Flow constituted by drops appears in a wide range of natural, biological and engineering situations. For example, liquid-liquid two phase flow inside capillaries constitutes a model commonly used to represent fluid flow in a petroleum reservoir. The typical modeling approach considers inertial forces negligible in comparison to viscous forces, allowing the use of Stokes equation to study flow dynamics. Very few numerical simulations have been made considering inertial effects. In this project, the flow of a periodic train of drops in a viscous suspending fluid, due to the influence of a fixed pressure gradient, was studied by numerical simulation considering the full Navier-Stokes equations.
A numerical approach based on a Volume of Fluid (VOF) formulation was employed using JADIM software, developed by the Institut de Mécanique des Fluides de Toulouse, France. JADIM solves Navier-Stokes equations using a VOF finite volume method, second order in space and time using structured mesh. This two-fluid approach without reconstruction of the interface allows simulating two-phase flows with complex interface shapes.
Densities of the drops equal to those of the suspending fluid and a constant interface tension were assumed. The effect of drop size, viscosity ratio, interfacial forces and system pressure gradient on the flow dynamics was studied. Parameters values were chosen to be representative for some particular viscous oil.
The result validation shows an excellent agreement between both numerical results. However, there are relative differences between them due to the increase in flow velocity when drop relative size increase and validity of Stokes approach is questionable.
Results show non-symmetric eddies in the continuum phase, in a referential frame fixed to the drop. The shape of eddies is strongly influenced by viscosity radio. Drop mobility decreases with increasing size. Additionally, drop mobility also decreases when the viscosity ratio increases. Extra pressure gradient of the system due to the presence of the drop shows a strong dependency on the size ratio between the drop and the pore. For size ratio lower than 0.5, the extra pressure gradient required to move the continuum phase is small. However, when drop to micro-channel ratio exceeds 0.5, the extra pressure gradient significantly increases when the drop size increases. Also, viscosity ratio affects on the system pressure loss, especially in cases where the viscosity ratio is high. The analysis of the capillary number effect on the dynamics of the two-phase system shows that it does not influence drop mobility for the drop sizes considered.