# Fluid Flows and Bernoulli’s Principle - West Virginia University · PDF file...

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Fluid Flows and Bernoulli’s Principle

Streamlines demonstrating laminar (smooth) and turbulent flows of an ideal “fluid”

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STRICT DEADLINE OF APRIL 28.

v = Δx/t Velocity

a = Δv/t Acceleration

= Volume through a surface per time

Volume flow rate

Thinking about rates…

I wanted to point out the concept of a rate: the change in something over time. We’ve previously talked about velocity (change in distance over time), acceleration (change in velocity over time). Today we’ll be talking about a VOLUME FLOW RATE, so volume per second (show different versions of this in terms of density and area and velocity).

Rates and fluid flow

A drooly ruminant

Cows sometimes eat small rocks and particulates! Water (an “ideal fluid”) moves rapidly, and water/saliva help flush these

materials from their stomachs.

A cow swallows about 100 Liters (0.1 m3) of saliva each day. Assuming cow swallows it all, what is the volume flow rate

(volume per unit time) of saliva into the cow?

A. 0.1 m3/s B. 1.2 x 10-3 m3/s C. 1.2 x 10-6 m3/s

Units of volume flow rate:

m3/s Q94

I know a lot of you are doing nutrition (either cows or humans). I encourage you to ask your profs about how fluid dynamics operates in cow nutrition, but here’s what I came up with.

Ruminants, or cows, produce tons of saliva. Published estimates for adult cows are in the range of 100 to 150 liters of saliva per day! Water/saliva flows through the rumen rapidly and appears to be critical in flushing particulate matter downstream.

This is about 1 cubic centimeter per second! They’d have to drool a lot or swallow a lot… No wonder cows are so drooly. Anyways the point is: volume flow rate is the amount of volume that goes through some barrier per unit time. We will use this later.

• Non-viscous fluid (no internal friction.) Note: Honey is viscous. Mud is viscous. Water is not. Blood SHOULDN’T be viscous!

• Density is constant.

• Fluid motion is steady.

• No turbulence in the fluid.

Assumptions Today

There are a few limiting assumptions we’ll use today.

All of these make our analysis of fluid flows a lot more simplified in terms of mathematics.

We’ll be talking about something called “Laminar” or “Streamline” flows.

The book (in unassigned reading the rest of this chapter) treats other more complex fluid flows, including viscosity.

Rate of mass in

= Rate of

mass out

time Volume

time Volume 21 =

If the flow rate is constant, the mass going in for each time interval has to equal the mass coming out. This leads to the realization that the VOLUME FLOW RATE at two different points in the pipe should be the same. So WHAT DOES THIS MEAN?

This means as much volume as you push in should come out in the same amount of time.

A1

Δx1

A2

Δx2

Rate of mass in

= Rate of

mass out

time Volume

time Volume 21 =

A1

Δx1

A2

Δx2

Av t xA =

Δ Av

t xA =

Δ

Av t xA =

Δ Av

t xA =

Δ

Therefore, the amount of volume going passing through one end of the pipe at a given time will be the same amount of volume coming out. You can see that we can write the volume passing through the tube at a given time as A delta x. [LIGHT BOARD DERIVATION]

THIS PRODUCT A v IS CALLED THE VOLUME FLOW RATE or the VOLUME FLUX, JUST LIKE WE DID EARLIER. Basically, it tells you that if the cross-sectional area of a channel or pipe is larger, you get slower flow. Smaller channels/pipes get faster flow.

“Continuity” Equation Flow rate is FASTER if pushed through a smaller

cross-sectional area.

A1v1 = A2v2

This is called the continuity equation, and it’s really cool! If you know the volume of fluid flowing into or out of a channel, you can determine the velocity of that fluid at any point along the channel. Big tubes have low velocity, small tubes have high velocity.

[See light board notes for proportionality]

There are a lot of applications where this applies! Plumbing, watering your garden, circulation, GI track.

What do you do if your garden hose does not reach all of your plants?

A1v1 = A2v2

Physics!

Covering part of the hose opening makes the water flow through a smaller cross-sectional area, so the water must flow faster. PHYSICS! It’s all around us if you take the time to think about it!

Aneurysms Is the blood flow faster in a normal blood

vessel or in a blood vessel with aneurysm?

saccular aneurysm

fusiform aneurysm

A. Normal blood vessel B. Aneurysm C. Same in healthy and aneurysed vessel D. Not enough information to determine

Q95

It’s also inside of us. In an aneurysm, your blood vessel gets bulged out. I’d like you to tell me, where is the velocity faster?

ANSWER: A.

Again I encourage you to ask your nutrition profs about fluid dynamics in the circulatory and GI system.

Bernoulli’s Principle If volume flow rate is constant,

and conservation of energy applies to fluids, then…

P1 + 1/2 ρv12 + ρgy1 =

P2 + 1/2 ρv22 + ρgy2

P1, v1, y1

P2, v2, y2

We’re not going to go through the derivation but you can see it in the book. If you take the continuity equation, and apply conservation of energy principles to the fluid, then you can show that for two different points in a fluid flow, the PRESSURE plus the K

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