Lecture 21: Potential Flow and Elastic Fluids
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Overview
Steven M. Wise develops potential-flow results for ideal fluids, then extends them to compressible elastic fluids with a pressure-density equation of state. He shows how vorticity preservation and Bernoulli relations reduce fluid motion to scalar-potential problems, and derives consequences for frame indifference, mechanical energy, and boundary-value formulations.
Key takeaways
- An ideal fluid that begins irrotational stays irrotational under conservative body forces; Wise proves this by evolving a material-coordinate vorticity variable built with the inverse deformation gradient.
- In a bounded connected region, an incompressible potential flow with impermeable walls must be stationary, because Laplace's equation and homogeneous Neumann data force the velocity potential to be constant.
- The half-space potential φ = a(x₁² - x₃²)/2 produces a nontrivial incompressible flow satisfying no penetration, illustrating why unbounded domains avoid the bounded-domain triviality result.
- An elastic fluid uses an increasing pressure law p = π(ρ); for π(ρ) = p* + c²(ρ - ρ*), the constant slope c² sets the linear pressure-wave speed.
- The elastic-fluid transform γ′(ρ) = π′(ρ)/ρ turns pressure forces into gradients; for the linear law, γ(ρ) = c² log(ρ/a), enabling both vorticity-preservation and Bernoulli arguments.
- Closed isothermal cycles in the elastic-fluid model return the deformation Jacobian to its initial value, making the integrated mechanical-energy inequality an equality.
Chapters
- An ideal fluid has constant density, mass conservation, and spherical Cauchy stress S = -pI, with no viscosity.
- For a conservative body force and initially zero vorticity, taking the curl of momentum gives an evolution equation for vorticity.
- Wise corrects a term omitted in the previous lecture, then uses a material-coordinate transformation and F inverse to show zero initial vorticity remains zero.
- With zero vorticity, the velocity-dependent vorticity term in the momentum equation vanishes.
- Writing velocity as the gradient of a scalar potential makes every remaining momentum term a gradient.
- On a connected domain, the resulting Bernoulli quantity is spatially constant, with the constant allowed to depend on time.
- For incompressible irrotational flow, velocity is the gradient of a potential and the potential satisfies Laplace's equation, Δφ = 0.
- Impermeability gives the homogeneous Neumann condition, ∇φ · n = 0, on the boundary.
- In a bounded connected region, multiplying Laplace's equation by φ and integrating by parts shows ∇φ = 0, so the only velocity field is V = 0.
- In the half-space x₃ > 0, the potential φ = a(x₁² - x₃²)/2 gives V = ax₁e₁ - ax₃e₃.
- The field is divergence-free, irrotational, and satisfies no penetration at x₃ = 0, while remaining nonuniform and nontrivial.
- Bernoulli's relation determines pressure up to an additive constant; the example has a stagnation point on the boundary.
- This slip-compatible boundary condition differs from the no-slip condition generally used for Navier-Stokes fluids.
- Elastic fluids retain spherical stress, S = -pI, but drop the incompressibility constraint and relate pressure to density.
- The equation of state p = π(ρ) must be increasing, with π′(s) > 0 for positive density.
- A fixed-temperature ideal-gas law has the form p = cρθ, which is linear and increasing in density when cθ > 0.
- A representative equation of state is π(ρ) = p* + c²(ρ - ρ*), using reference pressure p*, reference density ρ*, and c > 0.
- The model permits pressure to be positive or negative depending on whether ρ is above or below ρ*.
- The derivative π′(ρ) = c² is constant, and c is associated with the linear speed of pressure waves.
- The model couples the full continuity equation to momentum balance, with pressure eliminated using p = π(ρ).
- The stress divergence becomes -∇p = -π′(ρ)∇ρ, simplifying to -c²∇ρ for the linear equation of state.
- Angular momentum balance is automatic because spherical stress is symmetric; isothermal modeling omits the internal-energy equation.
- Two observers related by x* = Q(t)x + c(t) differ by a rigid rotation and translation, with Q orthogonal and det Q = 1.
- Assuming both observers measure the same mass in corresponding fluid regions, a change of variables gives ρ*(x*,t) = ρ(x,t).
- Since QIQᵀ = I, the stress S = -π(ρ)I transforms as S* = QSQᵀ, satisfying frame indifference.
- For elastic stress, the stress power S : L reduces to -π(ρ) div V because the trace of the symmetric velocity gradient equals div V.
- Mass conservation gives ρₘ det F = ρ₀, while the deformation identity gives ∂ₜ det F = (div V)ₘ det F.
- Together these identities rewrite material stress power in terms of π, ρ₀, and the time derivative of the deformation Jacobian.
- A function G can be chosen so its derivative with respect to the Jacobian parameter represents the elastic stress-power expression.
- The integrated mechanical-energy term becomes the change in G between the initial and final states.
- For a closed isothermal process that returns to the same motion and deformation gradient, the change is zero, so the mechanical energy inequality holds with equality.
- A well-posed elastic-fluid problem couples density and velocity equations with initial data and a boundary condition such as no penetration.
- Wise defines γ(s) as the integral of π′(s)/s from a reference point, so γ′(s) = π′(s)/s.
- Because π′(s) > 0 for positive density, γ is increasing and has an inverse, denoted ζ.
- For the linear law π′ = c², γ(s) = c² log(s/a) and ζ(r) = a exp(r/c²), where a is the reference point.
- For conservative body force and initially irrotational velocity, dividing momentum balance by density converts the pressure term into ∇γ(ρ).
- Taking the curl yields a vorticity equation; after the material-coordinate substitution, zero is the unique solution for the transformed vorticity field.
- Thus elastic-fluid motion also preserves irrotationality, despite being compressible.
- With velocity represented by a potential, momentum balance becomes a gradient equation and gives an elastic-fluid Bernoulli relation that is spatially constant on connected domains.
- For a fixed region, substituting V = ∇φ into continuity and no-penetration conditions gives equations for the flow potential.
- The Bernoulli relation can be inverted with ζ to express density as a function of φ and its time derivative, reducing the system to an equation for φ alone.
- For steady flow, the time derivative disappears and the potential equation becomes elliptic rather than heat-equation-like.
- A known flow state at a reference point determines the Bernoulli constant, completing a nonlinear boundary-value formulation.
Transcript excerpt
0:04 Welcome back to continuum mechanics. This is class number 21. We're in chapter 6, our second lecture. Talking about isothermal fluid mechanics. Some very basic models so far. Um, I still have a couple of things to clean up with respect to ideal fluids. So, these are fluids that to for which we have a mass conservation constraint and the stress tensor is is spherical. It's very simple. It's equal to minus the pressure times the identity. So, we call such fluids inviscid because there's no viscosity term that appears.
0:52 Later on when we talk about Newtonian fluids, we'll talk about viscous fluids and viscosity. I need to finish up a couple of results from ideal fluids, especially related to irrotational flows. So, we ran through these results very quickly last time and I want to take some time to go through them more carefully here. So, let's suppose that we have a motion of an ideal fluid with spatial velocity V and constant mass density rho naught. So, we have the mass conservation constraint. Suppose that the body force is
1:39 conservative in the sense that there's some potential. I take the gradient of that and get the body force per unit mass. Suppose that V is irrotational at T equals 0. That means its curl is equal to 0 at time equals 0. Then, it's it must be irrotational for all time. So, how do we prove this? We got through most of it. I did make a mistake in one part and I want to point that out. Um we got about halfway through before uh we had to abandon the proof. So, to start the proof, let's assume that W is equal to the vorticity of V. So, this uh So, our assumption then is going to be
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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.