Introduction
The existence result
Bifurcation: Turning point
Further research
Modeling of fluid flow through the system of pipes
with lid
Boris Muha
Zvonimir Tutek
Department of Mathematics
University of Zagreb
Croatia
Recent developments in the theory of elliptic partial differential
equations, 2009.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Geometry of problem
This is an example of interaction of a fluid and rigid body
(lid).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Geometry of problem
This is an example of interaction of a fluid and rigid body
(lid).
Lid can move only along the ”vertical” tube without rotations
(like a piston).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Geometry of problem
This is an example of interaction of a fluid and rigid body
(lid).
Lid can move only along the ”vertical” tube without rotations
(like a piston).
Lid is heavy.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Geometry of problem
This is an example of interaction of a fluid and rigid body
(lid).
Lid can move only along the ”vertical” tube without rotations
(like a piston).
Lid is heavy.
Our goal is to determine a stationary state and its dependence
on the geometry.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Ωαh is smooth domain (also technical assumption).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Ωαh is smooth domain (also technical assumption).
Σh (lid) has constant normal e2 .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Ωαh is smooth domain (also technical assumption).
Σh (lid) has constant normal e2 .
Domain is symmetric w.r.t. y axis (this is not a restriction)
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Ωαh is smooth domain (also technical assumption).
Σh (lid) has constant normal e2 .
Domain is symmetric w.r.t. y axis (this is not a restriction)
T = −pI + µsym(∇u) is stress tensor and u, p are velocity
and pressure of the fluid.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Ωαh is smooth domain (also technical assumption).
Σh (lid) has constant normal e2 .
Domain is symmetric w.r.t. y axis (this is not a restriction)
T = −pI + µsym(∇u) is stress tensor and u, p are velocity
and pressure of the fluid.
R
F (h) = Σh T h n · n is total force on the lid.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Notations and assumptions
We will consider 2D case (for technical simplicity).
Ωαh is smooth domain (also technical assumption).
Σh (lid) has constant normal e2 .
Domain is symmetric w.r.t. y axis (this is not a restriction)
T = −pI + µsym(∇u) is stress tensor and u, p are velocity
and pressure of the fluid.
R
F (h) = Σh T h n · n is total force on the lid.
Fluid is Newtonian and incompressible.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Statement of the problem
Problem: Find (u, p, h) such that
−µ4u + ∇p = −g ρe2 , in Ωαh ,
div u = 0 in Ωαh ,
u = 0, on Γ,
u = 0, on Σ0 ,
u × n = 0, p = Pp/k − g ρy , na Σp/k ,
F (h) = P0 .
(1)
µ is viscosity and ρ density of the fluid.
P0 is a constant which takes into account the weight of the
lid.
(1)6 is balance of the forces.
We can suppose Pp = −Pk .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Remark on boundary conditions
We have different possibilities for choosing boundary
conditions on Σp/k .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Remark on boundary conditions
We have different possibilities for choosing boundary
conditions on Σp/k .
Since we are interested in effects near the junction and the lid,
a natural choice would be to prescribe boundary conditions at
the infinity.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Remark on boundary conditions
We have different possibilities for choosing boundary
conditions on Σp/k .
Since we are interested in effects near the junction and the lid,
a natural choice would be to prescribe boundary conditions at
the infinity.
Because we also want to make numerical experiments, we
need ”artificial” boundary conditions on Σp/k .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Remark on boundary conditions
We have different possibilities for choosing boundary
conditions on Σp/k .
Since we are interested in effects near the junction and the lid,
a natural choice would be to prescribe boundary conditions at
the infinity.
Because we also want to make numerical experiments, we
need ”artificial” boundary conditions on Σp/k .
Some of the possible choices are Dirichlet BC and ”do
nothing” BC
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Remark on boundary conditions
We have different possibilities for choosing boundary
conditions on Σp/k .
Since we are interested in effects near the junction and the lid,
a natural choice would be to prescribe boundary conditions at
the infinity.
Because we also want to make numerical experiments, we
need ”artificial” boundary conditions on Σp/k .
Some of the possible choices are Dirichlet BC and ”do
nothing” BC
We have decided to use BC (1)5 (introduced by Conca,
Murat, Pironneau, ’94.) because they involve pressure (which
need to be fixed in our problem) and there are known
regularity results (Bernard, 2002.)
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Numerical example 1
Numerical experiments are made in FreeFem++.
In this example α is π4 and h is fixed.
Pictures represent profile of speed and isobars for pressure
respectively.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Numerical example 1
Numerical experiments are made in FreeFem++.
In this example α is π4 and h is fixed.
Pictures represent profile of speed and isobars for pressure
respectively.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Numerical example 2
In this example α is also π4 and g = 0, but we vary height h. For
every h, F (h) is computed.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Numerical example 2
In this example α is also π4 and g = 0, but we vary height h. For
every h, F (h) is computed.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Numerical example 2
In this example α is also π4 and g = 0, but we vary height h. For
every h, F (h) is computed.
This computation suggests non-uniqueness of solution for some P0
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 1
This example demonstrates why we need to consider the problem
in a gravity field.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 1
This example demonstrates why we need to consider the problem
in a gravity field.
We take g = 0, α = 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 1
This example demonstrates why we need to consider the problem
in a gravity field.
We take g = 0, α = 0.
It is easy to verify that than we have symmetries:
u1 (−x, y ) = u1 (x, y ),
u2 (−x, y ) = −u2 (−x, y , z), p(−x, y ) = −p(x, y ).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 1
This example demonstrates why we need to consider the problem
in a gravity field.
We take g = 0, α = 0.
It is easy to verify that than we have symmetries:
u1 (−x, y ) = u1 (x, y ),
u2 (−x, y ) = −u2 (−x, y , z), p(−x, y ) = −p(x, y ).
Further computation gives F (h) = 0, h ∈ R.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 1
This example demonstrates why we need to consider the problem
in a gravity field.
We take g = 0, α = 0.
It is easy to verify that than we have symmetries:
u1 (−x, y ) = u1 (x, y ),
u2 (−x, y ) = −u2 (−x, y , z), p(−x, y ) = −p(x, y ).
Further computation gives F (h) = 0, h ∈ R.
It this case solution exists iff P0 = 0 and if it exists there are
infinitely many solutions.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 2
We are considering same situation as in the analytical example 1,
only this time we take g 6= 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 2
We are considering same situation as in the analytical example 1,
only this time we take g 6= 0.
We define pH = −ρgy .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 2
We are considering same situation as in the analytical example 1,
only this time we take g 6= 0.
We define pH = −ρgy .
Then every solution has a form (u, q + pH ), where (u, q) is
the solution from example 1.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 2
We are considering same situation as in the analytical example 1,
only this time we take g 6= 0.
We define pH = −ρgy .
Then every solution has a form (u, q + pH ), where (u, q) is
the solution from example 1.
F (h) = −ρgh|Σ0 |.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Analytical example 2
We are considering same situation as in the analytical example 1,
only this time we take g 6= 0.
We define pH = −ρgy .
Then every solution has a form (u, q + pH ), where (u, q) is
the solution from example 1.
F (h) = −ρgh|Σ0 |.
We have unique solution given by formula
h=−
Boris Muha Zvonimir Tutek
P0
.
g ρ|Σ0 |
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Geometry and description of the problem
Statement of the problem
Few examples
Remark on symmetry of force
If we use symmetries of the solution from the previous example and
notify that this symmetry maps Ωαh 7→ Ω−α
h , we can easily get:
(F α (h) + ρgh) = −(F −α (h) + ρgh).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Statement of the result
Lemma
Let (u, p) be smooth solution of Stokes system in Ωαh with
boundary condition as in problem (1). Then for every h ≥ 0 we
have
Z
F (h) =
ph .
Σh
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Statement of the result
Lemma
Let (u, p) be smooth solution of Stokes system in Ωαh with
boundary condition as in problem (1). Then for every h ≥ 0 we
have
Z
F (h) =
ph .
Σh
Theorem
There exists P ≥ 0 such that for every P0 ≤ P problem (1) has at
least one solution.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
Notice that F (0) = 0 (Poiseuille solution)
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
Notice that F (0) = 0 (Poiseuille solution)
For fixed h we know that the solution of the Stokes problem
has a form (uh , ph + pH ).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
Notice that F (0) = 0 (Poiseuille solution)
For fixed h we know that the solution of the Stokes problem
has a form (uh , ph + pH ).
R
limh→∞ Σh pH = −∞.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
Notice that F (0) = 0 (Poiseuille solution)
For fixed h we know that the solution of the Stokes problem
has a form (uh , ph + pH ).
R
limh→∞ Σh pH = −∞.
Let (v, τ ) be the solution of corresponding Leray’s problem in
Ω∞ with prescribed flux 0 at the infinity.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
Notice that F (0) = 0 (Poiseuille solution)
For fixed h we know that the solution of the Stokes problem
has a form (uh , ph + pH ).
R
limh→∞ Σh pH = −∞.
Let (v, τ ) be the solution of corresponding Leray’s problem in
Ω∞ with prescribed flux 0 at the infinity.
It is well known that τ decays exponentially fast to some
constant C as y → ∞.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
One can prove estimate:
kph − τ kH 1 (Ωh ) → 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
One can prove estimate:
kph − τ kH 1 (Ωh ) → 0.
limh→∞ F (h) = −∞
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Sketch of the proof
One can prove estimate:
kph − τ kH 1 (Ωh ) → 0.
limh→∞ F (h) = −∞
Assertion of the theorem follows from the continuity of the
function F .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Comment on non-uniqueness
As we have seen in numerical example 2, one can not expect the
uniqueness of the solution.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Comment on non-uniqueness
As we have seen in numerical example 2, one can not expect the
uniqueness of the solution.
α 6= 0, F α = Fpα − ρgh|Σh |
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Comment on non-uniqueness
As we have seen in numerical example 2, one can not expect the
uniqueness of the solution.
α 6= 0, F α = Fpα − ρgh|Σh |
Let ε be such that
d α
dh Fp (ε)
Boris Muha Zvonimir Tutek
6= 0.
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Comment on non-uniqueness
As we have seen in numerical example 2, one can not expect the
uniqueness of the solution.
α 6= 0, F α = Fpα − ρgh|Σh |
d α
dh Fp (ε) 6= 0.
d α
that dh
Fp (ε) > 0
Let ε be such that
We can assume
Boris Muha Zvonimir Tutek
(otherwise we take Fp−α ).
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Comment on non-uniqueness
As we have seen in numerical example 2, one can not expect the
uniqueness of the solution.
α 6= 0, F α = Fpα − ρgh|Σh |
d α
dh Fp (ε) 6= 0.
d α
that dh
Fp (ε) > 0
Let ε be such that
We can assume
(otherwise we take Fp−α ).
If (uαε , pεα − ρgy ) is a solution of Stokes problem, then
(λuαε , λpεα − ρgy ) is also a solution, λ ∈ R (with different BC).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Comment on non-uniqueness
As we have seen in numerical example 2, one can not expect the
uniqueness of the solution.
α 6= 0, F α = Fpα − ρgh|Σh |
d α
dh Fp (ε) 6= 0.
d α
that dh
Fp (ε) > 0
Let ε be such that
We can assume
(otherwise we take Fp−α ).
If (uαε , pεα − ρgy ) is a solution of Stokes problem, then
(λuαε , λpεα − ρgy ) is also a solution, λ ∈ R (with different BC).
We can take λ sufficiently large, so that function F α is not
monotone.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
The problem is nonlinear in h. To derive a linearized problem we
need to reformulate problem in such a way that variable h is
expressed explicitly. We will do that by transforming the problem
π
into a fixed domain Ω12 =: Ω.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
The problem is nonlinear in h. To derive a linearized problem we
need to reformulate problem in such a way that variable h is
expressed explicitly. We will do that by transforming the problem
π
into a fixed domain Ω12 =: Ω.
Let βhα : Ω → Ωαh be smooth change of variable.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
The problem is nonlinear in h. To derive a linearized problem we
need to reformulate problem in such a way that variable h is
expressed explicitly. We will do that by transforming the problem
π
into a fixed domain Ω12 =: Ω.
Let βhα : Ω → Ωαh be smooth change of variable.
We also suppose some symmetry:
(βhα )x (x, y ) = −(βhα )x (−x, y ),
(βhα )y (x, y ) = (βhα )y (−x, y ).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
Now we can write problem (1) in a form:


−ρg e2
,
0
S(u, p, h; α, P0 ) = 
0
where α and P0 are parameters.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
Now we can write problem (1) in a form:


−ρg e2
,
0
S(u, p, h; α, P0 ) = 
0
where α and P0 are parameters.
Since the problem is nonlinear only in h, alternatively we can write:
S(u, p, h; α, P0 ) = S(h; α, P0 )(u, p),
where S(h; α, P0 ) is a linear operator.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
Now we can compute Frechet derivative of S, i.e. linearized
problem around stationary state (u0 , p0 , h0 ):
S0 (u0 , p0 , h0 )(u, p, h) = S(h0 )(u, p) + hS 0 (h0 )(u0 , p0 ).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
Now we can compute Frechet derivative of S, i.e. linearized
problem around stationary state (u0 , p0 , h0 ):
S0 (u0 , p0 , h0 )(u, p, h) = S(h0 )(u, p) + hS 0 (h0 )(u0 , p0 ).
S(h0 ) = (S1 (h0 ), S2 (h0 )) is Stokes operator in new variables.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
Now we can compute Frechet derivative of S, i.e. linearized
problem around stationary state (u0 , p0 , h0 ):
S0 (u0 , p0 , h0 )(u, p, h) = S(h0 )(u, p) + hS 0 (h0 )(u0 , p0 ).
S(h0 ) = (S1 (h0 ), S2 (h0 )) is Stokes operator in new variables.
S(h0 ) is regular.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Linearization
Now we can compute Frechet derivative of S, i.e. linearized
problem around stationary state (u0 , p0 , h0 ):
S0 (u0 , p0 , h0 )(u, p, h) = S(h0 )(u, p) + hS 0 (h0 )(u0 , p0 ).
S(h0 ) = (S1 (h0 ), S2 (h0 )) is Stokes operator in new variables.
S(h0 ) is regular.
S30 (h0 ) = 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
Theorem
S’ is either regular operator or Fredholm operator with index 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
Theorem
S’ is either regular operator or Fredholm operator with index 0.
Let (UJ , PJ ) be such that
0
S (u0 , p0 , h0 )(UJ , PJ , 1) = 0,
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
Theorem
S’ is either regular operator or Fredholm operator with index 0.
Let (UJ , PJ ) be such that
0
S (u0 , p0 , h0 )(UJ , PJ , 1) = 0,
Let (Uf , Pf ) be such that
0
S (u0 , p0 , h0 )(Uf , Pf , 0) = f .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
Theorem
S’ is either regular operator or Fredholm operator with index 0.
Let (UJ , PJ ) be such that
0
S (u0 , p0 , h0 )(UJ , PJ , 1) = 0,
Let (Uf , Pf ) be such that
0
S (u0 , p0 , h0 )(Uf , Pf , 0) = f .
We define uf ,h = hUJ + Uf and pf ,h = hPJ + Pf .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
We have
0
S (u0 , p0 , h0 )(uf ,h , pf ,h , h) = f .
Z
Z
0
S3 (u0 , p0 , h0 )(uf ,h , pf ,h , h) = h
PJ +
Pf .
Σ
Boris Muha Zvonimir Tutek
Σ
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
We have
0
S (u0 , p0 , h0 )(uf ,h , pf ,h , h) = f .
Z
Z
0
S3 (u0 , p0 , h0 )(uf ,h , pf ,h , h) = h
PJ +
Pf .
Σ
If
R
Σ PJ
6= 0 then
S0 (u0 , p0 , h0 )
Boris Muha Zvonimir Tutek
Σ
is a regular operator.
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
We have
0
S (u0 , p0 , h0 )(uf ,h , pf ,h , h) = f .
Z
Z
0
S3 (u0 , p0 , h0 )(uf ,h , pf ,h , h) = h
PJ +
Pf .
Σ
If
R
RΣ
PJ 6= 0 then
Σ
S0 (u0 , p0 , h0 ) is a regular operator.
Ker (S0 ) = [{(UJ , PJ , 1)}] and
If Σ PJ = 0 then
coKer (S0 ) = {0} × {0} × R.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Regularity of S’
We have
0
S (u0 , p0 , h0 )(uf ,h , pf ,h , h) = f .
Z
Z
0
S3 (u0 , p0 , h0 )(uf ,h , pf ,h , h) = h
PJ +
Pf .
Σ
If
R
RΣ
PJ 6= 0 then
Σ
S0 (u0 , p0 , h0 ) is a regular operator.
Ker (S0 ) = [{(UJ , PJ , 1)}] and
If Σ PJ = 0 then
coKer (S0 ) = {0} × {0} × R.
DP0 S(u0 , p0 , h0 ; α, P0 ) = (0, 0, −1) ∈
/ ImS0 (u0 , p0 , h0 ; P0 , α).
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Turning point
Lemma
There exists α and P0 with corresponding stationary state
(u0 , p0 , h0 ) such that S0 (u0 , p0 , h0 ; P0 , α) is Fredholm operator
with index 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Turning point
Lemma
There exists α and P0 with corresponding stationary state
(u0 , p0 , h0 ) such that S0 (u0 , p0 , h0 ; P0 , α) is Fredholm operator
with index 0.
Theorem
There exists α and P0 with corresponding stationary state
(u0 , p0 , h0 ) in which we have a turning point. More precisely,
(u0 , p0 , h0 ) is a solution of problem (1) and all solutions of this
problem in some neighbourhood belong to some curve
(X (s), P(S)) with X (0) = (u0 , p0 , h0 ) and P(0) = P0 .
Furthermore, tangent at (X (0), P(0) is (V , 0) and P 00 (0) 6= 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Remark on change of variables
Smooth change of variables is effectively very hard to
compute, so it is not useful in numerical computations.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Remark on change of variables
Smooth change of variables is effectively very hard to
compute, so it is not useful in numerical computations.
We can define:
βhα (x, y )
=
(x, y )
y < 0,
(x + hy tan α, hy ) y > 0.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Remark on change of variables
Smooth change of variables is effectively very hard to
compute, so it is not useful in numerical computations.
We can define:
βhα (x, y )
=
(x, y )
y < 0,
(x + hy tan α, hy ) y > 0.
βhα (x, y ) ∈ W 1,∞ , we have a problem with defining a trace
operator
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Linearized problem
Turning point
Remark on change of variables
Remark on change of variables
Smooth change of variables is effectively very hard to
compute, so it is not useful in numerical computations.
We can define:
βhα (x, y )
=
(x, y )
y < 0,
(x + hy tan α, hy ) y > 0.
βhα (x, y ) ∈ W 1,∞ , we have a problem with defining a trace
operator
We can ”fix” that with spaces of this type
V = {u ∈ H 1 (Ω); u ∈ H 2 (Ω+ ), u ∈ H 2 (Ω− )}.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Remark on Navier-Stokes case
In Navier-Stokes we can not impose boundary conditions on
pressure, but we must consider dynamical pressure p + 12 |u|2 .
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Remark on Navier-Stokes case
In Navier-Stokes we can not impose boundary conditions on
pressure, but we must consider dynamical pressure p + 12 |u|2 .
Another possibility is to impose Dirichlet boundary condition
on the velocity.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Remark on Navier-Stokes case
In Navier-Stokes we can not impose boundary conditions on
pressure, but we must consider dynamical pressure p + 12 |u|2 .
Another possibility is to impose Dirichlet boundary condition
on the velocity.
Existence result is also valid for this case with a restriction on
smallness of the data since we can solve Leray’s problem in
this case as well.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Few comments on the evolution case
In the evolution case, movement of the lid will be described
with Newton’s second law.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Few comments on the evolution case
In the evolution case, movement of the lid will be described
with Newton’s second law.
In this case we will not have smooth solution even in Stokes
case in smooth domain and with smooth data because
velocity will have discontinuity on the frontier of the lid.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Few comments on the evolution case
In the evolution case, movement of the lid will be described
with Newton’s second law.
In this case we will not have smooth solution even in Stokes
case in smooth domain and with smooth data because
velocity will have discontinuity on the frontier of the lid.
One needs to redefine the solution because trace operator is
not well defined due to lack of regularity.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Few comments on the evolution case
In the evolution case, movement of the lid will be described
with Newton’s second law.
In this case we will not have smooth solution even in Stokes
case in smooth domain and with smooth data because
velocity will have discontinuity on the frontier of the lid.
One needs to redefine the solution because trace operator is
not well defined due to lack of regularity.
First we consider the ”regularized” problem.
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid
Introduction
The existence result
Bifurcation: Turning point
Further research
Thank you for your attention!
Boris Muha Zvonimir Tutek
Modeling of fluid flow through the system of pipes with lid