## Conservation laws models in networks and multiscale flow optimization.

##### Abstract

The flow of fluids in a network is of practical importance in gas, oil and water
transport for industrial and domestic use. When the flow dynamics are understood,
one may be interested in the control of the flow formulated as follows: given some
fluid properties at a final time, can one determine the initial flow properties that
lead to the desired flow properties?
In this thesis, we first consider the flow of a multiphase gas, described by the drift flux
model, in a network of pipes and that of water, modeled by the shallow water
equations, in a network of rivers. These two models are systems of partial differential
equations of first order generally referred to as systems of conservation laws. In
particular, our contribution in this regard can be summed up as follows: For the
drift-flux model, we consider the flow in a network of pipes seen mathematically as an
oriented graph. We solve the standard Riemann problem and prove a well posedness
result for the Riemann problem at a junction. This result is obtained using coupling
conditions that describe the dynamics at the intersection of the pipes. Moreover, we
present numerical results for standard pipes junctions. The numerical results and
the analytical results are in agreement. This is an extension for multiphase flows of
some known results for single phase flows. Thereafter, the shallow water equations
are considered as a model for the flow of water in a network of canals. We analyze
coupling conditions at the confluence of rivers, precisely the conservation of mass
and the equality of water height at the intersection, and implement these results for
some classical river confluences. We also consider the case of pooled stepped chutes,
a geometry frequently utilized by dams to spill floodwater. Here we consider an
approach different from the engineering community in the sense that we resolve the
dynamics by solving a Riemann problem at the dam for the shallow water equations
with some suitable coupling conditions.
Secondly, we consider an optimization problem constrained by the Euler equations
with a flow-matching objective function. Differently from the existing approaches
to this problem, we consider a linear approximation of the flow equation
in the form of the microscopic Lattice Boltzmann Equations (LBE). We derive an
adjoint calculus and the optimality conditions from the microscopic LBE. Using
multiscale analysis, we obtain an equivalent macroscopic result at the hydrodynamic
limit. Our numerical results demonstrate the ability of our method to solve
challenging problems in fluid mechanics.