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Duality methods for variational inequalitiesand Non-Newtonian fluid mechanics
Enrique Fernández-Nieto?, Paul Vigneaux•
?Dpto. Matemática Aplicada I, Universidad de Sevilla•UMPA, Ecole Normale Supérieure de Lyon – France
PhD Course IMUSUniversidad de Sevilla.
October 2013
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This PhD Course at IMUS is made of two parts :
October 2013 – Paul Vigneaux
Introduction and Course’s motivation :Non-Newtonian Fluid Mechanics. Bingham flows.Formulations with variational inequalitiesMinimization, Lagrangians and duality
February 2014 – Enrique Fernández-Nieto
Augmented Lagrangian methods for numerical resolutionBermudez-Moreno algorithm for numerical resolutionTheoretical derivation of optimal parametersSpace discretization. Effective implementation.
Following slides are part of the material used for the Course’s motivation.
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Physical motivation Newtonian case Non-Newtonian Bingham in details
Course’s motivation and introduction
1 Motivations from natural/physical phenomena
2 Reminder : Newtonian case & Navier-Stokes
3 Generalization : Non-Newtonian
4 The Bingham case in more details
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Physical motivation Newtonian case Non-Newtonian Bingham in details
Course’s motivation and introduction
1 Motivations from natural/physical phenomena
2 Reminder : Newtonian case & Navier-Stokes
3 Generalization : Non-Newtonian
4 The Bingham case in more details
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The “first” flow :When one thinks about fluid flows,
first example which usually comes to mind is flow of water
Leonardo da Vinci. In Codex Atlanticus, drawings and writings from ∼1478 to ∼1518.
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The “first” flow :Though liquids such water can exhibit very complex flows :
e.g. turbulence,topological changes (free surface flows), etc
these liquids can be described by what is known as the mostsimple behaviour law : a linear law (explained later ,)
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Fluids more “complicated” than water : 1
For instance, let us have a look at this flow of honey : Movie
one can “feel” that this material is more “viscous” (again, aconcept to be defined later ,) and that some flow features(actually linked to non linear behaviour laws) can not beencountered with water
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Fluids more “complicated” than water : 2aIn other cases, “viscosity” can depend on “shear” applied on thefluid :
Corn flour (Maizena) Movie
mix gently : very “fluid”mix strong : very “viscous”
Source : Blkutter 2009, Wikimedia.
Same kind of feeling withsand/water at low tide on a beach :do not strike too hard, it can hurt ,
“viscosity”↗ with “shear”
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Fluids more “complicated” than water : 2bMore commonly, one encounters : “viscosity”↘ with “shear”
Very useful for wall paint :high shear when spread paint on wall : easy paintingno shear when pulling out the brush : no drip
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Fluids more “complicated” than water : 3Another striking effect is “the ability to NOT flow” :
toothpaste, mayonnaise, etcIf you gently turn an open tube upside down... material stays in the tube.→ Need to shake/squeeze the tube to let it flow
Conversely, some geophysical materials flow and then stopwith a shape different from water surface at rest :
Snow avalanches Movie
Mud flows (after heavy rains)
Lava flows (+ thermal effects)Movie
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Fluids more “complicated” than water : etc∃ other “exotic” situations not described here, including
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A local example : Alboran SeaSubmarine landslide triggers a tsunami,
Movie by EDANYA team, Malaga :
Non-Newtonian models are of interest for these materials.
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Physical motivation Newtonian case Non-Newtonian Bingham in details
Course’s motivation and introduction
1 Motivations from natural/physical phenomena
2 Reminder : Newtonian case & Navier-Stokes
3 Generalization : Non-Newtonian
4 The Bingham case in more details
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Newtonian case & Navier-StokesRecall the (Eulerian, incompressible) Navier-Stokes equations :
u(x1, x2, x3; t) : velocity, R3 × R→ R3
p(x1, x2, x3; t) : pressure, R3 × R→ Rf (x1, x2, x3; t) : body forces, R3 × R→ R
0 = ∇.u = div(u) =∑
i
∂ui
∂xi: mass conservation (1)
u.∇u :=
∂1u1 ∂2u1 ∂3u1∂1u2 ∂2u2 ∂3u2∂1u3 ∂2u3 ∂3u3
.
u1u2u3
D(u) =
∇u + (∇u)t
2: rate of deformation tensor
ρ (∂tu + u.∇u)︸ ︷︷ ︸inertia
−2η ∇.D(u)︸ ︷︷ ︸viscous
+ ∇p︸︷︷︸pressure
= f︸︷︷︸body force
(2)
Rk : Eulerian vs Lagrangian and Convected derivative Dtu
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Navier-Stokes asymptotics
Depending on the Reynolds number Re = ρULη ≡
inertia effectsviscous effects
we have formally
Re→∞ : Euler equations
ρ (∂tu + u.∇u) +∇p = f (3)
Re→ 0 : Stokes equations
− 2η ∇.D(u) +∇p = f (4)
Rk : transito to turbulence : laminar, recirculation, Von KarmanRk : No compressible effects hereRk : No energy equation here taken into account
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More on the momentum equations (say, Stokes)We need to recall a bit the origin of viscous/pressure
Newton’s 2nd law : acceleration =∑
forces
To write it, we consider an elementary volume dx dy dz (à laBatchelor) and we look for all the external forces :
Continuum Mechanics ensures :
∃ a stress tensor σ s. t.force
ds on a surface of normal n is
F̃ = σ.n with Cauchy tensor
σ =
σ11 σ12 σ13σ21 σ22 σ23σ31 σ32 σ33
"σ = tangential + normal stress"σ = σ′︸︷︷︸
viscous
−p Id︸ ︷︷ ︸pressure
, σ′ = σ′(D(u))
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The viscous stress tensor (Newtonian version)Invoking again Continuum Mechanics :
σ′ = 2η D(u) (5)
:= constitutive (or behaviour) equation. It’s linear ! Previous
Viscosity := link between stress and rate-of-deformation
Go back to momentum conservation (following a REV V)
DDt
[∫Vρu dv
]=
∫Vρf dv +
∫∂Vσ.n ds, and use Div Thm
(6)∫Vρ
DuDt
dv =
∫Vρf dv +
∫V∇.σ dv , for every V, (7)
which leads to the previous Navier-Stokes equations, since :
∇.σ = ∇.(σ′ − p Id) = 2η ∇.D(u)−∇p. (8)
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Blackboard interlude : a "1D case", Poiseuille flow (1835)
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Physical motivation Newtonian case Non-Newtonian Bingham in details
Course’s motivation and introduction
1 Motivations from natural/physical phenomena
2 Reminder : Newtonian case & Navier-Stokes
3 Generalization : Non-Newtonian
4 The Bingham case in more details
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Why linear ?
We know that σ′ is a function of D(u) [convenience σ′ ; σ]
But is this only linear ? Can it be more general ?
Actually, physical examples of Part 1 are possible due to this
type of non linearity, in particular η = η(D(u)) is variable ! ! !
⇒ these are so called Generalized Newtonian fluids
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A bit of history : rheology
Study of deviation (from the linear law) σ′ as a function of D(u)
belongs to the field called Rheology, studying complex fluids,
more precisely the deformation and flow of matter.
Term. due to E. C. Bingham in 1929, from greek "ρεω-"to flow"
Years "around" 1900’s saw a significant increase of activity onthese subjects, including :
Maxwell (1868), Boltzmann (1877), Bingham, Blair, Reiner,Herschel-Bulkley, Weissenberg (all between 1900–1930) ...
Then became a full field of activity, more vibrant today than ever
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Revisiting examples : corn flour
Corn flour is a material s.t.
constitutive eq. is like→
:= shear-thickening
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Revisiting examples : paint
Paint is a material s.t.
constitutive eq. is like→
:= shear-thinning
More precisely, such shear-thinning/shear-thickening fluids
are called power-law fluids (indeed).
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Revisiting examples : "threshold" fluids
And what about mayonnaise, snow, lava, etc ? ? ?
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Revisiting examples : "threshold" fluids
And what about mayonnaise, snow, lava, etc ? ? ?
Below a stress τc ,
nothing happens : γ̇ = 0,rigid/no deformation.
Above τc ,
material begins to flow ! ! !γ̇ > 0,
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Revisiting examples : "threshold" fluids
And what about mayonnaise, snow, lava, etc ? ? ?
Below a stress τc ,
nothing happens : γ̇ = 0,rigid/no deformation.
Above τc ,
material begins to flow ! ! !γ̇ > 0,
Constitutive equation :{|σ| < τc if |D(u)| = 0σ = 2ηD(u) + τc
D(u)|D(u)| if |D(u)| 6= 0
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"threshold" fluids : a bit of history
1916 ; 1922→
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Plasticity - The origin
“We may now defineplasticity as aproperty of solids invirtue of which theyhold their shapepermanently under theaction of smallshearing stresses butthey are readilydeformed, worked ormolded, undersomewhat largerstresses. [...]”
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The general case ...... combines power-law and plasticity
:= Herschel-Bulkley constitutive equation
Newtonian
Herschel-Bulkley Bingham
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Physical motivation Newtonian case Non-Newtonian Bingham in details
Course’s motivation and introduction
1 Motivations from natural/physical phenomena
2 Reminder : Newtonian case & Navier-Stokes
3 Generalization : Non-Newtonian
4 The Bingham case in more details
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Reminder : Poiseuille flow
Stress goes to zero
at channel’s center :
opens the door
for a central plug :=
moving plastic zone
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Blackboard interlude : 1D, Poiseuille-Bingham flow
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The square channel case
Phenomenology : presence of
“plug” zones : motion withconstant velocity
“dead” zones : no motion
“shear” zones : material deforms
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
spee
d(x)
x
Values of yield stress : 1.61.71.81.92.02.12.22.32.42.5
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Physical motivation Newtonian case Non-Newtonian Bingham in details
Let’s roll on’ !
Variational inequalities
Minimization with constraints
Duality Methods
Spatial discretization
...
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Physical motivation Newtonian case Non-Newtonian Bingham in details
ReferencesPapers and Books
G.K. Batchelor. An introduction to fluid dynamics.Cambridge University Press (2000)P. Oswald. Rheophysics. The Deformation and Flow ofMatter. Cambridge University Press (2009)R.I. Tanner, K. Walters. Rheology : an historicalperspective. Elsevier (1998)
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Physical motivation Newtonian case Non-Newtonian Bingham in details
ReferencesVideos
Honey extraction :http ://www.youtube.com/watch ?v=5XO1FEZhESkCorn flour :http ://www.youtube.com/watch ?v=XEEs4eoRPGwSnow avalanche (France) :https ://www.youtube.com/watch ?v=IlzT6VjXw-ELava Flow (Hawaii) :http ://www.youtube.com/watch ?v=HpNWLCmXyTEAlboran (Spain) :http ://www.youtube.com/user/grupoedanya