Topics in Fluid Dynamics: Classical physics and recent mathematics Toan T. Nguyen 1,2 Penn State University Graduate Student Seminar @ PSU Jan 18th, 2018 1 Homepage: http://math.psu.edu/nguyen 2 Math blog: https://nttoan81.wordpress.com Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 1 / 20
Fluid motion x t Ω u(x, t) Ω Figure : Fluid domain: Ω R 3 and unknown fluid trajectory: x t Ω (left) or unknown fluid velocity: u(x, t) R 3 (right). Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 2 / 20
Fluid motion x t Ω u(x, t) Ω Figure : Fluid domain: Ω R 3 and unknown fluid trajectory: x t Ω (left) or unknown fluid velocity: u(x, t) R 3 (right). Lagrangian description (left): trajectory of each fluid molecule x Ω ẋ t = u(x t, t), x 0 = x Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 2 / 20
Fluid motion x t Ω u(x, t) Ω Figure : Fluid domain: Ω R 3 and unknown fluid trajectory: x t Ω (left) or unknown fluid velocity: u(x, t) R 3 (right). Lagrangian description (left): trajectory of each fluid molecule x Ω ẋ t = u(x t, t), x 0 = x Eulerian description (right): velocity field u(x, t) R 3 at each position x Ω and time t 0. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 2 / 20
Fluid motion Classical fluid dynamics: Continuum Hypothesis: Each point in Ω corresponds to a fluid molecule (e.g., Hilbert s 6th open problem: continuum limit from N-particle system 3 ). 3 see my lecture notes on Kinetic Theory of Gases: https://nttoan81.wordpress.com Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 3 / 20
Fluid motion Classical fluid dynamics: Continuum Hypothesis: Each point in Ω corresponds to a fluid molecule (e.g., Hilbert s 6th open problem: continuum limit from N-particle system 3 ). Continuity equation: along particle trajectory, mass remains constant: ρ(x t, t) det( x x t )dx = ρ(x, 0)dx x t ρdy x ρdx Figure : Illustrated the Lagrangian map: x x t for each t 0. 3 see my lecture notes on Kinetic Theory of Gases: https://nttoan81.wordpress.com Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 3 / 20
Fluid motion Incompressibility: 4 volume preserving flows iff u = 0 (Exercise: d dt J = ( u)j). x1 u 1 + x2 u 2 = 0 4 water can be modeled by an incompressible flow, but air is compressible. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 4 / 20
Fluid motion Incompressibility: 4 volume preserving flows iff u = 0 (Exercise: d dt J = ( u)j). x1 u 1 + x2 u 2 = 0 In particular, fluid density ρ(x, t) remains constant along the flow. In what follows, ρ = 1 (continuity equation = incompressibility). z u u θ Figure : Illustrated shear flows (left) and circular flows (right), both are incompressible. 4 water can be modeled by an incompressible flow, but air is compressible. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 4 / 20
Fluid motion Momentum equation: Newton s law: F = ma or equivalently, D t u = F with D t := t + u with F being force acting on fluid parcel: Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 5 / 20
Fluid motion Momentum equation: Newton s law: F = ma or equivalently, D t u = F with D t := t + u with F being force acting on fluid parcel: No force: F = 0. Free particles satisfy Burgers equation (nonphysical: no particle interaction): u t x x 1 x 2 x Figure : Smooth solutions blow up in finite time (see, of course, the theory of entropy shock solutions: Bressan, Dafermos) Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 5 / 20
Euler equations Ideal fluid: 5 F = p, pressing normally inward on the fluid surface (called pressure gradient): F = p ndσ(x) O O 5 as opposed to viscous fluid. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 6 / 20
Euler equations Ideal fluid: 5 F = p, pressing normally inward on the fluid surface (called pressure gradient): F = p ndσ(x) O O This yields Euler equations (1757, very classical): D t u = p u = 0 posed on Ω R 3 with u n = 0 on Ω. NOTE: 4 equations and 4 unknowns: u, p. 5 as opposed to viscous fluid. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 6 / 20
Euler equations Examples (stationary): Laminar ( flows ) (shear or circular flows): U(z) u = for arbitrary U(z) 0 with zero pressure gradient. z U Couette flow: U(z) = z 6 Figure: internet Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 7 / 20
Euler equations Examples (stationary): Laminar ( flows ) (shear or circular flows): U(z) u = for arbitrary U(z) 0 with zero pressure gradient. z U Couette flow: U(z) = z Potential flows: u = φ and so φ is harmonic: φ = 0 (incompressible, irrotational flows) Streamlines of potential flows 6 6 Figure: internet Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 7 / 20
Euler equations Vorticity: to measure the rotation in fluids ω = u (anti-symmetric part of u, recalling ẋ = u u 0 + ( u 0 )x: translation, dilation, and rotation). Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 8 / 20
Euler equations Vorticity: to measure the rotation in fluids ω = u (anti-symmetric part of u, recalling ẋ = u u 0 + ( u 0 )x: translation, dilation, and rotation). Note that ω = [ω, u] (the Lie bracket), or explicitly D t ω = ω u Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 8 / 20
Euler equations Vorticity: to measure the rotation in fluids ω = u (anti-symmetric part of u, recalling ẋ = u u 0 + ( u 0 )x: translation, dilation, and rotation). Note that ω = [ω, u] (the Lie bracket), or explicitly Theorem (Helmholtz s vorticity law) D t ω = ω u Vorticity moves with the flow: ω(x, t) = x t # ω 0(x). (as a consequence, vortex remains a vortex). Hint: Compute d dt (ω x t # ω 0(x)). Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 8 / 20
Euler equations u γ t S t Theorem (Kelvin s circulation theorem) Vorticity flux through an oriented surface or circulation around an oriented curve is invariant under the flow: Γ γ = u ds = ω ds Hint: A direct computation. γ S Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 9 / 20
Euler equations 3D Euler: D t ω = ω u Vortex stretching: ω u, which appears quadratic in ω, and one could end up with d dt ω ω2 or even d dt ω ω1+ɛ, whose solutions blow up in finite time. However, Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 10 / 20
Euler equations 3D Euler: D t ω = ω u Vortex stretching: ω u, which appears quadratic in ω, and one could end up with d dt ω ω2 or even d dt ω ω1+ɛ, whose solutions blow up in finite time. However, Open problem: do smooth solutions to 3D Euler actually blow up in finite time? (no, if vorticity remains bounded, Beale-Kato-Majda 84). Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 10 / 20
Euler equations 3D Euler: D t ω = ω u Vortex stretching: ω u, which appears quadratic in ω, and one could end up with d dt ω ω2 or even d dt ω ω1+ɛ, whose solutions blow up in finite time. However, Open problem: do smooth solutions to 3D Euler actually blow up in finite time? (no, if vorticity remains bounded, Beale-Kato-Majda 84). Recent mathematics and then a proof of Onsager s conjecture 49: Isett, De Lellis, Székelyhidi, Buckmaster, Vicol,... Numerical proof of finite time blow up: Luo-Hou, Sverak,... Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 10 / 20
2D Euler equations In fact, many physical fluid flows are essentially 2D: Atmospheric and oceanic flows Flows subject to a strong magnetic field, rotation, or stratification. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 11 / 20
2D Euler equations In fact, many physical fluid flows are essentially 2D: Atmospheric and oceanic flows Flows subject to a strong magnetic field, rotation, or stratification. In 2D, vorticity is scalar and is transported by the flow: D t ω = 0 (no vortex stretching). In particular, vorticity remains bounded, smooth solutions remain smooth, and weak solutions with bounded vorticity are unique (Yudovich 63). Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 11 / 20
2D Euler equations An important problem: the large time dynamics of 2D Euler. Complete mixing: whether ω(t j ) 0 in L, as t j? No, due to energy conservation. However, Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 12 / 20
2D Euler equations An important problem: the large time dynamics of 2D Euler. Complete mixing: whether ω(t j ) 0 in L, as t j? No, due to energy conservation. However, Conjecture (2D inverse energy cascade, Kraichnan 67) Unlike 3D, energy transfers to larger and larger scales (low frequencies). Figure : Source: van Gogh and internet Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 12 / 20
2D Euler equations 2D Euler steady states: u 0 ω 0 = 0 With u 0 = φ 0, the stream function φ 0 and vorticity ω 0 have parallel gradient, hence (locally) ω 0 = F (φ 0 ), yielding φ 0 = F (φ 0 ) Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 13 / 20
2D Euler equations 2D Euler steady states: u 0 ω 0 = 0 With u 0 = φ 0, the stream function φ 0 and vorticity ω 0 have parallel gradient, hence (locally) ω 0 = F (φ 0 ), yielding φ 0 = F (φ 0 ) Major open problem: which F determines the large time dynamics of Euler? Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 13 / 20
2D Euler equations 2D Euler steady states: u 0 ω 0 = 0 With u 0 = φ 0, the stream function φ 0 and vorticity ω 0 have parallel gradient, hence (locally) ω 0 = F (φ 0 ), yielding φ 0 = F (φ 0 ) Major open problem: which F determines the large time dynamics of Euler? Theorem (Arnold 65) If F is strictly convex, then steady states u 0 are nonlinearly stable in H 1. Hint: Find casimir functional so that u 0 is a critical point. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 13 / 20
2D Euler equations A delicate question: whether Arnold stability implies asymptotic stability (recalling Euler is an Hamiltonian)? Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 14 / 20
2D Euler equations A delicate question: whether Arnold stability implies asymptotic stability (recalling Euler is an Hamiltonian)? Inviscid damping: Kelvin 1887, Orr 1907 Mathematics near Couette: Masmoudi, Bedrossian, Germain 14- z u Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 14 / 20
2D Euler equations Hydrodynamic stability: Rayleigh, Kelvin, Orr, Sommerfeld, Heisenberg,...the study of spectrum of shear flows: ( U(z) Z u = 0 ) U = 0 U U = 0 Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 15 / 20
2D Euler equations Hydrodynamic stability: Rayleigh, Kelvin, Orr, Sommerfeld, Heisenberg,...the study of spectrum of shear flows: ( U(z) Z u = 0 ) U = 0 U U = 0 Rayleigh (1880): U(z) that has no inflection point is spectrally stable. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 15 / 20
2D Euler equations Hydrodynamic stability: Rayleigh, Kelvin, Orr, Sommerfeld, Heisenberg,...the study of spectrum of shear flows: ( U(z) Z u = 0 ) U = 0 U U = 0 Rayleigh (1880): U(z) that has no inflection point is spectrally stable. Figure : Great interest in the early of 20th century (aerodynamics). Source: internet??? Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 15 / 20
2D Navier-Stokes equations Viscous fluid: F = p + ν u (Newtonian), with fluid viscosity ν > 0: ( t + u )u = p + ν u u = 0 posed on Ω R 3 with u = 0 on Ω. NOTE: u, p are unknown. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 16 / 20
2D Navier-Stokes equations Viscous fluid: F = p + ν u (Newtonian), with fluid viscosity ν > 0: ( t + u )u = p + ν u u = 0 posed on Ω R 3 with u = 0 on Ω. NOTE: u, p are unknown. Million-dollar open problem: whether smooth solutions to 3D Navier Stokes blow up in finite time. (Like 2D Euler, smooth solutions to 2D Navier Stokes remain smooth, Ladyzhenskaya 60s). Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 16 / 20
2D Navier-Stokes equations Viscous fluid: F = p + ν u (Newtonian), with fluid viscosity ν > 0: ( t + u )u = p + ν u u = 0 posed on Ω R 3 with u = 0 on Ω. NOTE: u, p are unknown. Million-dollar open problem: whether smooth solutions to 3D Navier Stokes blow up in finite time. (Like 2D Euler, smooth solutions to 2D Navier Stokes remain smooth, Ladyzhenskaya 60s)....back to Hydrodynamic Stability. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 16 / 20
2D Navier-Stokes equations The role of viscosity: d Alembert s paradox, 1752: Zero drag exerted on a body immersed in a potential flow (as momentum equation is in the divergence form). Birds can t fly! L. Prandtl, 1904: the birth of the boundary layer theory (viscous forces become significant near the boundary). This gave birth of Aerodynamics. Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 17 / 20
2D Navier-Stokes equations The role of viscosity, cont d: Lord Rayleigh, 1880: Viscosity may or may not destabilize the flow Reynolds experiment, 1885: All laminar flows become turbulent at a high Reynolds number: Re := inertial force viscous force = u u ν u = UL ν 10 4 Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 18 / 20
2D Navier-Stokes equations Recent mathematics: 7 Grenier (ENS Lyon)-Toan 2017-: Confirming the viscous destabilization (linear part with Y. Guo) Invalidating generic Prandtl s boundary layer expansion Disproving the Prandtl s boundary layer Ansatz 7 for more, see my blog: https://nttoan81.wordpress.com Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 19 / 20
2D Navier-Stokes equations Perspectives: Boundary layer cascades, bifurcation theory, stability of roll waves, fluid mixing, and much more! y u Euler x ν 1 2 : Prandtl s layer ν 3 4 : 1 st sublayer ν 5 8 : 2 nd sublayer ν: Kato s layer Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 20 / 20