Physical Revier E - doi: 10.1103/PhysRevE.102.023109

**Authors - **Soheil Esmaeilzadeh, Zhipeng Qin, Amir Riaz, and Hamdi A. Tchelepi

**Abstract - **We study the interfacial evolution of immiscible two-phase flow within a capillary tube in the partial wetting regime using direct numerical simulation. We investigate the flow patterns resulting from the displacement of a more viscous fluid by a less viscous one under a wide range of wettability conditions. We find that beyond a wettability dependent critical capillary number, a uniform displacement by a less viscous fluid can transition into a growing finger that eventually breaks up into discrete blobs by a series of pinch-off events for both wetting and nonwetting contact angles. This study validates previous experimental observations of pinch-off for wetting contact angles and extends those to nonwetting contact angles. We find that the blob length increases with the capillary number. We observe that the time between consecutive pinch-off events decreases with the capillary number and is greater for more wetting conditions in the displaced phase. We further show that the blob separation distance as a function of the difference between the inlet velocity and the contact line speed collapses into two monotonically decreasing curves for wetting and nonwetting contact angles. For the phase separation in the form of pinch-off, this work provides a quantitative study of the emerging length and timescales and their dependence on the wettability conditions, capillary effects, and viscous forces.

**Keywords: ** Displacement of immiscible fluids, Microfluidics, Porous media, Wetting

**Fig.1. **A capillary tube is illustrated in an axisymmetric computational domain where $z$ and $r$ are the axial and radial coordinates, respectively. The axis of symmetry is represented by $OO'$, $AB$ is the solid wall boundary, and $OA$ and $O'B$ are respectively the inlet and outlet boundaries. The interface is shown in red with the level-set function $\phi =0$ at the interface and $\phi>0$ and $\phi<0$ in the advancing and receding fluid phases with respective density and dynamic viscosity of $\rho_1, \mu_1$ and $\rho_2,\mu_2$. $\mathbf{n}$ represents the interface normal vector. $\theta$ is the interface contact angle at the triple point where the interface is in contact with the solid wall boundary. $L_t$, $R$, $D$ are respectively the length, radius, and diameter of the capillary tube. Throughout this work we consider a tube length to diameter ratio of $L_t/D = 10$ with $D= 0.00073\,[m]$.