Can numerical analysis help in understanding piezoviscous - - PowerPoint PPT Presentation

Can numerical analysis help in understanding piezoviscous hydrodynamic lubrication?

• M. Lanzend¨
• rfer

Mathematical Institute, Charles University in Prague Institute of Computer Science, AS CR Math Modelling A nalysis Computing

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 1 / 16

Hydrodynamic (thick-film) lubrication

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 2 / 16

Standard approach

◮ focus (oversimplified): isothermal, slow, steady, full-film

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 3 / 16

Standard approach

◮ focus (oversimplified): isothermal, slow, steady, full-film

(reality: strong thermal effects, significant inertial effects, dynamic FSI, cavitation)

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 3 / 16

Standard approach

◮ focus (oversimplified): isothermal, slow, steady, full-film

(reality: strong thermal effects, significant inertial effects, dynamic FSI, cavitation)

◮ compute lift (and drag) force

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 3 / 16

Standard approach

◮ focus (oversimplified): isothermal, slow, steady, full-film

(reality: strong thermal effects, significant inertial effects, dynamic FSI, cavitation)

◮ compute lift (and drag) force ⇐ compute pressure profile (and velocity field)

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 3 / 16

Standard approach

◮ focus (oversimplified): isothermal, slow, steady, full-film

(reality: strong thermal effects, significant inertial effects, dynamic FSI, cavitation)

◮ compute lift (and drag) force ⇐ compute pressure profile (and velocity field) ◮ dimensional reduction: Reynolds approximation

◮ a single equation for the pressure ◮ a number of assumptions on the flow characteristics

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 3 / 16

Standard approach

◮ focus (oversimplified): isothermal, slow, steady, full-film

(reality: strong thermal effects, significant inertial effects, dynamic FSI, cavitation)

◮ compute lift (and drag) force ⇐ compute pressure profile (and velocity field) ◮ dimensional reduction: Reynolds approximation

◮ a single equation for the pressure ◮ a number of assumptions on the flow characteristics

◮ focus: high pressures

(EHL: pressure ∼ 2-3 GPa, shear rate ∼ 106 ms−1)

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 3 / 16

Viscosity at large pressure and shear rate

Viscosity and volume variation with pressure for squalane

(“representing a low viscosity paraffinic mineral oil”, 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 (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 4 / 16

Viscosity at large pressure and shear rate

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

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 4 / 16

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 (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 5 / 16

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)

Cauchy stress tensor

T T T = −πI I I + 2ν(π, |D D D|2)D D D, trD D D = 0

◮ π is not the thermodynamical pressure ◮ π is the mean normal stress,

π = − 1

3 trT

T T,

◮ implicitely constituted model

T T T − 1

3(trT

T T)I I I − 2ν(− 1

3 trT

T T, |D D D|)D D D = 0 see Rajagopal, J. Fluid Mech., 2006 (and M´ alek, Rajagopal, 2006, 2007)

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 5 / 16

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 (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 5 / 16

So what. . .

. . . is the problem?

◮ the lubrication works and is used since before the invention of wheel ◮ the viscosity–pressure relation is present in the very basis of the theory of

elastohydrodynamic lubrication

◮ are there any fundamental questions left open?

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 6 / 16

So what. . .

◮ are there any fundamental questions left open?

First, one may answer. . .

◮ lubrication is used everywhere (transportation, electricity production) ◮ any optimization can save energy consumption and prolongate the lifespan ◮ more and more precize quantitative predictions are needed

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 6 / 16

So what. . .

◮ are there any fundamental questions left open?

First, one may answer. . .

◮ more and more precize quantitative predictions are needed

. . . in fact, it is worse than that. . . Bair, Gordon, 2006:

” . . . there has been relatively little progress since the classic Newtonian solutions . . . toward relating film thickness and traction to the properties of individual liquid lubricants and it not clear at this time that full numerical solutions can even be obtained for heavily loaded contacts using accurate models. One central issue is the validity of Reynolds equation, derived under the isoviscous

• assumption. . . ”
• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 6 / 16

So what. . .

◮ are there any fundamental questions left open?

First, one may answer. . .

◮ more and more precize quantitative predictions are needed

. . . in fact, it is worse than that. . . Bair, Gordon, 2006:

” . . . there has been relatively little progress since the classic Newtonian solutions . . . toward relating film thickness and traction to the properties of individual liquid lubricants and it not clear at this time that full numerical solutions can even be obtained for heavily loaded contacts using accurate models. One central issue is the validity of Reynolds equation, derived under the isoviscous

• assumption. . . ”

◮ Rajagopal, Szeri, Proc. R. Soc. Lond. A, 2003

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 6 / 16

So what. . .

◮ are there any fundamental questions left open?

First, one may answer. . .

◮ more and more precize quantitative predictions are needed

. . . in fact, it is worse than that. . . Bair, Gordon, 2006:

” . . . there has been relatively little progress since the classic Newtonian solutions . . . toward relating film thickness and traction to the properties of individual liquid lubricants and it not clear at this time that full numerical solutions can even be obtained for heavily loaded contacts using accurate models. One central issue is the validity of Reynolds equation, derived under the isoviscous

• assumption. . . ”

◮ Rajagopal, Szeri, Proc. R. Soc. Lond. A, 2003

. . . in fact, even worse. . . is the full system of governing equations well-posed?

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 6 / 16

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 (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 7 / 16

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—a wave of succesful 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, Bul´

ıˇ cek, Majdoub, . . .

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 7 / 16

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)

Problem well-posedness—a wave of succesful 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

◮ Franta, M´

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

◮ & Bul´

ıˇ cek, Fiˇ serov´ a, Kaplick´ y, Lanzend¨

• rfer, Stebel, Hirn, . . .
• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 7 / 16

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 the 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 uniquely 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

∂S S S(π,D D D(v v v)) ∂π dπ

• p′

≤ γ0 π1 − π2p′ where

• ∂S

S S ∂π

• ≤ γ0
• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 8 / 16

Two major questions

. . . concerned with Reynolds equation

Is it based on well-posed system of equations?

◮ no success for ν(π) ◮ successful analysis for a subclass of models within the subcritical case:

|∂S S S/∂π| < . . . < 1

◮ no success in the supercritical case |∂S

S S/∂π| > 1.

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 9 / 16

Hydrodynamic (thick-film) lubrication

Flow in a converging channel

Newtonian model ν = const

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 10 / 16

Hydrodynamic (thick-film) lubrication

Flow in a converging channel

Barus model ν = exp(απ), α = 0.306

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 10 / 16

Hydrodynamic (thick-film) lubrication

Flow in a converging channel

Barus model ν = exp(απ), α = 0.3061

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 10 / 16

Sensitivity on boundary data

0.5 1 1.5 2 2.5 3 x 10

−3

1 2 3 4 5 6 7 8 9 x 10 x coordinate / [m] pressure / [Pa]

Pressure for π(0) = π(L) = 0 (full), 1 MPa (dashed) a 10 MPa (dash dotted). pressure π viskosity ν(π, |D D D|)

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 11 / 16

Hydrodynamic (thick-film) lubrication

Discrete problem stability.

Numerical experiments for journal bearing.

0.7 0.75 0.8 0.85 0.9 0.95 1 10

−4

10

−3

10

−2

10

−1

10 10

1

relative eccentricity, ε maximal reached |dS/dp|

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 12 / 16

Two major questions

. . . concerned with Reynolds equation

Is it based on well-posed system of equations?

◮ no success for ν(π) ◮ successful analysis for a subclass of models within the subcritical case:

|∂S S S/∂π| < . . . < 1

◮ no success in the supercritical case |∂S

S S/∂π| > 1.

When do the assumptions on the flow characteristics hold?

◮ no success with FEM simulations in the supercritical case ◮ interesting phenomena (e.g. the modified Reynolds equation) arise in the

supercritical, = ⇒ validation of Reynolds approximation not possible

◮ in subcritical case, a priori error estimates for FEM derived

(Hirn, Lanzend¨

• rfer, Stebel, 2012)

◮ a posteriori error analysis and adaptive FEM needed for quantitative studies

(localized nonlinearity)

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 13 / 16

Thank you for your attention

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 14 / 16

Hydrodynamic (thick-film) lubrication

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 15 / 16

Poiseuille flow

the pressure π the viscosity ν(p) the velocity component v v v ·e e ex the velocity component v v v ·e e ey

• M. Lanzend¨
• rfer (Charles University; ICS CAS)

Piezoviscous lubrication November 30, 2012 16 / 16