  We investigate the behaviour of a system where a single phase fluid domain is
coupled to a biphasic poroelastic domain. The fluid domain consists of an
incompressible Newtonian viscous fluid while the poroelastic domain consists of
a linear elastic solid filled with the same viscous fluid. The properties of
the poroelastic domain, i.e. permeability and elastic parameters, depend on the
inhomogeneous initial porosity field. The theoretical framework highlights how
the heterogeneous material properties enter the linearised governing equations
for the poroelastic domain. To couple flows through this domain with a
surrounding Stokes flow, we show case a numerical implementation based on a new
mixed formulation where the equations in the poroelastic domain are rewritten
in terms of three fields: displacement, fluid pressure and total pressure.
  Coupling single phase and multiphase flow problems are ubiquitous in many
industrial and biological applications, and here we consider an example from
in-vitro tissue engineering. We consider a perfusion system, where a flow is
forced to pass from the single phase fluid to the biphasic poroelastic domain.
We focus on a simplified two dimensional geometry with small aspect ratio, and
perform an asymptotic analysis to derive analytical solutions for the
displacement, the pressure and the velocity fields. Our analysis advances the
quantitative understanding of the role of heterogeneous material properties of
a poroelastic domain on its mechanics when coupled with a fluid domain.
Specifically, (i) the analytical analysis gives closed form relations that can
be directly used in the design of slender perfusion systems; (ii) the numerical
method is validated by comparing its result against selected theoretical
solutions, opening towards the possibility to investigate more complex
geometrical configurations.
