On the hyperbolicity of certain models of polydisperse sedimentation
Corresponding Author
Raimund Bürger
CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Raimund Bürger, CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
E-mail: [email protected]
Search for more papers by this authorRosa Donat
Departament de Matemàtica Aplicada, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
Search for more papers by this authorPep Mulet
Departament de Matemàtica Aplicada, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
Search for more papers by this authorCarlos A. Vega
Departamento de Matemáticas y Estadística, División de Ciencias Básicas, Universidad del Norte, Barranquilla, Colombia
Search for more papers by this authorCorresponding Author
Raimund Bürger
CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile
Raimund Bürger, CI2MA and Departamento de Ingeniería Matemática, Facultad de Ciencias Físicas y Matemáticas, Universidad de Concepción, Casilla 160-C, Concepción, Chile.
E-mail: [email protected]
Search for more papers by this authorRosa Donat
Departament de Matemàtica Aplicada, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
Search for more papers by this authorPep Mulet
Departament de Matemàtica Aplicada, Universitat de València, Av. Dr. Moliner 50, E-46100 Burjassot, Spain
Search for more papers by this authorCarlos A. Vega
Departamento de Matemáticas y Estadística, División de Ciencias Básicas, Universidad del Norte, Barranquilla, Colombia
Search for more papers by this authorAbstract
The sedimentation of a polydisperse suspension of small spherical particles dispersed in a viscous fluid, where particles belong to N species differing in size, can be described by a strongly coupled system of N scalar, nonlinear first-order conservation laws for the evolution of the volume fractions. The hyperbolicity of this system is a property of theoretical importance because it limits the range of validity of the model and is of practical interest for the implementation of numerical methods. The present work, which extends the results of R. Bürger, R. Donat, P. Mulet, and C.A. Vega (SIAM Journal on Applied Mathematics 2010; 70:2186–2213), is focused on the fluxes corresponding to the models by Batchelor and Wen, Höfler and Schwarzer, and Davis and Gecol, for which the Jacobian of the flux is a rank-3 or rank-4 perturbation of a diagonal matrix. Explicit estimates of the regions of hyperbolicity of these models are derived via the approach of the so-called secular equation (J. Anderson. Linear Algebra and Applications 1996; 246:49–70), which identifies the eigenvalues of the Jacobian with the poles of a particular rational function. Hyperbolicity of the system is guaranteed if the coefficients of this function have the same sign. Sufficient conditions for this condition to be satisfied are established for each of the models considered. Some numerical examples are presented. Copyright © 2012 John Wiley & Sons, Ltd.
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