**1.3. Different numerical methods**

In numerical simulation of droplet dynamics or as a whole in the simulation of two-phase flow, there are several challenges which need to be carefully tackled to obtain reliable results. The main challenge is capturing the moving interface of the two phases accurately, which is not known priory. The accurate tracking of the interface and investigation of twophase flow topology should be the essentiality of a good numerical method. There are several numerical methods based on interface kinematics to track the interface in free surface flows. Among them are: volume of fluid methods, front tracking methods, level set methods, phase field formulations, continuum advection schemes, boundary integral methods, particle-based methods, and moving mesh methods.

Volume of fluid method (VOF), earlier known as the volume tracking method, were originally developed by Nichols and Hirt (1975), Noh and Woodward (1976) and further extended by Hirt and Nichols (1981). Since then, the method has been extensively used and significantly improved over the years (Rudman,1997; Rider and Kothe,1998). The VOF

method is based on the conservation of the volume fraction function *F* with respect to time and space, expressed as

$$\frac{\partial F}{\partial t} + (\boldsymbol{\upsilon}, \nabla)F = \mathbf{0} \tag{13}$$

Numerical Simulation of Droplet Dynamics in Membrane Emulsification Systems 423

(14) ݒ ൌ

material interfaces are automatically tracked. By using particle motion to approximate the convection terms, numerical diffusion across interfaces (where particles change identity) is

In moving mesh methods, the position history of discrete point's *x*i lying on the interface is

A moving mesh is Lagrangian if every point is moved, and mixed (Lagrangian-Eulerian) if grid points in a subset of the domain are moved. Mixed methods are used for mold filling simulations, where the mold computational domain can be held stationary and the molten

Besides these individual methods, there are some combined method such as coupled level set and volume of fluid method. This method has been developed to tackle the inefficiency of VOF method in calculating complex geometrical properties and problem of mass imbalance of level set method. In the coupled method, the LS function is used only to compute the geometric properties (normal and curvature) at the interface while the void fraction is calculated using the VOF approach. Ohta et al. (2007) and Sussman et al. (2007) developed a novel coupled LS-VOF method to determine the sharp interface for

Numerical investigation of active droplet formation is reasonably complex as it is potentially a multi-physics problem governed by a large number of partial differential equations (PDEs). Numerical simulation of droplet dynamics in membrane emulsion process requires accurate capturing of the evolving liquid–liquid interfaces. This gives sufficient challenge since the boundary between the two phases is not known priori and it is a part of the solution. In addition, the solution technique has to deal with different properties of both the phases such as density, viscosity, and velocity ratios. Numerical investigation of membrane emulsification process requires two main issues to be dealt with: one is permeation of the disperse phase through the membrane pores; the other is the mechanism of droplet detachment. Several earlier investigations have treated these issues separately; however both of them simultaneously contribute to droplet evolution and should be considered in the same framework. Both these issues have been considered in the present work. In the present work, numerical simulation of droplet dynamics in a membrane emulsification has been carried out considering multipore membrane. The hydrodynamic effects due to multipores and the effect

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tracked for all time by integrating the evolution equation, forward in time.

incompressible, immiscible two-phase flows for large value of density ratio.

of different operating parameters on the droplet dynamics have been investigated.

The flow configuration of membrane emulsification considered in the present work has been shown in Fig. 3. Here the membrane emulsification process with two pores has been

virtually zero; hence interface widths are well defined.

liquid is followed with a Lagrangian mesh.

**1.4. Issues of numerical simulation** 

**2. Problem description** 

In VOF method, the computational grid is kept fixed and the interface between the two fluids is tracked within each cell through which it passes. In a computational grid cell, the interface can be effectively represented by line of the slope. To reconstruct the interface, the piecewise linear interpolation calculation (PLIC) method developed by Youngs (1982) is used in the computation. The interfacial surface forces are incorporated as body forces per unit volume in the Navier-Stokes equations; hence no extra boundary condition is required across the interface.

The basic working principle of front tracking method is based on the marker and cell (MAC) formulation (Harlow and Welch, 1966; Daly, 1967). The interface is represented discretely by Lagrangian markers connected to form a front which lies within and moves through a stationary Eulerian mesh. As the front moves and deforms, interface points are added, deleted, and reconnected as necessary. Further details of the method may be found in (Glimm et al., 1985; Churn et al., 1986; Tryggvason et al., 1998).

Level set methods have been developed by Osher and Sethian (1988). This method can compute the geometrical properties of highly complicated interface without explicitly tracking the interface. The basic principle of the level set method is to embed the propagating interface Γ(t) as the zero level set of a higher dimensional function *φ* , defined as *φ* (*x*, *t* =0) = ±*d*, where *d* is the distance from *x* to *Γ* (*t* = 0). The function is chosen to be positive (negative) if *x* is outside (inside) the initial position of the interface *Γ* (*t* = 0) = *φ* (*x*, *t* =0)=0. Afterwards a dynamical equation for *φ* (*x*, *t* =0) that contains the embedded motion for *Γ*(*t*) as the level set *φ* =0 can be derived similarly as in the volume of fluid conservation equation (13)

In phase field method, interfacial forces are modeled as continuum forces by smoothing interface discontinuities and forces over thin but numerically resolvable layers. This smoothing allows conventional numerical approximations of interface kinematics on fixed grids. The method has been used for investigating the problems governed by Navier-Stokes equations (Antanovskii, 1995; Jacqumin, 1996).

Boundary integral methods are designed to track the interface explicitly, as in front tracking methods, although the flow solution in the entire domain is deduced solely from information possessed by discrete points along the interface. The advantage of these methods is the reduction of the flow problem by one dimension involving quantities of the interface only.

Particle-based methods use discrete "particles" to represent macroscopic fluid parcels. Here, Lagrangian coordinates are used to solve the Navier-Stokes equations on "particles" having properties such as mass, momentum, and energy. The nonlinear convection term is modeled simply as particle motion and by knowing the identity and position of each particle, material interfaces are automatically tracked. By using particle motion to approximate the convection terms, numerical diffusion across interfaces (where particles change identity) is virtually zero; hence interface widths are well defined.

In moving mesh methods, the position history of discrete point's *x*i lying on the interface is tracked for all time by integrating the evolution equation, forward in time.

$$\frac{d\mathbf{x}\_l}{dt} = \boldsymbol{\nu}\_l \tag{14}$$

A moving mesh is Lagrangian if every point is moved, and mixed (Lagrangian-Eulerian) if grid points in a subset of the domain are moved. Mixed methods are used for mold filling simulations, where the mold computational domain can be held stationary and the molten liquid is followed with a Lagrangian mesh.

Besides these individual methods, there are some combined method such as coupled level set and volume of fluid method. This method has been developed to tackle the inefficiency of VOF method in calculating complex geometrical properties and problem of mass imbalance of level set method. In the coupled method, the LS function is used only to compute the geometric properties (normal and curvature) at the interface while the void fraction is calculated using the VOF approach. Ohta et al. (2007) and Sussman et al. (2007) developed a novel coupled LS-VOF method to determine the sharp interface for incompressible, immiscible two-phase flows for large value of density ratio.
