Topics in Fluid Dynamics:Classical physics and recent mathematics
Toan T. Nguyen1,2
Penn State University
Graduate Student Seminar @ PSUJan 18th, 2018
1Homepage: http://math.psu.edu/nguyen2Math blog: https://nttoan81.wordpress.com
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 1 / 20
Fluid motion
xt
Ω
u(x , t)
Ω
Figure : Fluid domain: Ω ⊂ R3 and unknown fluid trajectory: xt ∈ Ω (left) orunknown fluid velocity: u(x , t) ∈ R3 (right).
• Lagrangian description (left): trajectory of each fluid molecule x ∈ Ω
xt = u(xt , t), x0 = x
• Eulerian description (right): velocity field
u(x , t) ∈ R3
at each position x ∈ Ω and time t ≥ 0.
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 2 / 20
Fluid motion
xt
Ω
u(x , t)
Ω
Figure : Fluid domain: Ω ⊂ R3 and unknown fluid trajectory: xt ∈ Ω (left) orunknown fluid velocity: u(x , t) ∈ R3 (right).
• Lagrangian description (left): trajectory of each fluid molecule x ∈ Ω
xt = u(xt , t), x0 = x
• Eulerian description (right): velocity field
u(x , t) ∈ R3
at each position x ∈ Ω and time t ≥ 0.
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 2 / 20
Fluid motion
xt
Ω
u(x , t)
Ω
Figure : Fluid domain: Ω ⊂ R3 and unknown fluid trajectory: xt ∈ Ω (left) orunknown fluid velocity: u(x , t) ∈ R3 (right).
• Lagrangian description (left): trajectory of each fluid molecule x ∈ Ω
xt = u(xt , t), x0 = x
• Eulerian description (right): velocity field
u(x , t) ∈ R3
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 fluidmolecule (e.g., Hilbert’s 6th open problem: continuum limit fromN-particle system3).
Continuity equation: along particle trajectory, mass remains constant:
ρ(xt , t) det(∇xxt)dx = ρ(x , 0)dx
x
xt
ρdx
ρdy
Figure : Illustrated the Lagrangian map: x 7→ xt for each t 6= 0.
3see my lecture notes on Kinetic Theory of Gases: https://nttoan81.wordpress.comToan 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 fluidmolecule (e.g., Hilbert’s 6th open problem: continuum limit fromN-particle system3).
Continuity equation: along particle trajectory, mass remains constant:
ρ(xt , t) det(∇xxt)dx = ρ(x , 0)dx
x
xt
ρdx
ρdy
Figure : Illustrated the Lagrangian map: x 7→ xt for each t 6= 0.
3see my lecture notes on Kinetic Theory of Gases: https://nttoan81.wordpress.comToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 3 / 20
Fluid motion
Incompressibility:4 volume preserving flows iff
∇ · u = 0
(Exercise: ddt J = (∇ · u)J).
∂x1u1 + ∂x2u2 = 0
In particular, fluid density ρ(x , t) remains constant along the flow. Inwhat follows, ρ = 1 (continuity equation = incompressibility).
z
u
uθ
Figure : Illustrated shear flows (left) and circular flows (right), both areincompressible.
4water 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: ddt J = (∇ · u)J).
∂x1u1 + ∂x2u2 = 0
In particular, fluid density ρ(x , t) remains constant along the flow. Inwhat follows, ρ = 1 (continuity equation = incompressibility).
z
u
uθ
Figure : Illustrated shear flows (left) and circular flows (right), both areincompressible.
4water 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,
Dtu = F with Dt := ∂t + u · ∇
with F being force acting on fluid parcel:
No force: F = 0. Free particles satisfy Burgers equation (nonphysical:no particle interaction):
x
ut
xx1 x2
Figure : Smooth solutions blow up in finite time (see, of course, the theoryof entropy shock solutions: Bressan, Dafermos)
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 5 / 20
Fluid motion
• Momentum equation: Newton’s law: F = ma or equivalently,
Dtu = F with Dt := ∂t + u · ∇
with F being force acting on fluid parcel:
No force: F = 0. Free particles satisfy Burgers equation (nonphysical:no particle interaction):
x
ut
xx1 x2
Figure : Smooth solutions blow up in finite time (see, of course, the theoryof 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 fluidsurface (called pressure gradient):
F = −∫∂O
p ~ndσ(x)O
This yields Euler equations (1757, very classical):
Dtu = −∇p∇ · u = 0
posed on Ω ⊂ R3 with u · n = 0 on ∂Ω. NOTE: 4 equations and 4unknowns: u, p.
5as 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 fluidsurface (called pressure gradient):
F = −∫∂O
p ~ndσ(x)O
This yields Euler equations (1757, very classical):
Dtu = −∇p∇ · u = 0
posed on Ω ⊂ R3 with u · n = 0 on ∂Ω. NOTE: 4 equations and 4unknowns: u, p.
5as 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 =
(U(z)
0
)for arbitrary U(z)
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 flows6
6Figure: internetToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 7 / 20
Euler equations
• Examples (stationary):
Laminar flows (shear or circular flows):
u =
(U(z)
0
)for arbitrary U(z)
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 flows6
6Figure: internetToan 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 x = u ≈ u0 + (∇u0)x : translation,dilation, and rotation).
Note that ω = [ω, u] (the Lie bracket), or explicitly
Dtω = ω · ∇u
Theorem (Helmholtz’s vorticity law)
Vorticity moves with the flow: ω(x , t) = x t#ω0(x).(as a consequence, vortex remains a vortex).
Hint: Compute ddt (ω − x t#ω0(x)).
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 x = u ≈ u0 + (∇u0)x : translation,dilation, and rotation).
Note that ω = [ω, u] (the Lie bracket), or explicitly
Dtω = ω · ∇u
Theorem (Helmholtz’s vorticity law)
Vorticity moves with the flow: ω(x , t) = x t#ω0(x).(as a consequence, vortex remains a vortex).
Hint: Compute ddt (ω − x t#ω0(x)).
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 x = u ≈ u0 + (∇u0)x : translation,dilation, and rotation).
Note that ω = [ω, u] (the Lie bracket), or explicitly
Dtω = ω · ∇u
Theorem (Helmholtz’s vorticity law)
Vorticity moves with the flow: ω(x , t) = x t#ω0(x).(as a consequence, vortex remains a vortex).
Hint: Compute ddt (ω − x t#ω0(x)).
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 8 / 20
Euler equations
γt St
u
Theorem (Kelvin’s circulation theorem)
Vorticity flux through an oriented surface or circulation around an orientedcurve is invariant under the flow:
Γγ =
∮γu · ds =
∫∫Sω · dS
Hint: A direct computation.
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 9 / 20
Euler equations
3D Euler:Dtω = ω · ∇u
• Vortex stretching: ω · ∇u, which appears “quadratic” in ω, and onecould end up with d
dtω ≈ ω2 or even d
dtω ≈ ω1+ε, whose solutions blow up
in finite time. However,
• Open problem: do smooth solutions to 3D Euler actually blow up infinite time? (no, if vorticity remains bounded, Beale-Kato-Majda ’84).
• Recent mathematics and then a proof of Onsager’s conjecture ’49: Isett,De Lellis, Szekelyhidi, Buckmaster, Vicol,.... Numerical proof of finite timeblow up: Luo-Hou, Sverak,....
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 10 / 20
Euler equations
3D Euler:Dtω = ω · ∇u
• Vortex stretching: ω · ∇u, which appears “quadratic” in ω, and onecould end up with d
dtω ≈ ω2 or even d
dtω ≈ ω1+ε, whose solutions blow up
in finite time. However,
• Open problem: do smooth solutions to 3D Euler actually blow up infinite time? (no, if vorticity remains bounded, Beale-Kato-Majda ’84).
• Recent mathematics and then a proof of Onsager’s conjecture ’49: Isett,De Lellis, Szekelyhidi, Buckmaster, Vicol,.... Numerical proof of finite timeblow up: Luo-Hou, Sverak,....
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 10 / 20
Euler equations
3D Euler:Dtω = ω · ∇u
• Vortex stretching: ω · ∇u, which appears “quadratic” in ω, and onecould end up with d
dtω ≈ ω2 or even d
dtω ≈ ω1+ε, whose solutions blow up
in finite time. However,
• Open problem: do smooth solutions to 3D Euler actually blow up infinite time? (no, if vorticity remains bounded, Beale-Kato-Majda ’84).
• Recent mathematics and then a proof of Onsager’s conjecture ’49: Isett,De Lellis, Szekelyhidi, Buckmaster, Vicol,.... Numerical proof of finite timeblow 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.
• In 2D, vorticity is scalar and is transported by the flow:
Dtω = 0
(no vortex stretching). In particular, vorticity remains bounded, smoothsolutions remain smooth, and weak solutions with bounded vorticity areunique (Yudovich ’63).
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:
Dtω = 0
(no vortex stretching). In particular, vorticity remains bounded, smoothsolutions remain smooth, and weak solutions with bounded vorticity areunique (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. Completemixing: whether ω(tj)
∗ 0 in L∞, as tj →∞? 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
• An important problem: the large time dynamics of 2D Euler. Completemixing: whether ω(tj)
∗ 0 in L∞, as tj →∞? 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:u0 · ∇ω0 = 0
With u0 = ∇⊥φ0, the stream function φ0 and vorticity ω0 have parallelgradient, hence (locally) ω0 = F (φ0), yielding
∆φ0 = F (φ0)
• Major open problem: which F determines the large time dynamics ofEuler?
Theorem (Arnold ’65)
If F is strictly convex, then steady states u0 are nonlinearly stable in H1.
Hint: Find casimir functional so that u0 is a critical point.
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 13 / 20
2D Euler equations
• 2D Euler steady states:u0 · ∇ω0 = 0
With u0 = ∇⊥φ0, the stream function φ0 and vorticity ω0 have parallelgradient, hence (locally) ω0 = F (φ0), yielding
∆φ0 = F (φ0)
• Major open problem: which F determines the large time dynamics ofEuler?
Theorem (Arnold ’65)
If F is strictly convex, then steady states u0 are nonlinearly stable in H1.
Hint: Find casimir functional so that u0 is a critical point.
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 13 / 20
2D Euler equations
• 2D Euler steady states:u0 · ∇ω0 = 0
With u0 = ∇⊥φ0, the stream function φ0 and vorticity ω0 have parallelgradient, hence (locally) ω0 = F (φ0), yielding
∆φ0 = F (φ0)
• Major open problem: which F determines the large time dynamics ofEuler?
Theorem (Arnold ’65)
If F is strictly convex, then steady states u0 are nonlinearly stable in H1.
Hint: Find casimir functional so that u0 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)?
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
• 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 =
(U(z)
0
) U = 0
U = 0
Z
U
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 Euler equations
• Hydrodynamic stability: Rayleigh, Kelvin, Orr, Sommerfeld,Heisenberg,....the study of spectrum of shear flows:
u =
(U(z)
0
) U = 0
U = 0
Z
U
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 Euler equations
• Hydrodynamic stability: Rayleigh, Kelvin, Orr, Sommerfeld,Heisenberg,....the study of spectrum of shear flows:
u =
(U(z)
0
) U = 0
U = 0
Z
U
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 ⊂ R3 with u = 0 on ∂Ω. NOTE: u, p are unknown.
• Million-dollar open problem: whether smooth solutions to 3D NavierStokes blow up in finite time. (Like 2D Euler, smooth solutions to 2DNavier 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
• Viscous fluid: F = −∇p + ν∆u (Newtonian), with fluid viscosity ν > 0:
(∂t + u · ∇)u = −∇p + ν∆u
∇ · u = 0
posed on Ω ⊂ R3 with u = 0 on ∂Ω. NOTE: u, p are unknown.
• Million-dollar open problem: whether smooth solutions to 3D NavierStokes blow up in finite time. (Like 2D Euler, smooth solutions to 2DNavier 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
• Viscous fluid: F = −∇p + ν∆u (Newtonian), with fluid viscosity ν > 0:
(∂t + u · ∇)u = −∇p + ν∆u
∇ · u = 0
posed on Ω ⊂ R3 with u = 0 on ∂Ω. NOTE: u, p are unknown.
• Million-dollar open problem: whether smooth solutions to 3D NavierStokes blow up in finite time. (Like 2D Euler, smooth solutions to 2DNavier 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 immersedin 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 (viscousforces become significant near the boundary). This gave birth ofAerodynamics.
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 ahigh Reynolds number:
Re :=inertial force
viscous force=
u · ∇uν∆u
=UL
ν& 104
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
7for more, see my blog: https://nttoan81.wordpress.comToan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 19 / 20
2D Navier-Stokes equations
Perspectives: Boundary layer cascades, bifurcation theory, stability of rollwaves, fluid mixing, and much more!
x
yuEuler
ν12 : Prandtl’s layer
ν34 : 1st sublayer
ν58 : 2nd sublayer· · ·
ν: Kato’s layer
Toan T. Nguyen (Penn State) Fluid Dynamics PSU, Jan 18th, 2018 20 / 20