## Abstract

The lattice Boltzmann method for multi-component immiscible fluids is applied to simulations of the behaviour of a drop in a square pipe flow for various Reynolds numbers of 10<*Re*<500, Weber numbers of 0<*We*<250 and viscosity ratios of *η*=1/5, 1, 2 and 5. It is found that, for low Weber numbers, the drop moves straight along a stable position on the diagonal line of the pipe section, and it moves along the centre axis of the pipe for larger Weber numbers. We obtain the boundary of the two different behaviours of the drop in terms of Reynolds number, Weber number and viscosity ratio.

## 1. Introduction

The behaviour of particles and drops in Poiseuille flows is of interest not only in many engineering fields, such as the handling of slurries, colloids and ceramics, but also in biological fields in connection with blood flow in capillaries. A particularly important experimental study on the motion of spherical particles in a pipe flow was performed by Segré & Silberberg [1]. They discovered that neutrally buoyant particles in a pipe flow migrate laterally away from both the wall and the centreline and reach a certain equilibrium lateral position. Since then, many researches on the motions of spherical particles in pipe flows have been performed using experimental (e.g. [2,3]), theoretical (e.g. [4,5]) and numerical approaches (e.g. [6,7]).

On the other hand, it is found that deformable drops in pipe flows always migrate to the axis of the pipe [8]. That is, the behaviour of drops in pipe flows is very different from that of spherical particles in pipe flows. A schematic of the equilibrium positions of solid and deformable particles is illustrated in fig. 7 of the review by Munn & Dupin [9]. Recently, Kaoui *et al.* [10,11] investigated the lateral migration of a two-dimensional vesicle in a Poiseuille flow at vanishing Reynolds number (the Stokes limit) and found the bifurcation of the equilibrium lateral position of the vesicle as a function of its degree of deflation. However, the effects of drop deformability and of Reynolds number on drop migration have not yet been made clear.

In the present paper, in order to investigate the effects of drop deformability and of Reynolds number on drop migration, we apply the lattice Boltzmann method (LBM) for multi-component immiscible fluids with the same density [12] to simulations of the behaviour of a drop in a square pipe flow.

## 2. Numerical method

Non-dimensional variables are used as in Inamuro [13]. In the LBM, a modelled fluid, composed of identical particles whose velocities are restricted to a finite set of *N* vectors *c*_{i} (*i*=1,2,…,*N*), is considered. The 15-velocity model (*N*=15) is used in the present paper. The velocity vectors of this model are given by *c*_{i}=(0,0,0), (0,0,±1), (0,±1,0), (±1,0,0), (±1,±1,±1) for *i*=1,2,…,15.

In order to distinguish each of the two components in a fluid with the same density (*ρ*=1), we use an order parameter *ϕ* that has different constant values in each component and is changed inside the interface of two components. In the formulation of the method, the lattice kinetic scheme (LKS) proposed in Inamuro [14], which is an extension method of LBMs, is used. In the LKS, macroscopic variables are calculated without particle velocity distribution functions, and thus the scheme can save computer memory. The physical space is divided into a cubic lattice, and the order parameter *ϕ*, the pressure *p*(** x**,

*t*) and the velocity

**(**

*u***,**

*x**t*) of the whole fluid at the lattice point

**and at time**

*x**t*are computed as follows: 2.1 2.2 2.3 where and

*g*

^{c}

_{i}are equilibrium distribution functions, Δ

*x*is the spacing of the cubic lattice and Δ

*t*is the time step during which the particles travel the lattice spacing.

The equilibrium distribution functions in equations (2.1)–(2.3) are given by
2.4
and
2.5
where
2.6
and
2.7
with *α*,*β*,*γ*=*x*,*y*,*z* (subscripts *α*,*β* and *γ* represent Cartesian coordinates and the summation convention is used). In the above equations, *δ*_{αβ} is the Kronecker delta, *κ*_{f} is a constant parameter determining the width of the interface, *κ*_{g} is a constant parameter determining the strength of the surface tension and *A* is a constant parameter related to fluid viscosity as shown below. In equation (2.4), *p*_{0}(*ϕ*) is given by
2.8
where *a*,*b* and *T*_{ϕ} are free parameters determining the maximum and minimum values of the order parameter, and , respectively.

The kinematic viscosity *ν* and the surface tension *σ* are given by
2.9
and
2.10
with *ξ* being the coordinate normal to the interface.

## 3. Results and discussion

### (a) Computational conditions

A drop with an initial diameter of *D*_{d}=24Δ*x* is placed in a square pipe with a width of *D*=60Δ*x* and a length of *L*_{x}=120Δ*x* as shown in figure 1, and the periodic boundary with a pressure difference Δ*p* [6] is used at the inlet and the outlet of the pipe. The no-slip boundary condition is used on the side walls. Note that we consider a line of drops in an infinite long square pipe. The initial fluid velocity is zero, and the drop is placed at an initial position.

The dimensionless parameters for this problem are Reynolds number *Re*=*V* *D*/*ν*_{f} (fluid’s inertia/viscous force), the Weber number *We*=*ρV*^{2}*D*/*σ* (fluid’s inertia/surface tension) and the viscosity ratio *η*=*ν*_{d}/*ν*_{f}, where *V* is the time- and space-averaged velocity at the inlet after a steady flow is obtained. In the following calculations, the parameters are fixed at *a*=9/49, *b*=2/21, *T*_{ϕ}=0.55 ( and ) and *κ*_{f}=0.01(Δ*x*)^{2}. The other parameters, Δ*p*, *A* and *κ*_{g}, are changed so that 10<*Re*<500, 0<*We*<220 and *η*=1/5,1,2 and 5.

### (b) Equilibrium position of drop

After the pressure difference Δ*p* is applied, fluid flow is induced in the square pipe, and the drop is carried and deformed by the flow. Figure 2 shows the trajectories of the centre of mass of the drop put at four different initial positions on the *yz* plane for *Re*=100, *We*=12 and *η*=1. Considering the symmetry of the domain, we choose the initial positions in a quarter domain. It is found from figure 2 that the drop migrates to the same equilibrium position in spite of the initial positions. Therefore, we focus on the final equilibrium position of the drop on the *yz* plane.

Hereafter, the equilibrium positions are classified into three types. The first type is named ‘centre’, in which the equilibrium position (*y*_{c},*z*_{c}) of the centre of mass of the drop exists in the region of around the axis of the pipe. The second type is named ‘diagonal’, in which the equilibrium position (*y*_{c},*z*_{c}) of centre of mass of the drop exists in the region of *z*_{c}−Δ*x*≤*y*_{c}≤*z*_{c}+Δ*x* near the diagonal line of the pipe section excluding the centre region. The third type is named ‘transition’, in which the equilibrium position exists outside both the centre and the diagonal regions.

### (c) Effects of Reynolds and Weber numbers on equilibrium positions

We perform calculations for various Reynolds and Weber numbers with *η*=1 and investigate the equilibrium position of the drop on the *yz* plane. Figure 3 shows the calculated results classified into three types of equilibrium positions. In the figure, the broken line indicates the boundary between the centre and diagonal equilibrium positions. It is seen that, in the region of *Re*>100, the drop migrates to the centre of the pipe for roughly *We*>40, while the drop stays on the diagonal line of the pipe for roughly *We*<40. In addition, the critical Weber number separating the centre and diagonal equilibrium positions decreases with decreasing Reynolds number in the region of 10<*Re*<100 (i.e. in this region, the capillary number, *Ca*=*We*/*Re*=*μ*_{f}*V*/*σ*, becomes a key parameter). It indicates that, for low Reynolds numbers of *Re*<10, the drop might migrate to the centre of the pipe in spite of its deformability.

### (d) Deformed shape of drop in pipe flow

An initially spherical drop is carried on the induced flow and is deformed into a final shape at the equilibrium position. Figure 4 shows the time variations of the drop shape for *Re*=105, *We*=53.8 and *η*=1, which are the conditions of the centre equilibrium position. It is seen that finally the drop is deformed into a bullet shape at the axis. Figure 5 shows the time variations of the drop shape for *Re*=105, *We*=15.7 and *η*=1, which are the conditions of the diagonal equilibrium position. In this case, the drop is deformed into an ellipsoidal shape on the diagonal line.

### (e) From diagonal to centre

Figure 6 shows the relation between the distance of the equilibrium position from the axis and the Weber number for *Re*≈100 and *η*=1. It is found that the distance decreases as the Weber number increases and suddenly approaches zero (i.e. the drop approaches the axis) at around *We*=40. Figure 7 shows the effect of the viscosity ratio on the distance of the equilibrium position from the axis for *Re*≈100. The Weber number where the drop suddenly approaches the axis increases as the viscosity ratio becomes large.

## 4. Concluding remarks

The LBM for multi-component immiscible fluids has been applied to the simulations of the behaviour of a drop in a square pipe flow for various Reynolds numbers of 10<*Re*<500, Weber numbers of 0<*We*<250 and viscosity ratios of *η*=1/5, 1, 2 and 5. It is found that, for *Re*>100 and *η*=1, the drop moves straight along a stable position on the diagonal line of the pipe section for roughly *We*<40, and it moves along the centre axis of the pipe for roughly *We*>40. In addition, the critical Weber number separating the centre and diagonal equilibrium positions decreases with decreasing Reynolds number in the region of 10<*Re*<100. It indicates that, for low Reynolds numbers of *Re*<10, the drop might migrate to the centre of the pipe in spite of its deformability. Finally, we find that the critical Weber number increases with increasing viscosity ratio.

The ratio of the drop diameter to the width of the pipe is another key parameter of the problem. Also, the investigation of the behaviour of a drop in a circular pipe remains for future work.

## Footnotes

One contribution of 25 to a Theme Issue ‘Discrete simulation of fluid dynamics: applications’.

- This journal is © 2011 The Royal Society