Korean, Edit

Lesson 1. Properties of Fluids

Recommended Reading: 【Physics】 Fluid Mechanics Table of Contents

1. Overview

2. Density, Specific Weight, Specific Gravity, Specific Volume

3. Viscosity

4. Pressure

5. Buoyancy

6. Surface Tension

7. Types of Fluids

1. Overview

⑴ Fluid : Substance that continuously deforms under even the slightest shear stress

⑵ Instruments for fluid measurement

① Lagrangian Accelerometer (Lagrangian Pressure Tube)

② Pitot Tube

⑶ Instruments for flow rate measurement

① Pipe : Venturi Meter, Orifice, Rotameter

② Weir : V-notch Weir, Rectangular Weir

2. Density, Specific Weight, Specific Gravity, Specific Volume

⑴ Density : Mass per unit volume. Represented by ρ

① Compressible Fluid : Fluid accompanying density changes

○ Examples : Common gases, liquids with shock waves or water hammering

② Incompressible Fluid : Fluid assumed to have no compressibility

○ Examples : Gases flowing at low velocities, typical liquids

③ Density of various substances

Table. 1. Density of Various Substances

⑵ Specific Weight : Weight per unit volume. Represented by γ

⑶ Specific Gravity

① Definition : Density of a substance relative to water. Represented by s

② Measurement Methods : Principle of buoyancy, Hydrometer, Pascal’s principle

⑷ Specific Volume : Reciprocal of density

3. Viscosity

⑴ Definition

① Viscosity : Measure of stickiness. Scale of intermolecular forces

② Shear Rate : Rate of velocity gradient

③ Absolute Viscosity Coefficient (Dynamic Viscosity) : Proportional constant of shear strain rate to shear stress. Represented by μ

○ Dimension : μ = FTL^-2 = ML^-1T^-1

○ Unit : 1 poise = 100 centipoise = 1 dyne·s / cm² = 0.1 N·s / m²

④ Kinematic Viscosity Coefficient : Physical quantity obtained by dividing absolute viscosity coefficient by density. Represented by ν

○ Dimension : ν = L²T^-1

○ Unit : 1 stokes = 1 cm² / s = 10^-4 m² / s

⑤ Ideal Fluid : Fluid with no viscosity

⑥ Real Fluid : Fluid with viscosity

⑵ Derivation of viscosity coefficient : According to the kinetic theory for gases, viscosity for an ideal gas is derived as follows:

① m : Molecular weight of gas

② R : Gas constant

③ T : Absolute temperature

④ d : Diameter of gas molecule

⑶ Methods of measuring viscosity coefficient

① Viscometers applying Newton’s Law of Viscosity : MacMichael Viscometer, Stormer Viscometer

② Viscometers applying Hagen-Mowzel Equation : Ostwald Viscometer, Saybolt Viscometer

③ Viscometers applying Stokes’ Law : Falling Ball Viscometer

⑷ Applications

① Question: When colloidal particles of a solid (particle size < 1 μm, specific gravity = 1.20) are dispersed in water with volume fractions Φ1 and Φ2 (Φ1 > Φ2), which side of the colloidal particles will precipitate first?

○ Source : Application and Solution of Transport Phenomena, Transport Phenomena Division Committee (Hyun-Gu Lee), Korean Institute of Chemical Engineers, 2002, p. 14

② Answer: The effective viscosity μ of a suspension with fine particles, when the volume fraction Φ of the particles is not large (i.e., Φ « 1) and the Reynolds number is very small (i.e., Re « 1)

is given. Here, μ0 represents the viscosity of pure water. Therefore, as Φ increases, the effective viscosity μ of the solution increases, and the settling velocity of particles in the colloidal solution decreases. Therefore, when Φ1 > Φ2, the particles with smaller volume fraction Φ2 will precipitate first. Also, as the particle concentration increases, the settling velocity is also reduced due to the hindering effect caused by interactions between particles.

⑸ Prandtl’s Mixing Length

4. Pressure

⑴ Definition : Magnitude of force per unit area

P = F / A

① Scalar quantity, does not include direction

⑵ Units : N / m², Pa, bar, hPa, atm, mmHg, torr, Aq

① 1 Pa = 1 N / m²

② 1 bar = 100 kPa

③ 1 hPa = 100 Pa

④ 1 atm = 101,325 Pa

⑤ 1 atm = 760 mmHg

⑥ 1 torr = 1 mmHg

⑦ 1 Pa = 0.00010204 mAq

⑶ Pressure at a point h depth from the fluid surface

① Formulation

② Proof

Figure. 1. Derivation of buoyancy formula

○ Force Representation

○ By force equilibrium, F_up + Weight = F_down holds

○ If ρ is constant, integrating the above equation gives the following:

○ P(z) represents pressure at a vertical z (m) with the seabed as 0 ↔ Pº(z) represents pressure below a vertical z (m) with sea level as 0

○ If ρ is not constant, the equation transforms as follows

⑷ Pascal’s Principle

① Overview

○ Definition: Law stating that pressure is evenly transmitted throughout a fluid

○ Theorems:

○ Theorem 1. Pressure at any point is the same in all directions

○ Theorem 2. Pressure acts perpendicular to the surface

○ Theorem 3. Pressure on the same vertical line is the same

② Proof

Figure. 2. Proof of Pascal’s Principle

○ Assuming no shear stress

○ Horizontal force equilibrium

○ Vertical force equilibrium

○ Any underwater structure is a combination of countless triangles → pressure in the vertical direction at any point is uniform

③ Gain of force with no gain of work

④ Application: Siphon Principle

⑸ Pressure Measurement Instruments

① Micromanometer: Measures minute pressure

② Piezometer Hole: Unperturbed static pressure measurement

③ Pitot Tube: Instrument for stagnation pressure measurement

④ Bourdon Gauge: Measures gauge pressure by balancing

○ Intended to measure gas pressure inside a tank rather than precise vacuum pressure

5. Buoyancy

⑴ Definition: Force exerted by a fluid on an object submerged in it, opposite to the object’s gravity

⑵ Magnitude

⑶ Induction

① Sum of pressures exerted on the front, back, left, and right of an object within a gravitational field is zero

② The pressure below is greater than above in a gravitational field → a force lifts the object from below

③ Rigorous mathematical proof: Utilization of Divergence Theorem

⑷ Archimedes’ Principle

① Alternative expressions of buoyancy

② Expression 1: A submerged object experiences buoyant force equal to the weight of the fluid displaced

③ Expression 2: Buoyant force in a vessel filled with fluid equals the weight of the displaced fluid

④ Interpretation:

○ Volume of pure gold and fake gold remains constant → Buoyant force on pure gold equals that on fake gold

○ Since the density of pure gold is greater than that of fake gold, gravity is stronger on pure gold

○ Net gravity is greater on pure gold → When placed in water, an equilibrium balance tilts toward the pure gold side

⑸ Calculation of volume of submerged iceberg: Approximately 87% ~ 88.5%

① Typical density of ice: 0.9 g/cm³

② Density of iceberg ice: 0.917 g/cm³ (compressed)

③ Density of seawater: 1.035 g/cm³

④ Formulation

6. Surface Tension

⑴ Definition: Force that tends to minimize the surface area of a liquid due to intermolecular forces

① Reason for the spherical shape of water droplets

② Stronger intermolecular forces result in stronger cohesion, especially in substances like water that engage in hydrogen bonding

⑵ Simple Understanding

① Molecules inside experience equal forces in all directions, resulting in zero net force

② Molecules on the surface experience a net inward force

⑶ Complex Understanding

① Surface has higher potential energy than the interior

② Molecules strive to minimize their presence on the surface

③ This results in a force trying to reduce the surface area

⑷ Laplace-Young Equation: Describes the magnitude of surface tension

⑸ Phenomenon 1: Expansion of a liquid film

Figure. 3. Expansion of a liquid film and surface tension

The size of the liquid film is exaggerated.

Particles 2 and 3 in equilibrium do not exert resistance.

However, particle 1 experiences a net force in the ← direction.

Because other liquids try to pull it down. (Indicated by V)

Connecting particles 1 forms a line, so surface tension is proportional to length, not area.

F = 2 × b × σ_s + F_{friction, motion}. (Where width is ignored.)

The coefficient 2 arises from having two sides of the film, thus two edges.

⑹ Phenomenon 2: Coin floating on the surface of a liquid: Example of a salt shaker

Figure. 4. Coin floating on the surface of a liquid

In this case, particles under the coin experience no upward force (i.e., vertical force = 0).

Water molecules at the edge of the coin are pulled by the water molecules on the inclined surface by an angle of θ; this force lifts the coin.

⑺ Phenomenon 3: Capillary action and meniscus

① Capillary action

○ Definition: Formation of a fluid column by the forces of a thin tube

○ Adhesion: Attraction between different substances

○ Cohesion: Attraction between the same substances

② Meniscus

○ Surface shape of the fluid column in a capillary

○ Water’s meniscus: Formation of a ∪-shaped meniscus due to the adhesive force between water and the tube exceeding the cohesive force between water molecules

○ Mercury’s meniscus: Formation of a ∩-shaped meniscus due to the adhesive force between mercury and the tube being less than the cohesive force between mercury molecules

⑻ Contact Angle

① Definition: Angle formed between the tangent at the contact point and the solid surface

Figure. 5. Definition of contact angle

② Net force at the contact point is zero

③ Classification of materials based on wetting properties

○ θ = 0°: Hydrophilic

○ 0° < θ < 90°: Partially hydrophilic

○ 90° < θ: Hydrophobic

④ Critical surface tension

○ Definition: Minimum surface tension of a liquid when the contact angle is 0, denoted as γLG. Proposed by Zisman

○ 1¹. Measure contact angles of liquids with different surface tensions on a single surface

○ 2^nd. Plot γ_LG-cos θ graph

Figure. 6. Plot γ_LG-cos θ graph

○ 3^rd. Extend the graph with a linear trendline to estimate γLG at θ = 0

○ Importance

○ Unique property of materials

○ Prediction of contact angles through critical surface tension

⑤ Methods for measuring contact angle: Static contact angle measurement, Dynamic contact angle measurement

⑥ Determining surface energy from contact angle

○ 1¹. Measure contact angle

○ 2^nd. Determine dispersive and polar components

○ 3^rd. Calculate Owens-wendt surface energy

⑼ Surfactants: Lower surface tension

① Strong surfactants: Soap, SDS, Triton X

② Weak surfactants: Bile salt

7. Types of Fluids

⑴ Power Law Model

① Assuming fluid flows along the z-axis.

② Fluid motion is governed by the following power law:

○ τ : Shear stress (Pa)

○ k : Consistency coefficient (Pa·sⁿ)

○ η : Flow behavior index or apparent viscosity

○ Shear rate

⑵ Type 1: Newtonian Fluid (e.g., water, air)

① Definition: A fluid with constant viscosity regardless of stress. n = 1

② Shear stress and velocity gradient are linear.

③ Velocity distribution in a conduit is parabolic.

Figure. 7. Types of Newtonian Fluids

⑶ Type 2: Non-Newtonian Fluid

① Type 2-1: Time-independent behavior fluid: Follows the power law model well.

② Type 2-2: Time-dependent behavior fluid: Corresponds to a few non-Newtonian fluids.

Figure. 8. Types of Non-Newtonian Fluids

⑷ Type 2-1-1: Pseudoplastic Fluid (e.g., syrup, paint)

① Definition: A fluid whose viscosity decreases with increasing stress. n < 1

② Also known as shear-thinning fluid.

③ Velocity distribution in a conduit:

○ Near the wall, viscosity decreases, resulting in a relatively large velocity gradient.

○ Near the conduit’s center, velocity gradient is very small.

⑸ Type 2-1-2: Dilatant Fluid (e.g., cornstarch)

① Definition: A fluid whose viscosity increases with increasing stress. n > 1

○ Also known as shear-thickening fluid.

○ Under sudden impact, it becomes solid.

② Velocity distribution in a conduit:

○ Near the wall, viscosity increases significantly.

○ Near the conduit’s center, viscosity increases gradually.

③ Utilized in applications like bulletproof vests.

⑹ Type 2-1-3: Bingham Plastic Fluid (e.g., toothpaste, mayonnaise)

① Definition: A fluid requiring finite yield stress before flow starts.

○ τ₀ : Yield stress required to initiate flow (Pa)

② Also known as ideal plastic fluid.

③ Qualitative description:

○ Bingham fluid refers to fluids with considerable yield stress τ0.

○ For conduit flow, if τwall < τ0, the fluid cannot flow due to static equilibrium of forces.

○ For conduit flow, if τwall ≥ τ0, the fluid flows naturally due to dynamic equilibrium of pressure gradient and shear stress.

○ (Note) Static equilibrium is similar to static friction, and dynamic equilibrium is similar to kinetic friction.

⑺ Type 2-2-1: Thixotropic Fluid

① Definition: A fluid whose apparent viscosity decreases with time under constant shear stress.

② Examples: Gelatin, shortening, cream, paint, etc.

⑻ Type 2-2-2: Rheopectic Fluid

① Definition: A fluid whose apparent viscosity increases with time under constant shear stress.

② Example: Highly concentrated starch solution.

Input : 2016.12.01 22:28