Validation of the N-component method

Background/experimental observations

The behaviour of suspensions of solid and deformable particles is well documented experimentally. On the computer side, some models can deal with a large number of solid particles. However, deformable particles remained untouched so far. This chapter proposes to investigate deformability of particles by comparing the results of our N-component algorithm with the experimental data by Goldsmith. It validates by the same token our model against experiment.

A straight channel driven by pressure gradient is filled with suspended particles with fixed solidity

Hereafter, the solidity of our droplets is access by their surface tension and the ratio of their viscosity to the surrounding fluid's: phi. Goldsmith work consisted in determining the effect of deformability on the flow in this configuration. His motivation was to apply his results to the effect of the red blood cells deformability to the flow in capillaries and veinules. Several types of particles were considered, from rigid spheres to liquid droplets. He measured each time the flow profile

and came to a couple of conclusions, on the dependence of the velocity flow profile's blunting upon deformability of the suspended particles

• ‘In the case of solid particle, the velocity profile is determined solely by the suspension concentration and relative particle size and is independent of the flow rate’
• ‘Liquid drops and deformable particle suspensions, on the other hand, show a dependence of velocity profile upon flow rate’

We propose in this chapter to investigate how our N-component model reacts in this configuration and to ultimately, validate against Goldsmith's experimental observations.

Our blunted profiles have noticeable departure from parabolic profiles. We measure the blunting by the averaged departure, at normalised cross-duct distance (1/4 and 3/4). We call it $\beta$:
\begin{equation}
\label{beta} \beta \equiv \frac{
\bar{v}(\bar{y}=0.25)+\bar{v}(\bar{y}=0.75) }{2\times 0.75 }
\end{equation}
%
in which, 0.75$ is the height of a normalised parabola.

With this definition, a flat velocity profile is characterised by $\beta$=1.33 and a parabolic velocity profile by $\beta=1.00$.

Solid and deformable particles

In our simulations, solid particles are modelled as droplets with high viscosity phi=50 and high surface tension. Note that the validity and applicability of this assumption is discussed in section C(iii) and E of this page.

We investigate the dependence of beta upon the particle concentration Phi and the flow rate, taken to be quantified by the pressure difference deltaP.

i) Solid particles and flow rate

Goldsmith found that Beta does not depend upon delta P for solid particles. The following graph shows beta upon delta P for our 'solid particles':

Parameters of the simulation: phi=0.55, lambda=30 and high surface tension.

In this case, beta depends slightly on delta P. It can be explained

Solid particles show a slight decrease of beta with increasing delta P. This departure from experimental results shows the limit of the assumption used to model solid particles. After this graph, we consider the assumption as reasonable up to flow rates of 1e-5 (where the blunting is 'constant'). Values of deltaP above that, the flow rate is too high for the particles to dominate the flow. Surface tension or viscosity ration should then be used to make the particle even more 'solid'.

ii) Solid particles and particle volume fraction

The same configuration was used than for section i), with deltaP=1e-5. We varied the volume fraction phi from 0.15 to 0.6:

The expected increase of beta with increasing Phi is clear for solid particles. Note asymptotic behaviour of beta towards phi=0. Packing considerations restricted the maximum concentration of the suspension.

iii) Deformable particles

Deformable particles were modelled using Lambda=1 and low surface tension. All the other parameters remained the same. Recalling the non-dependance of solid particles upon delta P (figure of section C.(i)), deformable particles on the other hand should show significant dependence upon flow rate (delta P).

The expected decrease of beta with increasing deltaP validates the behaviour of the deformable particles in this configuration.

In-between deformable and solid particles

The previous results show a significant difference between the blunting of a suspension of solid particle and the blunting of a suspension of deformable particles with increasing flow rate. We recall that in our model, particles considered as solid and particles considered as deformable only differ by their viscosity and surface tension. This is consequently this parameter space that we propose to investigate here.

i) Simulation parameters

We need to carefully chose the parameter of this series of simulations to get a maximum resolution of beta against surface tension sigma and lambda. Ideally, for the given flow rate, we need beta=1.00 (parabolic profile) for solid particles and beta=1.33 for solid particles.

We consider Phi=0.6 (its maximum achievable value with packing restrictions), in order to ensure to recover large values of beta for solid particle (see section Cii above).
bulk behaviour \S 3\ref{rigid} above). With $\Lambda$ and $\sigma$ large, we limit $\Delta P $
to lie within the constant $\beta$ \lq solid particle' regime (\S 3\ref{rigid}). For $\Lambda$ and $\sigma$
small, the choice of $\Delta P$ should allow $\beta \rightarrow 1.00$ (deformable suspension bulk behaviour) to
be observable ( \S 3\ref{flexible}). Figure 1 shows the departure of $\beta$ from a parabolic as a function of $\Lambda$ and $\sigma$.
Data of this figure was compiled after the method outlined at the start of this section.

ii) In between: results

This gives us a calibration of the solidity of our droplets by the contours of 'iso-solidity' for the colloidal suspension's behaviour in a straight pipe.

This calibration provides us very useful information for future applications, on the top of validating our N-component model. It could be used for example to model solid particles in a pipe where deformability is and should not be important.

Chaining of the particles

Experimental results show that nano-particles in this configuration exhibit some alignment along the direction of the channel, what we denote here as chaining. Chaining is clearly visible in the following picture, taken from M. Frank et al.

Martin Frank, Douglas Anderson, Eric R. Weeks, and Jeffrey F. Morris, J. Fluid Mech. 493, 363-378 (2003)

This chaining effect can be explained by the considering the relative motion of each particle against a transversally neighbouring one. Since the flow is more or lest parabolic (depending on beta, see previous section),neighbouring particles across the channel a a significantly different velocity. This induces particles to pass each other and ultimately, to rearrange each other. This rearranging is done along two directions in 3D (along a section of the channel) and only 1 direction (direction perpendicular to the pipe) in 2D. For this reason, chaining in 2D is much more significant than in 3D as our model demonstrates (see below).

This picture depict the random initial configuration of the colloids.

This system evolves towards the following configuration

where chaining can clearly be observed.

To illustrate this phenomenon, the following couple of pictures explicit the path of each particle with time and there steady state distribution.