A particle-in-Burgers model: theory and numerics - Laboratoire de ...

Model and motivation Auxiliary steps Results h = 0: coupling h = 0: definition, uniqueness h = 0: numerics, existence The coupled problem
...Model and motivation...
The 1D case : the Lagoutière-Seguin-Takahashi model for the
interaction, via a drag force, of a point particle with a Burgers fluid
writes
∂tu +∂x(u 2 /2) = λ D(h ′ (t)−u) δ0(x − h(t)),
here
mh ′′ (t) = λ D(u(t, h(t)) − h ′ (t)).
• u, the velocity of the fluid, is unknown
• h, the position of the solid particle, is unknown
(then h ′ and h ′′ respectively denote its velocity and acceleration);
• the parameters are λ (the drag coefficient) and m (the mass of the
solid particle); both are positive.
• the function D which intervenes in the drag force is an increasing
odd function.
Actually, we will suppose that
either D(v) = v (the linear case)
or D(v) = v|v| (the quadratic case).