Momentum Transport eq material Acceleration add continuity Residual
FluidDynamics.FluidFlow.momentumTransport_eq_materialAcceleration_add_continuityResidual
Plain-language statement
The conservative momentum transport terms equal mass density times material acceleration plus the continuity residual times velocity.
Exact Lean statement
lemma momentumTransport_eq_materialAcceleration_add_continuityResidual
(d : ℕ) (fluid : FluidFlow d)
(t : Time) (x : Space d)
(hRhoTime : DifferentiableAt ℝ (fluid.rho · x) t)
(hVelocityTime : DifferentiableAt ℝ (fluid.velocity · x) t)
(hMomentumDensity : Differentiable ℝ (momentumDensity d fluid t))
(hVelocitySpace : Differentiable ℝ (fluid.velocity t)) :
∂ₜ (momentumDensity d fluid · x) t + matrixDiv d (momentumFlux d fluid t) x =
fluid.rho t x • materialAcceleration d fluid t x +
continuityResidual d fluid t x • fluid.velocity t xFormal artifact
Lean source
lemma momentumTransport_eq_materialAcceleration_add_continuityResidual (d : ℕ) (fluid : FluidFlow d) (t : Time) (x : Space d) (hRhoTime : DifferentiableAt ℝ (fluid.rho · x) t) (hVelocityTime : DifferentiableAt ℝ (fluid.velocity · x) t) (hMomentumDensity : Differentiable ℝ (momentumDensity d fluid t)) (hVelocitySpace : Differentiable ℝ (fluid.velocity t)) : ∂ₜ (momentumDensity d fluid · x) t + matrixDiv d (momentumFlux d fluid t) x = fluid.rho t x • materialAcceleration d fluid t x + continuityResidual d fluid t x • fluid.velocity t x := by rw [continuityResidual] rw [timeDeriv_momentumDensity d fluid t x hRhoTime hVelocityTime] rw [matrixDiv_momentumFlux d fluid t x hMomentumDensity hVelocitySpace] ext i simp [materialAcceleration, convectiveTerm, div, momentumDensity, smul_eq_mul] ring_nf- Project
- Physlib
- License
- Apache-2.0
- Commit
- dd43e9e65791
- Source
- Physlib/FluidDynamics/FluidFlow/Momentum.lean:160-175
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