Fluids Fluid Dynamics, Lecture notes of Fluid Dynamics

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Fluids

Fluid Dynamics

Lana Sheridan De Anza College

April 15, 2020

Last time

• Pascal’s principle
• measurements of pressure
• (^) introduced fluid dynamics

Overview

• fluid dynamics
• (^) the continuity equation
• (^) Bernoulli’s equation
• Torricelli’s law (?)

Fluid Dynamics

We will make some simplifying assumptions:

1 the fluid is nonviscous, ie. not sticky, it has no internal friction between layers 2 the fluid is incompressible, its density is constant 3 the flow is laminar, ie. the streamlines are constant in time 4 the flow is irrotational, there is no curl

In real life no fluids actually have the second property, and almost none have the first.

Flows can have the second two properties, in the right conditions.

Bernoulli’s Principle

Why should this principle hold? Where does it come from?

(^1) Something similar can be argued for compressible fluids also.

Bernoulli’s Principle

Why should this principle hold? Where does it come from?

Actually, it just comes from the conservation of energy, and an assumption that the fluid is incompressible.^1

Consider a fixed volume of fluid, V.

In a narrower pipe, this volume flows by a particular point 1 in time ∆t. However, it must push the same volume of fluid past a point 2 in the same time. If the pipe is wider at point 2, it flows more slowly.

(^1) Something similar can be argued for compressible fluids also.

Bernoulli’s Principle

V = A 1 v 1 ∆t also, V = A 2 v 2 ∆t

This means

A 1 v 1 = A 2 v 2

The “Continuity equation”.

Bernoulli’s Equation

Bernoulli’s equation is just the conservation of energy for this fluid. The system here is all of the fluid in the pipe shown.

Both light blue cylinders of fluid have the same volume, V , and same mass m.

We imagine that in a time ∆t, volume V of fluid enters the left end of the pipe, and another V exits the right.

Bernoulli’s Equation

8a and 14.18b are equal because the fluid is incompressible.) This work ecause the force on the segment of fluid is to the left and the displace - point of application of the force is to the right. Therefore, the net work segment by these forces in the time interval D t is W 5 ( P 1 2 P 2 ) V y 1 y 2 The pressure at point 1 is P 1. P 1 A 1 i The pressure at point 2 is P 2. (^) v 2 ! x 1^ v^1 ! x 2 Point 2 a Point 1 S S " P 2 A 2 i ˆ ˆ b The work done is the sum of the work done on each end of the fluid by more fluid that is on either side of it: W = F 1 ∆x 1 − F 2 ∆x 2 = P 1 A 1 ∆x 1 − P 2 A 2 ∆x 2 (The “environment fluid” just to the right of the system fluid does negative work on the system as it must be pushed aside by the system fluid.) (^1) Diagram from Serway & Jewett.

Bernoulli’s Equation

Notice that V = A 1 ∆x 1 = A 2 ∆x 2 W = P 1 A 1 ∆x 1 − P 2 A 2 ∆x 2 = (P 1 − P 2 )V Conservation of energy: W = ∆K + ∆U

Bernoulli’s Equation

Notice that V = A 1 ∆x 1 = A 2 ∆x 2 W = P 1 A 1 ∆x 1 − P 2 A 2 ∆x 2 = (P 1 − P 2 )V Conservation of energy: W = ∆K + ∆U

(P 1 − P 2 )V =

2 m(v^ 22 −^ v^12 ) +^ mg^ (h^2 −^ h^1 )

Dividing by V :

P 1 − P 2 = 1 2

ρv 22 + ρg (h 2 − h 1 )

P 1 + 1 2

ρv 12 + ρgh 1 = P 2 + 1 2

ρv 22 + ρgh 2

Bernoulli’s Equation

P 1 + 1

ρv 12 + ρgh 1 = P 2 + 1 2

ρv 22 + ρgh 2 This expression is true for any two points along a streamline.

Therefore,

P + 12 ρv 2 + ρgh = const

is constant along a streamline in the fluid.

This is Bernoulli’s equation.

Bernoulli’s Principle from Bernoulli’s Equation

For two different points in the fluid, we have: 1 2 ρv^ 12 +^ ρgh^1 +^ P^1 =^1 2 ρv^ 22 +^ ρgh^2 +^ P^2

Bernoulli’s Principle from Bernoulli’s Equation

For two different points in the fluid, we have: 1 2 ρv^ 12 +^ ρgh^1 +^ P^1 =^1 2 ρv^ 22 +^ ρgh^2 +^ P^2

Suppose the height of the fluid does not change, so h 1 = h 2 : 1 2

ρv 12 + P 1 = 1 2

ρv 22 + P 2