General Basset–Boussinesq–Oseen equation: existence, uniqueness, approximation and regularity of solutions

Abstract

The Basset–Boussinesq–Oseen equation (BBO equation) is the mathematical formulation of the Lagrangian acceleration of a spherical particle, moving with Lagragian velocity, in an unsteady flow as the sum of the viscous, gravitational, buoyancy, virtual mass, and Basset forces acting respectively on the particle. Despite the widespread use of the equation in applications, the basic properties of its solutions have remained unexplored. Here we fill this gap by proving global existence and uniqueness of solutions in an appropriate partially ordered Banach space. The method also gives an approximate solution to the problem with accurate result which is easy to implement. Meanwhile, regularity properties of the solutions are proved under some conditions which gives a representation of the solutions. Illustrative examples exhibit the efficiency of our method.

Keywords:

• Basset–Boussinesq–Oseen equation
• existence
• uniqueness
• approximation
• regularity

2010 Mathematics Subject Classifications:

• 26A33
• 34A08
• 34A12

Acknowledgments

The authors are thankful to the Editor(s) and reviewers of the manuscript for their helpful comments.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author would like to thank University of Tabriz for the financial support of this research. The work of J.J. Nieto has been partially supported by the Agencia Estatal de Innovaci\'on (AEI) of Spain and co-financed by the European Community fund FEDER under grant MTM2016-75140-P,