Lecture 16: Balance Laws: Mass, Momentum, and Energy

Overview

Steven M. Wise develops the integral balance laws for a moving continuum: mass and momentum conservation, the first law for internal and kinetic energy, and the second law through entropy production. He then derives the spatial continuity equation from mass conservation, connects it to its material form using the deformation Jacobian, and uses Reynolds transport theorem to prepare local energy and entropy equations for a later lecture.

Key takeaways

Mass conservation for every material region is equivalent to the local continuity equation ∂ρ/∂t + div(ρv) = 0.
In material coordinates, local mass conservation takes the form ρmJ = ρ₀; differentiating it and using J̇ = J div(v) produces the spatial mass balance.
Reynolds transport theorem plus continuity gives d/dt ∫Ω(t) ρQ = ∫Ω(t) ρDQ/Dt for any spatial scalar Q, a key tool for deriving local energy and entropy laws.
The first-law balance tracks internal and kinetic energy: their combined rate of change equals net heating plus external power.
The second law constrains entropy production to be nonnegative, but entropy in a non-isolated body can decrease when heat is removed through its boundary.
Resultant force includes body-force and traction contributions, while resultant torque weights each contribution by its moment arm from the chosen reference point.

Chapters

0:00 Mass, Linear Momentum, and Angular Momentum of a Moving Region

For a spatial region Ω(t), mass is the volume integral of density ρ and linear momentum is the integral of ρv.
Angular momentum about a reference point z is the integral of (x − z) × ρv.
The spatial velocity field v is central to expressing balance laws for a deforming body.

6:00 Conservation of Mass for Material Regions

A material region Ω(t) contains the same particles throughout the motion, so its total mass remains constant.
The mass balance assumes no chemical reactions that change the tracked species and neglects relativistic mass-energy conversion.
The integral statement is that the time derivative of the mass of Ω(t) is zero, equivalently its mass equals that of Ω(0).

10:00 Body Forces, Surface Traction, and Resultant Torque

The resultant force combines the volume integral of ρb, where b is force per unit mass, with surface traction t.
Traction has units of force per area, so integrating it over the boundary gives a force.
Resultant torque about z integrates (x − z) × ρb over the volume and (x − z) × t over the surface.

14:00 Linear and Angular Momentum Balance Laws

The rate of change of a moving region's linear momentum equals its resultant applied force.
Angular momentum about the origin changes at the rate of the resultant torque about the origin.
Wise recommends Section 5.1 of the text for the particle-system motivation and derivation of these continuum axioms.

19:00 Internal Energy and the Non-Equilibrium Thermodynamics Framework

Wise distinguishes these moving-body balance laws from equilibrium thermodynamics and describes them as non-equilibrium laws.
Internal energy U(Ω(t)) is the volume integral of ρe, where e is specific internal energy per unit mass.
Specific internal energy may depend on temperature and, for mixtures, on chemical-species concentrations.

24:00 Net Heating from Sources and Boundary Heat Flux

Net heating combines a volume source ρr, with r the heat supply per unit mass, and the boundary flux term involving heat flux q and outward normal n.
Heat can enter or leave through the boundary, for example when a hot or cold object contacts the body.
The sign of the boundary contribution distinguishes heat entering the region from heat leaving it.

29:00 External Power, Kinetic Energy, and Net Working

External power is the sum of the volume integral of ρb · v and the surface integral of t · v.
Kinetic energy is defined as the volume integral of ½ρ|v|², the continuum counterpart of summing ½mv² over particles.
Net working is defined as external power minus the rate of change of kinetic energy, allowing mechanical input to affect more than bulk motion.

34:00 First Law: Heat and Mechanical Work Change Total Energy

For thermal and mechanical energy, the rate of change of internal energy equals net heating plus net working.
Rearranging gives the rate of change of internal plus kinetic energy as net heating plus external power.
Mechanical work can become internal energy: compressing gas in a container can raise its temperature and pressure.

39:00 Potential Energy Form and the Introduction to Entropy

If external power can be represented through a potential-energy rate, the first-law balance can include internal, kinetic, and potential energy.
For a simple particle with no heat transfer, the combined kinetic and potential energy is conserved.
Entropy is introduced as the state quantity behind the second law and the spontaneous transfer of heat from hot bodies to cold bodies.

44:00 The Second Law and Entropy Production

Total entropy is the integral of ρs, where s is specific entropy per unit mass, and the temperature field is assumed positive.
The second-law balance compares the entropy rate with heating divided by temperature and requires nonnegative entropy production.
Entropy need not increase within every non-isolated region: boundary heat removal can reduce its entropy without violating the second law.

50:00 Clausius-Duhem Inequality and the Hot-to-Cold Heat Flow

The localized entropy-production condition is identified as the Clausius-Duhem inequality.
For an insulated system with no internal heat supply, entropy cannot decrease over time.
When hot and cold bodies are isolated together, heat flows from hot to cold while total energy stays fixed and entropy production drives the process.

55:00 From Integral Mass Balance to the Continuity Equation

The continuity equation is the spatial local form of mass conservation: ∂ρ/∂t + div(ρv) = 0.
Wise first converts the integral mass balance to material coordinates, using the deformation Jacobian J and the pulled-back density ρm.
The resulting material mass relation is ρmJ = ρ₀, obtained by applying localization to arbitrary material regions.

1:00:00 Differentiating the Material Mass Relation

Differentiating ρmJ = ρ₀ in time uses the product rule because both material density and J depend on time.
The Jacobian identity is J̇ = J div(v) in material coordinates; since J is positive, it can be divided out.
Returning to spatial coordinates yields the material-derivative form of mass balance, which expands to ∂ρ/∂t + div(ρv) = 0.

1:05:00 A Short Reynolds Transport Proof and Mass-Weighted Fields

A shorter continuity-equation proof applies Reynolds transport theorem directly to the conserved mass integral, then localizes the result.
For any spatial scalar Q, Wise derives a time-derivative rule for the integral of ρQ over a moving region.
The derivation uses the continuity equation to cancel the density and velocity-divergence terms left by Reynolds transport theorem.

1:10:00 Preparing Local Energy and Entropy Balances

The mass-weighted transport identity reduces the rate of change of the integral of ρQ to the integral of ρ times the material derivative of Q.
The identity applies to specific internal energy and specific entropy, the quantities used in the first- and second-law balances.
Wise closes by noting that this result will enable local forms of the energy and entropy laws in the next lecture.

Transcript excerpt

0:04 Welcome back to continuum mechanics class number 15 part one of our discussion on balance laws and chapter five. And I imagine we'll do at least three parts from chapter five. And um chapter five is really giving us everything that we need to formulate our equations of motion uh and even equilibrium equations for describing continuum bodies that are under stress, strain, um the elasticity equations that we're going to be interested in discussing and of course Navier-Stokes equations. We will actually see the Navier-Stokes equations today.

0:52 Um they're kind of hidden and they're not in their fullest um they're not they're not complete. But uh you'll you will see the the two equations which become the Navier-Stokes equations starting today. So I want to first talk about mass and momentum conservation and we're thinking about um in this case we're not in equilibrium anymore. We're talking about motions and and we're talking about mass and momentum conservation in a deforming body during a motion. All right. So let me tell you or remind you what we mean by the time dependent mass and the time dependent linear

1:38 momentum. Mass and momentum conservation. Well, I need to tell you what the mass and momentum of the system are. Let phi be a motion with mass density rho and spatial velocity field v. Remember we we have these options of defining certain variables in a material description or spatial description. The most important thing for us, it turns out, in describing these um these balance laws is the velocity field in the spatial description, the so-called spatial velocity field. Let omega t be an arbitrary set from B of t. Remember B of t is just the mapping under phi t of B, the our initial set.

2:27 This is where the body lived in its

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In this course

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Summary, takeaways, and chapters were generated by AI from the video's transcript and may contain errors. The video belongs to its creator, Steven M. Wise.