Lecture 18: Entropy Production and Lagrangian Balance Laws

Overview

Steven M. Wise develops the entropy-production and free-energy balances, then translates continuum-mechanics balance laws from the spatial (Eulerian) description to the material (Lagrangian) description. He explains how conservation of mass, momentum, angular momentum, and energy produce equations involving the Jacobian, the first and second Piola-Kirchhoff stresses, and a transformed heat flux, while emphasizing that constitutive laws are needed to close the system.

Key takeaways

Specific free energy is psi = phi - theta eta; when temperature is constant and uniform, its dissipation provides a useful way to assess thermodynamic consistency.
The entropy-production rate sigma is nonnegative, so the free-energy balance contains a nonpositive contribution -theta sigma for positive absolute temperature.
The 22 listed Eulerian fields are not determined by the 11 balance equations alone; constitutive relations for stress, heat flux, and state variables are needed to close the model.
The Lagrangian mass balance is rho_m J = rho_0, where J = det F converts current volume to reference volume.
The first Piola-Kirchhoff stress P = J sigma_m F^{-T} converts spatial traction into a reference-configuration force, while the second Piola-Kirchhoff stress S = F^{-1}P is symmetric.
The material first law uses the referential heat flux Q = J F^{-1}q_m and is derived by transforming the integral energy balance, applying the divergence theorem, and localizing over an arbitrary reference region.

Chapters

0:00 Why Derive Balance Laws in Both Eulerian and Lagrangian Forms

Steven M. Wise distinguishes the spatial, or Eulerian, description from the material, or Lagrangian, description.
The integral balance axioms lead to distinct local equations in each description, including separate forms for mass and linear momentum.
The lecture first completes entropy-production and free-energy results, then derives Lagrangian local balance laws.

2:20 Entropy Production and the Nonnegative Second-Law Term

The local Eulerian entropy balance includes specific entropy, temperature, heat supply, heat flux, and the entropy-production rate sigma.
Temperature is treated as absolute and positive, while sigma is nonnegative at every point and time.
Heat supplied to a system can change its entropy, while internal irreversible processes are represented by entropy production.

4:00 Multiplying the Entropy Balance by Temperature

Wise multiplies the local entropy equation by temperature theta to remove theta from the heat-source term.
He expands the divergence of heat flux divided by temperature using the scalar-vector product rule.
The identity grad(1/theta) = -grad(theta)/theta^2 produces a heat-flux contribution involving q dotted with grad(theta), divided by theta.

7:00 Collecting Terms into the Temperature-Weighted Entropy Balance

The transformed entropy equation groups theta times the material entropy rate, the heat source, and the divergence of heat flux into a quantity called delta.
The remaining heat-flux gradient term and theta sigma express the local entropy constraint.
Because theta is positive and sigma is nonnegative, the entropy-production contribution remains nonnegative.

9:48 Helmholtz Free Energy as Internal Energy Minus Temperature-Weighted Entropy

Specific free energy is defined as psi = phi - theta eta, where phi is specific internal energy and eta is specific entropy.
Multiplying psi by mass density gives free energy per unit volume.
For reversible processes at constant temperature, free energy represents the portion of internal energy available to perform work.

13:30 Deriving the Eulerian Free-Energy Balance

Wise differentiates psi = phi - theta eta using the product rule for material time derivatives.
He combines the local energy balance with the temperature-weighted entropy balance to eliminate the entropy-rate terms.
The resulting balance contains stress power, the temperature-rate term, heat-flux transport, and the dissipative contribution -theta sigma.

20:30 The Reduced Clausius-Duhem Inequality for Isothermal Flow

Since theta sigma is nonnegative, its negative contribution constrains the rate of free-energy change.
For uniform, constant temperature, free energy dissipates in the way entropy is produced.
Wise identifies this reduced inequality as especially useful for studying the isothermal Navier-Stokes equations in the next chapter.

21:30 Counting Eulerian Unknowns and Balance Equations

The formulation has 22 listed fields, including motion, velocity, density, nine stress components, temperature, three heat-flux components, internal energy, and entropy.
The kinematic relation, mass balance, linear momentum, angular momentum, and energy balance provide 11 equations in the count presented.
Constitutive laws, such as relations for stress, heat flux, and pressure, are required to close the system; entropy production acts as a constraint rather than an independent balance equation.

27:30 Lagrangian Mass Balance and the Reference Density

The reference density rho_0 is fixed in the initial configuration, where the motion map is the identity.
Changing variables from the current configuration introduces the Jacobian J = det F.
Mass conservation gives rho_m J = rho_0, linking spatial density to its material description.

34:30 Piola-Kirchhoff Stress and Lagrangian Linear Momentum

The first Piola-Kirchhoff stress is defined by P = J sigma_m F^{-T}, converting spatial stress and surface forces to the reference configuration.
Changing variables in the momentum balance transforms its volume and boundary integrals, with the boundary normal adjusted by the deformation map.
The divergence theorem and localization over an arbitrary reference region yield rho_0 times the material acceleration equal to Div P plus rho_0 times the material body force.

48:00 Angular Momentum Makes the Second Piola-Kirchhoff Stress Symmetric

The Eulerian angular-momentum balance requires the Cauchy stress tensor to be symmetric.
The second Piola-Kirchhoff stress is S = F^{-1}P and is symmetric as a consequence of that spatial stress symmetry.
The first Piola-Kirchhoff stress commonly appears in material motion equations, but it is not generally symmetric.

52:00 Converting External Power into Material-Configuration Terms

The first-law derivation begins by rewriting external power as body-force and traction contributions alongside kinetic energy.
Changing variables converts spatial volume and boundary integrals to the reference configuration, introducing J and the transformed normal.
The first Piola-Kirchhoff stress emerges in the transformed traction integral; the divergence theorem and a tensor-vector product rule simplify the expression.

1:03:00 The Lagrangian First Law and Referential Heat Flux

Specific internal energy becomes a material field, while the referential heat flux is defined as Q = J F^{-1}q_m.
The transformed heat-source term uses the reference density rho_0, and the heat-flux boundary term becomes a material divergence.
Localization yields the material energy balance, with the same overall structure as the Eulerian law but using P, Q, and material derivatives.

1:10:00 Next Topics: Frame Indifference and Isothermal Navier-Stokes

Wise previews frame indifference as the final topic in Chapter 5.
The following chapter begins the isothermal fluid-flow equations and develops the isothermal Navier-Stokes model.
The reduced free-energy inequality will be useful for analyzing thermodynamic consistency in that setting.

Transcript excerpt

0:04 All right, welcome back to continue mechanics. This is class number 18. We're still in chapter five. This is part three in chapter five where we're talking about balance laws. I expect to have at least one more part to this part four. I don't expect to do a part five, but that may happen for chapter five. Today we're going to talk about um a couple of familiar items. We're going to work a little bit further with this entropy production equation that we developed last time. Then I want to summarize all the equations that we've derived in the Oilerian formulation that is the moving frame formulation. And then I want to derive kind of rederive everything in terms of the lrronian uh formulations that is

0:54 formulations where the equations are written over the reference configuration. Um so you'll notice that our axioms our balance laws were given in terms of integrals and they didn't really have an oilarian or a lrangeian formulation. the uh I guess if you wanted to call it you could say they're kind of a mixture of the two in some sense but when we derive the local laws from the integration laws we derive differential equations we have these two ways of deriving or two ways of writing functions that is in a material description also called a lranchian description or a spatial description also called or oilarian so that's why we get two of each equation

1:42 mass balance you get two um the balance of linear momentum you get two angular momentum you get two and so on. Uh so why go through all why go through both of them? While the derivations reinforce the definitions, they also reinforce the fact that these um local laws are in fact um distinct from each other. Even

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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.