Can numerical analysis help in understanding piezoviscous hydrodynamic lubrication? M. Lanzend¨ orfer Mathematical Institute, Charles University in Prague Institute of Computer Science, AS CR M odelling Math A nalysis C omputing M. Lanzend¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 1 / 16
Hydrodynamic (thick-film) lubrication M. Lanzend¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 2 / 16
Standard approach ◮ focus (oversimplified): isothermal, slow, steady, full-film M. Lanzend¨ orfer (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¨ orfer (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¨ orfer (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¨ orfer (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¨ orfer (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 ∼ 10 6 ms − 1 ) M. Lanzend¨ orfer (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). 10 8 10 7 1 10 6 0.8 viscosity [mPa s] 5 10 V / V 0 0.6 4 10 0.4 3 10 0.2 2 10 10 1 0 0 200 400 600 800 1000 1200 0 100 200 300 400 p [MPa] p [MPa] M. Lanzend¨ orfer (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¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 4 / 16
Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v = 0 v ∂ τ v v v + div( v v v ⊗ v v v ) − div S S = −∇ π + f f f , S v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) M. Lanzend¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 5 / 16
Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v = 0 v ∂ τ v v v + div( v v v ⊗ v v v ) − div S S = −∇ π + f f f , S v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) Cauchy stress tensor D | 2 ) D T = − π I T I I + 2 ν ( π, | D D D D , tr D D = 0 D T ◮ π is not the thermodynamical pressure ◮ π is the mean normal stress, π = − 1 3 tr T T T , ◮ implicitely constituted model T − 1 I − 2 ν ( − 1 T T 3 (tr T T T ) I I 3 tr T T T , | D D D | ) D D D = 0 see Rajagopal, J. Fluid Mech. , 2006 (and M´ alek, Rajagopal, 2006, 2007) M. Lanzend¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 5 / 16
Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v = 0 v ∂ τ v v v + div( v v v ⊗ v v v ) − div S S = −∇ π + f f f , S v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v D D ( v v v ) Viscosity formulas used in applications � ∼ exp( απ ) , v ) | 2 ) = ν = ν ( π, | D D D ( v v p − 2 v ) | 2 ) 2 , ∼ (1 + | D D ( v D v 1 < p < 2 M. Lanzend¨ orfer (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¨ orfer (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¨ orfer (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¨ orfer (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¨ orfer (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¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 6 / 16
Incompressible fluids with viscosity depending on pressure and shear rate Mathematical formulation inside (0 , T ) × Ω: div v v = 0 v ∂ τ v v + div( v v v ⊗ v v v v ) − div S S = −∇ π + f f f , S v ) | 2 ) D S S S = 2 ν ( π, | D D D ( v v 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¨ orfer (Charles University; ICS CAS) Piezoviscous lubrication November 30, 2012 7 / 16
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