Laminar shear flow Free surface film flow Dynamic b.c.. Reynolds .... Ritz. FEM. Comparison with FEM soluti- ons of original NavierâStokes equations for h = 0.2, ...
New solution methods for coating flows based on first integrals of Navier-Stokes equations M. Scholle1 , F. Marner1 , A. Haas2 , and P. H. Gaskell3 1
Institute for Automotive Technology and Mechatronics Heilbronn University, Germany 2
Manufacture Franaise des Pneumatiques Michelin Centre of Technology Ladoux, France 3
School of Engineering and Computing Sciences University of Durham, UK
10th European Coating Symposium, Sep. 11–13, 2013, Mons. .
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Contents 1.
Motivation
2.
Steady 2D–flow
3.
Examples
4.
Steady 3D–flow
5.
Outlook
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
2 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Motivation: Foundations of fluid mechanics
Inviscid & irrotational
Scalar potential: ⃗u = ∇φ
Inviscid, ∇ × ⃗u ̸= ⃗0
A. Clebsch (1859): ⃗u = ∇φ + α∇β
Integration
Integration
Bernoulli’s equation ⃗u 2 φ˙ + +P +U =0 2
Gen. Bernoulli’s equation and 2 additional transport equations
Scholle (HHN)
First Integral of Navier–Stokes Equations
Viscous flow
?
ECS 2013
3 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Motivation: Foundations of fluid mechanics
Inviscid & irrotational
Scalar potential: ⃗u = ∇φ
Inviscid, ∇ × ⃗u ̸= ⃗0
A. Clebsch (1859): ⃗u = ∇φ + α∇β
Integration
Integration
Bernoulli’s equation ⃗u 2 φ˙ + +P +U =0 2
Gen. Bernoulli’s equation and 2 additional transport equations
Scholle (HHN)
First Integral of Navier–Stokes Equations
Viscous flow
?
ECS 2013
4 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Motivation: Foundations of fluid mechanics
Inviscid & irrotational
Scalar potential: ⃗u = ∇φ
Inviscid, ∇ × ⃗u ̸= ⃗0
A. Clebsch (1859): ⃗u = ∇φ + α∇β
Integration
Integration
Bernoulli’s equation ⃗u 2 φ˙ + +P +U =0 2
Gen. Bernoulli’s equation and 2 additional transport equations
Scholle (HHN)
First Integral of Navier–Stokes Equations
Viscous flow
?
ECS 2013
5 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Restriction to steady 2D–flow
Complex variables:
Scholle (HHN)
ξ := x2 + ix1
First Integral of Navier–Stokes Equations
u := u1 + iu2
ECS 2013
6 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Restriction to steady 2D–flow
ξ := x2 + ix1
Complex variables: Navier–Stokes equations:
Scholle (HHN)
u := u1 + iu2
[ [ ] ] ¯u ∂ ∂ u u2 ∂u iϱ + η p+U +ϱ =i 2 ∂ξ ∂ξ 2 ∂ξ
First Integral of Navier–Stokes Equations
ECS 2013
6 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Restriction to steady 2D–flow
ξ := x2 + ix1
Complex variables: Navier–Stokes equations: New scalar ’potential’ Φ: (Ranger 1994)
Scholle (HHN)
u := u1 + iu2
[ [ ] ] ¯u ∂ ∂ u u2 ∂u iϱ + η p+U +ϱ =i 2 ∂ξ ∂ξ 2 ∂ξ p + U + Re
¯u u ∂ ∂ = Φ 2 ∂ξ ∂ξ
First Integral of Navier–Stokes Equations
ECS 2013
6 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Restriction to steady 2D–flow
ξ := x2 + ix1
Complex variables: Navier–Stokes equations: New scalar ’potential’ Φ: (Ranger 1994)
integrable equation:
Scholle (HHN)
u := u1 + iu2
[ [ ] ] ¯u ∂ ∂ u u2 ∂u iϱ + η p+U +ϱ =i 2 ∂ξ ∂ξ 2 ∂ξ p + U + Re
¯u u ∂ ∂ = Φ 2 ∂ξ ∂ξ
] [ ∂u ∂2Φ ∂ u2 −i 2 =0 iϱ + η 2 ∂ξ ∂ξ ∂ξ
First Integral of Navier–Stokes Equations
ECS 2013
6 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Restriction to steady 2D–flow
ξ := x2 + ix1
Complex variables: Navier–Stokes equations: New scalar ’potential’ Φ: (Ranger 1994)
integrable equation:
Integration and gauging of Φ Scholle (HHN)
u := u1 + iu2
[ [ ] ] ¯u ∂ ∂ u u2 ∂u iϱ + η p+U +ϱ =i 2 ∂ξ ∂ξ 2 ∂ξ p + U + Re
¯u u ∂ ∂ = Φ 2 ∂ξ ∂ξ
] [ ∂u ∂2Φ ∂ u2 −i 2 =0 iϱ + η 2 ∂ξ ∂ξ ∂ξ
iϱ
u2 ∂u ∂2Φ +η − i 2 = f (ξ) = 0 2 ∂ξ ∂ξ
First Integral of Navier–Stokes Equations
ECS 2013
6 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Introducing a streamfunction
Continuity equation identically fulfilled by:
Scholle (HHN)
First Integral of Navier–Stokes Equations
u=
∂ψ ∂ξ
ECS 2013
7 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Potential representation Introducing a streamfunction
Continuity equation identically fulfilled by:
u=
∂ψ ∂ξ
First integral of Navier–Stokes equations iϱ 2
(
∂ψ ∂ξ
)2
+η
∂2ψ ∂2Φ − i =0 ∂ξ 2 ∂ξ 2
Remark: 2nd order PDE instead of 3rd order PDE
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
7 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Reynolds number effects on shear flow between wavy walls
y
✻
✲ U0
✻
h
❄ ✛
a ✻ ❄ 2π
Scholle (HHN)
✲
x
✲
First Integral of Navier–Stokes Equations
ECS 2013
8 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Reynolds number effects on shear flow between wavy walls
y
✻
✲ U0
✻
h
❄ ✛
a ✻ ❄ 2π
Scholle (HHN)
✲
x
✲
∂2φ ∂2ψ ∂2ψ uv − − 2 = Re ∂x∂y h ∂y 2 ∂x 2 2
∂2φ ∂2φ u 2 −v 2 ∂2ψ − + = Re ∂x∂y 2h ∂x 2 ∂y 2
with Reynolds number Re = U0 H/η.
First Integral of Navier–Stokes Equations
ECS 2013
8 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Reynolds number effects on shear flow between wavy walls
y
✻
✲ U0
✻
h
❄ ✛
a ✻ ❄ 2π
No–slip–conditions:
Scholle (HHN)
✲
x
✲
∂2φ ∂2ψ ∂2ψ uv − − 2 = Re ∂x∂y h ∂y 2 ∂x 2 2
∂2φ ∂2φ u 2 −v 2 ∂2ψ − + = Re ∂x∂y 2h ∂x 2 ∂y 2
with Reynolds number Re = U0 H/η. ψ (x, −a cos x) = 0 v (x, −a cos x) = 0
First Integral of Navier–Stokes Equations
u (x, h) = 1 v (x, h) = 0
ECS 2013
8 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Reynolds number effects on shear flow between wavy walls
y
✻
✲ U0
✻
h
❄ ✛
a ✻ ❄ 2π
No–slip–conditions:
✲
x
✲
∂2φ ∂2ψ ∂2ψ uv − − 2 = Re ∂x∂y h ∂y 2 ∂x 2 2
∂2φ ∂2φ u 2 −v 2 ∂2ψ − + = Re ∂x∂y 2h ∂x 2 ∂y 2
with Reynolds number Re = U0 H/η. ψ (x, −a cos x) = 0 v (x, −a cos x) = 0
u (x, h) = 1 v (x, h) = 0
Approximations:
◮
lubrication approximation for Stokes flow case (Re = 0)
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
8 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Reynolds number effects on shear flow between wavy walls
y
✻
✲ U0
✻
h
❄ ✛
a ✻ ❄ 2π
No–slip–conditions:
✲
x
✲
∂2φ ∂2ψ ∂2ψ uv − − 2 = Re ∂x∂y h ∂y 2 ∂x 2 2
∂2φ ∂2φ u 2 −v 2 ∂2ψ − + = Re ∂x∂y 2h ∂x 2 ∂y 2
with Reynolds number Re = U0 H/η. ψ (x, −a cos x) = 0 v (x, −a cos x) = 0
u (x, h) = 1 v (x, h) = 0
Approximations:
◮ ◮
lubrication approximation for Stokes flow case (Re = 0) variational formulation for Re–corrections Scholle (HHN)
First Integral of Navier–Stokes Equations
Ritz’s direct method ECS 2013
8 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Reynolds number effects on shear flow between wavy walls
y
✻
✲ U0
✻
h
❄ ✛
a ✻ ❄ 2π
No–slip–conditions:
✲
x
✲
∂2φ ∂2ψ ∂2ψ uv − − 2 = Re ∂x∂y h ∂y 2 ∂x 2 2
∂2φ ∂2φ u 2 −v 2 ∂2ψ − + = Re ∂x∂y 2h ∂x 2 ∂y 2
with Reynolds number Re = U0 H/η. ψ (x, −a cos x) = 0 v (x, −a cos x) = 0
u (x, h) = 1 v (x, h) = 0
Approximations:
◮ ◮ ◮
lubrication approximation for Stokes flow case (Re = 0) variational formulation for Re–corrections discretization with 20 coefficients Scholle (HHN)
Ritz’s direct method
low computational costs
First Integral of Navier–Stokes Equations
ECS 2013
8 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Results: Reynolds number study at h = 0.2, a = 0.08 and validation
Re = 0
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
9 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Results: Reynolds number study at h = 0.2, a = 0.08 and validation
Re = 0
Re = 40
Re = 80
Re = 120 Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
9 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Results: Reynolds number study at h = 0.2, a = 0.08 and validation
Re = 0
Re = 40
Comparison with FEM solutions of original Navier–Stokes equations for h = 0.2, a = 0.1 and Re = 100:
Ritz
Re = 80 FEM Re = 120 Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
9 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation
⃗g
❄
z
❏ ❪ ❏
❏
❏
✑ ❏❛✑ ✑ ✑ ✑ ✰ ✑ ✑ x ✑α
Scholle (HHN)
✑ ✑ ✑ ✑
✑ ✑ ✑
✑ ✑
First Integral of Navier–Stokes Equations
ECS 2013
10 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation
Field equations for Stokes’ flow: ( ∂2φ y) ∂2ψ ∂2ψ − − 2 = 2 1 − ∂x∂y h ∂y 2 ∂x 2
⃗g
❄
z
❏ ❪ ❏
❏
❏
✑ ❏❛✑ ✑ ✑ ✑ ✰ ✑ ✑ x ✑α
Scholle (HHN)
✑ ✑ ✑ ✑
✑ ∂2ψ ∂2φ ∂2φ ✑ 2 + = 0 − ✑ ∂x∂y ∂x 2 ∂y 2 ✑ ✑
First Integral of Navier–Stokes Equations
ECS 2013
10 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation
Field equations for Stokes’ flow: ( ∂2φ y) ∂2ψ ∂2ψ − − 2 = 2 1 − ∂x∂y h ∂y 2 ∂x 2
⃗g
❄
z
❏ ❪ ❏
❏
❏
✑ ❏❛✑ ✑ ✑ ✑ ✰ ✑ ✑ x ✑α
Scholle (HHN)
✑ ✑ ✑ ✑
✑ ∂2ψ ∂2φ ∂2φ ✑ 2 + = 0 − ✑ ∂x∂y ∂x 2 ∂y 2 ✑ ✑
Boundary conditions:
◮ ◮ ◮
ψ = 0 and ∂n ψ = 0 at bottom ∂n ψ = 0 at free surface Dynamic b.c. at free surface
First Integral of Navier–Stokes Equations
ECS 2013
10 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation
Field equations for Stokes’ flow: ( ∂2φ y) ∂2ψ ∂2ψ − − 2 = 2 1 − ∂x∂y h ∂y 2 ∂x 2
⃗g
❄
z
❏ ❪ ❏
❏
❏
✑ ❏❛✑ ✑ ✑ ✑ ✰ ✑ ✑ x ✑α
✑ ✑ ✑ ✑
✑ ∂2ψ ∂2φ ∂2φ ✑ 2 + = 0 − ✑ ∂x∂y ∂x 2 ∂y 2 ✑ ✑
Boundary conditions:
◮ ◮ ◮
ψ = 0 and ∂n ψ = 0 at bottom ∂n ψ = 0 at free surface Dynamic b.c. at free surface
Reformulation of dynamic b.c. required: replace pressure p by its potential representation! Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
10 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation of dynamic boundary condition
[
Original form: (p0 − p)δij + η
Scholle (HHN)
(
∂uj ∂ui + ∂xj ∂xi
)]
nj (s) = σ xk =fk (s)
First Integral of Navier–Stokes Equations
dti ds
ECS 2013
11 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation of dynamic boundary condition
[
Original form: (p0 − p)δij + η
(
∂uj ∂ui + ∂xj ∂xi
)]
nj (s) = σ xk =fk (s)
dti ds
substitute:
∂2Φ u u −ϱ k k −U ∂xk ∂xk 2 ( [ ] [ 2 ] ) δij ∂uj ∂ui ∂ Φ ∂ 2 Φ δij η = ϱ ui uj − uk uk +2 + − ∂xj ∂xi 2 ∂xi ∂xj ∂xk ∂xk 2 p =
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
11 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Formulation of dynamic boundary condition
[
Original form: (p0 − p)δij + η
(
∂uj ∂ui + ∂xj ∂xi
)]
nj (s) = σ xk =fk (s)
dti ds
substitute:
∂2Φ u u −ϱ k k −U ∂xk ∂xk 2 ( [ ] [ 2 ] ) δij ∂uj ∂ui ∂ Φ ∂ 2 Φ δij η = ϱ ui uj − uk uk +2 + − ∂xj ∂xi 2 ∂xi ∂xj ∂xk ∂xk 2 p =
Condition for potential Φ of 2nd order ] [ 2 ∂2Φ ∂ Φ − δij U (fk (s)) ni (s) + 2 ∂xi ∂xj ∂xk ∂xk x
nj (s) = σ k =fk (s)
dti ds
This condition is integrable! Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
11 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Resulting integrals of dynamic boundary conditions
Final form: Dirichlet/Neumann boundary conditions!
Φ = ∂n Φ =
Scholle (HHN)
[
] dσ [xn − fn ] Uδni + εni ti ds ds ] ∫ [ σ ti dσ Uni + − ti ds 2 2 ds
1 2
∫
First Integral of Navier–Stokes Equations
ECS 2013
12 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Complete formulation
Field equations for Stokes’ flow: ( ∂2φ y) ∂2ψ ∂2ψ − − 2 = 2 1 − ∂x∂y h ∂y 2 ∂x 2
⃗g
❄
z
❏ ❪ ❏
❏
❏
✑ ❏❛✑ ✑ ✑ ✑ ✰ ✑ ✑ x ✑α
Scholle (HHN)
✑ ✑ ✑ ✑
✑ ✑ ✑
∂2ψ ∂2φ ∂2φ 2 + = 0 − ✑ ∂x∂y ∂x 2 ∂y 2 ✑ Boundary conditions:
◮ ◮ ◮
ψ = 0 and ∂n ψ = 0 at bottom ∂n ψ = 0 at the free surface φ = · · · and ∂n φ = · · · at the free surface
First Integral of Navier–Stokes Equations
ECS 2013
13 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Complete formulation
Field equations for Stokes’ flow: ( ∂2φ y) ∂2ψ ∂2ψ − − 2 = 2 1 − ∂x∂y h ∂y 2 ∂x 2
⃗g
❄
z
❏ ❪ ❏
❏
❏
✑ ❏❛✑ ✑ ✑ ✑ ✰ ✑ ✑ x ✑α
✑ ✑ ✑ ✑
✑ ✑ ✑
∂2ψ ∂2φ ∂2φ 2 + = 0 − ✑ ∂x∂y ∂x 2 ∂y 2 ✑ Boundary conditions:
◮ ◮ ◮
ψ = 0 and ∂n ψ = 0 at bottom ∂n ψ = 0 at the free surface φ = · · · and ∂n φ = · · · at the free surface
Solution method? Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
13 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. FOSLS - First Order System Least-Squares
◮
Rewrite field equations as FOS with solution vector W = (u1 , u2 , φx , φy )T pure Dirichlet boundaries for W
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
14 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. FOSLS - First Order System Least-Squares
◮
◮
Rewrite field equations as FOS with solution vector W = (u1 , u2 , φx , φy )T pure Dirichlet boundaries for W 4 ∑ Minimization of LS-functional J (W ; f ) = ∥Li W − fi ∥20 i=1
yields variational formulation: ∫ ∫ LVh f dΩ LWh LVh dΩ = Ω
˜h ∀Vh ∈ V
Ω
→ FEM discretization, Newton linearization: spd linear systems efficient scalable solvers
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
14 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. FOSLS - First Order System Least-Squares
◮
◮
Rewrite field equations as FOS with solution vector W = (u1 , u2 , φx , φy )T pure Dirichlet boundaries for W 4 ∑ Minimization of LS-functional J (W ; f ) = ∥Li W − fi ∥20 i=1
yields variational formulation: ∫ ∫ LVh f dΩ LWh LVh dΩ = Ω
◮
˜h ∀Vh ∈ V
Ω
→ FEM discretization, Newton linearization: spd linear systems efficient scalable solvers Free surface: Eulerian grid-adaptive method
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
14 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. FOSLS - First Order System Least-Squares
◮
◮
Rewrite field equations as FOS with solution vector W = (u1 , u2 , φx , φy )T pure Dirichlet boundaries for W 4 ∑ Minimization of LS-functional J (W ; f ) = ∥Li W − fi ∥20 i=1
yields variational formulation: ∫ ∫ LVh f dΩ LWh LVh dΩ = Ω
◮ ◮
˜h ∀Vh ∈ V
Ω
→ FEM discretization, Newton linearization: spd linear systems efficient scalable solvers Free surface: Eulerian grid-adaptive method Problem: mass conservation → appropriate weighting, augmented LS . . . Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
14 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Varied film thickness
h = 0.15
Scholle (HHN)
h = 0.3
First Integral of Navier–Stokes Equations
h = 0.8
ECS 2013
15 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Varied Reynolds number
Re = 0.3
Scholle (HHN)
Re = 10
First Integral of Navier–Stokes Equations
Re = 30
ECS 2013
16 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Laminar shear flow Free surface film flow Dynamic b.c. Varied Reynolds number
Re = 0.3
Re = 30 Scholle (HHN)
Re = 10
Re = 50 First Integral of Navier–Stokes Equations
Re = 30
Re = 100 ECS 2013
16 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition
Complex representation: not available for 3D! ¯u u2 u ∂2Φ ∂u ∂2Φ p + U + Re , iϱ + η = =i 2 2 2 ∂ξ ∂ξ ∂ξ∂ξ
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
17 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition
Complex representation: not available for 3D! ¯u u2 u ∂2Φ ∂u ∂2Φ p + U + Re , iϱ + η = =i 2 2 2 ∂ξ ∂ξ ∂ξ∂ξ Tensor representation (with Levi-Cevita Symbol ε): [ ] ∂uj ∂ui ∂ 2 (2Φ) ϱui uj + pδij − η + +Uδij = εik εjp ∂xj ∂xi ∂xk ∂xp | {z } −Tij
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
17 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition
Complex representation: not available for 3D! ¯u u2 u ∂2Φ ∂u ∂2Φ p + U + Re , iϱ + η = =i 2 2 2 ∂ξ ∂ξ ∂ξ∂ξ Tensor representation (with Levi-Cevita Symbol ε): [ ] ∂uj ∂ui ∂ 2 (2Φ) ϱui uj + pδij − η + +Uδij = εik εjp ∂xj ∂xi ∂xk ∂xp | {z } −Tij
Generalisation: Replace 1. Tij −→ stress tensor of any non-Newtonian fluid.
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
17 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition
Complex representation: not available for 3D! ¯u u2 u ∂2Φ ∂u ∂2Φ p + U + Re , iϱ + η = =i 2 2 2 ∂ξ ∂ξ ∂ξ∂ξ Tensor representation (with Levi-Cevita Symbol ε): [ ] ∂uj ∂ui ∂ 2 (2Φ) ϱui uj + pδij − η + +Uδij = εik εjp ∂xj ∂xi ∂xk ∂xp | {z } −Tij
Generalisation: Replace 1. Tij −→ stress tensor of any non-Newtonian fluid. ∂ 2 alq ∂ 2 (2Φ) −→ εik l εjpq ∂xk ∂xp ∂xk ∂xp with symmetric tensor potential alq = aql . 2. εik εjp
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
17 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition First integral of equations of motion, non-Newtonian, 3D, steady
ϱui uj − Tij + Uδij ∂ui ∂xi
Scholle (HHN)
= εikl εjpq
∂ 2 alq ∂xk ∂xp
with Tij = Tij
= 0
First Integral of Navier–Stokes Equations
(
∂uk ∂xl
)
− pδij
ECS 2013
18 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition First integral of equations of motion, non-Newtonian, 3D, steady
ϱui uj − Tij + Uδij ∂ui ∂xi
= εikl εjpq
∂ 2 alq ∂xk ∂xp
with Tij = Tij
= 0
Gauging of the tensor potential:
alq → alq +
(
∂uk ∂xl
)
− pδij
∂βq ∂βl + ∂xl ∂xq
with arbitrary vector field βq .
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
18 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition First integral of equations of motion, non-Newtonian, 3D, steady
ϱui uj − Tij + Uδij ∂ui ∂xi
= εikl εjpq
∂ 2 alq ∂xk ∂xp
with Tij = Tij
= 0
Gauging of the tensor potential:
alq → alq +
(
∂uk ∂xl
)
− pδij
∂βq ∂βl + ∂xl ∂xq
with arbitrary vector field βq . Reasonable gauging criteria: Choose βq such that ◮ 3 of 6 potentials vanish, or
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
18 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition First integral of equations of motion, non-Newtonian, 3D, steady
ϱui uj − Tij + Uδij ∂ui ∂xi
= εikl εjpq
∂ 2 alq ∂xk ∂xp
with Tij = Tij
= 0
Gauging of the tensor potential:
alq → alq +
(
∂uk ∂xl
)
− pδij
∂βq ∂βl + ∂xl ∂xq
with arbitrary vector field βq . Reasonable gauging criteria: Choose βq such that ◮ 3 of 6 potentials vanish, or ◮ field equations become self-adjoint, or
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
18 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition First integral of equations of motion, non-Newtonian, 3D, steady
ϱui uj − Tij + Uδij ∂ui ∂xi
= εikl εjpq
∂ 2 alq ∂xk ∂xp
with Tij = Tij
= 0
Gauging of the tensor potential:
alq → alq +
(
∂uk ∂xl
)
− pδij
∂βq ∂βl + ∂xl ∂xq
with arbitrary vector field βq . Reasonable gauging criteria: Choose βq such that ◮ 3 of 6 potentials vanish, or ◮ field equations become self-adjoint, or ◮ theory can be extended toward unsteady flows, or ◮ ... Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
18 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition Reformulation in terms of the tensor potential:
Original form of dynamic b.c. at the free surface F (xk ) = 0: Tij nj = σκni
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
19 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition Reformulation in terms of the tensor potential:
Original form of dynamic b.c. at the free surface F (xk ) = 0: Tij nj = σκni First integal and kinematic b.c. boundary condition for the tensor potential −εikl εjpq ∂k ∂p alq nj = [σκ − U] ni
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
19 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
From 2D to 3D General theory Dynamic boundary condition Reformulation in terms of the tensor potential:
Original form of dynamic b.c. at the free surface F (xk ) = 0: Tij nj = σκni First integal and kinematic b.c. boundary condition for the tensor potential −εikl εjpq ∂k ∂p alq nj = [σκ − U] ni Integration of b.c.
Scholle (HHN)
Dirichlet/Neumann type b.c.
First Integral of Navier–Stokes Equations
ECS 2013
19 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
◮
Development of efficient methods for film flows
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
◮
Development of efficient methods for film flows
2. 3D flow:
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
◮
Development of efficient methods for film flows
2. 3D flow: ◮
Generalisation of the theory using tensor calculus
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
◮
Development of efficient methods for film flows
2. 3D flow: ◮
Generalisation of the theory using tensor calculus
◮
Further investigations necessary
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
◮
Development of efficient methods for film flows
2. 3D flow: ◮
Generalisation of the theory using tensor calculus
◮
Further investigations necessary
More details: Proc. Roy. Soc. Lond. A 467 (2011) 127–143.
Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
Motivation
Steady 2D–flow
Examples
Steady 3D–flow
Outlook
Summary and Outlook
1. 2D flow: ◮
Reduction of dynamic b.c. to Dirichlet/Neumann type
◮
Development of efficient methods for film flows
2. 3D flow: ◮
Generalisation of the theory using tensor calculus
◮
Further investigations necessary
More details: Proc. Roy. Soc. Lond. A 467 (2011) 127–143. Acknowledgement: The authors are grateful to the Deutsche Forschungsgemeinschaft (DFG) for their financial support. Scholle (HHN)
First Integral of Navier–Stokes Equations
ECS 2013
20 / 20
