*blows*.) (Video link: "hose collapse.")

*dynamic*fluids--more specifically, ideal fluid flow. First we'll define what we mean by "ideal" fluids, then we'll see how two conservation laws are applied simultaneously to flowing ideal fluids.

*incompressible*, which water is to some extent. Note that while air is

*not*incompressible, there are situations where we can make the crude approximation that it is.

*laminar*flow, where the adjacent particles flow smoothly past each other, as opposed to turbulent flow, where particles swirl around in a chaotic manner. Like many fluids water can undergo both laminar and turbulent flow, so we'll restrict our attention to certain conditions where water undergoes laminar flow.

*non-viscous*, that is, flow without appreciable frictional losses, as opposed to a viscous fluid that, well, looks and is literally, "gooey." (Video link: "Viscosity.")

*V*/∆

*t*through the pipe is constant. To convince yourself of this you'll need a friend to watch this animation with you. Every time you see fluid particles entering the pipe, say "in." "In. In. In..." Keep doing that. Convince your friend to say "out" every time fluid particles are leaving the pipe. "Out. Out. Out..." If the two of you do this correctly, each time you say "in," your friend immediately follows-up by saying "out." This means that rate of fluid volume going in (represented here by three dots) must continuously be equal to the rate of fluid volume going out. "Stuff in, stuff out," right?

Then from the continuity equation:

*A*

_{1}·

*v*

_{1}=

*A*

_{2}·

*v*

_{2},

since the cross-sectional areas of where the fluid goes in and where it goes out are the same (

*A*

_{1}=

*A*

_{2}), then

*v*

_{1}=

*v*

_{2}, so the speed of the fluid flowing through this pipe must be constant as well.

*V*/∆

*t*through the pipe is also constant. Let's do the same "in and out" exercise as above. Every time you see fluid particles entering the narrow end of the pipe, say "in," while your friend says "out" every time fluid particles are leaving the wider end of the pipe. "In. Out. In. Out. In. Out..." As before, since each time you say "in," your friend immediately follows-up by saying "out," so the rate of fluid volume going in the narrow end (represented here by three dots) must continuously be equal to the rate of fluid volume going out the wider end. "Stuff in, stuff out," right?

Then from the continuity equation:

*A*

_{1}·

*v*

_{1}=

*A*

_{2}·

*v*

_{2},

since the cross-sectional area of where the fluid goes in is smaller than the cross-sectional area of where it goes out (

*A*

_{1}<

*A*

_{2}), then

*v*

_{1}>

*v*

_{2}, so the speed of the fluid flowing through this pipe slows down.

*per*volume will be conserved.)

^{3}, such that they can transfer to/from each other, as long as the net balance of exchanges is zero.

*v*

^{2}) term increase, decrease, or have no change? Let's refer back to the continuity equation discussion above and recall that while the volume flow rate doesn't change, the speed changes, where the fluid slows down travels through this pipe. This means that the kinetic energy density will

*decrease*(as it depends on the square of the speed), and this term will be negative.

Does the gravitational potential energy density term ρ·

*g*·∆

*y*term increase, decrease, or have no change? Since the center of the pipe has no change in height (even though the cross-sectional areas are different, they are still "horizontally aligned" with each other), there is

*no*change in the gravitational potential energy density term, and this term will be zero.

Then as a result, does the pressure of the ideal fluid flowing through this pipe increase, decrease, or have no change? Note the steps in determining the changes (if any) in pressure for this ideal flowing fluid--first we apply the continuity equation to determine the change in speeds (if any), which tells us the change (if any) in the kinetic energy density. We then look at the change in height of the centerline of the pipe (if any), which tells us the change (if any) in the gravitational potential energy density.

*Then*we can look at Bernoulli's equation:

0 = ∆

*P*+ ρ·

*g*·∆

*y*+ (1/2)·ρ·∆(

*v*

^{2}),

and look at the increases (+) or decreases (–) (or no changes) of the terms we know so far:

0 = ∆

*P*+ (0) + (–).

In order to balance out this equation to equal zero on the left-hand side, the pressure of the fluid as it flows through this pipe must

*increase*, making ∆

*P*positive, such that:

0 = (+) + (0) + (–),

and both sides of Bernoulli's equation are balanced. So for this ideal fluid flowing through this widening pipe, the pressure will increase. This is why a weakened, enlarged blood vessel (an aneurysm) is dangerous, as blood flowing through this damaged, wider section will temporarily experience an increase in pressure as it slows down, and may widen the blood vessel even more and cause it to eventually rupture.

*v*

^{2}) term increase, decrease, or have no change? (Refer back to the continuity equation discussion to determine this.) Does the gravitational potential energy density term ρ·

*g*·∆

*y*term increase, decrease, or have no change?

Then as a result, does the pressure of the ideal fluid flowing through this pipe increase, decrease, or have no change?

(Note that the pressure should

*decrease*in the narrow portion of this pipe, which is why the crimped hose at the start of this presentation collapsed--as air flowed through the narrow crimped portion of the hose, its speed increased, which made the pressure decrease inside the hose, and the surrounding atmospheric pressure then flattened the crimped portion even further.)

*v*

^{2}) term increase, decrease, or have no change? (Refer back to the continuity equation discussion to determine this.) Does the gravitational potential energy density term ρ·

*g*·∆

*y*term increase, decrease, or have no change?

Then as a result, does the pressure of the ideal fluid flowing through this pipe increase, decrease, or have no change?

## 2 comments:

I don't know how you can make people read these before class, but make people READ THESE BEFORE CLASS. Read these before the reading. READ THE BLOG PEOPLE!

Stunning.

I truly like the way you passed on your message and clarified the procedure.

GIF pictures are truly useful.

Keep it up .

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