NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF - PowerPoint PPT Presentation

NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF FLUIDS Pedro Lima CENTRO DE MATEM´ ¸ ˜ ATICA E APLICAC OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ECNICA DE LISBOA PORTUGAL October 11, 2011 Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 1 / 28

Joint work with: Luisa Morgado Department of Mathematics, Universidade de Tras-os-Montes e Alto Douro, Portugal G. Hastermann and E. Weinmuller Institute for Analysis and Scientific Computing, Vienna University of Technology, Austria Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 2 / 28

Outline of the talk 1 Introduction 2 Existence of Solution 3 The singularities of the problem and the associated one-parameter families of solutions 4 Shooting method based on asymptotic expansions 5 Collocation method 6 Numerical results 7 Conclusions and future work Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 3 / 28

INTRODUCTION Physical interpretation: The behavior of mixtures of fluids (for example: liquid-gas) is described by the Cahn-Hillard theory. Free volume energy: E ( ρ , |∇ ρ | 2 ) = E 0 ( ρ ) + γ 2 |∇ ρ | 2 , γ > 0, where ρ - density of the fluid. E 0 ( ρ ) - classical volume free energy γ - surface tension coefficient (independent from |∇ ρ | ). Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 4 / 28

Generalized Model If we allow that the surface tension depends on ∇ ρ , the free volume energy takes the form E ( ρ , |∇ ρ | ) = E 0 ( ρ ) + c p |∇ ρ | p , γ > 0, p > 1; in this case we obtain the following PDE: c div ( |∇ ρ | p − 2 ∇ ρ ) = µ ( ρ ) − µ 0 ; The operator in the left-hand side is the p-laplacian, where p > 1 (if p = 2 we obtain the classical laplacian). In the case of spherical bubbles, we obtain the radial ODE: � ′ r 1 − N � r N − 1 | ρ ′ ( r ) | p − 2 ρ ′ ( r ) = f p ( ρ ) , ( 0 < r < ∞ ) , where f p is a function with three real roots, whose specific form depends on p . Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 5 / 28

Right-Hand Side In the classical laplacian case ( p = 2), f 2 is a third degree polynomial f 2 ( ρ ) = 4 λ 2 ( ρ − ξ )( ρ + 1 ) ρ , where ξ is a real parameter; In the degenerate laplacian case ( p � = 2), f p has the form f p ( ρ ) = 2 p λ 2 ( ρ − ξ )( ρ + 1 ) ρ | ρ − ξ | α | ρ + 1 | α , where α = 0 in the case p ≤ 2; for p > 2 the value of α will be discussed later. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 6 / 28

Boundary Conditions � � � � r → 0 + r ρ ′ ( r ) = 0, � lim r → 0 + ρ ( r ) � < ∞ , lim � � r → ∞ ρ ′ ( r ) = 0. r → ∞ ρ ( r ) = ξ , lim lim In the bubble case (if ξ > 0) , we search for a strictly increasing solution. In the droplet case (if ξ < − 1), we search for a strictly decreasing solution. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 7 / 28

References F.dell’Isola, H.Gouin and P.Seppecher, ”Radius and Surface Tension of Microscopic Bubbles by Second Gradient Theory”, C.R.Acad. Sci. Paris, 320 (Serie IIb), 211–216 (1995). F.dell’Isola, H.Gouin and G.Rotoli, ”Nucleation of Spherical Shell–Like Interfaces by Second Gradient Theory: Numerical Simulations”, Eur. J. Mech. B / Fluids 15 , 545–568 (1996). H.Gouin and G.Rotoli, ”An Analytical Approximation of Density Profile and Surface Tension of Microscopic Bubbles for Van der Waals Fluids”, Mechanics Research Communications 24 , 255–260 (1997). N. Kim, L. Consiglieri and J.F.Rodrigues, On non-newtonian incompressible fluids with phase transitions, Mathematical Methods in Applied Sciences, 29 1523–1541 . Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 8 / 28

Existence of Solution Existence results for problems of this type can be found in: F. Gazzola, J. Serrin and M. Tang, Existence of ground states and free boundary problems for quasilinear elliptic operators, Adv. Diff. Equ., 5 (2000) 1-30. From this work, it follows that, when p ≤ 2, for 0 < ξ < 1, the considered problem (choosing α = 0) has a bubble-type solution. For p > 2,existence of solution is guaranteed only if we choose α = p − 2 in the right hand side function. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 9 / 28

The Singularity at r = 0 Initial conditions: r → 0 + r ρ ′ ( r ) = 0. r → 0 + ρ ( r ) = ρ 0 lim lim (1) We assume that in the neighborhood of r = 0 the solution can be represented as r → 0 + , ρ ( r ) = ρ 0 + Cr k ( 1 + o ( 1 )) , as (2) Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 10 / 28

Asymptotic approximation close to the origin Proposition 3.1. Let N > 1 and p > 1. For each ρ 0 , the considered singular Cauhy problem has, in the neighborhood of r = 0, a unique holomorphic solution that can be represented by 1 ρ ( x , ρ 0 ) = ρ 0 + p − 1 � f p ( ρ 0 ) � p − 1 p � p � p �� p − 1 + o 1 + y 1 r r x , (3) p − 1 p − 1 p N where y 1 can be determined analytically. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 11 / 28

Singularity at Infinity As r → ∞ we introduce the variable substitution 1 − N ( p − 1 ) 2 z ( r ) . ρ ( r ) = ξ + r (4) In the new variable z we obtain an asymptotically autonomous equation. In order to analyse the asymptotic behavior of the solutions, we can consider the autonomous equation: ∞ ( r ) = 2 p λ 2 z ∞ ( r ) p − 1 ξ p − 1 ( ξ + 1 ) ( p − 1 ) z ′′ . (5) z ′ ∞ ( r ) p − 2 We search for a solution of (5) in the form z ∞ ( r ) = c exp ( τ r ) , (6) where c and τ are constants. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 12 / 28

Asymptotic Expansion at Infinity Subsituting in the equation, we obtain:   � 2 p λ 2 ( 1 + ξ ) ξ p − 1  . z ∞ ( r ) = c exp  − p r (7) p − 1 Then, the solution of the non-autonomous equation can be expressed in the form of a Lyapunov series: ∞ b k C k ( r ) e − τ kr , ∑ z ( r ) = (8) k = 1 where the functions C k can be determined by solving a set of linear ODEs. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 13 / 28

Asymptotic Expansion at Infinity We have obtained an asypmptotic expression of the solutions which satisfy the condition r → ∞ z ′ ( r ) = 0. r → ∞ z ( r ) = lim lim (9) In the old variable ρ , we obtain the asymptotic expression ρ ( r ) = ξ − bC 1 ( r ) r a e − τ r ( 1 + o ( 1 )) , r → ∞ . (10) We must compute the value of b for which the solution satisfies the prescribed boundary condition close to 0. Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 14 / 28

Numerical Approximation - Shooting Method R - bubble radius ( ρ ( R ) = 0). r 0 - initial approximation of R . First Auxiliary Problem ρ − ( r ) - monotone solution on [ δ , r 0 ] , which satisfies the boundary conditions 1 ρ ( δ ) = ρ 0 + p − 1 � f p ( ρ 0 ) � p − 1 p p � � 1 + y 1 δ , (11) δ p − 1 p − 1 p N ρ ( r 0 ) = 0. (12) Second Auxiliary Problem ρ + ( r ) - monotone solution on [ r 0 , r ∞ ] , which satisfies the boundary conditions (12) and ∞ C 1 ( r ∞ ) e − τ r ∞ . ρ ( r ∞ ) = ξ − br a (13) Pedro Lima (CENTRO DE MATEM´ ¸ ˜ OES INSTITUTO SUPERIOR T´ ECNICO UNIVERSIDADE T´ ATICA E APLICAC NUMERICAL SOLUTION OF THE DENSITY PROFILE EQUATION FOR MIXTURES OF October 11, 2011 ECNICA DE LISBO 15 / 28