wpressure W = K Conservation of Energy for Ideal Fluid
Lecture 21 Act 2 Bernoulli's Principle A housing contractor saves v1/2 some money by reducing the size of a pipe from1” diameter to 1/2 diameter at some point in your house . )What is the pressure in the 1 /2 pipe relative to the 1”pipe? a smaller b same c larger Physics 121: Lecture 21, Pg 10
Physics 121: Lecture 21, Pg 10 Lecture 21 Act 2 Bernoulli’s Principle A housing contractor saves some money by reducing the size of a pipe from 1” diameter to 1/2” diameter at some point in your house. 2) What is the pressure in the 1/2” pipe relative to the 1” pipe? a) smaller b) same c) larger v1 v1/2