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Diffusion in a fluid flow generated by a source at the apex of a wedge

P. L. Krapivsky

  • Department of Physics, Boston University, Boston, Massachusetts 02215, USA and Santa Fe Institute, Santa Fe, New Mexico 87501, USA

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Phys. Rev. Fluids 7, 064502 – Published 21 June, 2022

DOI: https://doi.org/10.1103/PhysRevFluids.7.064502

Abstract

We consider a particle diffusing inside a wedge with absorbing boundaries and driven by a radial flow of incompressible fluid generated by a source at the apex. The survival probability decays as (time)−β with β depending on the opening angle of the wedge and the Reynolds number associated with the hydrodynamic flow. The computation of the exponent β reduces to finding the ground-state energy of the quantum particle in an infinitely deep potential well with a shape determined by the radial flow velocity.

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References (65)

  1. A. Sommerfeld, Zur analytischen theorie der wärmeleitung, Math. Ann. 45, 263 (1894).
  2. S. Redner, A Guide to First-Passage Processes (Cambridge University Press, Cambridge, UK, 2001).
  3. A. Comtet and J. Desbois, Brownian motion in wedges, last passage time and the second arc-sine law, J. Phys. A: Math. Gen. 36, L255 (2003).
  4. T. Lagache and D. Holcman, Effective motion of a virus trafficking inside a biological cell, SIAM J. Appl. Math. 68, 1146 (2008).
  5. Diandrew Lexter L. Dy and J. P. Esguerra, First-passage-time distribution for diffusion through a planar wedge, Phys. Rev. E 78, 062101 (2008).
  6. D. L. Dy and J. P. Esguerra, First-passage characteristics of biased diffusion in a planar wedge, Phys. Rev. E 88, 012121 (2013).
  7. M. Chupeau, O. Bénichou, and S. N. Majumdar, Survival probability of a Brownian motion in a planar wedge of arbitrary angle, Phys. Rev. E 91, 032106 (2015).
  8. M. E. Fisher, Walks, walls, wetting, and melting, J. Stat. Phys. 34, 667 (1984).
  9. D. A. Huse and M. E. Fisher, Commensurate melting, domain walls, and dislocations, Phys. Rev. B 29, 239 (1984).
  10. M. E. Fisher and M. P. Gelfand, The reunions of three dissimilar vicious walkers, J. Stat. Phys. 53, 175 (1988).
  11. P. L. Krapivsky and S. Redner, Kinetics of a diffusive capture process: Lamb besieged by a pride of lions, J. Phys. A: Math. Gen. 29, 5347 (1996).
  12. S. Redner and P. L. Krapivsky, Capture of the lamb: Diffusing predators seeking a diffusing prey, Am. J. Phys. 67, 1277 (1999).
  13. D. J. Grabiner, Brownian motion in a Weyl chamber, non-colliding particles, and random matrices, Ann. Inst. H. Poincaré. Probab. Statist. 35, 177 (1999).
  14. D. ben-Avraham, B. M. Johnson, C. A. Monaco, P. L. Krapivsky, and S. Redner, Ordering of random walks: The leader and the laggard, J. Phys. A: Math. Gen. 36, 1789 (2003).
  15. J. Cardy and M. Katori, Families of vicious walkers, J. Phys. A: Math. Gen. 36, 609 (2003).
  16. E. Ben-Naim and P. L. Krapivsky, Kinetics of first passage in a cone, J. Phys. A: Math. Theor. 43, 495007 (2010).
  17. E. Ben-Naim and P. L. Krapivsky, First-passage exponents of multiple random walks, J. Phys. A: Math. Theor. 43, 495008 (2010).
  18. D. ben-Avraham, Computer simulation methods for diffusion-controlled reactions, J. Chem. Phys. 88, 941 (1988).
  19. P. L. Krapivsky, S. Redner, and E. Ben-Naim, A Kinetic View of Statistical Physics (Cambridge University Press, Cambridge, UK, 2010).
  20. B. J. Kirby, Micro- and Nanoscale Fluid Mechanics: Transport in Microfluidic Devices (Cambridge University Press, Cambridge, UK, 2010).
  21. P. Tabeling, Introduction to Microfluidics (Oxford University Press, Oxford, UK, 2010).
  22. Z. Torquato, Random Heterogeneous Materials: Microstructure and Macroscopic Properties (Springer, Berlin, 2002).
  23. D. Holcman and Z. Schuss, Stochastic Narrow Escape in Molecular and Cellular Biology (Springer, Berlin, 2015).
  24. D. A. Lauffenburger and J. J. Linderman, Receptors: Models for Binding, Trafficking, and Signaling (Oxford University Press, Oxford, UK, 1996).
  25. R. J. Hunter, Foundations of Colloid Science (Oxford University Press, Oxford, UK, 2001).
  26. N. A. Mortensen, F. Okkels, and H. Bruus, Reexamination of hagen-poiseuille flow: Shape dependence of the hydraulic resistance in microchannels, Phys. Rev. E 71, 057301 (2005).
  27. M. Z. Bazant, Exact solutions and physical analogies for unidirectional flows, Phys. Rev. Fluids 1, 024001 (2016).
  28. S. Chevalier, Semianalytical modeling of the mass transfer in microfluidic electrochemical chips, Phys. Rev. E 104, 035110 (2021).
  29. G. K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, Cambridge, UK, 1967).
  30. L. D. Landau and E. M. Lifshitz, Fluid Mechanics (Pergamon Press, New York, 1987).
  31. C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, Annu. Rev. Fluid Mech. 23, 159 (1991).
  32. P. G. Drazin and N. Riley, The Navier-Stokes Equations: A Classification of Flows and Exact Solutions (Cambridge University Press, Cambridge, UK, 2006).
  33. G. B. Jeffery, The two-dimensional steady motion of a viscous fluid, Lond. Edinb. Dubl. Philos. Mag. J. Sci. 29, 455 (1915).
  34. G. Hamel, Spiralförmige bewegungen zäher flüssigkeiten, Jahresber. Deutsch. Math. 25, 34 (1917).
  35. W. R. Dean, Note on the divergent flow of fluid, Lond. Edinb. Dubl. Philos. Mag. J. Sci. 18, 759 (1934).
  36. L. Rosenhead, The steady two-dimensional radial flow of viscous fluid between two inclined plane walls, Proc. R. Soc. Lond. A 175, 436 (1940).
  37. A. J. Bray, Random walks in logarithmic and power-law potentials, nonuniversal persistence, and vortex dynamics in the two-dimensional XY model, Phys. Rev. E 62, 103 (2000).
  38. P. L. Krapivsky and S. Redner, Probing non-integer dimensions, J. Phys.: Condens. Matter 19, 065119 (2007).
  39. P. L. Krapivsky, Reaction-diffusion process driven by a localized source: First-passage properties, Phys. Rev. E 85, 031124 (2012).
  40. P. L. Krapivsky and S. Redner, First-passage duality, J. Stat. Mech. (2018) 093208.
  41. G. Carrier, M. Krook, and C. E. Pearson, Functions of a Complex Variable (Hod Books, Ithaca, NY, 1983).
  42. J. Koplik, S. Redner, and E. J. Hinch, Tracer dispersion in planar multipole flows, Phys. Rev. E 50, 4650 (1994).
  43. M. Z. Bazant, Conformal mapping of some non-harmonic functions in transport theory, Proc. R. Soc. Lond. A 460, 1433 (2004).
  44. L. D. Landau and E. M. Lifshitz, Quantum Mechanics (Hod Books, Ithaca, NY, 1983).
  45. L. E. Fraenkel, Laminar flow in symmetrical channels with slightly curved walls, I. On the Jeffery-Hamel solutions for flow between plane walls, Proc. R. Soc. Lond. A 267, 119 (1962).
  46. W. H. H. Banks, P. G. Drazin, and M. B. Zaturska, On perturbations of Jeffery-Hamel flow, J. Fluid Mech. 186, 559 (1988).
  47. I. J. Sobey and P. G. Drazin, On perturbations of Jeffery-Hamel flow, J. Fluid Mech. 171, 263 (1986).
  48. M. Hamadiche, J. Scott, and D. Jeandel, Temporal stability of Jeffery-Hamel flow, J. Fluid Mech. 268, 71 (1994).
  49. S. C. R. Dennis, W. H. H. Banks, P. G. Drazin, and M. B. Zaturska, Temporal stability of Jeffery-Hamel flow, J. Fluid Mech. 336, 183 (1997).
  50. C. M. Bender and S. A. Orszag, Advanced Mathematical Methods for Scientists and Engineers (McGraw–Hill, New York, 1978).
  51. P. M. Morse and H. Feshbach, Methods of Theoretical Physics, Part 1 (McGraw–Hill, Boston, NA, 1953).
  52. H. J. W. Müller-Kirsten, Introduction to Quantum Mechanics: Schrödinger Equation and Path Integral (World Scientific, Singapore, 2012).
  53. L. E. Fraenkel, On corner eddies in plane inviscid shear flow, J. Fluid Mech. 11, 400 (1961).
  54. H. K. Moffatt and B. R. Duffy, Local similarity solutions and their limitations, J. Fluid Mech. 96, 299 (1980).
  55. E. Sternberg and W. T. Koiter, The wedge under a concentrated couple: A paradox in the two-dimensional theory of elasticity, J. Appl. Mech. 25, 575 (1958).
  56. H. K. Moffatt, Viscous and resistive eddies near a sharp corner, J. Fluid Mech. 18, 1 (1964).
  57. N. A. Slezkin, On an exact solution of the equations of viscous flow, Sci. Rec. Moscow State Univ. 2, 89 (1934).
  58. L. D. Landau, A new exact solution of the Navier-Stokes equations, Dokl. Akad. Nauk SSSR 43, 286 (1944).
  59. H. B. Squire, The round laminar jet, Q. J. Mech. Appl. Math. 4, 321 (1951).
  60. R. Metzler, G. Oshanin, and S. Redner, A First-Passage Phenomena and Their Applications (World Scientific, Singapore, 2014).
  61. J.-F. Rupprecht, O. Bénichou, D. S. Grebenkov, and R. Voituriez, Exit time distribution in spherically symmetric two-dimensional domains, J. Stat. Phys. 158, 192 (2015).
  62. D. S. Grebenkov, Universal formula for the mean first passage time in planar domains, Phys. Rev. Lett. 117, 260201 (2016).
  63. D. S. Grebenkov and A. T. Skvortsov, Mean first-passage time to a small absorbing target in an elongated planar domain, New J. Phys. 22, 113024 (2020).
  64. A. Skvortsov, Mean first passage time for a particle diffusing on a disk with two absorbing traps at the boundary, Phys. Rev. E 102, 012123 (2020).
  65. M. J. Simpson, D. J. VandenHeuvel, J. M. Wilson, S. W. McCue, and E. J. Carr, Mean exit time for diffusion on irregular domains, New J. Phys. 23, 043030 (2021).

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Phys. Rev. Fluids 7, 064502– Published 21 June, 2022
  • Received 4 January 2022
  • Accepted 24 May 2022
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