On a nonlocal dispersive equation modeling particle suspensions
Author:
Kevin Zumbrun
Journal:
Quart. Appl. Math. 57 (1999), 573-600
MSC:
Primary 35L65; Secondary 45K05, 76T99
DOI:
https://doi.org/10.1090/qam/1704419
MathSciNet review:
MR1704419
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Abstract: We study a nonlocal, scalar conservation law ${u_t} + {\left ( \left ( {K_a} * u \right )u \right )_x} = 0$, modeling sedimentation of particles in a dilute fluid suspension, where ${K_a}\left ( x \right ) = {a^{ - 1}}K\left ( x/a \right )$ is a symmetric smoothing kernel, and $\ast$ represents convolution. We show this to be a dispersive regularization of the Hopf equation, ${u_t} + {\left ( {u^2} \right )_x} = 0$, analogous to KdV and certain dispersive difference schemes. Using the smoothing property of convolution and the physical principle of conservation of mass, we establish the global existence of smooth solutions.
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G. K. Batchelor, Sedimentation in a dilute suspension of spheres, J. Fluid Mech. 52, 245 (1972)
C. Beenakker and P. Mazur, Phys. Fluids 28, 3203 (1985)
R. Caflisch and G. C. Papanicolaou, Dynamic theory of suspensions with Brownian effects, SIAM J. Appl. Math. 43, No. 4, 885–906 (1983)
S. Childress, Viscous flow past a random array of spheres, J. Chem. Phys. 56, 2527–2539 (1972)
J. Goodman and P. Lax, On dispersive difference schemes. I, Comm. Pure Appl. Math. 41 591–613 (1988)
T. Hou and P. Lax, Dispersive approximations in fluid dynamics, Comm. Pure Appl. Math. 44, No. 1, 1–40 (1991)
H. Hasimoto, On the periodic fundamental solutions of the Stokes’ equations and their application to viscous flow past a cubic array of spheres, J. Fluid Mech. 5, 317–328 (1959)
J. Happel and H. Brenner, Slow viscous flow past a sphere in a cylindrical tube, J. Fluid Mech. 4, 195–213 (1958)
E. J. Hinch, J. Fluid Mech. 83, 695 (1977)
M. Khodja, Thesis, University of Michigan.
M. Khodja and K. Zumbrun, Stability of an oscillatory shock arising in fluid suspensions, to appear.
P. Lax, Hyperbolic Systems of Conservation Laws, SIAM Reg. Conf. Ser. 77, 1973
S. R. Merchant and E. A. Rosauer, Clays Clay Miner 17, 289 (1969)
R. W. O’Brien, A method for the calculation of the effective transport properties of suspensions of interacting particles, J. Fluid Mech. 91, 17–39 (1979)
J. Rubinstein, Evolution equations for stratified dilute suspensions, Phys. Fluids A, vol. 2, No. 1, 3–6 (1990)
J. Rubinstein and J. B. Keller, Sedimentation of a dilute suspension, Phys. Fluids A 1, 637–643 (1989)
J. Rubinstein and J. B. Keller, Particle distribution functions in suspensions, Phys. Fluids A 1, 1632–1641 (1989)
A. S. Sangani and A. Acrivos, J. Multiphase Flow 8, 193 (1982)
D. B. Siano, Layered sedimentation in suspensions of monodispersive spherical colloidal particles, J. Colloid. Interface Sci. 68, 111 (1979)
G. Strang, Accurate partial difference methods, II. Nonlinear problems, Numer. Math. 6, 37–46 (1964)
T. J. Tobias and A. P. Ruotsala, Clays Clay Miner 13, 395 (1966)
A. I. Volpert, The spaces BV and quasilinear equations, Math. USSR Sb. 2, 257–267 (1967)
A. A. Zick and G. M. Homsy, J. Fluid Mech. 115, 13 (1982)
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© Copyright 1999
American Mathematical Society