Basics of Fluid Mechanics

🧭 Overview

🧠 One-sentence thesis

Mass conservation in fluid mechanics establishes that the rate of mass change within a control volume equals the net mass flow across its boundaries, forming the foundation for analyzing fluid systems through the continuity equation and Reynolds Transport Theorem.

📌 Key points (3–5)

• Control volume concept: a defined region in space (fixed or moving) through which fluid flows, used to apply conservation laws.
• Continuity equation: mathematical expression of mass conservation relating mass accumulation inside the control volume to net mass flow in/out.
• Special cases simplify analysis: non-deformable control volumes and constant-density (incompressible) fluids lead to simpler forms of the continuity equation.
• Reynolds Transport Theorem: general framework connecting system properties (moving with the fluid) to control volume analysis (fixed in space).
• Common confusion: distinguishing between a system (a fixed collection of fluid particles moving through space) and a control volume (a fixed region in space through which different fluid particles pass).

🧩 Control volume framework

🧩 What is a control volume

Control volume: a defined region in space, either fixed or moving, through which fluid can flow; used as the basis for applying conservation laws in fluid mechanics.

• Unlike a system (which follows the same fluid particles), a control volume stays in one place while fluid moves through it.
• The boundary of the control volume is called the control surface.
• Mass, momentum, and energy can cross the control surface.
• Example: analyzing flow through a pipe section—you draw a box around a segment of the pipe; fluid enters one end and exits the other, but the box itself doesn't move.

🔍 Why use control volumes

• In fluid mechanics, tracking individual fluid particles (the system approach) is often impractical because fluids deform continuously.
• A control volume allows you to analyze what happens at fixed locations (e.g., inlets, outlets, walls) without tracking every particle.
• This approach is especially useful for steady flows where conditions at each point don't change over time, even though different fluid particles pass through.

📐 Continuity equation

📐 General form of mass conservation

The continuity equation states:

• Rate of mass accumulation inside the control volume = Net mass flow rate into the control volume
• In words: if more mass flows in than out, mass inside increases; if more flows out than in, mass inside decreases.
• This is the mathematical expression of "mass cannot be created or destroyed."

🔧 Non-deformable control volume

• When the control volume does not change shape or size over time, the equation simplifies.
• The volume itself is constant, so only the density inside and the flow rates at the boundaries matter.
• Example: a rigid pipe section—its volume is fixed, so any change in mass inside must come from differences between inflow and outflow.

💧 Constant density (incompressible) fluids

• For fluids with constant density (liquids under most conditions), the continuity equation simplifies further.
• Because density doesn't change, volume flow rate in = volume flow rate out for steady flow.
• This is the most common form used in introductory fluid mechanics: the sum of volumetric flow rates entering equals the sum leaving.
• Example: water flowing through a pipe—if the pipe narrows, the velocity must increase to keep the volume flow rate constant (since density is constant).
• Don't confuse: "incompressible" means constant density, not that the fluid cannot be squeezed; it means density changes are negligible for the problem at hand.

🔄 Reynolds Transport Theorem

🔄 Bridging system and control volume

Reynolds Transport Theorem: a mathematical tool that relates the rate of change of any extensive property (mass, momentum, energy) for a system (moving fluid particles) to the corresponding rate of change within a control volume (fixed region) plus the net flux of that property across the control surface.

• It generalizes the control volume approach to any conserved or transported quantity, not just mass.
• The theorem has two parts:
  • Accumulation term: how much the property changes inside the control volume over time.
  • Flux term: how much of the property flows in or out across the boundaries.
• This theorem is the foundation for deriving the integral forms of momentum and energy equations from their system-based (Lagrangian) forms.

🧭 Why it matters

• Allows you to use control volumes (which are easier to work with in practice) while still applying physical laws that are naturally stated for systems.
• Example: Newton's second law applies to a system (a fixed mass); Reynolds Transport Theorem lets you apply it to a control volume where mass is flowing in and out.

🧮 Examples and applications

🧮 Velocity–area relationship

• For incompressible flow in a pipe, continuity implies: if the cross-sectional area decreases, velocity must increase, and vice versa.
• Mathematically: (velocity₁ × area₁) = (velocity₂ × area₂) for any two cross-sections.
• This relationship is used constantly in pipe flow analysis, nozzle design, and duct systems.
• Example: a garden hose with a nozzle—the nozzle reduces the area, so water exits at higher velocity.

📊 Practical problem-solving

The excerpt mentions "Examples For Mass Conservation" and "More Examples for Mass Conservation," indicating:

• Application to real systems: tanks filling or draining, branching pipes, mixing flows.
• Combining continuity with other information (geometry, boundary conditions) to find unknown flow rates or velocities.
• Steady vs. unsteady cases: steady flow (no change over time at any point) vs. unsteady (conditions change with time, e.g., a tank draining).

Scenario	Key simplification	Typical unknown
Steady flow, constant density	Volume in = volume out	Velocity or area at outlet
Unsteady flow, rigid tank	Mass accumulation ≠ 0	Rate of level change
Compressible flow	Density varies	Density or mass flow rate

🔍 Common pitfalls

• Don't confuse system vs. control volume: a system is a fixed collection of particles; a control volume is a fixed region in space.
• Don't assume incompressibility for gases: gases are compressible; constant-density simplifications usually apply only to liquids.
• Remember the sign convention: flow into the control volume is typically positive for the accumulation side, flow out is negative (or vice versa, depending on convention—be consistent).

🧭 Overview

🧠 One-sentence thesis

This table of contents excerpt introduces momentum conservation as a core topic within differential fluid mechanics analysis, bridging mass conservation and energy principles to describe fluid motion through differential equations and boundary conditions.

📌 Key points (3–5)

• Momentum conservation in differential form: appears as part of a systematic differential analysis framework alongside mass and energy conservation.
• Derivation and equations: the section covers derivations of the momentum equation (Navier-Stokes) and their application to fluid flow problems.
• Boundary conditions and driving forces: momentum equations require proper boundary conditions to solve real flow problems.
• Connection to broader analysis: momentum conservation sits within a larger structure that includes integral methods, dimensional analysis, and applications to internal/external flows.
• Common confusion: differential vs. integral approaches—this chapter focuses on the differential (point-by-point) formulation, distinct from the integral (control volume) methods covered earlier.

📚 Context within the textbook structure

📚 Where momentum conservation fits

The excerpt shows momentum conservation appears in Chapter 8: Differential Analysis, which is part of Part II: Differential Analysis of the textbook.

• The chapter sequence is:
  • Mass conservation (continuity equation)
  • Conservation of a general quantity
  • Momentum conservation (section 8.4)
  • Derivations of the momentum equation (section 8.5)
  • Boundary conditions and driving forces (section 8.6)
  • Examples using the Navier-Stokes equations (section 8.7)

📚 Relationship to other topics

Topic	Location	Relationship to momentum conservation
Mass conservation	Section 8.2	Precedes momentum; provides continuity equation needed for momentum analysis
Energy equation	Chapter 7 (Part I)	Parallel conservation principle; integral form covered earlier
Dimensional analysis	Chapter 9	Follows differential analysis; uses dimensionless forms of momentum equations
Navier-Stokes examples	Section 8.7	Applications of the momentum equations derived in sections 8.4–8.5

📚 Part II structure

Part II is titled Differential Analysis and includes:

• Chapter 8: Differential Analysis (includes momentum conservation)
• Chapter 9: Dimensional Analysis
• Chapter 10: External Flow
• Chapter 11: Internal Flow
• Chapter 12: Potential Flow

This structure indicates momentum conservation is foundational for all subsequent flow analysis topics.

🔧 What the momentum conservation section covers

🔧 Core content (section 8.4)

The excerpt shows Section 8.4: Momentum Conservation appears on page 290.

• This section is positioned after:
  • Mass conservation examples and simplified continuity equations
  • A general framework for conservation of any quantity (section 8.3)

🔧 Derivations (section 8.5)

Section 8.5: Derivations of the Momentum Equation (page 295) follows immediately.

• This suggests section 8.4 introduces the concept and governing principles, while section 8.5 provides the mathematical derivation.
• The derivations likely lead to the Navier-Stokes equations, which are referenced in section 8.7.

🔧 Supporting material

The momentum conservation topic is supported by:

Section	Page	Purpose
8.6 Boundary Conditions and Driving Forces	307	Specifies how to apply momentum equations to real problems
8.6.1 Boundary Conditions Categories	307	Classifies types of boundary conditions needed
8.7 Examples for Differential Equation (Navier-Stokes)	311	Worked problems applying momentum conservation
8.7.1 Interfacial Instability	324	Advanced application of momentum principles

🧮 Methodology and approach

🧮 Generalized conservation framework

Section 8.3 (page 288) presents a "Conservation of General Quantity" framework before momentum.

• Section 8.3.1: "Generalization of Mathematical Approach for Derivations"
• Section 8.3.2: "Examples of Several Quantities"

This indicates the textbook teaches a unified method for deriving conservation laws, then applies it specifically to momentum.

🧮 From integral to differential

The textbook structure shows a pedagogical progression:

1. Part I (not fully shown): Integral analysis methods
2. Chapter 7: Integral energy conservation (referenced on page xxxiv)
3. Part II, Chapter 8: Differential analysis, including differential momentum conservation

Don't confuse: Integral methods analyze finite control volumes (total forces, total mass flow); differential methods analyze infinitesimal fluid elements (point-by-point variation). The same physical principle (momentum conservation) appears in both forms.

🧮 Mathematical tools

The appendix (Chapter A, page 765) provides mathematical background:

• Vector algebra and differential operators (A.1.1–A.1.2)
• Ordinary and partial differential equations (A.2–A.3)

These tools are essential for working with the momentum equations in differential form.

🌊 Applications and examples

🌊 Navier-Stokes applications

Section 8.7 (page 311) provides "Examples for Differential Equation (Navier-Stokes)".

• The Navier-Stokes equations are the momentum conservation equations for viscous fluids.
• Example 8.7.1 covers interfacial instability (page 324), showing momentum conservation at fluid-fluid boundaries.

🌊 Downstream applications

Momentum conservation principles feed into later chapters:

Chapter	Topic	How momentum conservation is used
9	Dimensional Analysis	Dimensionless forms of momentum equations (section 9.7, page 389)
10	External Flow	Boundary layer theory requires momentum equations (page 396)
11	Internal Flow	Pipe flow analysis uses momentum principles (page 399)
12	Potential Flow	Inviscid momentum equations (Euler equations) (page 436)

🌊 Compressible and multi-phase extensions

The table of contents shows momentum conservation extends to:

• Part III: Compressible Flow (chapters 14–15): momentum equations for gases at high speed
• Part IV, Chapter 16: Multi-Phase Flow (page 697): momentum conservation when multiple phases interact

Example: Section 14.5 "Normal Shock" (page 563) applies momentum conservation across shock waves in compressible flow.

⚠️ Common confusions and distinctions

⚠️ Integral vs. differential momentum

• Integral form: applies to a control volume; gives total force on a finite region (covered earlier in the textbook).
• Differential form: applies to an infinitesimal element; gives the momentum equation at every point in the flow field (Chapter 8).
• Both are valid; the choice depends on the problem type.

⚠️ Momentum vs. mass conservation

• Mass conservation (continuity equation, section 8.2): describes how density and velocity relate to ensure mass is neither created nor destroyed.
• Momentum conservation (section 8.4): describes how forces (pressure, viscosity, body forces) change the momentum of fluid elements.
• Don't confuse: continuity is a scalar equation; momentum is a vector equation (three components in 3D).

⚠️ Boundary conditions matter

Section 8.6 emphasizes boundary conditions and driving forces (page 307).

• The momentum equations alone are not enough; you must specify conditions at walls, inlets, outlets, and interfaces.
• Example: no-slip condition at a solid wall (velocity equals wall velocity) is a momentum boundary condition.

⚠️ Simplified vs. full equations

• Section 8.2.2 mentions "Simplified Continuity Equation" (page 283).
• Similarly, momentum equations can be simplified for special cases (e.g., incompressible flow, steady flow, inviscid flow).
• Don't assume the full Navier-Stokes equations are always needed; many problems use reduced forms.

🧭 Overview

🧠 One-sentence thesis

This excerpt contains only a list of figures from a fluid mechanics textbook and does not present substantive content on energy conservation principles.

📌 Key points (3–5)

• The excerpt is a table of contents listing figure captions and page numbers.
• Figure topics span fluid statics, control volumes, momentum, energy work, dimensional analysis, boundary layers, and pipe flow.
• No explanatory text, definitions, derivations, or conceptual discussions are provided.
• The actual content of Chapter 7 "Energy Conservation" cannot be reviewed from this list alone.

📋 Content assessment

📋 What the excerpt contains

The provided text is a List of Figures section (pages xxxix–xlv) from what appears to be a fluid mechanics or hydraulics textbook. It includes:

• Figure numbers (e.g., 1.14, 3.1, 7.1)
• Brief descriptive captions (e.g., "Rotating disc in steady state," "The work on the control volume")
• Page references

📋 What is missing

The excerpt does not include:

• Definitions or explanations of energy conservation principles
• Mathematical relationships or equations
• Conceptual discussions of work, kinetic energy, potential energy, or energy balance
• Examples, derivations, or problem-solving approaches
• Any prose or instructional content

📋 Observations about Chapter 7

From the figure captions alone, Chapter 7 appears to cover:

Figure	Topic hint
7.1	Work on control volume
7.2	Discharge from container
7.3	Kinetic energy and averaged velocity
7.4	Outlet configuration resistance
7.5–7.7	Flow scenarios (oscillating manometer, pressure differences, tank discharge)

However, without the accompanying text, the principles, derivations, and applications of energy conservation in fluid systems cannot be extracted or reviewed.

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Note: To create meaningful review notes on energy conservation, the actual chapter text—not just the figure list—would be required.

🧭 Overview

🧠 One-sentence thesis

This excerpt is a list of figures from a fluid mechanics textbook and does not contain substantive technical content about differential analysis.

📌 Key points (3–5)

• The excerpt consists entirely of a table of contents listing figure captions and page numbers.
• Topics referenced include potential flow, vortex flow, compressible flow (Fanno, Rayleigh, oblique shocks), multiphase flow, and open channel flow.
• No definitions, explanations, derivations, or conceptual discussions are present.
• A copyright notice at the end mentions a modified GNU Free Documentation License.

📋 Content summary

📋 What the excerpt contains

The provided text is a list of figures (pages xlvi–xlviii) from a fluid mechanics or fluid dynamics textbook. It enumerates:

• Figure numbers (e.g., 12.8, 14.1, 15.3, 17.5)
• Brief descriptive captions (e.g., "Vortex free flow," "Accelerating ball in a close container," "Oblique shock around a cone")
• Page numbers

The list spans multiple chapters:

Chapter	Topics referenced in figure captions
12	Source, sink, vortex, doublet, uniform flow, Magnus effect, wing flow
13	Added mass, ship motion, rolling, heave, coupling
14	Compressible flow, Mach number, shock waves, Fanno flow, Rayleigh flow, isothermal flow
15	Oblique shocks, Prandtl–Meyer expansion, supersonic flow
16	Multiphase flow, stratified flow, plug flow, vertical flow, counter-current flow
17	Open channel flow, hydraulic jump, subcritical/supercritical flow, energy lines
A	Vector calculus, coordinate systems (Cartesian, cylindrical, spherical, parabolic)

📋 What is missing

• No conceptual explanations: the excerpt does not define or explain any fluid mechanics principles.
• No equations or derivations: figure captions mention topics like "Mach number," "shock," "vortex," but do not present formulas or methods.
• No worked examples or problem-solving guidance.
• No pedagogical content: the list is purely navigational, intended to help readers locate diagrams in the full textbook.

📜 Copyright notice

📜 License terms

At the end of the excerpt, a copyright notice states:

This document is published under modified FDL [Free Documentation License].

Key modifications mentioned:

• Printing requirement: anyone printing more than 200 copies must provide the author with two free copies within one month.
• Electronic conversion requirement: anyone converting the book to another electronic format must provide the author with a copy within one month.
• The base license is GNU Free Documentation License, Version 1.2 (November 2002), published by the Free Software Foundation.

Don't confuse: this is a legal notice about distribution rights, not technical content about differential analysis or fluid mechanics.

---

Note for review: Because the excerpt contains no substantive technical material—only a figure index and a copyright statement—there are no concepts, mechanisms, or methods to extract for study purposes. To learn about differential analysis in fluid mechanics, you would need to consult the actual chapter text and figures referenced in this list.

🧭 Overview

🧠 One-sentence thesis

The excerpt contains no substantive content on dimensional analysis; it consists solely of a list of figures from a fluid mechanics textbook and the GNU Free Documentation License text.

📌 Key points (3–5)

• The excerpt is a list of figure captions (numbered 17.14–A.7) related to open channel flow, hydraulic jumps, coordinate systems, and other fluid mechanics topics.
• A copyright notice and the full text of the GNU Free Documentation License (modified) occupy the remainder of the excerpt.
• No definitions, explanations, or technical content about dimensional analysis are present.
• The excerpt does not contain teaching material, derivations, or examples that can be reviewed.

📄 What the excerpt contains

📄 Figure list only

• The first portion lists figure numbers and captions from a textbook chapter (likely Chapter 17 on open channel flow and Appendix A on coordinate systems).
• Topics referenced in captions include:
  • Downstream flow height and step height
  • Energy diagrams for various flow rates
  • Subcritical and supercritical flow in contractions
  • Hydraulic jumps
  • Triangular and non-rectangular channel cross sections
  • Cartesian, cylindrical, spherical, and parabolic coordinate systems
• These are references to visual aids, not explanatory text.

📜 License text

• The second portion reproduces the GNU Free Documentation License (GFDL) Version 1.2.
• A modification requires anyone printing more than 200 copies to provide two copies to the author within one month.
• Definitions of terms like "Document," "Modified Version," "Transparent," and "Opaque" are provided in the context of copyright and distribution.
• This material is legal boilerplate, not technical content.

⚠️ Note for review

⚠️ No dimensional analysis content

• The title "9 Dimensional Analysis" does not match the excerpt.
• The excerpt does not define dimensional analysis, discuss the Buckingham π theorem, explain dimensionless groups, or provide any related examples.
• Readers seeking review notes on dimensional analysis will not find relevant material here.

🧭 Overview

🧠 One-sentence thesis

The excerpt contains only licensing and document formatting terms from the GNU Free Documentation License and does not present substantive content related to external flow or any technical subject matter.

📌 Key points (3–5)

• The excerpt is composed entirely of legal license text (GNU FDL) covering copying, modification, and distribution rules.
• No technical, scientific, or educational content about "external flow" or any related engineering/physics topic is present.
• The text defines terms like "Title Page," "Transparent" and "Opaque" formats, and "Entitled XYZ" sections purely for document licensing purposes.
• Common confusion: this appears to be front matter or appendix material mistakenly included instead of the chapter content.

📄 What the excerpt contains

📄 License and format definitions

The excerpt provides definitions and rules from the GNU Free Documentation License:

Transparent formats: formats designed for human modification (e.g., PNG, XCF, JPG, standard HTML, PostScript, or PDF).

Opaque formats: proprietary formats readable only by specific tools, or machine-generated output not intended for editing.

Title Page: for printed books, the title page and following pages with required license material; for other formats, text near the work's title before the body.

Entitled XYZ: a named subsection whose title is precisely XYZ or contains XYZ in parentheses with translation text.

📋 Copying and modification rules

The text outlines conditions for:

• Verbatim copying (section 2): reproduction in any medium with license notices intact, no technical obstruction of further copying.
• Copying in quantity (section 3): requirements for printed copies over 100, including cover texts and transparent copy availability.
• Modifications (section 4): conditions for distributing modified versions, including title page requirements, author attribution, copyright notices, and preservation of invariant sections.

⚠️ Content assessment

⚠️ No substantive material

• The excerpt does not contain any information about external flow, fluid dynamics, aerodynamics, or any related technical subject.
• All content pertains exclusively to document licensing, copyright, and distribution terms.
• This appears to be licensing boilerplate from a textbook's front or back matter, not the chapter titled "10 External Flow."

⚠️ What is missing

• Definitions, principles, or concepts related to external flow.
• Technical explanations, mechanisms, or applications.
• Examples, comparisons, or problem-solving approaches for the stated topic.

Note: The excerpt lacks the substantive content expected for a chapter on external flow; only legal licensing text is present.

🧭 Overview

🧠 One-sentence thesis

The excerpt contains only licensing and document modification instructions from the GNU Free Documentation License and does not present substantive content related to internal flow or any technical subject matter.

📌 Key points (3–5)

• The excerpt consists entirely of legal text from the GNU Free Documentation License (GFDL).
• It covers rules for modifying documents, combining documents, translations, and license termination.
• No technical, scientific, or educational content about "internal flow" or any related engineering/physics topic is present.
• The text appears to be boilerplate licensing material that precedes or follows actual chapter content.

📄 Content analysis

📄 What the excerpt contains

The provided text is purely administrative and legal in nature. It includes:

• Sections 4–10 of the GNU Free Documentation License
• Instructions for preserving document sections, acknowledgements, and warranty disclaimers
• Rules for combining multiple licensed documents
• Guidelines for translations and aggregations with other works
• License termination conditions
• An addendum explaining how to apply the license to new documents

❌ What is missing

No content related to the chapter title "11 Internal Flow" appears in this excerpt. The text does not discuss:

• Fluid mechanics or flow dynamics
• Internal flow phenomena in pipes, ducts, or channels
• Engineering principles or physical concepts
• Any technical definitions, equations, or applications

🔍 Conclusion

🔍 Substantive content assessment

This excerpt lacks any meaningful educational or technical material for review or study purposes. It represents only the licensing framework under which a document may be distributed and modified, not the actual chapter content on internal flow that would be expected given the title.

🧭 Overview

🧠 One-sentence thesis

The excerpt provided contains only licensing information, author biography, and book preface material, with no substantive content on the topic of potential flow.

📌 Key points (3–5)

• The excerpt consists entirely of GNU Free Documentation License terms, author credentials, and book history notes.
• No technical content, definitions, concepts, or explanations related to potential flow are present in this excerpt.
• The material describes the author's background in fluid mechanics and ship stability but does not explain potential flow theory.
• The excerpt mentions that the author has shown "the potential method has limitations because stability is compartmental" in the context of ship stability, but provides no details.

📄 What the excerpt contains

📜 License and copyright information

The first portion of the excerpt covers:

• GNU Free Documentation License termination clauses (section 9)
• Future revisions policy (section 10)
• Instructions for applying the license to documents
• Copyright notice templates with placeholders for year, name, and optional invariant sections

👤 Author biography

The excerpt includes biographical information about Genick Bar-Meir:

• Educational background: Ph.D. in Mechanical Engineering from University of Minnesota, Master in Fluid Mechanics from Tel Aviv University
• Claims of contributions to ship stability theory, die casting, dimensional analysis, and compressible flow
• Brief mention that "the potential method has limitations" in stability calculations, but no explanation of what potential flow is or how it works

📖 Book history notes

• Brief notes from 2023, 2022, 2021, and initial writing periods
• Mentions the book started as an introduction to a compressible flow book
• No technical content about potential flow itself

⚠️ Missing content

⚠️ No potential flow theory

The excerpt does not contain:

• A definition of potential flow
• Mathematical formulations or governing equations
• Physical principles or assumptions
• Applications or examples
• Comparisons with other flow models

⚠️ What would be needed

To write meaningful review notes on potential flow, the excerpt would need to include:

• Core concepts (irrotational flow, velocity potential, stream function)
• Governing equations (Laplace equation, boundary conditions)
• Physical meaning and limitations
• Practical applications and examples

🧭 Overview

🧠 One-sentence thesis

Floating bodies possess not only added mass properties but also transfer properties that govern the transfer mechanism between the various modes of movement of the floating body.

📌 Key points (3–5)

• New discovery: floating bodies have transfer properties in addition to the previously known added mass properties.
• What transfer properties do: they are responsible for the transfer mechanism between the various modes of movement of the floating body.
• Historical context: until recently, ship stability understanding was based on the metacenter concept established 300 years ago.
• Common confusion: the old metacenter-based theory prevented the ability to write the correct governing equations of ship movement; the new approach built the correct governing equations by including transfer properties.
• Practical impact: this work revolutionized the ship stability field by explaining phenomena that the old theory could not address.

🚢 The revolution in ship stability theory

🚢 What was wrong with the old theory

• For 300 years, ship stability understanding relied on the metacenter concept.
• The excerpt states that the previous theory "prevented the ability to write the correct governing equations of ship movement."
• In other words: the old framework was incomplete and blocked progress toward accurate mathematical descriptions of how floating bodies move.

🔧 What the new theory adds

• The new approach recognizes that floating bodies have two kinds of properties:
  1. Added mass properties (already known)
  2. Transfer properties (newly identified)
• The excerpt emphasizes that transfer properties are distinct and were missing from the old theory.

🔄 Transfer properties explained

🔄 What transfer properties are

Transfer properties: properties responsible for the transfer mechanism between the various modes of movement of the floating body.

• "Modes of movement" refers to different ways a floating body can move (e.g., rolling, pitching, heaving, etc.).
• Transfer properties govern how motion in one mode affects or couples with motion in another mode.
• Example: if a floating body rolls, transfer properties describe how that rolling motion influences or transfers energy/momentum to pitching or other modes.

🔄 Why they matter

• Without accounting for transfer properties, the governing equations are incomplete.
• The excerpt states that the new theory "built the governing equations" by including these properties.
• This allows for correct prediction and analysis of floating body behavior.

🆚 Old vs new approach

Aspect	Old theory (metacenter-based)	New theory (added mass + transfer properties)
Foundation	Metacenter concept (300 years old)	Added mass properties + transfer properties
Governing equations	Could not write correct equations	Built the correct governing equations
Understanding of movement	Incomplete; blocked progress	Explains transfer mechanism between modes
Field impact	Standard for centuries	Revolutionized ship stability field

🆚 Don't confuse

• Added mass properties are not the same as transfer properties.
• Added mass properties were already known; transfer properties are the new discovery.
• Transfer properties specifically handle the coupling and transfer mechanism between different movement modes, not just the mass effects.

🧪 Analytical solutions and practical examples

🧪 Floating cylinder stability

• The excerpt mentions that "the stability of floating cylinder is for the first time was solved analytically."
• This is a concrete example of what the new theory enables: problems that could not be solved with the old metacenter approach can now be solved analytically.
• Example: a floating cylinder's stability can now be calculated exactly using the new governing equations that include transfer properties.

🧪 Why analytical solutions matter

• Analytical solutions provide exact answers and deeper insight into the physics.
• The old theory's limitations meant such solutions were not possible.
• The new theory's correct governing equations unlock these solutions.

🧭 Overview

🧠 One-sentence thesis

Viscosity—the resistance of a fluid to shear stress—varies widely across materials and conditions, requiring different models (Newtonian, non-Newtonian, power-law, Bingham) and estimation methods (Sutherland's equation, reduced-property charts, mixture formulas) to predict fluid behavior in engineering applications.

📌 Key points (3–5)

• Newtonian vs non-Newtonian fluids: Newtonian fluids show linear shear stress–velocity relationships; non-Newtonian fluids exhibit power-law, thixotropic, rheopectic, or Bingham behavior.
• Temperature and pressure effects: absolute viscosity of gases increases with temperature; kinematic viscosity (viscosity divided by density) also rises; pressure effects are minor for most liquids but significant for oils.
• Estimation tools: Sutherland's equation for gases (−40 °C to 1600 °C), reduced-property charts (analogous to compressibility charts), and Wilke's correlation for low-density gas mixtures.
• Common confusion: kinematic viscosity (units: m²/sec, like acceleration) vs absolute viscosity (units: N·sec/m²); kinematic viscosity = absolute viscosity / density.
• Why it matters: viscosity governs flow resistance, torque in rotating machinery, lubrication, and the behavior of complex fluids (concrete, drilling mud, molten metals).

🧪 Newtonian and non-Newtonian models

🧪 Newtonian fluids

Newtonian fluid: a material in which shear stress is directly proportional to the velocity gradient (shear rate), with a constant viscosity coefficient μ.

• The relationship is τ = μ (dU/dx), where τ is shear stress, μ is dynamic viscosity, and dU/dx is the velocity gradient.
• Water, air, and most simple liquids and gases are Newtonian.
• The viscosity coefficient μ is always positive and does not depend on the shear rate.

⚙️ Power-law (non-Newtonian) fluids

• Many materials follow τ = K (dU/dx)^n, where K and n are constants.
• When n = 1, the fluid is Newtonian (K becomes μ).
• Dilatant fluids: n > 1 (viscosity increases with shear rate).
• Pseudoplastic fluids: n < 1 (viscosity decreases with shear rate).
• These are called purely viscous fluids because they have no time-dependent hysteresis.

🕰️ Time-dependent non-Newtonian fluids

• Thixotropic fluids: viscosity decreases over time at constant shear stress (e.g., bentonite clay suspensions used in drilling).
• Rheopectic fluids: viscosity increases over time at constant shear stress (e.g., printer inks, gypsum pastes).
• Both exhibit hysteresis loops—the stress–strain curve depends on loading history.

🧱 Bingham plastics

• Materials that behave as solids below a yield stress τ₀ and as liquids above it.
• Simple Bingham model: τ = −μ ± τ₀ if |τ| > τ₀; dU/dy = 0 if |τ| < τ₀.
• Example: concrete (though more sophisticated models replace the Newtonian part with a power-law for better accuracy).
• Don't confuse: Bingham fluids are not simply "high-viscosity" fluids—they require a threshold stress to start flowing.

🌡️ Temperature and pressure effects

🌡️ Temperature dependence

• Gases: absolute viscosity increases with temperature (molecular activity rises); kinematic viscosity also increases because density drops faster than viscosity rises.
• Liquids: absolute viscosity typically decreases with temperature (e.g., water, oils).
• Figures in the excerpt show air and water viscosity vs temperature at atmospheric pressure.

🔧 Pressure dependence

• For most fluids, absolute viscosity changes little with pressure (isothermal flow can assume constant viscosity).
• Exception: oils show the greatest viscosity increase with pressure, which is beneficial for lubrication.
• The excerpt notes that "viscosity in the dome is meaningless"—referring to two-phase regions where structure matters more than a single viscosity value.

📐 Kinematic viscosity

Kinematic viscosity ν = μ / ρ (units: m²/sec).

• Combines viscosity and density into one property.
• Called "kinematic" because its dimensions (length²/time) resemble acceleration units.
• Useful when experimental data are reported in this form or when density variations are significant.

📊 Estimation methods

📊 Sutherland's equation (gases)

• For gases from −40 °C to 1600 °C:
  • μ = μ₀ × [0.555 T₀ + Suth] / [0.555 T + Suth] × (T / T₀)^(3/2)
  • Where μ₀ and T₀ are reference values, Suth is Sutherland's constant (tabulated for air, CO₂, H₂, N₂, O₂, etc.).
• Example: air at 800 K has viscosity ≈ 2.51×10⁻⁵ N·sec/m² (observed ≈ 3.7×10⁻⁵), showing ~40% increase from reference conditions.
• The excerpt notes the author is "ambivalent" about the accuracy claim.

📈 Reduced-property charts

• Analogous to compressibility charts: plot reduced viscosity μᵣ = μ / μ꜀ vs reduced temperature Tᵣ = T / T꜀, with lines of constant reduced pressure Pᵣ = P / P꜀.
• Critical viscosity μ꜀ can be obtained from tables, back-calculated from known data, or estimated:
  • μ꜀ = √M T꜀ / ṽ꜀^(2/3) or μ꜀ = √M P꜀^(2/3) T꜀^(−1/6)
  • Where M is molecular weight, P꜀ and T꜀ are critical pressure and temperature.
• Example: oxygen at 100 °C and 20 bar → Tᵣ ≈ 2.41, Pᵣ ≈ 0.4 → μᵣ ≈ 1.2 → μ ≈ 21.6 (observed 24).

🧬 Gas mixture viscosity (Wilke's correlation)

• For low-density gas mixtures:
  • μₘᵢₓ = Σ [xᵢ μᵢ / Σ xⱼ Φᵢⱼ]
  • Where xᵢ is mole fraction, μᵢ is component viscosity, and Φᵢⱼ = [1/√8] [1 + Mᵢ/Mⱼ]^(−1/2) [1 + √(μᵢ/μⱼ) (Mⱼ/Mᵢ)^(1/4)]²
  • Φᵢᵢ = 1.
• Example: air (20% O₂, 80% N₂) at 20 °C → μₘᵢₓ ≈ 0.0000181 N·sec/m² (observed 0.0000182).
• Don't confuse: this is highly nonlinear—you cannot simply average component viscosities by mole fraction.

🧪 Liquid mixture viscosity

• No general "silver bullet" formula; experimental data usually required.
• For some two-liquid mixtures, Reiner–Philippoff formula models shear-dependent behavior:
  • dU/dy = [1/μ∞ + (μ₀ − μ∞) / (1 + (τ/τₛ)²)] τ
  • Where μ∞ is high-shear viscosity, μ₀ is low-shear viscosity, τₛ is characteristic shear stress.
• Example: molten sulfur at 120 °C has μ∞ = 0.0215, μ₀ = 0.00105, τₛ = 0.0000073 (valid only up to τ = 0.001).

🌡️ Oils and Arrhenius-type relationships

• Kinematic viscosity of oils follows ν = A exp(Eₐ / R T), where A is a pre-factor, Eₐ is activation energy, R is gas constant, T is absolute temperature.
• For wider temperature ranges, use Vogel–Fulcher–Tammann (VFT) equation: ln ν = A_VFT + B_VFT / (T − T₀).
• Cotton seed oil viscosity is shown as a 3D function of pressure and temperature.

🛠️ Practical applications

🛠️ Rotating cylinders (Example 1.6)

• Inner cylinder (radius 0.1 m) rotates at 31.4 rev/sec inside fixed outer cylinder (radius 0.101 m), length 0.2 m.
• Torque M = 1 N·m required.
• Shear stress τ = μ (dU/dr); velocity gradient ≈ 100 sec⁻¹.
• Torque M = 2π rᵢ² h μ (dU/dr) → solve for μ.

🛠️ Sliding block (Example 1.7)

• 1.0 kN block on 20° incline, oil film thickness 1×10⁻⁶ m, oil kinematic viscosity 3×10⁻⁵ m²/sec.
• Shear stress τ = μ (U / δ); friction force f = τ A.
• At steady state, gravity component = friction → solve for speed U.

🛠️ Rotating disc (Example 1.8)

• Disc of radius R rotates at ω in gap δ.
• Shear stress τ(r) = μ ω r / δ (increases with radius).
• Torque T = ∫₀ᴿ r τ dA = ∫₀ᴿ r (μ ω r / δ) 2π r dr = π μ ω R⁴ / (2δ).
• Don't confuse: torque depends on R⁴, not R²—outer regions contribute much more.

🧊 Density and bulk modulus

🧊 Fluid density

• Density ρ is related to temperature and pressure through the equation of state.
• Water density vs temperature and pressure is plotted; note the non-monotonic behavior near 4 °C at low pressure.

🧊 Thermal expansion in closed tanks (Example 1.9)

• Steel tank filled with water, heated from 10 °C to 50 °C; steel linear expansion coefficient 8×10⁻⁶ per °C.
• Tank volume change: V₂/V₁ = (1 + α ΔT)³.
• Water density change: ρ₂ = ρ₁ (1 + α ΔT)³ (thermal) and ρ₂ ∝ 1 / (1 − ΔP/E)³ (pressure).
• Equate to find final pressure: P₂ = P₁ − E α ΔT.
• If P₂ becomes very small or negative, the assumption breaks down and water evaporates.
• Pressure change can also be written as ΔP = βᵥ Δv + E ΔT, where βᵥ and E are partial derivatives of pressure with respect to volume and temperature.

🧭 Overview

🧠 One-sentence thesis

The bulk modulus quantifies how liquids resist volume change under pressure and is critical for hydraulic systems and deep-ocean applications, while surface tension arises from sharp density changes between phases rather than from unbalanced molecular forces.

📌 Key points (3–5)

• Density is a state property related to temperature and pressure through equations of state; thermal expansion and pressure changes both affect density.
• Bulk modulus measures how much pressure is needed to compress a liquid by a given volume fraction; it increases with pressure and decreases with temperature.
• Common confusion: Many sources incorrectly attribute surface tension to "unbalanced molecular cohesive forces," which conflicts with Newton's second law; surface tension actually results from sharp density changes between adjoining phases.
• When bulk modulus matters: hydraulic systems, deep ocean, geological systems, and cosmology—especially when rapid pressure or temperature changes occur.
• Mixture bulk modulus: for emulsions or multi-liquid systems, the effective bulk modulus is the harmonic mean weighted by volume fractions.

🧪 Fluid Density and Thermal Effects

🌡️ Density as a state property

Density is a property related to other state properties such as temperature and pressure through the equation of state or similar relationships.

• Density changes when temperature or pressure changes.
• Water density varies with temperature (see the excerpt's reference to a figure showing water density from 0 to 60 °C at various pressures).
• The excerpt emphasizes that density is "simple to analyze and understand" but requires careful attention to thermal expansion and pressure effects.

🔥 Thermal expansion in a steel tank (Example 1.9)

• A steel tank filled with water is heated from 10 °C to 50 °C.
• The steel undergoes linear thermal expansion of 8 × 10⁻⁶ per °C.
• The new water density after heating is:
  • ρ₂ = ρ₁ (1 + α ΔT)³ (thermal expansion term)
• The tank volume also expands: V₂/V₁ = (1 + α ΔT)³
• Pressure change is calculated using Young's modulus (or bulk modulus for water): P₂ = P₁ − E α ΔT
• Don't confuse: If P₁ − E α ΔT becomes very small or negative, the basic assumption fails and water may evaporate.

📐 Pressure–volume–temperature relationships

The excerpt shows that pressure change can be written as:

• dP = (∂P/∂v) dv + (∂P/∂T) dT
• Approximated as: ΔP = βᵥ Δv + E ΔT
• For water: ΔP ≈ 0.0002 Δρ + 2.15 × 10⁹ ΔT
• Notice that density change Δρ < 0 when volume decreases.

🔧 Bulk Modulus: Definition and Behavior

🔧 What bulk modulus measures

Bulk modulus is defined as B_T = −v (∂P/∂v)_T or equivalently B_T = ρ (∂P/∂ρ)_T

• It describes volume change as a result of pressure change at constant temperature.
• Analogous to a spring coefficient but in three dimensions instead of one.
• It is a measure of the energy that can be stored in the liquid.
• The excerpt notes it is "not the result of the equation of state but related to it."

📊 Bulk modulus values for common liquids

The excerpt provides a table (Table 1.5) with bulk modulus values (in 10⁹ N/m²):

Liquid	Bulk Modulus (10⁹ N/m²)	Critical Temperature	Critical Pressure
Water	2.15–2.174	647.096 K	22.064 MPa
Mercury	26.2–28.5	1750 K	172.00 MPa
Glycerol	4.03–4.52	850 K	7.5 Bar
Gasoline	1.3	not found	not found
SAE 30 Oil	1.5	not available	not available
Seawater	2.34	not available	not available

• Mercury has the highest bulk modulus (least compressible).
• Acetone has one of the lowest (0.80).

🌡️ How temperature and pressure affect bulk modulus

• Pressure increase → bulk modulus increases (molecules are closer, rejecting forces are stronger).
• Temperature increase → bulk modulus decreases (molecules are farther apart).
• Example: Commercial hydraulic oil can change temperature by 50 °C due to friction, causing bulk modulus to change by more than 60%.

🔗 Related thermal coefficients

The excerpt defines three related parameters:

1. Thermal expansion coefficient: β_P = (1/v)(∂v/∂T)_P
   Indicates volume change due to temperature change at constant pressure.

2. Coefficient of tension: β_v = (1/P)(∂P/∂T)_v
   Indicates pressure change due to temperature change at constant volume.

3. Relationship: β_T = − β_v / β_P
   This relationship is sometimes used to measure bulk modulus indirectly.

🧮 Bulk Modulus Calculations and Examples

🧮 Simple compression (Example 1.10)

• A liquid is reduced by 0.035% of its volume by applying 5 Bar pressure.
• Using the definition: B_T ≈ v (ΔP / Δv) = 5 / 0.00035 ≈ 14,285.714 Bar

💧 Compressing water (Example 1.11)

• To reduce water volume by 1% at 20 °C:
• ΔP ≈ B_T (Δv/v) ≈ 2.15 × 10⁹ × 0.01 = 2.15 × 10⁷ N/m² = 215 Bar
• Don't confuse: This is a large pressure—water is relatively incompressible.

🏗️ Two-layer liquid system (Example 1.12)

• Two liquids (oil and water) in a solid tank are compressed from P₀ to P₁.
• Volume change for any liquid: Δh = h ΔP / B_T
• Total height change (ignoring hydrostatic pressure): Δh₁₊₂ = ΔP (h₁/B_T₁ + h₂/B_T₂)
• Each layer compresses according to its own bulk modulus.

🔬 Pushka equation and pressure vessel (Example 1.13)

• A cylindrical steel pressure vessel (volume 1.31 m³) is filled with water.
• Pressure increases by 1000 kPa, then a safety plug bursts.
• Simple calculation: ΔV = − V ΔP / B_T = −1.31 × 1000 / (0.2 × 10¹⁰) = 6.55 × 10⁻⁷ m³ ≈ 0.655 liters
• Better approach: The excerpt notes this process can be isothermal or isentropic; the simple calculation needs correction for temperature change.
• Full differential: dρ = ρ dP / B_T − ρ P β_v dT / B_T
• The online calculation needs to subtract the second (temperature) term.

🧪 Bulk Modulus of Mixtures

🧪 Averaging for multi-liquid systems

When two or more liquids (or phases) are exposed to pressure together:

• Total volume change is the sum of individual changes: ∂V = ∂V₁ + ∂V₂ + ⋯ + ∂Vᵢ
• For each liquid: ∂Vᵢ = Vᵢ ∂P / B_Tᵢ
• Total volume: V = x₁ V + x₂ V + ⋯ + xᵢ V, where xᵢ = Vᵢ / V (volume fraction)

📐 Mixture bulk modulus formula

B_T_mix = 1 / (x₁/B_T₁ + x₂/B_T₂ + ⋯ + xᵢ/B_Tᵢ)

• This is a harmonic mean weighted by volume fractions.
• Assumption: pressure change is uniform for all phases, and total volume change is the sum of individual changes.
• The excerpt notes this applies to emulsions (suspensions of small globules of one liquid in another).
• Don't confuse: If the mixture has special interactions or energy-volume effects, another approach may be needed.

🏭 When Bulk Modulus Matters

🏭 Hydraulic systems

• Hydraulic systems use liquid to transmit forces (pressure) from one piston to another.
• In a theoretical (incompressible) liquid, moving one piston results in predictable movement of the other.
• Real liquids: density and volume are functions of pressure and temperature.
• Rapid systems experience significant temperature and pressure variations during operation.
• Commercial hydraulic fluid can change by 50 °C due to friction, causing bulk modulus to change by more than 60%.
• This large change affects response time significantly, so analysis must account for these effects.

🌊 Other applications

The excerpt lists situations where bulk modulus is important:

• Deep ocean: pressure increases with depth.
• Geological systems: Earth's interior, where pressures are extreme.
• Cosmology: the excerpt notes it is "not aware of any special issues" but mentions it as a potential application.
• The Pushka equation (relating pressure, volume, and bulk modulus) normally addresses deep ocean and geological systems.

🫧 Surface Tension

🫧 What surface tension is

Surface tension is force per length, measured in [N/m], and acts to stretch the surface.

• It manifests as a rise or depression of liquid at the free surface edge.
• Responsible for creation of drops and bubbles.
• Responsible for breakage of a liquid jet into drops (atomization).
• Results from a sharp change in density between two adjoined phases or materials.

❌ Common misconception (Example 1.14)

The excerpt strongly criticizes a widespread explanation:

• Erroneous explanation: "Cohesive forces between molecules down into a liquid are shared with all neighboring atoms. Those on the surface have no neighboring atoms above, and exhibit stronger attractive forces upon their nearest neighbors on the surface."
• Why it's wrong: This explanation conflicts with Newton's second law.
• The excerpt traces this error to Adam's book (early source, possibly earlier).
• Correct explanation: Surface tension results from a sharp change in density between two adjoined phases, not from "unbalanced molecular cohesive forces."
• The excerpt notes this misconception is "prevalent in physics and chemistry" and compares it to outdated ideas like "mountains were created by cooling lava."

🔬 Surface tension control volume

The excerpt includes a figure (1.17) showing surface tension control volume analysis with two principal radii (R₁ and R₂) and infinitesimal lengths (dℓ₁ and dℓ₂) and angles (dβ₁ and dβ₂).

• This suggests a more rigorous mechanical analysis of surface tension forces.
• The excerpt does not provide the full derivation but indicates that surface tension can be analyzed using control volume principles.

🧪 Disciplines affected by misconceptions

The excerpt notes that "die casting and ship stability disciplines and others are plagued such nonsense."

• Example in die casting: it was believed that for critical plunger velocity, physics could be ignored (over 300 research teams worked on this).
• The excerpt emphasizes the importance of correcting fundamental misunderstandings in engineering disciplines.

🧭 Overview

🧠 One-sentence thesis

Surface tension creates pressure differences across curved interfaces that drive capillary phenomena like liquid rise in tubes, with the magnitude depending on interface curvature, contact angle, and fluid properties.

📌 Key points (3–5)

• Pressure difference from curvature: Surface tension creates higher pressure on the concave side of a curved interface, with magnitude proportional to surface tension and inversely proportional to radius of curvature.
• Wetting vs non-wetting: Contact angle below 90° indicates wetting fluid; above 90° indicates non-wetting; this angle depends on all three materials (solid, liquid, gas) and can change with surface treatment.
• Capillary rise mechanism: Surface tension forces at the contact line balance against the weight of the raised liquid column, determining equilibrium height.
• Common confusion: Contact angle is not a fixed property of the liquid alone—it depends on the solid surface, gas medium, temperature, and even surface cleanliness; published data for the same material pairs often vary widely.
• Shape of free surfaces: The curved meniscus shape results from balancing surface tension forces with gravitational pressure differences, described by a nonlinear differential equation.

💧 Pressure differences across curved interfaces

💧 General curvature formula

Pressure difference across a curved interface: ΔP = σ(1/R₁ + 1/R₂), where R₁ and R₂ are the two principal radii of curvature and σ is surface tension.

• This equation predicts that pressure difference increases with the inverse of radius.
• The pressure is higher on the concave side (inside) of the interface.
• Two principal radii account for curvature in two perpendicular directions.

🔵 Special case: infinite cylinder

For an infinitely long cylinder (one radius is infinite):

• ΔP = σ/R
• Only one radius matters because curvature exists in only one direction.
• Example: A very long cylindrical bubble or droplet.

⚪ Special case: sphere

For a sphere (both radii equal):

• ΔP = 2σ/R
• Both principal radii equal the sphere radius.
• For a soap bubble with two layers (inner and outer surface): ΔP = 4σ/R

🧪 Work and bubble compression

When compressing bubbles reversibly (very slowly, maintaining constant temperature):

• Work = 4πσ(r_f² - r_0²) for one bubble
• Work is negative (done on the system) because pressure increases.
• The relationship uses the sphere pressure formula ΔP = 2σ/r throughout the process.

📏 Capillary rise in tubes and gaps

📏 Two-dimensional parallel plates

For liquid rising between two parallel plates separated by distance ℓ:

• Balance: 2σ cos(0°) = ρ g h ℓ (surface tension on two sides vs. weight of liquid column)
• Height: h = 2σ/(ℓ ρ g)
• The contact angle is assumed 0° (maximum possible upward force).
• Example: With σ = 0.05 N/m, ρ = 1000 kg/m³, ℓ = 0.001 m, the rise can be calculated directly.

🔄 Concentric cylinders

For liquid between two concentric cylinders (inner radius r_i, outer radius r_o):

• Balance: σ · 2π(r_i cos θ_i + r_o cos θ_o) = ρ g h · π(r_o² - r_i²)
• Height: h = 2σ(r_i cos θ_i + r_o cos θ_o) / [ρ g(r_o² - r_i²)]
• Maximum rise (when both contact angles = 0°): h = 2σ / [ρ g(r_o - r_i)]
• The inner cylinder adds extra upward force and reduces the volume of liquid to lift.

🧪 Mercury tube depression

For a glass tube inserted into mercury:

• Surface tension acts on both inner diameter D_i and outer diameter D_o.
• Upward force: F = σ · 2π cos(55°) · (D_i + D_o)
• If diameters differ considerably, horizontal components don't fully cancel: F_horizontal = σ · 2π sin(55°) · (D_o - D_i)
• Pressure balance: P · πr² = σ · 2πr (approximately, neglecting weight)
• Depression depth: h = 2σ/(g ρ r)

🎯 Wetting and contact angles

🎯 Three-phase contact and force balance

At the point where solid (S), liquid (L), and gas (G) meet:

• Three surface tensions act: σ_gs (gas-solid), σ_ls (liquid-solid), σ_lg (liquid-gas).
• Along the solid boundary: σ_gs - σ_ls - σ_lg cos β = 0
• Perpendicular to solid: F_solid = σ_lg sin β
• A contact angle β forms to balance these forces because straight-line geometry cannot balance them otherwise.

💧 Wetting vs non-wetting definition

Wetting fluid: contact angle < 90°
Non-wetting fluid: contact angle > 90°

Condition	Contact angle	Behavior
Wetting	< 90°	Liquid spreads on surface; meniscus curves upward at edges
Non-wetting	> 90°	Liquid beads up; meniscus curves downward at edges

• The angle depends on properties of the liquid, gas medium, and solid surface.
• Small changes to the solid surface (e.g., coating) can flip wetting to non-wetting.
• Example: Water is usually wetting, but certain commercial sprays coat surfaces to make water non-wetting.

⚠️ Unreliability of contact angle data

The excerpt emphasizes that no reliable universal data exists except for pure substances and perfect geometries:

• Table 1.6 shows contact angles for distilled water on copper ranging from 9.6° to 90° across different studies.
• For nickel: π/4.74 to π/3.83 (roughly 38° to 47°).
• Don't confuse: "Water is a wetting fluid" is not an absolute statement—it depends on the solid surface.
• Temperature also significantly affects contact angle (order-of-magnitude changes observed).

📐 Shape of liquid surfaces

📐 Governing differential equation

For a two-dimensional liquid surface h(x):

• Pressure difference below the surface: ρ g h(x) (due to gravity)
• Surface tension creates curvature: ρ g h(x) = σ/R(x)
• Radius of curvature: R(x) = [1 + (dh/dx)²]^(3/2) / (d²h/dx²)
• Combining gives the 1-D surface tension equation:
  g h ρ [1 + (dh/dx)²]^(3/2) - σ (d²h/dx²) = 0

This is a nonlinear differential equation describing the meniscus shape.

🔢 Laplace capillarity constant

Define Laplace's capillarity constant: L_p = σ/(ρ g), with units of length squared.

• Integrating the differential equation yields: h²/(2L_p) + constant = -1/√(1 + (dh/dx)²)
• At infinity (flat surface): h = 0 and dh/dx = 0, so constant = -1.
• Further manipulation gives: dh/dx = √[(1/(1 - h²/(2L_p)))² - 1]
• This can be separated and integrated: ∫ dh/√[...] = x + constant
• The constant is determined by boundary conditions (e.g., height at x = 0).

🧩 Capillarity definition

Capillarity: Surface tension causes liquid to rise or penetrate into areas (volumes) where it otherwise would not be.

• The curved meniscus shape results from balancing surface tension forces against gravitational pressure differences.
• The analysis is "studied extensively in classes on surface tension" and relates to dimensionless parameters (covered in Chapter 9).
• An analytical solution exists but is complex (involves inverse hyperbolic cosine functions).

🧭 Overview

🧠 One-sentence thesis

Surface tension creates pressure differences across curved liquid surfaces that can cause liquids to rise or fall in narrow tubes, but these effects are significant only when the tube radius is very small and gravity is negligible.

📌 Key points (3–5)

• What surface tension does: it balances pressure differences across free surfaces and can cause liquid to rise (or depress) in tubes.
• The governing equation: the pressure difference due to surface tension equals surface tension divided by the radius of curvature; gravity opposes this by pulling the liquid down.
• Capillarity range: simplified equations (like h = 2σ cos β / (g Δρ r)) predict height reasonably only in a certain range of tube radii (roughly 1–5 mm for many liquids); outside this range the model breaks down.
• Common confusion: the contact angle β is needed for accurate predictions, but reliable contact-angle data are often unavailable or conflicting, limiting practical use of the simple formulas.
• When it matters: surface tension is important only when the radius is very small; for large radii gravity dominates and the liquid surface approaches a straight line.

🧩 Pressure balance and the governing equation

🧩 How surface tension balances pressure

Surface tension reduces the pressure in the liquid above the liquid line; the pressure just below the surface is −g h(x) ρ, while the gas side is at atmospheric pressure.

• The liquid surface curves because surface tension pulls inward, creating a pressure difference.
• The pressure difference across the curved surface must be balanced by surface tension.
• The relationship is: g h(x) ρ = σ / R(x), where σ is surface tension and R(x) is the radius of curvature.

📐 Radius of curvature for a continuous surface

• For any continuous height function h = h(x), the radius of curvature is:
  • R(x) = (1 + [ḣ(x)]²)^(3/2) / ḧ(x)
  • ḣ is the first derivative (slope), ḧ is the second derivative (curvature).
• Substituting this into the pressure balance gives a nonlinear differential equation:
  • g h(x) ρ = σ (1 + [ḣ(x)]²)^(3/2) / ḧ(x)
• This equation describes the one-dimensional surface shape due to surface tension.

🔧 Solving the differential equation

• The equation can be rewritten and integrated step by step.
• A key constant is Laplace's capillarity constant Lp = σ / (ρ g), with units of meter squared.
• Boundary conditions (e.g., symmetry at x = r, or height at a given point) determine the specific solution.
• The excerpt notes that the full analytical solution exists but is complex; the book limits discussion because it is introductory.

Don't confuse: the differential equation is nonlinear, so simple integration is not straightforward; variable separation and dummy variables (ξ = ḣ) are used to transform it into a solvable form.

💧 Capillarity: liquid rising in tubes

💧 What capillarity means

Capillary forces refer to the fact that surface tension causes liquid to rise or penetrate into an area (volume) where it otherwise would not be.

• Surface tension can pull liquid upward in a narrow tube against gravity.
• The height the liquid rises depends on the tube radius, surface tension, density difference, and the contact angle.

📏 The simplified capillarity equation

• The height that liquid rises in a tube is given by:
  • h = 2 σ cos β / (g Δρ r)
  • Δρ is the density difference between liquid and gas.
  • r is the tube radius.
  • β is the contact angle.
• Maximum height occurs when β = 0 (perfect hemisphere, cos β = 1):
  • h_max = 2 σ / (g Δρ r)

Limitation: this equation is "unusable and useless" unless the contact angle is known, and in reality reliable contact-angle data are often unavailable or conflicting.

📊 Range of validity

Tube radius	What happens	Why
Very small	Equation (1.69) overpredicts height; negative pressure may cause vaporization; continuous model breaks down	Gravity effect is small, but extreme conditions invalidate assumptions
Small (1–5 mm range)	Equation (1.57) (the full differential equation) gives better results; surface approaches hemispherical	Gravity effect is moderate; simplified equation is reasonable
Large	Liquid surface approaches a straight line; gravity dominates	Surface tension effect becomes negligible

• The excerpt emphasizes that the actual working range for many liquids (including water) is about 1–5 mm.
• Figure 1.25 (described in the text) shows the theoretical height (blue line) vs. actual height (red line), illustrating that the simple equation works only in a certain range.

🔽 Depression of liquid (negative contact angle)

• When the contact angle is "negative" (non-wetting), the liquid is depressed rather than raised.
• The depression height is similar to equation (1.69) but with a minus sign.
• Gravity works against surface tension in this case, reducing the range and quality of predictions.

Example (from the excerpt): Measurements of distilled water and mercury (Figure 1.26) agree with the discussion—mercury depresses, water rises, and the simple equation is accurate only in a limited range of radii.

🔬 Practical implications and examples

🔬 When surface tension matters

• Surface tension is important only when the radius is very small and gravity is negligible.
• For large radii, gravity dominates and the liquid surface is nearly flat.
• The surface tension depends on the two materials (or mediums) that the interface separates.

🧪 Worked examples from the excerpt

Example: Water Droplet (pressure difference of 1000 N/m²)

• Problem: Calculate the diameter of a water droplet at 20°C to achieve a pressure difference of 1000 N/m².
• Solution: Using the pressure formula for a droplet, D = 2R = 4σ / ΔP.
  • With σ = 0.0728 N/m (water at 20°C), D ≈ 2.912 × 10⁻⁴ m (about 0.29 mm).

Example: Droplet Pressure (diameter 0.02 cm)

• Problem: Calculate the pressure difference for a water droplet at 20°C with diameter 0.02 cm (radius 0.0002 m).
• Solution: ΔP = 2σ / r ≈ 2 × 0.0728 / 0.0002 ≈ 728 N/m².

Don't confuse: the pressure inside a droplet is higher than outside because the curved surface pulls inward; smaller droplets have higher pressure differences.

📋 Surface tension data

• The excerpt mentions that surface tension values for selected materials are given in Table 1.7 (not shown in the excerpt).
• Contact-angle information exists but is often conflicting and unreliable (see Table 1.6, also not shown).