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  • Microfluidica

    Difusin

    1

  • 2

    Microfluidica Ecuacin de difusin

    En microfluidica y aplicaciones (lab-on-a-chip; TAS): Transporte de especies qumicas, nanoparGculas y parGculas coloidales (transporte de masa) En muchos casos puede suponerse que no afectan el movimiento del fluido (escalares pasivos) Dos mecanismos de transporte independientes: Conveccin (adveccion) y Difusin (movimiento trmico o Browniano) Mtodos de descripcin: Macroscpico (teora del conMnuo): Ecuacin de transporte para la concentracin. Microscpico (caminata aleatoria): Ecuacin estocsMca para la densidad de probabilidad

    Ecuacin de conservacin de un escalar C(x,t) (e. g. concentracin):

    Ct + ! = 0 J(x,t) es el flujo del escalar C

  • 3

    Microfluidica Ecuacin de conveccindifusin

    J(x,t): el flujo del escalar C

    Transporte por conveccin: !! = !!! Transporte por difusin: Ecuacin de Fick (1856) (fenomenolgica y por analoga a la ecuacin del calor de Fourier (1822))

    !! = !!!! Ct + !! + !! =

    Ct + !!! !!! = 0

    DM : Coeficiente de Difusin Molecular

    Ct + ! ! =

    !"!" = !!

    !!

  • 4

    Microfluidica Ecuacin de conveccindifusin

    Ct + ! ! = !!

    !!

    v' = vv!;! x' = xl!

    ;! t' =tt!= v!tl!

    ;!

    Ecuacin Adimensional: Elegir magnitudes caractersticas del problema: velocidad v0 y longitud l0

    Concentracin?

    ! = !C!!

    v!!!

    !t +

    v!!!

    !! ! = 1!!!!!!!

    v!!!!!

    !t + !! ! =

    !!

    Pe = v!!!!! Nmero de Peclet:

  • 5

    Pe !t + !! ! = !!

    Microfluidica Ecuacin de conveccindifusin

    Nmero de Peclet: Pe = v!!!!! Transporte convectivo

    Transporte difusivo

    Ley de escala en microfluidos ?

    Pe#~!!!!

    (Igual que el nmero de Reynolds)

    En microfluidica, a medida que reducimos el tamao, la difusin es ms importante y el nmero de Peclet 0

    longitud l0 100m velocidad v0 100m/s agua DM 1000 m2/s

    Peclet 5

    Todava es bastante alto! Depende de cada caso

    Ej.1: v0 10m/s; l0 10m Pe > 1

  • 6

    Microfluidica Solucin fundamental con una fuente puntual (1D)

    Ct = !!

    !!! Condicin Inicial t=0

    !!!(!)!

    !(!, !) = !!4!!!!

    !exp !!

    4!!!

    !!~! 2!!!!

    !~! 2!!!!

    Distribucin Gaussiana (distribucin Normal en probabilidades)

    Varianza

    Radio de una gota de Mnta

  • 7

    Microfluidica Algunos ejemplos de difusin y Mempos caractersMcos

    P1: JZP0521849101c07 CUFX064/Leal Printer: cupusbw 0 521 84910 1 April 25, 2007 13:17

    Creeping Flows Two-Dimensional and Axisymmetric Problems

    up

    (a)

    up

    (b)

    Figure 73. A schematic representation of the proof that a spherical particle cannot undergo lateral migra-tion in either 2-D or axisymmetric Poiseuille flow if the disturbance flow is a creeping flow. In (a) we supposethat the undisturbed flow moves from left to right and the sphere migrates inward with velocity up . Then,in the creeping-flow limit, if direction of the undisturbed flow is reversed, the signs of all velocities includingthat of the sphere would also have to be reversed, as shown in (b). Because the problems (a) and (b) areidentical other than the direction of the flow through the channel or tube, we conclude that up = 0.

    body without actually solving the full fluid mechanics problem and calculating the force byintegrating the stress vector n T over the sphere surface.

    3. Lateral Migration of a Sphere in Poiseuille FlowOne of the best-known experimental results for particle motion in viscous flows is theobservation by Segre and Silberberg2 of lateral migration for a small, neutrally buoyantsphere (sphere = fluid) that is immersed in Poiseuille flow through a straight, circular tubeor in the pressure-driven parabolic flow (sometimes called 2D Poiseuille flow) betweentwo parallel plane boundaries. The experiments of Segre and Silberberg, and many laterinvestigators, show that a freely suspended sphere in these circumstances will slowly moveperpendicular to the main direction of flow until it reaches an equilibrium position that isapproximately 60% of the way from the centerline (or central plane) to the wall. Hence asuspension of such spheres flowing in Poiseuille flow through a tube of radius R will tend toaccumulate in an annular ring at r = 0.6R. Because the Reynolds number for many of theexperimental observations was quite small, one might assume that a theoretical explanationcould be achieved by using detailed solutions of the creeping-flow equations with suitableboundary conditions. However, in view of the complexity of the geometry (an eccentricallylocated sphere inside a circular tube), this theoretical problem is extremely complex anddifficult to solve, even in the creeping-flow limit. Thus, before actually trying to solvethe problem, it is prudent to determine whether lateral migration is possible at all in thecreeping-flow limit.

    The fact is that a theory based entirely on the creeping-flow approximation will leadto the result that lateral migration is impossible, at least for a single sphere immersed inaxisymmetric or two-dimensional Poiseuille flow. To see that this is true, we can refer toFig. 73. Here is a sketch of the hypothetical situation of a sphere that is undergoing lateralmigration in Poiseuille flow through a tube. The undisturbed flow in part (a) of Fig. 73is shown moving from left to right, and the sphere is assumed to be migrating radiallyinward toward the center of the tube. Now, however, if the creeping-motion approximationis valid, the governing equations and boundary conditions are linear in the velocity andpressure, and we can change the signs of all velocities and the pressure and still have asolution of the same problem but with the direction of the undisturbed flow reversed, asshown in Fig. 73(b). However, because all the velocities have the opposite sign, the inwardmigration velocity from configuration (a) must now become an outward migration velocityfor configuration (b). But there is now a clear contradiction. The problems in (a) and (b)are clearly indistinguishable in all respects. Thus, if the sphere undergoes a lateral motion,it should be in the same direction in both cases. Because the preceding argument, based onthe linearity of the problem, shows that a nonzero migration velocity in case (a) must lead

    438

    P1: JZP0521849101c07 CUFX064/Leal Printer: cupusbw 0 521 84910 1 April 25, 2007 13:17

    Creeping Flows Two-Dimensional and Axisymmetric Problems

    up

    (a)

    up

    (b)

    Figure 73. A schematic representation of the proof that a spherical particle cannot undergo lateral migra-tion in either 2-D or axisymmetric Poiseuille flow if the disturbance flow is a creeping flow. In (a) we supposethat the undisturbed flow moves from left to right and the sphere migrates inward with velocity up . Then,in the creeping-flow limit, if direction of the undisturbed flow is reversed, the signs of all velocities includingthat of the sphere would also have to be reversed, as shown in (b). Because the problems (a) and (b) areidentical other than the direction of the flow through the channel or tube, we conclude that up = 0.

    body without actually solving the full fluid mechanics problem and calculating the force byintegrating the stress vector n T over the sphere surface.

    3. Lateral Migration of a Sphere in Poiseuille FlowOne of the best-known experimental results for particle motion in viscous flows is theobservation by Segre and Silberberg2 of lateral migration for a small, neutrally buoyantsphere (sphere = fluid) that is immersed in Poiseuille flow through a straight, circular tubeor in the pressure-driven parabolic flow (sometimes called 2D Poiseuille flow) betweentwo parallel plane boundaries. The experiments of Segre and Silberberg, and many laterinvestigators, show that a freely suspended sphere in these circumstances will slowly moveperpendicular to the main direction of flow until it reaches an equilibrium position that isapproximately 60% of the way from the centerline (or central plane) to the wall. Hence asuspension of such spheres flowing in Poiseuille flow through a tube of radius R will tend toaccumulate in an annular ring at r = 0.6R. Because the Reynolds number for many of theexperimental observations was quite small, one might assume that a theoretical explanationcould be achieved by using detailed solutions of the creeping-flow equations with suitableboundary conditions. However, in view of the complexity of the geometry (an eccentricallylocated sphere inside a circular tube), this theoretical problem is extremely complex anddifficult to solve, even in the creeping-flow limit. Thus, before actually trying to solvethe problem, it is prudent to determine whether lateral migration is possible at all in thecreeping-flow limit.

    The fact is that a theory based entirely on the creeping-flow approximation will leadto the result that lateral migration is impossible, at least for a single sphere immersed inaxisymmetric or two-dimensional Poiseuille flow. To see that this is true, we can refer toFig. 73. Here is a sketch of the hypothetical situation of a sphere that is undergoing lateralmigration in Poiseuille flow through a tube. The undisturbed flow in part (a) of Fig. 73is shown moving from left to right, and the sphere is assumed to be migrating radiallyinward toward the center of the tube. Now, however, if the creeping-motion approximationis valid, the governing equations and boundary conditions are linear in the velocity andpressure, and we can change the signs of all velocities and the pressure and still have asolution of the same problem but with the direction of the undisturbed flow reversed, asshown in Fig. 7