Geodesic
Three models for rectilinear particle motion with the Basset history force
Electronic Journal of Differential Equations, Tome 2015 (2015).

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Summary: We consider three model problems that describe rectilinear particle motion in a viscous fluid under the influence of the Basset history force. These problems consist of sedimentation starting from rest, impulsive motion in a quiescent fluid, and oscillatory sliding motion. The equations of motion are integro-differential equations with a weakly singular kernel. We derive analytical solutions to all three problems using Laplace transforms and discuss the mathematical relation between the sedimentation and impulsive start problems. We also compare several numerical schemes for solving the integro-differential equations and benchmark them against the analytical results.
Classification : 76T20, 45J05, 65L05
Keywords: particle motion, history force, integro-differential equations, Laplace transforms, numerical methods, multiphase flow 
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     author = {Xu, Shujing and Nadim, Ali},
     title = {Three models for rectilinear particle motion with the {Basset} history force},
     journal = {Electronic Journal of Differential Equations},
     publisher = {mathdoc},
     volume = {2015},
     year = {2015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a44/}
}
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Xu, Shujing; Nadim, Ali. Three models for rectilinear particle motion with the Basset history force. Electronic Journal of Differential Equations, Tome 2015 (2015). http://geodesic.mathdoc.fr/item/EJDE_2015__2015__a44/