2.  Consider a fluid-filled tube that has two diameters:  Imagine dying a portion of the fluid green. Topic 2.2 Extended L – Fluid dynamics  If we apply a force F 1 on the left volume to push it through the pipe, the fluid beyond the right volume will apply a force F 2 in opposition to this fluid motion: F1F1 F2F2  If the fluid is incompressible, the volumes will remain equal. Thus x1x1 x2x2 A 1  x 1 = A 2  x 2 A 1 v 1  t = A 2 v 2  t Why? Why? A 1 v 1 = A 2 v 2 Equation of Continuity for Incompressible Fluid FYI: The reason this is called the equation of "continuity" is because no fluid is lost. There can be no leaks in the system.  Imagine now taking a movie of the dyed fluid as it is pushed from left to right. Two images are shown here: T HE E QUATION OF C ONTINUITY

3.  If the fluid IS compressible, then the densities change  Then, instead of having the volume remain constant, we have the mass remain constant: Topic 2.2 Extended L – Fluid dynamics  1 A 1  x 1 =  2 A 2  x 2 Why?  1 A 1 v 1 =  2 A 2 v 2 Why? Equation of Continuity for Compressible Fluid  We can solve the equation of continuity for v 2 : v 2 = v 1 1A12A21A12A2 The Flow Rate Equation for Compressible Fluid  Of course, if the fluid is incompressible, this reduces to v 2 = v 1 A1A2A1A2 The Flow Rate Equation for Incompressible Fluid FYI: It is best to learn the more general equations for COMPRESSIBLE fluids. They reduce to the simpler equations if the fluid is incompressible, because  1 =  2, and therefore cancels. T HE E QUATION OF C ONTINUITY

4.  When the railways began to spread, and busy trunk lines had tracks parallel to one another, a phenomenon called "railcar sway" was observed when oncoming trains passed on close tracks.  For some inexplicable reason train cars swayed toward each other when the trains passed at high relative speeds, sometimes even causing damage. Topic 2.2 Extended L – Fluid dynamics B ERNOULLI'S E QUATION  In 1738 the Swiss mathematician Daniel Bernoulli found the reason.  Recall that the net work done on a body was equal to the change in kinetic energy of that body (the work- energy theorem).

5.  Observe our fluid in the pipe like before, only this time in the vicinity of the constriction: Topic 2.2 Extended L – Fluid dynamics B ERNOULLI'S E QUATION F1F1 F2F2 x1x1 x2x2  The net work done on the fluid is given by W = F 1  x 1 - F 2  x 2 W = p 1 A 1  x 1 - p 2 A 2  x 2  Since pressure p = F/A,  The mass of the fluid is given by m =  V  For an incompressible fluid we know that A 1  x 1 = A 2  x 2. W = V(p 1 - p 2 )  But this is just the volume of the fluid. Thus so that W = (p 1 - p 2 ) mm

6. Topic 2.2 Extended L – Fluid dynamics B ERNOULLI'S E QUATION  Invoking the work-kinetic energy theorem we have W =  K W = mv 2 2 - mv 1 2 1212 1212  Putting it all together... (p 1 - p 2 ) = mm mv 2 2 - mv 1 2 1212 1212 (p 1 - p 2 ) =  v 2 2 -  v 1 2 1212 1212 p 1 +  v 1 2 = p 2 +  v 2 2 1212 1212 Bernoulli's Equation (On the level)  If our pipe has a level change from one end to the other like this, then we simply add a potential energy term: p 1 +  v 1 2 +  gy 1 = p 2 +  v 2 2 +  gy 2 1212 1212 Bernoulli's Equation (General)

7. Topic 2.2 Extended L – Fluid dynamics B ERNOULLI'S E QUATION  We can also write Bernoulli's equation in this form: p +  v 2 +  gy = constant 1212 Bernoulli's Equation (General) How does Bernoulli's equation explain the railcar sway phenomenon? When two trains pass each other head on parallel tracks, the relative speed of the near sides is twice as fast.  If we look at the equation we see that p +  v 2 +  gy = constant 1212  The height is not changing, so the equation reduces to p +  v 2 = constant 1212  If v increases (as it does) then p has to decrease, in order to keep the sum constant.  Less pressure on the near sides causes the cars to tip toward one another.

8. Topic 2.2 Extended L – Fluid dynamics B ERNOULLI'S E QUATION  The same phenomenon is taken advantage of in the design of airfoils.  Consider the following airfoil, cutting through the air: Two adjacent air molecules are shown. Airfoil  The top air molecule covered more distance along the wing surface than the bottom one and thus travels faster:  Thus the pressure on the top of the wing is LESS than the pressure on the bottom of the wing.  This generates aerodynamic lift, as long at the plane is traveling forward.