CE360 – Week 1

WEEK 1: DIMENSIONS, PHYSICAL PROPERTIES

TOPICS: I. What is a fluid?

a. Key Properties b. The Continuum Assumption

II. Dimensions—Units a. Dimensional Homogeneity b. In-Class Exercise c. Systems of Units

III. Measures of Fluid Mass & Weight I. What is a fluid? Substance that deforms (or flows) continuously when acted on by a shearing stress (force/area) of any magnitude. This is the definition of a Newtonian fluid. In this class, we are interested in measuring and predicting fluid behavior. Can you think of some good examples where fluid mechanics are important? • Pipe flow (scales ranging from Alaskan pipeline to blood

vessels) • Flight (airfoil design) • Design of bridges, dams, and culverts • Drag (vehicle design)

CE360 – Week 1

Comparing Solids & Fluids

CE360 – Week 1

More Examples of Fluid Mechanics In art…..(see movie: hydraulic opera) and engineering…

A Grand Dam: The Hoover Dam built during the Depression

CE360 – Week 1

I.A. Key Properties We must define and consider key properties used to describe fluids: • Density (Mass per Volume) • Pressure (Force per Area) • Velocity (Length per Time)

We assume that these properties are continuous within a fluid (i.e., we assume the fluid is a “continuum”). 1.B. The Continuum Assumption What is a continuum? From the dictionary: a continuum is defined to be a continuous nonspatial whole or extent or succession in which no portion is distinguishable from adjacent parts. (http://www.wordreference.com/definition/continuum) We can conceptualize the “continuum” idea using a hypothetical thought experiment illustrated below.

CE360 – Week 1

Assume we have a perfect measuring device that can determine the density of a fluid.

How would the density of a fluid change with different scales of analysis? • The length scale of single molecules • The length scale of molecules are uniformly distributed

CE360 – Week 1

We can see from the following figure that the “continuum” scale represents a spatial scale where fluid properties are constant

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II. Dimensions—Units We measure fluid properties with the fundamental dimensions of • L: length • T: time • M: mass

Table 1.1 provides dimensional definitions for key physical properties for fluids. You need to know these.

CE360 – Week 1

II.A. Dimensional Homogeneity Equations must comply with the concept of “dimensional homogeneity” which can be defined as follows: The dimensions on the left side of the equation must be the same as those on the right side, and all additive separate terms must have the same dimensions. II.B. In-Class Exercise A commonly used equation for determining the volume rate of flow, Q, of a liquid through an orifice located in the side of a tank is

0.61 2Q A gh=

Where A is the area of the orifice, g is the acceleration of gravity, and h is the height of the liquid above the orifice. Is this equation dimensionally homogeneous?

CE360 – Week 1

Solution: Q = volume/time = L3T-1 A = area = L2 g = acceleration of gravity = LT-2 h = height = L

( )11

3 1 2 2 22

3 1 3 1

( ) 0.61( )( 2) ( )

( ) [(0.61)( 2)]( )

LT L LT L

LT LT

− −

− −

=

=

The equation is dimensionally homogeneous

II.C. Systems of Units Dimensions represent qualitative measures of system properties. We use two primary unit systems to quantitatively characterize fluids: • British Gravitational System (BGS) • International System (SI)

CE360 – Week 1

(i) British Gravitational System L = foot (ft) T = second (s) F = force = pound (lb) M = mass = slug Weight (lb) = Mass (slugs) * Gravity (32.2 ft/s2) (ii) International System L = meter (m) T = second (s) F = force = Newton (N) M = mass = Kilogram (kg) Weight (N) = Mass (kg) * Gravity (9.81m/s2)

CE360 – Week 1

The following tables provide some common conversions (also see Appendix A).

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III. Measures of Fluid Mass and Weight • Density (ρ) “rho”

3 3 3

mass M slugs kgor

volume L ft mρ ⎛ ⎞= = ⎜ ⎟

⎝ ⎠

Density is temperature dependent! (see Figure 1.1 pg. 8)

• Specific Weight (γ) “gamma”

3 2 2 3 3 3 3

1M L ML Force F lb Ng or

L T T L Volume L ft mγ ρ ⎛ ⎞⎛ ⎞ ⎛ ⎞ ⎛ ⎞= = ⋅ = ⋅ = = ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ • Specific Gravity (SG)

@4 @39.2

fluid fluid

water C water F

SG orρ ρ

ρ ρ

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟=⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠

What does waterSG γ⋅ represent?

fluid fluid fluid

water water water

gSG

gρ ρ γρ ρ γ

⋅= = =

⋅

fluid water

water fluidwater

SGγ γ

γ γγ⋅

⇒ ⋅ = =

CE360 – Week 1

Physical Properties of Common Fluids (also see Appendix B).

CE360 – Week 1

III. Class Exercise Fluid Forces & Associated Physical Properties

Do gravity forces affect motion of the fluid particles in the following cases? Why? Yes No

a) waves breaking on the beach ____ ____ b) flow of water down a river ____ ____ c) flow of air mass before a storm ____ ____ d) airplane at cruising altitude ____ ____

Give practical examples where the other 5 forces are important. Note that in most cases there are several possibilities.