Coarse-Grained Continuum Mechanics – Conservation Laws

Here, we only consider the continuum picture.
See 2.5-Conservation-Laws for first principle derivations

Key Topics

Bridging Microscopic to Continuum Description

Introduction to Coarse-Graining:

• How we average over many atoms/molecules to treat a material as continuous fields
• Density, velocity, etc. on scales large compared to molecular size
• Motivation for continuum view in soft matter
• Example: A polymer solution or foam modeled as a continuous medium despite discrete entities

Fundamental Conservation Laws

Derived via coarse-graining:

1. Conservation of Mass (continuity equation)
2. Conservation of Momentum (Newton’s second law in continuum form → Euler or Navier-Stokes equations)
3. Conservation of Energy (first law of thermodynamics in continuum form)

Physical Meaning of Conservation Laws

Integral Form:

• What goes in minus what goes out = accumulation

Differential Form:

• Local divergence form
• Changes at a point due to flows or sources at that point

The Continuum Hypothesis

Assumptions:

• Material properties vary smoothly
• Can be defined at a “point” much larger than molecular scale
• Point must be small compared to system size
• Scales much larger than the mean free path of the molecules .

Field Variables

Result of coarse-graining:

• Velocity field:
• Pressure:
• Density:
• etc.

Learning Outcomes

1. State the three main conservation laws

   • Mass, momentum, energy in words
   • Recognize their mathematical forms

2. Derive a basic continuity equation

   • Starting from a small fixed volume
   • Show that mass change = (mass in) – (mass out)
   • Translate to:

3. Identify terms in momentum equation

   • Understand:
   • In words: mass×acceleration = forces from stress gradient + external forces
   • Grasp where inertia, pressure, and viscosity enter

4. Connect to physical reasoning

   • Apply conservation laws to simple scenarios
   • Example: Squeezing toothpaste tube (mass conservation + incompressibility)
   • Example: Fluid speeding up in narrow pipe (continuity with constant flow rate)
   • Articulate local vs global conservation (nothing “teleports”)

Conservation Law Details

1. Conservation of Mass

Physical Statement: Mass cannot be created or destroyed (for systems without nuclear reactions or relativistic effects).

Integral Form:

Differential Form (Continuity Equation):

For incompressible fluids (constant density):

2. Conservation of Momentum

Physical Statement: Newton’s second law applied to a fluid element: Rate of change of momentum equals sum of forces.

Differential Form:

Where:

• = material derivative (following fluid parcel)
• = stress tensor
• = external body forces (e.g., gravity)

For Newtonian fluids (Navier-Stokes):

3. Conservation of Energy

Physical Statement: First law of thermodynamics: Change in energy equals heat added plus work done.

Differential Form:

Where:

• = specific internal energy
• = heat flux
• = viscous dissipation term

Examples and Case Studies

Mass Conservation Example

Pipe Flow:

• Water flowing through a pipe that splits into two branches
• If 2 L/min enters, sum out of branches must be 2 L/min (steady state, incompressible)
• Illustrates:

Variable Cross-Section:

• If one branch is partially closed, water velocity increases there
• Smaller area → higher speed to satisfy continuity
• Analogous to blood flow speeding up in a narrowed artery

Momentum Conservation Example

Soft Gel in Slingshot:

• When stretched and released, momentum is transferred to the gel
• Causes it to fly
• Continuum terms: Internal elastic forces (stress) accelerated the gel’s mass

Stone in Pond:

• Dropping a stone in a pond
• Momentum from gravity and impact is redistributed via:
  • Pressure waves (sound)
  • Fluid motion
• Violating momentum conservation locally leads to observable flows

Energy Conservation Example

Silly Putty Ball:

• Dropped from height, deforms and warms slightly on impact
• Potential energy → deformation work + heat
• Illustrates energy bookkeeping

Mixing Fluids:

• Slow mixing of glycerol and water
• Mechanical work (stirring) dissipated as heat via viscous stress
• Internal energy increase consistent with energy conservation

Key Equations Summary

Conservation Law	Differential Form	Key Insight
Mass		Local density change = net flux
Momentum		Acceleration from stress + body forces
Energy		Energy change = heat flux + work

Key Concepts to Remember

• Coarse-graining: Averaging microscopic behavior to obtain continuum fields
• Field variables: Density, velocity, pressure as functions of position and time
• Local conservation: Changes at a point due to local fluxes/sources only
• Continuum hypothesis: Valid when system >> molecular scale
• Incompressibility: Common assumption for liquids ()

References

1. Continuity equation - Wikipedia: https://en.wikipedia.org/wiki/Continuity_equation
2. Batchelor, G.K. “An Introduction to Fluid Dynamics” (1967)
3. Landau & Lifshitz “Fluid Mechanics” (1987)

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