On isothermal steady flows of incompressible, pressure-thickening - - PowerPoint PPT Presentation

On isothermal steady flows of incompressible, pressure-thickening and shear-thinning fluids and their Galerkin approximation.

• M. Lanzend¨
• rfer 1,5

presenting what he understood from

• J. M´

alek1,2 and M. Bul´ ıˇ cek1,2, and collaborated on with

• A. Hirn3 and J. Stebel2,4.

1Faculty of Mathematics and Physics, Charles University in Prague 2Jindˇ

rich Neˇ cas Center for Mathematical Modeling

3University of Heidelberg, 4Mathematical Institute, AS CR 5Institute of Computer Science, AS CR

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 1 / 14

➊ Motivated by several practical applications, we consider ➋ incompressible fluids with viscosity depending on shear rate and on pressure. The latter dependence, in particular, leads to difficulties in both theory and numerical simulations. We will focus on isothermal steady flows of a subclass of such fluids; ➌ briefly discuss the known results on existence of weak solutions; ➍ show the connection of the viscosity–pressure relation with the inf–sup inequality and the stable Galerkin discretization; ➎ mention the relation of inf–sup inequality to the pressure boundary conditions. ➏ We will advert to open problems and disclose some troubles occurring in numerical experiments (motivated by the lubrication problems).

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 2 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside (0, T)× Ω: divv v v = ∂τv v v + div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 4 / 14

Applications: lubrication problems, journal bearing

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 5 / 14

Applications: lubrication problems, journal bearing

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 5 / 14

Applications: lubrication problems, journal bearing

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 5 / 14

Viscosity and volume variation with pressure for squalane

(representing a low viscosity paraffinic mineral oil, see S. Bair, Tribology Letters, 2006).

200 400 600 800 1000 1200 10

1

10

2

10

3

10

4

10

5

10

6

10

7

10

8

p [MPa] viscosity [mPa s]

100 200 300 400 0.2 0.4 0.6 0.8 1

p [MPa] V / V0

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 6 / 14

Viscosity for SAE 10W/40 reference oil RL 88/1, (partly) by Hutton, Jones, Bates, SAE, 1983

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 8 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside (0, T)× Ω: divv v v = ∂τv v v + div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

Viscosity formulas used in applications

ν = ν(π, |D D D(v v v)|2) = ∼ exp(απ), ∼ (1 + |D D D(v v v)|2)

p−2 2 ,

1 < p < 2

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside (0, T)× Ω: divv v v = ∂τv v v + div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

Problem well-posedness—first observations

ν = ν(π)

◮ M. Renardy, Comm. Part. Diff. Eq., 1986. ◮ F. Gazzola, Z. Angew. Math. Phys., 1997. ◮ F. Gazzola, P. Secchi, Navier–Stokes eq.: th. and num. meth. 1998.

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside (0, T)× Ω: divv v v = ∂τv v v + div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

Problem well-posedness—first positive results

∂S S S ∂D D D ∼ (1 + |D D D|2)

p−2 2

• ∂S

S S ∂π

• ≤ γ0 (1 + |D

D D|2)

p−2 4

1 < p < 2

◮ M´

alek, Neˇ cas, Rajagopal, Arch. Rational Mech. Anal., 2002.

◮ Hron, M´

alek, Neˇ cas, Rajagopal, Math. Comput. Simulation, 2003.

◮ M´

alek, Rajagopal, Handbook of mathematical fluid dynamics, 2007.

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside (0, T)× Ω: divv v v = ∂τv v v + div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

• n the boundary (0, T)× ∂Ω = ΓD ∪ ΓN ∪ ΓP:

v v v · n n n = 0 and − T T Tn n n = σ v v v

• n ΓN

v v v = v v v D v v v D v v v D

• n ΓD

if ΓP = ∅, −T T Tn n n = b b b(v v v)

• n ΓP

then ´

Ω0π dx

x x = 0

◮ Bul´

ıˇ cek, M´ alek, Rajagopal, Indiana Univ. Math. J., 2007

◮ Bul´

ıˇ cek, M´ alek, Rajagopal, SIAM J. Math. Anal., 2009

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside Ω: divv v v = div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

• n the boundary ∂Ω = ΓD ∪ ΓN ∪ ΓP:

v v v · n n n = 0 and − T T Tn n n = σ v v v

• n ΓN

v v v = v v v D v v v D v v v D

• n ΓD

if ΓP = ∅, −T T Tn n n = b b b(v v v)

• n ΓP

then ´

Ω0π dx

x x = 0

◮ Franta, M´

alek, Rajagopal, Proc. Royal Soc. A, 2005

◮ M. L., Nonlin. Anal.: Real World App., 2009

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside Ω: divv v v = div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

• n the boundary ∂Ω = ΓD ∪ ΓN ∪ ΓP:

v v v · n n n = 0 and − T T Tn n n = σ v v v

• n ΓN

v v v = v v v D v v v D v v v D

• n ΓD

if ΓP = ∅, −T T Tn n n = b b b(v v v)

• n ΓP

then ´

Ω0π dx

x x = 0

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Incompressible fluids with viscosity depending

• n pressure and shear rate

Mathematical formulation

inside Ω: divv v v = div(v v v ⊗ v v v) − divS S S = −∇π + f f f , S S S = 2 ν(π, |D D D(v v v)|2)D D D(v v v)

• n the boundary ∂Ω = ΓD ∪ ΓN ∪ ΓP:

v v v · n n n = 0 and − T T Tn n n = σ v v v

• n ΓN

v v v = v v v D v v v D v v v D

• n ΓD

if ΓP = ∅, −T T Tn n n = b b b(v v v)

• n ΓP

then ´

Ω0π dx

x x = 0

◮ Stebel & M. L., Appl. Mat.–Czech., in print; 2009 preprint NCMM ◮ Stebel & M. L., Math. Comput. Simulat., submitted 2009 preprint NCMM

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 9 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 ([∇v v v]v v v,w w w)Ω + (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b(v v v),w w w) ΓP

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

Test eq. by solution

(S S S(π,D D D(v v v)),D D D(v v v))Ω ∼ |D D D(v v v)|p ± 1 = ⇒ D D D(v v v)p ≤ K = ⇒ v v v1,p + S S Sp′ ≤ K

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

Inf–sup inequality and boundedness of ∂πS S S

0 < β ≤ inf

q∈Lp′

b.c.(Ω)

sup

w w w∈W1,p

b.c.(Ω)

(q, divw w w)Ω qp′w w w1,p = ⇒ β πp′ ≤ S S S(π,D D D(v v v))p′ + f f f + b b b ≤ K

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

Inf–sup inequality and boundedness of ∂πS S S

0 < β ≤ inf

q∈Lp′

b.c.(Ω)

sup

w w w∈W1,p

b.c.(Ω)

(q, divw w w)Ω qp′w w w1,p

Pressure uniquelly determined by velocity?

β π1 − π2p′ ≤ S S S(π1,D D D(v v v)) − S S S(π2,D D D(v v v))p′ ≤

• ˆ π2

π1 ∂πS

S S(π,D D D(v v v))dπ

• p′

≤ γ0 π1 − π2p′

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

Inf–sup inequality and boundedness of ∂πS S S

0 < β ≤ inf

q∈Lp′

b.c.(Ω)

sup

w w w∈W1,p

b.c.(Ω)

(q, divw w w)Ω qp′w w w1,p

Pressure and velocity uniquelly determined?

β π1 − π22 ≤ S S S(π1,D D D(v v v 1)) − S S S(π2,D D D(v v v 2))2

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

∂S S S ∂D D D ∼ (1 + |D D D|2)

p−2 2

• ∂S

S S ∂π

• ≤ γ0 (1 + |D

D D|2)

p−2 4

1 < p < 2

write

S S Si := S S S(πi,D D D(v v v i)), i = 1, 2 d(v v v 1,v v v 2) := ˆ

Ω

ˆ 1 (1 + |D D D(v v v 1) + s D D D(v v v 2 − v v v 1)|2)

p−2 2 |D

D D(v v v 1 − v v v 2)|2 ds dx x x

then the assumptions imply

S S S1 − S S S22 ≤ σ1 d(v v v 1,v v v 2) + γ0π1 − π22, d(v v v 1,v v v 2)2 ≤ 2 σ0

• S

S S1 − S S S2,D D D1 − D D D2

Ω + γ2

σ2 π1 − π22

2

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

Inf–sup inequality and boundedness of ∂πS S S

0 < β ≤ inf

q∈Lp′

b.c.(Ω)

sup

w w w∈W1,p

b.c.(Ω)

(q, divw w w)Ω qp′w w w1,p

Pressure and velocity uniquelly determined?

β π1 − π22 ≤ S S S1 − S S S22 ≤ σ1 d(v v v 1,v v v 2) + γ0π1 − π22

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Basic a priori estimates

Weak formulation

(q, divw w w)Ω = 0 (S S S(π,D D D(v v v)),D D D(w w w))Ω − (π, divw w w)Ω = (f f f ,w w w)Ω − (b b b ,w w w) ΓP

Inf–sup inequality and boundedness of ∂πS S S

0 < β ≤ inf

q∈Lp′

b.c.(Ω)

sup

w w w∈W1,p

b.c.(Ω)

(q, divw w w)Ω qp′w w w1,p

Pressure and velocity uniquelly determined?

β π1 − π22 ≤ S S S1 − S S S22 ≤ σ1 d(v v v 1,v v v 2) + γ0π1 − π22 ≤ γ0 σ1 σ0 + 1

• π1 − π22
• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 10 / 14

Discrete problem—numerical approximation

Galerkin approximation

V V V h ⊂ W1,p

b.c.(Ω),

Qh ⊂ Lp′

b.c.(Ω).

Find v v v h ∈ V V V h, πh ∈ Qh such that: (qh, divwh wh wh)Ω = 0 (S S S(πh,D D D(v v v h)),D D D(w w w h))Ω − (πh, divw w w h)Ω = (f f f − b b b,w w w h)Ω hold for all w w w h ∈ V V V h, qh ∈ Qh.

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 11 / 14

Discrete problem—numerical approximation

Galerkin approximation

V V V h ⊂ W1,p

b.c.(Ω),

Qh ⊂ Lp′

b.c.(Ω).

Find v v v h ∈ V V V h, πh ∈ Qh such that: (qh, divwh wh wh)Ω = 0 (S S S(πh,D D D(v v v h)),D D D(w w w h))Ω − (πh, divw w w h)Ω = (f f f − b b b,w w w h)Ω hold for all w w w h ∈ V V V h, qh ∈ Qh.

Discrete inf–sup condition

0 < ˜ β ≤ inf

q∈Qh

sup

w w w∈V V V h

(q, divw w w)Ω qp′w w w1,p

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 11 / 14

Summary

◮

∂S S S ∂D D D ∼ (1 + |D

D D|2)

p−2 2 , p < 2 shear-thinning setting.

◮ inf–sup inequality (β > 0), discrete inf–sup condition (˜

β > 0), boundary conditions

◮ ˛

˛ ∂S

S S ∂π

˛ ˛ ≤ γ0 (1 + |D D D|2)

p−2 4

to deal with pressure non-linearity

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 12 / 14

Summary

◮

∂S S S ∂D D D ∼ (1 + |D

D D|2)

p−2 2 , p < 2 shear-thinning setting.

◮ inf–sup inequality (β > 0), discrete inf–sup condition (˜

β > 0), boundary conditions

◮ ˛

˛ ∂S

S S ∂π

˛ ˛ ≤ γ0 (1 + |D D D|2)

p−2 4

to deal with pressure non-linearity

◮ ˜

β > 0 = ⇒ ∃ discrete solution (πh,v v v h) and vh vh vh1,p + S S S(πh,v v v h)p′ + βπhp′ ≤ K

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 12 / 14

Summary

◮

∂S S S ∂D D D ∼ (1 + |D

D D|2)

p−2 2 , p < 2 shear-thinning setting. (σ0, σ1)

◮ inf–sup inequality (β > 0), discrete inf–sup condition (˜

β > 0), boundary conditions

◮ ˛

˛ ∂S

S S ∂π

˛ ˛ ≤ γ0 (1 + |D D D|2)

p−2 4

to deal with pressure non-linearity

◮ ˜

β > 0 = ⇒ ∃ discrete solution (πh,v v v h) and vh vh vh1,p + S S S(πh,v v v h)p′ + βπhp′ ≤ K

◮ γ0 < ˜

β

σ0 σ0+σ1 =

⇒ (πh,v v v h) unique, γ0 < β

σ0 σ0+σ1 =

⇒ (π,v v v) at most one

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 12 / 14

Summary

◮

∂S S S ∂D D D ∼ (1 + |D

D D|2)

p−2 2 , p < 2 shear-thinning setting. (σ0, σ1)

◮ inf–sup inequality (β > 0), discrete inf–sup condition (˜

β > 0), boundary conditions

◮ ˛

˛ ∂S

S S ∂π

˛ ˛ ≤ γ0 (1 + |D D D|2)

p−2 4

to deal with pressure non-linearity

◮ ˜

β > 0 = ⇒ ∃ discrete solution (πh,v v v h) and vh vh vh1,p + S S S(πh,v v v h)p′ + βπhp′ ≤ K

◮ γ0 < ˜

β

σ0 σ0+σ1 =

⇒ (πh,v v v h) unique, γ0 < β

σ0 σ0+σ1 =

⇒ (π,v v v) at most one

◮ γ0 < β

σ0 σ0+σ1 =

⇒ ∃ weak solution (π,v v v) and v v v1,p + S S S(π,v v v)p′ + βπp′ ≤ K

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 12 / 14

Summary

◮

∂S S S ∂D D D ∼ (1 + |D

D D|2)

p−2 2 , p < 2 shear-thinning setting. (σ0, σ1)

◮ inf–sup inequality (β > 0), discrete inf–sup condition (˜

β > 0), boundary conditions

◮ ˛

˛ ∂S

S S ∂π

˛ ˛ ≤ γ0 (1 + |D D D|2)

p−2 4

to deal with pressure non-linearity

◮ ˜

β > 0 = ⇒ ∃ discrete solution (πh,v v v h) and vh vh vh1,p + S S S(πh,v v v h)p′ + βπhp′ ≤ K

◮ γ0 < ˜

β

σ0 σ0+σ1 =

⇒ (πh,v v v h) unique, γ0 < β

σ0 σ0+σ1 =

⇒ (π,v v v) at most one

◮ γ0 < β

σ0 σ0+σ1 =

⇒ ∃ weak solution (π,v v v) and v v v1,p + S S S(π,v v v)p′ + βπp′ ≤ K

◮ γ0 < ˜

β

σ0 σ0+σ1

= ⇒ a priori error estimates v v v − v v v h1,p inf

w w wh∈V V V h

v v v − w w w h1,p + inf

qh∈Qh π − qhp′

π − πhp′ inf

qh∈Qh π − qhp′ + v

v v − v v v h2/p′

1,p

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 12 / 14

Applications: lubrication problems, journal bearing

Flow in a converging channel

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 13 / 14

Applications: lubrication problems, journal bearing

Flow in a converging channel

Barus model ν = exp(απ), α = 0.306

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 13 / 14

Applications: lubrication problems, journal bearing

Flow in a converging channel

Barus model ν = exp(απ), α = 0.3061

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 13 / 14

Applications: lubrication problems, journal bearing

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 13 / 14

Applications: lubrication problems, journal bearing

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 13 / 14

Thank you for your attention!

• M. Lanzend¨
• rfer et al. (ICS AS CR)

Incompressible piezoviscous fluids January 13, 2011 14 / 14