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AACIMP 2010 Summer School lecture by Vsevolod Vladimirov. "Applied Mathematics" stream. "Selected Models of Transport Processes. Methods of Solving and Properties of Solutions" course. Part 2.More info at http://summerschool.ssa.org.ua
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Nonlinear transport phenomena:models, method of solving and unusual
features
Vsevolod Vladimirov
AGH University of Science and technology, Faculty of AppliedMathematics
Krakow, August 10, 2010
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 1 / 29
Burgers equation
Consider the second law of Newton for viscous incompressiblefluid:
∂ ui
∂ t+ uj ∂ u
i
∂ xj+
1
ρ
∂ P
∂ xi= ν∆ui, i = 1, ...n, n = 1, 2 or 3,
~u(t, x) is the velocity field,∂∂ t + uj ∂
∂ xj is the time ( substantial) derivative;ρ is the constant density ;P is the pressure ;ν is the viscosity coefficient;∆ =
∑ni=1
∂2
∂ x2i
is the Laplace operator.
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x. (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29
Burgers equation
Consider the second law of Newton for viscous incompressiblefluid:
∂ ui
∂ t+ uj ∂ u
i
∂ xj+
1
ρ
∂ P
∂ xi= ν∆ui, i = 1, ...n, n = 1, 2 or 3,
~u(t, x) is the velocity field,∂∂ t + uj ∂
∂ xj is the time ( substantial) derivative;ρ is the constant density ;P is the pressure ;ν is the viscosity coefficient;∆ =
∑ni=1
∂2
∂ x2i
is the Laplace operator.
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x. (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29
Burgers equation
Consider the second law of Newton for viscous incompressiblefluid:
∂ ui
∂ t+ uj ∂ u
i
∂ xj+
1
ρ
∂ P
∂ xi= ν∆ui, i = 1, ...n, n = 1, 2 or 3,
~u(t, x) is the velocity field,∂∂ t + uj ∂
∂ xj is the time ( substantial) derivative;ρ is the constant density ;P is the pressure ;ν is the viscosity coefficient;∆ =
∑ni=1
∂2
∂ x2i
is the Laplace operator.
For P = const, n = 1, we get the Burgers equation
ut + u ux = ν ux x. (1)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 2 / 29
Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t+ τ, x)
∂ t+ u(t, x)ux(t, x) = ν ux x(t, x).
Applying to the term ∂ u(t+τ, x)∂ t the Taylor formula, we get, up
to O(τ2) the equation called the hyperbolic generalization ofthe Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x. (2)
GBE appears when modeling transport phenomena in mediapossessing internal structure: granular media,polymers, cellularstructures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t+ τ, x)
∂ t+ u(t, x)ux(t, x) = ν ux x(t, x).
Applying to the term ∂ u(t+τ, x)∂ t the Taylor formula, we get, up
to O(τ2) the equation called the hyperbolic generalization ofthe Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x. (2)
GBE appears when modeling transport phenomena in mediapossessing internal structure: granular media,polymers, cellularstructures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t+ τ, x)
∂ t+ u(t, x)ux(t, x) = ν ux x(t, x).
Applying to the term ∂ u(t+τ, x)∂ t the Taylor formula, we get, up
to O(τ2) the equation called the hyperbolic generalization ofthe Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x. (2)
GBE appears when modeling transport phenomena in mediapossessing internal structure: granular media,polymers, cellularstructures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
Hyperbolic generalization to Burgers equation
Let us consider delayed equation
∂ u(t+ τ, x)
∂ t+ u(t, x)ux(t, x) = ν ux x(t, x).
Applying to the term ∂ u(t+τ, x)∂ t the Taylor formula, we get, up
to O(τ2) the equation called the hyperbolic generalization ofthe Burgers equation (GBE to abbreviate):
τ utt + ut + u ux = ν ux x. (2)
GBE appears when modeling transport phenomena in mediapossessing internal structure: granular media,polymers, cellularstructures in biology.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
Various generalizations of Burgers equation
Convection-reaction diffusion equation
ut + u ux = ν [un ux]x + f(u), (3)
and its hyperbolic generalization
τ ut t + ut + u ux = ν [un ux]x + f(u) (4)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 4 / 29
Various generalizations of Burgers equation
Convection-reaction diffusion equation
ut + u ux = ν [un ux]x + f(u), (3)
and its hyperbolic generalization
τ ut t + ut + u ux = ν [un ux]x + f(u) (4)
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 4 / 29
Solution to BELemma 1. BE is connected with the equation
ψt +1
2ψ2
x = ν ψx x (5)
by means of the transformation
ψx = u, ψt = ν ux −u2
2. (6)
Lemma 2. The equation (5) is connected with the heattransport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
Solution to BELemma 1. BE is connected with the equation
ψt +1
2ψ2
x = ν ψx x (5)
by means of the transformation
ψx = u, ψt = ν ux −u2
2. (6)
Lemma 2. The equation (5) is connected with the heattransport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
Solution to BELemma 1. BE is connected with the equation
ψt +1
2ψ2
x = ν ψx x (5)
by means of the transformation
ψx = u, ψt = ν ux −u2
2. (6)
Lemma 2. The equation (5) is connected with the heattransport equation
Φt = ν Φx x
by means of the transformation
ψ = −2 ν log Φ.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
Corollary. Solution to the initial value problem
ut + uux = ν ux x, (7)u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φx x, (8)
Φ(0, x) = exp
[− 1
2 ν
∫ x
0F (z) d z
]:= θ(x)
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
Corollary. Solution to the initial value problem
ut + uux = ν ux x, (7)u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φx x, (8)
Φ(0, x) = exp
[− 1
2 ν
∫ x
0F (z) d z
]:= θ(x)
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
Corollary. Solution to the initial value problem
ut + uux = ν ux x, (7)u(0, x) = F (x)
is connected with the solution to the initial value problem
Φt = ν Φx x, (8)
Φ(0, x) = exp
[− 1
2 ν
∫ x
0F (z) d z
]:= θ(x)
via the transformation
u(t, x) = −2 ν {log[Φ(t, x)]}x .
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
Let us remind, that solution to the initial value problem (8) canbe presented by the formula
Φ(t, x) =1√
4π ν t
∫ ∞
−∞θ(ξ) e−
(x−ξ)2
4 ν t d ξ.
Corollary. Solution to the initial value problem (7) is given bythe formula
u(t, x) =
∫∞−∞
x−ξt e−
f(ξ;t, x)2 ν d ξ∫∞
−∞ e− f(ξ;t, x)
2 ν d ξ, (9)
where
f(ξ; t, x) =
∫ ξ
0F (z) d z +
(x− ξ)2
2 t.
So, the formula (9)completely defines the solution to Cauchyproblem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
Let us remind, that solution to the initial value problem (8) canbe presented by the formula
Φ(t, x) =1√
4π ν t
∫ ∞
−∞θ(ξ) e−
(x−ξ)2
4 ν t d ξ.
Corollary. Solution to the initial value problem (7) is given bythe formula
u(t, x) =
∫∞−∞
x−ξt e−
f(ξ;t, x)2 ν d ξ∫∞
−∞ e− f(ξ;t, x)
2 ν d ξ, (9)
where
f(ξ; t, x) =
∫ ξ
0F (z) d z +
(x− ξ)2
2 t.
So, the formula (9)completely defines the solution to Cauchyproblem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
Let us remind, that solution to the initial value problem (8) canbe presented by the formula
Φ(t, x) =1√
4π ν t
∫ ∞
−∞θ(ξ) e−
(x−ξ)2
4 ν t d ξ.
Corollary. Solution to the initial value problem (7) is given bythe formula
u(t, x) =
∫∞−∞
x−ξt e−
f(ξ;t, x)2 ν d ξ∫∞
−∞ e− f(ξ;t, x)
2 ν d ξ, (9)
where
f(ξ; t, x) =
∫ ξ
0F (z) d z +
(x− ξ)2
2 t.
So, the formula (9)completely defines the solution to Cauchyproblem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
Let us remind, that solution to the initial value problem (8) canbe presented by the formula
Φ(t, x) =1√
4π ν t
∫ ∞
−∞θ(ξ) e−
(x−ξ)2
4 ν t d ξ.
Corollary. Solution to the initial value problem (7) is given bythe formula
u(t, x) =
∫∞−∞
x−ξt e−
f(ξ;t, x)2 ν d ξ∫∞
−∞ e− f(ξ;t, x)
2 ν d ξ, (9)
where
f(ξ; t, x) =
∫ ξ
0F (z) d z +
(x− ξ)2
2 t.
So, the formula (9)completely defines the solution to Cauchyproblem to BE!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
Example: solution of the ”point explosion” problem
Letu(0, x) = F (x) = Aδ(x)H(x),
δ(x) = limt→+0
1√4π ν t
e−(x−ξ)2
4 ν t , H(x) =
{1 if x ≥ 0,
0 if x < 0.
Figure:KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29
Example: solution of the ”point explosion” problem
Letu(0, x) = F (x) = Aδ(x)H(x),
δ(x) = limt→+0
1√4π ν t
e−(x−ξ)2
4 ν t , H(x) =
{1 if x ≥ 0,
0 if x < 0.
Figure:KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29
Performing simple but tedious calculations, we finally get thefollowing solution to the point explosion problem:
u(t, x) =
√ν
t
(eR − 1
)e−
x2
4 ν t
√π
2
[(eR + 1) + erf( x√
4 ν t) (1− eR)
] ,where
erf(z) =2√π
∫ z
0e−x2
d x,
R = A2 ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
Performing simple but tedious calculations, we finally get thefollowing solution to the point explosion problem:
u(t, x) =
√ν
t
(eR − 1
)e−
x2
4 ν t
√π
2
[(eR + 1) + erf( x√
4 ν t) (1− eR)
] ,where
erf(z) =2√π
∫ z
0e−x2
d x,
R = A2 ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
Performing simple but tedious calculations, we finally get thefollowing solution to the point explosion problem:
u(t, x) =
√ν
t
(eR − 1
)e−
x2
4 ν t
√π
2
[(eR + 1) + erf( x√
4 ν t) (1− eR)
] ,where
erf(z) =2√π
∫ z
0e−x2
d x,
R = A2 ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
Performing simple but tedious calculations, we finally get thefollowing solution to the point explosion problem:
u(t, x) =
√ν
t
(eR − 1
)e−
x2
4 ν t
√π
2
[(eR + 1) + erf( x√
4 ν t) (1− eR)
] ,where
erf(z) =2√π
∫ z
0e−x2
d x,
R = A2 ν plays the role of the Reynolds number!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
Suppose now, that ν becomes very large. Then
R→ 0 eR ≈ 1 +R, erf(
x√4 ν t
)≈ 0,
and
u(t, x) =
√ν
t
A2 ν
e−x2
4 ν t
√π
+ O(R2) ≈ A√4 π ν t
e−x2
4 ν t .
Corollary.Solution to the ”point explosion” problem for the BEapproaches solution to the ”heat explosion” problem for thelinear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
Suppose now, that ν becomes very large. Then
R→ 0 eR ≈ 1 +R, erf(
x√4 ν t
)≈ 0,
and
u(t, x) =
√ν
t
A2 ν
e−x2
4 ν t
√π
+ O(R2) ≈ A√4 π ν t
e−x2
4 ν t .
Corollary.Solution to the ”point explosion” problem for the BEapproaches solution to the ”heat explosion” problem for thelinear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
Suppose now, that ν becomes very large. Then
R→ 0 eR ≈ 1 +R, erf(
x√4 ν t
)≈ 0,
and
u(t, x) =
√ν
t
A2 ν
e−x2
4 ν t
√π
+ O(R2) ≈ A√4 π ν t
e−x2
4 ν t .
Corollary.Solution to the ”point explosion” problem for the BEapproaches solution to the ”heat explosion” problem for thelinear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
Suppose now, that ν becomes very large. Then
R→ 0 eR ≈ 1 +R, erf(
x√4 ν t
)≈ 0,
and
u(t, x) =
√ν
t
A2 ν
e−x2
4 ν t
√π
+ O(R2) ≈ A√4 π ν t
e−x2
4 ν t .
Corollary.Solution to the ”point explosion” problem for the BEapproaches solution to the ”heat explosion” problem for thelinear heat transport equation, when ν becomes large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
For large R the way of getting the approximating formula is less clear, sowe restore to the results of the numerical simulation. Below it is shown thesolution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
u(t, x) =
xt
if t > 0, 0 < x <√
2 A t,
0 if t > 0, x < 0 or x >√
2 A t,
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29
For large R the way of getting the approximating formula is less clear, sowe restore to the results of the numerical simulation. Below it is shown thesolution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
u(t, x) =
xt
if t > 0, 0 < x <√
2 A t,
0 if t > 0, x < 0 or x >√
2 A t,
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29
For large R the way of getting the approximating formula is less clear, sowe restore to the results of the numerical simulation. Below it is shown thesolution to ”point explosion” problem obtained for ν = 0.05 and R = 35:
Figure:
It reminds the shock wave profile
u(t, x) =
xt
if t > 0, 0 < x <√
2 A t,
0 if t > 0, x < 0 or x >√
2 A t,
which the BE ”shares” with the hyperbolic-type equation
ut + u ux = 0,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 11 / 29
Figure:
A common solution
u(t, x) =
{xt if t > 0, 0 < x <
√2A t,
0 if t > 0, x < 0 or x >√
2A t,
to the Burgers and the Euler equations
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 12 / 29
So the solutions to the point explosion problem for BE arecompletely different in the limiting cases: whenR = A/(2 ν) → 0 it coincides with the solution of the heatexplosion problem,while for large R it reminds the shock wave solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29
So the solutions to the point explosion problem for BE arecompletely different in the limiting cases: whenR = A/(2 ν) → 0 it coincides with the solution of the heatexplosion problem,while for large R it reminds the shock wave solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29
The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolicgeneralization of BE:
τ utt + ut + uux = ν ux x, (10)u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x,
we can conclude, that the parameter C =√ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supportedfunction, and D = max ϕ(x), then the number M = D/C (the”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29
The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolicgeneralization of BE:
τ utt + ut + uux = ν ux x, (10)u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x,
we can conclude, that the parameter C =√ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supportedfunction, and D = max ϕ(x), then the number M = D/C (the”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29
The hyperbolic generalization of BE
Let us consider the Cauchy problem for the hyperbolicgeneralization of BE:
τ utt + ut + uux = ν ux x, (10)u(0, x) = ϕ(x).
Considering the linearization of (10)
τ utt + ut + u0 ux = ν ux x,
we can conclude, that the parameter C =√ν/τ is equal to the
velocity of small (acoustic) perturbations.
If the initial perturbation ϕ(x) is a smooth compactly supportedfunction, and D = max ϕ(x), then the number M = D/C (the”Mach number”) characterizes the evolution of nonlinear wave.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 14 / 29
Results of the numerical simulation: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 15 / 29
Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 16 / 29
Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 17 / 29
Figure: M = 0.3
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 18 / 29
Figure: M = 0.3
The solution of the initial perturbation reminds the evolution ofthe point explosion problem for BE in the case whenR = A/(2 ν) is large.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 19 / 29
Results of the numerical simulation: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 20 / 29
Figure: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 21 / 29
Figure: M = 1.45
For M = 1 + ε a formation of the blow-up regime is observed atthe beginning of evolution,
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 22 / 29
Figure: M = 1.45
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 23 / 29
Figure: M = 1.45
but for larger t it is suppressed by viscosity and returns to theshape of the BE solution!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 24 / 29
Results of the numerical simulation: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 25 / 29
Figure: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 26 / 29
Figure: M = 1.8
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 27 / 29
Figure: M = 1.8
For M = 1.8 (and larger ones) a blow-up regime is formed atthe wave front in finite time!
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 28 / 29
Appendix 1. Calculation of point explosion problemfor BE
Since,∫ ξ
0+F (x) d x = −A lim
B→+0
∫ ∞
−∞δ(x)φB(x)H(x) d x =
{−A, if ξ < 0,0, if ξ > 0,
where φB(x) is any C∞0 function such that φ(x)|<B, ξ> ≡ 1, andsuppφ ⊂< B/2, ξ +B/2 > then
f(ξ; t, x) =
{(x−ξ)2
2 t −A if ξ < 0,(x−ξ)2
2 t , if ξ > 0.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29
Appendix 1. Calculation of point explosion problemfor BE
Since,∫ ξ
0+F (x) d x = −A lim
B→+0
∫ ∞
−∞δ(x)φB(x)H(x) d x =
{−A, if ξ < 0,0, if ξ > 0,
where φB(x) is any C∞0 function such that φ(x)|<B, ξ> ≡ 1, andsuppφ ⊂< B/2, ξ +B/2 > then
f(ξ; t, x) =
{(x−ξ)2
2 t −A if ξ < 0,(x−ξ)2
2 t , if ξ > 0.
KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29