The cross-stream migration of a deformable drop in two-dimensional Hagen–Poiseuille
flow at finite Reynolds numbers is studied numerically. In the limit of a small Reynolds
number (< 1), the motion of the drop depends strongly on the ratio of the viscosity
of the drop fluid to the viscosity of the suspending fluid. For viscosity ratio 0.125 a
drop moves toward the centre of the channel, while for ratio 1.0 it moves away from
the centre until halted by wall repulsion. The rate of migration increases with the
deformability of the drop. At higher Reynolds numbers (5–50), the drop either moves
to an equilibrium lateral position about halfway between the centreline and the wall –
according to the so-called Segre–Silberberg effect or it undergoes oscillatory motion.
The steady-state position depends only weakly on the various physical parameters of
the flow, but the length of the transient oscillations increases as the Reynolds number
is raised, or the density of the drop is increased, or the viscosity of the drop is
decreased. Once the Reynolds number is high enough, the oscillations appear to persist
forever and no steady state is observed. The numerical results are in good agreement
with experimental observations, especially for drops that reach a steady-state lateral
position. Most of the simulations assume that the flow is two-dimensional. A few
simulations of three-dimensional flows for a modest Reynolds number (Re = 10), and
a small computational domain, confirm the behaviour seen in two dimensions. The
equilibrium position of the three-dimensional drop is close to that predicted in the
simulations of two-dimensional flow.
In this paper, a fully coupled thermo-hydro-mechanical model is presented for two-phase fluid flow and heat transfer in fractured/fracturing porous media using the extended finite element method. In the fractured porous medium, the traction, heat, and mass transfer between the fracture space and the surrounding media are coupled. The wetting and nonwetting fluid phases are water and gas, which are assumed to be immiscible, and no phase-change is considered. The system of coupled equations consists of the linear momentum balance of solid phase, wetting and nonwetting fluid continuities, and thermal energy conservation. The main variables used to solve the system of equations are solid phase displacement, wetting fluid pressure, capillary pressure, and temperature. The fracture is assumed to impose the strong discontinuity in the displacement field and weak discontinuities in the fluid pressure, capillary pressure, and temperature fields. The mode I fracture propagation is employed using a cohesive fracture model. Finally, several numerical examples are solved to illustrate the capability of the proposed computational algorithm. It is shown that the effect of thermal expansion on the effective stress can influence the rate of fracture propagation and the injection pressure in hydraulic fracturing process. Moreover, the effect of thermal loading is investigated properly on fracture opening and fluids flow in unsaturated porous media, and the convective heat transfer within the fracture is captured successfully. It is shown how the proposed computational model is capable of modeling the fully coupled thermal fracture propagation in unsaturated porous media.
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