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# Nonlinear transport phenomena: models, method of solving and unusual features (2)

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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.

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### Nonlinear transport phenomena: models, method of solving and unusual features (2)

1. 1. Nonlinear transport phenomena: models, method of solving and unusual features Vsevolod Vladimirov AGH University of Science and technology, Faculty of Applied Mathematics ´ Krakow, August 10, 2010 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 1 / 29
2. 2. Burgers equation Consider the second law of Newton for viscous incompressible ﬂuid: ∂ ui ∂ ui 1∂P + uj j + = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3, ∂t ∂x ρ ∂ xi u(t, x) is the velocity ﬁeld, ∂ j ∂ ∂ t + u ∂ xj is the time ( substantial) derivative; ρ is the constant density ; P is the pressure ; ν is the viscosity coeﬃcient; 2 ∆ = n ∂∂x2 is the Laplace operator. i=1 i 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
3. 3. Burgers equation Consider the second law of Newton for viscous incompressible ﬂuid: ∂ ui ∂ ui 1∂P + uj j + = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3, ∂t ∂x ρ ∂ xi u(t, x) is the velocity ﬁeld, ∂ j ∂ ∂ t + u ∂ xj is the time ( substantial) derivative; ρ is the constant density ; P is the pressure ; ν is the viscosity coeﬃcient; 2 ∆ = n ∂∂x2 is the Laplace operator. i=1 i 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
4. 4. Burgers equation Consider the second law of Newton for viscous incompressible ﬂuid: ∂ ui ∂ ui 1∂P + uj j + = ν ∆ ui , i = 1, ...n, n = 1, 2 or 3, ∂t ∂x ρ ∂ xi u(t, x) is the velocity ﬁeld, ∂ j ∂ ∂ t + u ∂ xj is the time ( substantial) derivative; ρ is the constant density ; P is the pressure ; ν is the viscosity coeﬃcient; 2 ∆ = n ∂∂x2 is the Laplace operator. i=1 i 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
5. 5. Hyperbolic generalization to Burgers equation Let us consider delayed equation ∂ u(t + τ, x) + u(t, x) ux (t, x) = ν ux x (t, x). ∂t Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up ∂t to O(τ 2 ) the equation called the hyperbolic generalization of the Burgers equation (GBE to abbreviate): τ utt + ut + u ux = ν ux x . (2) GBE appears when modeling transport phenomena in media possessing internal structure: granular media,polymers, cellular structures in biology. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
6. 6. Hyperbolic generalization to Burgers equation Let us consider delayed equation ∂ u(t + τ, x) + u(t, x) ux (t, x) = ν ux x (t, x). ∂t Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up ∂t to O(τ 2 ) the equation called the hyperbolic generalization of the Burgers equation (GBE to abbreviate): τ utt + ut + u ux = ν ux x . (2) GBE appears when modeling transport phenomena in media possessing internal structure: granular media,polymers, cellular structures in biology. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
7. 7. Hyperbolic generalization to Burgers equation Let us consider delayed equation ∂ u(t + τ, x) + u(t, x) ux (t, x) = ν ux x (t, x). ∂t Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up ∂t to O(τ 2 ) the equation called the hyperbolic generalization of the Burgers equation (GBE to abbreviate): τ utt + ut + u ux = ν ux x . (2) GBE appears when modeling transport phenomena in media possessing internal structure: granular media,polymers, cellular structures in biology. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
8. 8. Hyperbolic generalization to Burgers equation Let us consider delayed equation ∂ u(t + τ, x) + u(t, x) ux (t, x) = ν ux x (t, x). ∂t Applying to the term ∂ u(t+τ, x) the Taylor formula, we get, up ∂t to O(τ 2 ) the equation called the hyperbolic generalization of the Burgers equation (GBE to abbreviate): τ utt + ut + u ux = ν ux x . (2) GBE appears when modeling transport phenomena in media possessing internal structure: granular media,polymers, cellular structures in biology. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 3 / 29
9. 9. 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
10. 10. 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
11. 11. Solution to BE Lemma 1. BE is connected with the equation 1 2 ψt + ψx = ν ψ x x (5) 2 by means of the transformation u2 ψx = u, ψt = ν ux − . (6) 2 Lemma 2. The equation (5) is connected with the heat transport equation Φt = ν Φx x by means of the transformation ψ = −2 ν log Φ. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
12. 12. Solution to BE Lemma 1. BE is connected with the equation 1 2 ψt + ψx = ν ψ x x (5) 2 by means of the transformation u2 ψx = u, ψt = ν ux − . (6) 2 Lemma 2. The equation (5) is connected with the heat transport equation Φt = ν Φx x by means of the transformation ψ = −2 ν log Φ. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
13. 13. Solution to BE Lemma 1. BE is connected with the equation 1 2 ψt + ψx = ν ψ x x (5) 2 by means of the transformation u2 ψx = u, ψt = ν ux − . (6) 2 Lemma 2. The equation (5) is connected with the heat transport equation Φt = ν Φx x by means of the transformation ψ = −2 ν log Φ. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 5 / 29
14. 14. Corollary. Solution to the initial value problem ut + u ux = ν u x x , (7) u(0, x) = F (x) is connected with the solution to the initial value problem Φt = ν Φ x x , (8) x 1 Φ(0, x) = exp − F (z) d z := θ(x) 2ν 0 via the transformation u(t, x) = −2 ν {log[Φ(t, x)]}x . KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
15. 15. Corollary. Solution to the initial value problem ut + u ux = ν u x x , (7) u(0, x) = F (x) is connected with the solution to the initial value problem Φt = ν Φ x x , (8) x 1 Φ(0, x) = exp − F (z) d z := θ(x) 2ν 0 via the transformation u(t, x) = −2 ν {log[Φ(t, x)]}x . KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
16. 16. Corollary. Solution to the initial value problem ut + u ux = ν u x x , (7) u(0, x) = F (x) is connected with the solution to the initial value problem Φt = ν Φ x x , (8) x 1 Φ(0, x) = exp − F (z) d z := θ(x) 2ν 0 via the transformation u(t, x) = −2 ν {log[Φ(t, x)]}x . KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 6 / 29
17. 17. Let us remind, that solution to the initial value problem (8) can be presented by the formula ∞ (x−ξ)2 1 Φ(t, x) = √ θ(ξ) e− 4ν t d ξ. 4πν t −∞ Corollary. Solution to the initial value problem (7) is given by the formula ∞ x−ξ − f (ξ;t, x) −∞ t e dξ 2ν u(t, x) = f (ξ;t, x) , (9) ∞ − −∞ e 2ν d ξ where ξ (x − ξ)2 f (ξ; t, x) = F (z) d z + 0 2t . So, the formula (9)completely deﬁnes the solution to Cauchy problem to BE! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
18. 18. Let us remind, that solution to the initial value problem (8) can be presented by the formula ∞ (x−ξ)2 1 Φ(t, x) = √ θ(ξ) e− 4ν t d ξ. 4πν t −∞ Corollary. Solution to the initial value problem (7) is given by the formula ∞ x−ξ − f (ξ;t, x) −∞ t e dξ 2ν u(t, x) = f (ξ;t, x) , (9) ∞ − −∞ e 2ν d ξ where ξ (x − ξ)2 f (ξ; t, x) = F (z) d z + 0 2t . So, the formula (9)completely deﬁnes the solution to Cauchy problem to BE! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
19. 19. Let us remind, that solution to the initial value problem (8) can be presented by the formula ∞ (x−ξ)2 1 Φ(t, x) = √ θ(ξ) e− 4ν t d ξ. 4πν t −∞ Corollary. Solution to the initial value problem (7) is given by the formula ∞ x−ξ − f (ξ;t, x) −∞ t e dξ 2ν u(t, x) = f (ξ;t, x) , (9) ∞ − −∞ e 2ν d ξ where ξ (x − ξ)2 f (ξ; t, x) = F (z) d z + 0 2t . So, the formula (9)completely deﬁnes the solution to Cauchy problem to BE! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
20. 20. Let us remind, that solution to the initial value problem (8) can be presented by the formula ∞ (x−ξ)2 1 Φ(t, x) = √ θ(ξ) e− 4ν t d ξ. 4πν t −∞ Corollary. Solution to the initial value problem (7) is given by the formula ∞ x−ξ − f (ξ;t, x) −∞ t e dξ 2ν u(t, x) = f (ξ;t, x) , (9) ∞ − −∞ e 2ν d ξ where ξ (x − ξ)2 f (ξ; t, x) = F (z) d z + 0 2t . So, the formula (9)completely deﬁnes the solution to Cauchy problem to BE! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 7 / 29
21. 21. Example: solution of the ”point explosion” problem Let u(0, x) = F (x) = Aδ(x)H(x), 1 (x−ξ)2 1 if x ≥ 0, δ(x) = lim √ e− 4 ν t , H(x) = . t→ +0 4πν t 0 if x < 0 Figure: KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29
22. 22. Example: solution of the ”point explosion” problem Let u(0, x) = F (x) = Aδ(x)H(x), 1 (x−ξ)2 1 if x ≥ 0, δ(x) = lim √ e− 4 ν t , H(x) = . t→ +0 4πν t 0 if x < 0 Figure: KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 8 / 29
23. 23. Performing simple but tedious calculations, we ﬁnally get the following solution to the point explosion problem: x2 ν eR − 1 e− 4 ν t u(t, x) = √ , t π x (eR + 1) + erf( √4 ν t ) (1 − eR ) 2 where z 2 2 erf(z) = √ e−x d x, π 0 A R= 2ν plays the role of the Reynolds number! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
24. 24. Performing simple but tedious calculations, we ﬁnally get the following solution to the point explosion problem: x2 ν eR − 1 e− 4 ν t u(t, x) = √ , t π x (eR + 1) + erf( √4 ν t ) (1 − eR ) 2 where z 2 2 erf(z) = √ e−x d x, π 0 A R= 2ν plays the role of the Reynolds number! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
25. 25. Performing simple but tedious calculations, we ﬁnally get the following solution to the point explosion problem: x2 ν eR − 1 e− 4 ν t u(t, x) = √ , t π x (eR + 1) + erf( √4 ν t ) (1 − eR ) 2 where z 2 2 erf(z) = √ e−x d x, π 0 A R= 2ν plays the role of the Reynolds number! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
26. 26. Performing simple but tedious calculations, we ﬁnally get the following solution to the point explosion problem: x2 ν eR − 1 e− 4 ν t u(t, x) = √ , t π x (eR + 1) + erf( √4 ν t ) (1 − eR ) 2 where z 2 2 erf(z) = √ e−x d x, π 0 A R= 2ν plays the role of the Reynolds number! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 9 / 29
27. 27. Suppose now, that ν becomes very large. Then x R→ 0 eR ≈ 1 + R, erf √ ≈ 0, 4ν t and x2 ν A e− 4 ν t A x2 u(t, x) = 2ν √ + O(R2 ) ≈ √ e− 4 ν t . t π 4πν t Corollary.Solution to the ”point explosion” problem for the BE approaches solution to the ”heat explosion” problem for the linear heat transport equation, when ν becomes large. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
28. 28. Suppose now, that ν becomes very large. Then x R→ 0 eR ≈ 1 + R, erf √ ≈ 0, 4ν t and x2 ν A e− 4 ν t A x2 u(t, x) = 2ν √ + O(R2 ) ≈ √ e− 4 ν t . t π 4πν t Corollary.Solution to the ”point explosion” problem for the BE approaches solution to the ”heat explosion” problem for the linear heat transport equation, when ν becomes large. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
29. 29. Suppose now, that ν becomes very large. Then x R→ 0 eR ≈ 1 + R, erf √ ≈ 0, 4ν t and x2 ν A e− 4 ν t A x2 u(t, x) = 2ν √ + O(R2 ) ≈ √ e− 4 ν t . t π 4πν t Corollary.Solution to the ”point explosion” problem for the BE approaches solution to the ”heat explosion” problem for the linear heat transport equation, when ν becomes large. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
30. 30. Suppose now, that ν becomes very large. Then x R→ 0 eR ≈ 1 + R, erf √ ≈ 0, 4ν t and x2 ν A e− 4 ν t A x2 u(t, x) = 2ν √ + O(R2 ) ≈ √ e− 4 ν t . t π 4πν t Corollary.Solution to the ”point explosion” problem for the BE approaches solution to the ”heat explosion” problem for the linear heat transport equation, when ν becomes large. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 10 / 29
31. 31. For large R the way of getting the approximating formula is less clear, so we restore to the results of the numerical simulation. Below it is shown the solution to ”point explosion” problem obtained for ν = 0.05 and R = 35: Figure: It reminds the shock wave proﬁle  x √ t if t > 0, 0 < x < 2√ t, A u(t, x) = , 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
32. 32. For large R the way of getting the approximating formula is less clear, so we restore to the results of the numerical simulation. Below it is shown the solution to ”point explosion” problem obtained for ν = 0.05 and R = 35: Figure: It reminds the shock wave proﬁle  x √ t if t > 0, 0 < x < 2√ t, A u(t, x) = , 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
33. 33. For large R the way of getting the approximating formula is less clear, so we restore to the results of the numerical simulation. Below it is shown the solution to ”point explosion” problem obtained for ν = 0.05 and R = 35: Figure: It reminds the shock wave proﬁle  x √ t if t > 0, 0 < x < 2√ t, A u(t, x) = , 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
34. 34. Figure: A common solution x √ t if t > 0, 0 < x < 2√ t, A u(t, x) = 0 if t > 0, x < 0 or x > 2 A t, to the Burgers and the Euler equations KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 12 / 29
35. 35. So the solutions to the point explosion problem for BE are completely diﬀerent in the limiting cases: when R = A/(2 ν) → 0 it coincides with the solution of the heat explosion problem, while for large R it reminds the shock wave solution! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29
36. 36. So the solutions to the point explosion problem for BE are completely diﬀerent in the limiting cases: when R = A/(2 ν) → 0 it coincides with the solution of the heat explosion problem, while for large R it reminds the shock wave solution! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 13 / 29
37. 37. The hyperbolic generalization of BE Let us consider the Cauchy problem for the hyperbolic generalization of BE: τ utt + ut + u ux = 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 supported function, 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
38. 38. The hyperbolic generalization of BE Let us consider the Cauchy problem for the hyperbolic generalization of BE: τ utt + ut + u ux = 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 supported function, 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
39. 39. The hyperbolic generalization of BE Let us consider the Cauchy problem for the hyperbolic generalization of BE: τ utt + ut + u ux = 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 supported function, 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
40. 40. Results of the numerical simulation: M = 0.3 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 15 / 29
41. 41. Figure: M = 0.3 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 16 / 29
42. 42. Figure: M = 0.3 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 17 / 29
43. 43. Figure: M = 0.3 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 18 / 29
44. 44. Figure: M = 0.3 The solution of the initial perturbation reminds the evolution of the point explosion problem for BE in the case when R = A/(2 ν) is large. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 19 / 29
45. 45. Results of the numerical simulation: M = 1.45 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 20 / 29
46. 46. Figure: M = 1.45 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 21 / 29
47. 47. Figure: M = 1.45 For M = 1 + ε a formation of the blow-up regime is observed at the beginning of evolution, KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 22 / 29
48. 48. Figure: M = 1.45 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 23 / 29
49. 49. Figure: M = 1.45 but for larger t it is suppressed by viscosity and returns to the shape of the BE solution! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 24 / 29
50. 50. Results of the numerical simulation: M = 1.8 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 25 / 29
51. 51. Figure: M = 1.8 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 26 / 29
52. 52. Figure: M = 1.8 KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 27 / 29
53. 53. Figure: M = 1.8 For M = 1.8 (and larger ones) a blow-up regime is formed at the wave front in ﬁnite time! KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 28 / 29
54. 54. Appendix 1. Calculation of point explosion problem for BE Since, ξ ∞ −A, if ξ < 0, F (x) d x = −A lim δ(x) φB (x) H(x) d x = 0+ B→+0 −∞ 0, if ξ > 0, ∞ where φB (x) is any C0 function such that φ(x)|<B, ξ> ≡ 1, and supp φ ⊂ < B/2, ξ + B/2 > then (x−ξ)2 2t − A if ξ < 0, f (ξ; t, x) = (x−ξ)2 2 t , if ξ > 0. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29
55. 55. Appendix 1. Calculation of point explosion problem for BE Since, ξ ∞ −A, if ξ < 0, F (x) d x = −A lim δ(x) φB (x) H(x) d x = 0+ B→+0 −∞ 0, if ξ > 0, ∞ where φB (x) is any C0 function such that φ(x)|<B, ξ> ≡ 1, and supp φ ⊂ < B/2, ξ + B/2 > then (x−ξ)2 2t − A if ξ < 0, f (ξ; t, x) = (x−ξ)2 2 t , if ξ > 0. KPI, 2010 Nonlinear transport phenomena, Burgers Eqn. 29 / 29