Skew Brownian Motion and Applications in Fluid Dispersion Ed Waymire Department of Mathematics Oregon State University Corvallis, OR 97331 * Based on joint work with Thilanka Appuhamillage, Vrushali Bokil, Enrique Thomann, Brian Wood, and Jorge Ramirez, and supported by a grant from the National Science Foundation. Workshop on Computational Methods with Applications in Finance, Insurance and the Life Sciences AND Stochastic Methods in Partial Differential Equations and Applications of Deterministic and Stochastic PDEs, November 17 - 21, 2008
CONCENTRATION EQUATION DIVERGENCE FORM ∂t = 1 ∂c (F-P) 2 ∇ ( D ∇ c ) − ∇ · ( vc ) c ( y , 0) = c 0 ( y ) ( D ( y ) ∇ y c ) | I · n y = 0 � c ( t, y ) = R k c 0 ( x ) p ( t, x , y ) d x Nash (1958)
CONCENTRATION EQUATION DIVERGENCE FORM ∂t = 1 ∂c (F-P) 2 ∇ ( D ∇ c ) − ∇ · ( vc ) c ( y , 0) = c 0 ( y ) ( D ( y ) ∇ y c ) | I · n y = 0 � c ( t, y ) = R k c 0 ( x ) p ( t, x , y ) d x Nash (1958) y (//-interface) D + v → I x D −
CONCENTRATION EQUATION DIVERGENCE FORM ∂t = 1 ∂c (F-P) 2 ∇ ( D ∇ c ) − ∇ · ( vc ) c ( y , 0) = c 0 ( y ) ( D ( y ) ∇ y c ) | I · n y = 0 � c ( t, y ) = R k c 0 ( x ) p ( t, x , y ) d x Nash (1958) ( -interface ) y ⊥ y (//-interface) I D + v → D + I D − x x D −
OUTLINE OF TALK • THE CLASSIC TAYLOR-ARIS PROBLEM AND EXTENSION TO //-INTERACES -- THE ANSWER • SKEW BROWNIAN MOTION AND LOCAL TIME -- THE REASON • RELATED STOCHASTIC PARTICLE TRACKING QUESTIONS • ORTHOGONAL INTERFACES • (PSTN, LOCAL TIME, OCCUPATION TIME) AND ELASTIC SKEW BROWNIAN MOTION • SOME RELATED FUTURE DIRECTIONS
CLASSICAL TAYLOR-ARIS No HOMOGENEOUS ! Interface --- D v → D � b D = D + 8( b − a ) 2 v 2 v ( y ) dy v = 945 D b − a a
(NONHOMOGENEOUS) // INTERFACE ! --- D v → D + ----------------------------------- D − D =?
NONHOMOGENEOUS // INTERFACE ! --- D v → D + ----------------------------------- D − D = D a + 8 v 2 ( b − a ) 2 w/ J. Ramirez, E. Thomann, R. Haggerty, B.Wood 945 D h SIAM Multiscale Modeling &Simulation D a = D + + D − 2006 2 1 1 1 = D + + D h D −
∂t = 1 ∂c c ( y , 0) = c 0 ( y ) 2 ∇ ( D ∇ c ) − ∇ · ( vc ) Nash (1958) y Stochastic Particle Motion (//-interface) v → X = { ( X ( t ) , Y ( t )) : t ≥ 0 } D + � x dX ( t ) = v ( Y ( t )) dt + D ( Y ( t ) dB 1 ( t ) D −
∂t = 1 ∂c c ( y , 0) = c 0 ( y ) 2 ∇ ( D ∇ c ) − ∇ · ( vc ) Nash (1958) y Stochastic Particle Motion (//-interface) v → X = { ( X ( t ) , Y ( t )) : t ≥ 0 } D + � x dX ( t ) = v ( Y ( t )) dt + D ( Y ( t ) dB 1 ( t ) D − √ D + α ∗ = Y ( t ) = f ( B α ∗ ( t )) D + + √ √ D − B α denotes skew Brownian motion with parameter α
∂t = 1 ∂c c ( y , 0) = c 0 ( y ) 2 ∇ ( D ∇ c ) − ∇ · ( vc ) Nash (1958) y Stochastic Particle Motion (//-interface) v → X = { ( X ( t ) , Y ( t )) : t ≥ 0 } D + � x dX ( t ) = v ( Y ( t )) dt + D ( Y ( t ) dB 1 ( t ) D − √ D + α ∗ = Y ( t ) = f ( B α ∗ ( t )) D + + √ √ D − B α denotes skew Brownian motion with parameter α Ito-Tanaka ⇒ dY ( t ) = D + − D − � D + + D − d� (0 , t ) + D ( Y ( t )) dB 2 ( t )
∞ = f ( B α ∗ ( t )) � B α ( t ) = Y t 1 J n ( t ) A n | B ( t ) | n =1 D + Ito-McKean (1963) t 0 D {
= f ( B α ∗ ( t )) Y t D + t Ito-McKean (1963) 0 D { 2 πt e − ( y − y 0)2 2 πt e − ( y + y 0)2 + (2 α − 1) 1 if y 0 > 0 , y > 0 2 t 2 t √ √ 2 πt e − ( y − y 0)2 2 πt e − ( y + y 0)2 − (2 α − 1) 1 if y 0 < 0 , y < 0 2 t 2 t √ √ p ( α ) ( y 0 , y ; t ) = 2 πt e − ( y − y 0)2 2 α if y 0 ≤ 0 , y > 0 2 t √ 2 πt e − ( y − y 0)2 Walsh (1978) 2(1 − α ) if y 0 ≥ 0 , y < 0 . 2 t √
= f ( B α ∗ ( t )) Y t D + t Ito-McKean (1963) 0 D { 2 πt e − ( y − y 0)2 2 πt e − ( y + y 0)2 + (2 α − 1) 1 if y 0 > 0 , y > 0 2 t 2 t √ √ 2 πt e − ( y − y 0)2 2 πt e − ( y + y 0)2 − (2 α − 1) 1 if y 0 < 0 , y < 0 2 t 2 t √ √ p ( α ) ( y 0 , y ; t ) = 2 πt e − ( y − y 0)2 2 α if y 0 ≤ 0 , y > 0 2 t √ 2 πt e − ( y − y 0)2 Walsh (1978) 2(1 − α ) if y 0 ≥ 0 , y < 0 . 2 t √ √ √ � � − ( y − y 0 ) 2 � � − ( y + y 0 ) 2 �� 1 D + − D − exp + D + exp if y 0 > 0 , y > 0 √ √ √ 4 D + t 4 D + t 4 πD + t D − + √ √ � � − ( y − y 0 ) 2 � � − ( y + y 0 ) 2 �� 1 D + − D − exp D + exp if y 0 < 0 , y < 0 − √ √ √ 4 D − t 4 D − t 4 πD − t D − + p ∗ ( y 0 , y ; t ) = √ √ � � D + ) 2 − ( y D − − y 0 1 1 πt exp if y 0 ≤ 0 , y > 0 √ √ √ 4 D − D + t D + + D − √ √ � D − ) 2 � − ( y D + − y 0 1 1 πt exp if y 0 ≥ 0 , y < 0 . √ √ √ 4 D − D + t D + + D −
COMPUTATION OF EFFECTIVE DISPERSION RATE � t � t � X ( t ) = x 0 + v ( Y ( s )) ds + D ( Y ( s )) dB ( s ) 0 0
COMPUTATION OF EFFECTIVE DISPERSION RATE � t � t � X ( t ) = x 0 + v ( Y ( s )) ds + D ( Y ( s )) dB ( s ) 0 0 Bhattacharaya (1982) g ( y ) = U ( y ) − U, g ∈ Ran( A ) . g ( y ) = v ( y ) − v, g ∈ Ran ( A ) � 1 � t � t � t � 1 √ ∴ V ar g ( Y s ) ds = E [ g ( Y s 1 ) g ( Y s 2 )] ds 1 ds 2 t t 0 0 0 � t � s 2 ds � ds � � � � Y u , u � s − s � �� = g ( Y s − s � ) E g ( Y s � ) E t 0 0 � t � s 2 E { g ( Y s − s � ) T s � g ( Y s − s � ) } ds � ds. = t 0 0 � s � ∞ E { g ( Y s − s � ) T s � g ( Y s − s � ) } ds � = 2 � g, T s � g � π ds � − − − − → 2 lim t →∞ s →∞ 0 0 � ∞ � � T s � g ds � = 2 g, = 2 � g, h � π . 0 π
FINDING h Ah = − g Solving Poisson equation: � ∞ g ∈ Ran( A ) = 1 ⊥ , T s � g ds � . h ( y ) := 0 T t h ( y ) − h ( y ) Ah ( y ) = lim t t → 0 � ∞ 1 T t + s � g ( y ) − T s � g ( y ) ds � = lim t t → 0 0 � ∞ d T s � g ( y ) ( s � ) ds � = d s � 0 = s � → 0 T s � g ( y ) − T 0 g ( y ) lim = E π g − g ( y ) = 0 − g ( y ) = − g ( y ) lishers B.V., isterdam, The Physica A 168 (1990) 677-696 North-Holland 1\02139, USA 9 I 405 Orsay, I Avenue, Los Utrecht, The ,TAYLOR DISPERSION REVISITED 12215, USA - C. VAN DEN BROECK* Department of Chemistry, University of California at San Diego, B-040, La Jolla, :shire San Diego, CA 92093, USA ~ Park rlington lIer Received 21 May 1990 ,vile neiro Review Jtrecht ¡, Leiden \1urray Hil The theory and applications of Taylor dispersion are reviewed. The connection between problems of this type and the theory of stochastic processes and kinetic theory is clarified. 1. Introduction In 1954, Taylor (1) investigated, both theoretically and experimentally, the terdam, The motion of particles suspended in a fluid in Poiseuille flow through a cylindrical ELSEVIER; tube. He observed that, while the particles are being carried downstream (that is, along the tube's axis x) with an average velocity u equal to that of the fluid, the currency they are also being dispersed in a diffusion-like manner, around this average :ed at a total motion. More precisely, he proved that the probability density P(x, t) for :ed at a total observing a suspended particle at position x at time t, obeys the following :ed at a total convection-diffusion (or Fokker-Planck) equation, for (asymptotically) large times t: :ed at a total lumes in all) a,p(x, t) + u axP(X, t) = K a;p(x, t) . II, 1000 AE (1) ace Airlifted To his own surprise, Taylor found that the effective diffusion coefficient K is idia, Brazil, are available inversely proportional to the molecular diffusion coeffcient D of the suspended pply missing partic es , . i #1 transmitted in -2 2 isher, Elsevier U a K = 48D ' ~ article to the (2) considerations * Permanent address: L.V.C., B-36lO Diepenbeek, Belgium. j in the Royal #1 In his derivation, Taylor neglectëd the molecular diffusive motion of the suspended particles ion therewith. ~n for copying pays through in the direction of the flow. The correct result for K reads: K= u2a2/48D + D. For particles 08 of the U.S. suspended in fluids, the second term is usually negligible as compared to the first one. ~nter. Inc.. 27 ission to copy yable through as for general 0378-4371/90/$03.50 (Q 1990 - Elsevier Science Publishers BV (North-Holland) Lined from the negligence or
PARTICLE TRACKING EXPERIMENTS (MCMC) and THE HMYLA NUMERICAL SCHEME α ∗ -EXPERIMENT Hoteit, Mose, Younes, Lehmann, Ackerer (2002) Q2: SINGLE PARTICLE MOTION ? (``PARTICLE PATH PROBLEM’’)
PARTICLE TRACKING: Continuous Coefficients σ 2 0 = sup x D ( x ) 0 < inf x D ( x ) ≤ SPATIAL GRID ∆ > 0 � = ∆ 2 TEMPORAL GRID σ 2 0 p i,i ± 1 = D ( i ∆) � ± v ( i ∆) � (BD) 2∆ 2 2∆ p i,i = 1 − D ( i ∆ ) � ∆ 2 D. Stroock, S.R. Varadhan (1997): Chapter 11
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