Fluid Mechanics Lecture: Properties of Fluids, Density, and Laboratory Procedures

Conceptual Overview of Buoyancy:
- The ability of an object to float (buoyancy) is dependent on the density of the fluid it is submerged in.
- Needle vs. Bucket: The instructor poses a question regarding whether a needle or a bucket will float/sink based on their relative densities. If an object's density is higher than the fluid, it sinks; if it is lower, it floats.
- Reference standard: Water is the universal standard for comparing densities of liquids and solids.
Mathematical Definition of Density:
- Density ( $\rho$ ) is defined as mass per unit volume: $\rho = \frac{m}{V}$
- If the volume of an object is too large relative to its mass (e.g., Lily/Lily density example), the resulting density will be low.
Standard Reference Values for Water:
- In the International System of Units (SI), the density of water is exactly $1,000\,kg/m^3$ .
- Floating Rule: If a material's density is less than the density of water (< 1,000\,kg/m^3), it will float. If it is high (> 1,000\,kg/m^3), it will sink.
- Example Scenario: A material with a density of $800\,kg/m^3$ will float because its density is less than water.

Mass Units:
- SI System: Expressed in kilograms ( $kg$ ).
- English System: Expressed in slugs ( $slags$ ).
Volume Units:
- SI System: Typically expressed in cubic meters ( $m^3$ ). While centimeters cubed ( $cm^3$ ) and millimeters cubed ( $mm^3$ ) exist, fluid mechanics primarily utilizes cubic meters due to large fluid volumes.
- Alternative SI Unit: Liters ( $L$ ).
- SI Conversion: $1\,m^3 = 1,000\,L$ . (Mnemonic reminder: a cubic meter is literally a cube measuring $1\,m \times 1\,m \times 1\,m$ ).
- English System: Expressed in cubic feet ( $ft^3$ ) or cubic inches ( $in^3$ ).
- English Alternative Unit: Gallons ( $gal$ ).
- English Conversion: $1\,m^3 = 264\,gallons$ .
- Mnemonic for 264: To remember the value 264, use the sum: $2 + 4 = 6$ , placing the sum in the middle to get the digits $2-6-4$ .
Density Units Summary:
- SI: Kilograms per cubic meter ( $kg/m^3$ ).
- English: Slugs per cubic foot ( $slugs/ft^3$ ).

Definition: Specific unit weight (denoted by the Greek letter gamma, $\gamma$ ) is the weight per unit volume of a substance.
Formulas:
- $\gamma = \rho \times g$ (where $\rho$ is density and $g$ is acceleration due to gravity).
- $\gamma = \frac{W}{V}$ (where $W$ is weight and $V$ is volume).
SI Units: Expressed in Newtons per cubic meter ( $N/m^3$ ) or kilonewtons per cubic meter ( $kN/m^3$ ).
- Verification: Multiplying $\rho \text{ (kg/m}^3\text{)}$ by $g \text{ (m/s}^2\text{)}$ results in $kg \cdot m/s^2$ over $m^3$ . Since $1\,N = 1\,kg \cdot m/s^2$ , the units resolve to $N/m^3$ .
English Units: Expressed in pounds per cubic foot ( $lb/ft^3$ ).
Standard Values for Water Unit Weight:
- SI: $\gamma_{\text{water}} = 9.81\,kN/m^3$ .
- English: $\gamma_{\text{water}} = 62.4\,lb/ft^3$ .

SI Value: $g = 9.81\,m/s^2$ .
English Value: $g = 32.2\,ft/s^2$ .
Calculating Density in the English System:
- Density can be derived from unit weight and gravity: $\rho = \frac{\gamma}{g}$
- For water in the English system: $\frac{62.4}{32.2} \approx 1.94\,slugs/ft^3$ . This is the standard value used in calculations.

Definition: Specific gravity ( $sg$ ) or relative density is a dimensionless ratio comparing a fluid's density to a standard reference density.
Reference Standards:
- For Liquids and Solids: The reference is Water.
- For Gases: The reference is Air.
Mathematical Representation:
- $sg = \frac{\rho_{\text{liquid}}}{\rho_{\text{water}}}$
- $sg = \frac{\gamma_{\text{liquid}}}{\gamma_{\text{water}}}$
Deriving Properties:
- $\rho_{\text{liquid}} = sg_{\text{liquid}} \times \rho_{\text{water}}$
- $\gamma_{\text{liquid}} = sg_{\text{liquid}} \times \gamma_{\text{water}}$
Specific Gravity Values for Common Liquids:
- Water: $1.0$ (Standard reference, calculated as $\frac{1,000}{1,000}$ ).
- Seawater: $1.03$ .
- Oil: $0.8$ (Expected to be less than water as it floats).
- Mercury: $13.6$ (Highly dense, standard value to memorize).

Title: Basic Fluid Mechanics Laboratory Equipment and Application.
Primary Objectives:
- Identify and describe functions of laboratory equipment.
- Understand applications of apparatuses in measuring fluid behavior.
- Demonstrate proper handling and basic operations.
Materials and Methodology:
- Students utilize internet searches, books, and drawing materials to identify five or more apparatuses.
- Requirement per Member: Each group member must submit one sheet of band paper containing a drawing/illustration of an apparatus, labeled parts, and a discussed function.
- Group Size Adjustments: Group 1 (7 members) submits 7 apparatuses; other groups (6 members) submit 6.
Deliverables:
- A draft compiled by the group must be submitted by 12:30 PM today for approval.
- Observation and Conclusion sections are written as a collaborative group effort.
- Approved drafts will later be converted into a Final Report.

Q: Which provides more buoyancy: Freshwater or Seawater?
- A: Seawater. Because seawater has a larger density ( $sg = 1.03$ ), it provides greater buoyant force, making it easier to float compared to freshwater.
Q: If we mix oil, water, seawater, and mercury, what is the layer sequence from top to bottom?
- A: The liquids will layer according to their specific gravity (lightest on top):
  1. Oil ( $sg = 0.8$ )
  2. Water ( $sg = 1.0$ )
  3. Seawater ( $sg = 1.03$ )
  4. Mercury ( $sg = 13.6$ )
Q: What is a quart in fluid mechanics?
- A: A quart is a unit of volume, used in examples such as "a quart of SAE 30 oil." Students are advised to search for the specific volume equivalent.
Logistics:
- Canvas Groupings: Groups are set to self-sign up. Groups are capped around 6-7 members.
- Plate Number 1: The deadline for Plate Number 1 is usually after the major preliminary exam.
- Friday Schedule: Lecture is scheduled for 07:30 to 09:30. Laboratory activities or field work status will be announced via the inbox.
- Calculators: Students are instructed to always have calculators and scratch paper ready for solving derivation problems like $\rho = \frac{62.4}{32.2}$ .