# Coarse-Grained Continuum Mechanics – Conservation Laws > Here, we only consider the continuum picture. <br> > 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 $\lambda$. ### Field Variables Result of coarse-graining: - Velocity field: $\boldsymbol{v}(\boldsymbol{r}, t)$ - Pressure: $p(\boldsymbol{r}, t)$ - Density: $\rho(\boldsymbol{r}, t)$ - etc. ## Learning Outcomes 1. State the three main conservation laws - Mass, momentum, energy in words - Recognize their mathematical forms 1. Derive a basic continuity equation - Starting from a small fixed volume - Show that mass change = (mass in) – (mass out) - Translate to: $\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \boldsymbol{v}) = 0$ 1. Identify terms in momentum equation - Understand: $\rho \frac{D\boldsymbol{v}}{Dt} = \nabla\cdot \boldsymbol{\sigma} + \boldsymbol{f}_{\text{ext}}$ - In words: mass×acceleration = forces from stress gradient + external forces - Grasp where inertia, pressure, and viscosity enter 1. 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: $\frac{d}{dt}\int_V \rho \, dV = -\int_S \rho \boldsymbol{v} \cdot \boldsymbol{n} \, dS$ Differential Form (Continuity Equation): $\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \boldsymbol{v}) = 0$ For incompressible fluids (constant density): $\nabla\cdot\boldsymbol{v} = 0$ ### 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: $\rho \frac{D\boldsymbol{v}}{Dt} = \nabla\cdot \boldsymbol{\sigma} + \boldsymbol{f}_{\text{ext}}$ Where: - $\frac{D}{Dt}$ = material derivative (following fluid parcel) - $\boldsymbol{\sigma}$ = stress tensor - $\boldsymbol{f}_{\text{ext}}$ = external body forces (e.g., gravity) For Newtonian fluids (Navier-Stokes): $\rho \left(\frac{\partial \boldsymbol{v}}{\partial t} + \boldsymbol{v}\cdot\nabla\boldsymbol{v}\right) = -\nabla p + \eta \nabla^2 \boldsymbol{v} + \boldsymbol{f}_{\text{ext}}$ ### 3. Conservation of Energy Physical Statement: First law of thermodynamics: Change in energy equals heat added plus work done. Differential Form: $\rho \frac{De}{Dt} = -\nabla\cdot\boldsymbol{q} + \boldsymbol{\sigma}:\nabla\boldsymbol{v}$ Where: - $e$ = specific internal energy - $\boldsymbol{q}$ = heat flux - $\boldsymbol{\sigma}:\nabla\boldsymbol{v}$ = 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: $\dot{m}_{\text{in}} = \dot{m}_{\text{out}}$ 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 | $\frac{\partial \rho}{\partial t} + \nabla\cdot(\rho \boldsymbol{v}) = 0$ | Local density change = net flux | | Momentum | $\rho \frac{D\boldsymbol{v}}{Dt} = \nabla\cdot \boldsymbol{\sigma} + \boldsymbol{f}_{\text{ext}}$ | Acceleration from stress + body forces | | Energy | $\rho \frac{De}{Dt} = -\nabla\cdot\boldsymbol{q} + \boldsymbol{\sigma}:\nabla\boldsymbol{v}$ | 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 ($\nabla\cdot\boldsymbol{v} = 0$) ## 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) ---