Simulating the HeleShaw flow in the presence of various obstacles and moving particles
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Abstract
A generalization of the approach developed in the recent papers by the authors is presented. It aims to provide a description of the HeleShaw cell in the presence of multiple small obstacles/moving particles. We perform an asymptotic analysis of the dynamics of the moving boundary and the moving particles. For this, a modification of Maz’ya–Movchan–Nieves uniform asymptotic formula for the Green’s function of the mixed boundary value problem for the Laplace equation in a multiply connected domain is utilized. The paper contains extensive numerical analysis, accounting for various physical mechanisms of particle movement in the HeleShaw flow.
Keywords
HeleShaw flow Point source/sink Moving obstacles Green’s function Neumann function Mixed boundary value problem Asymptotic analysis Numerical simulation1 Introduction
We consider a slow viscous flow in a narrow space between two parallel plates in presence of various fixed or moving obstacles. The model is a modification of the wellknown HeleShaw moving boundary value problem [1, 2] which in turn is an analogous to a onephase Stefan problem [3] and is a special case of the Navier–Stokes problem (see, e.g., [2]).
It is supposed that the driving mechanism for the flow is a onepoint source/sink. The movement of particles in the flow depends not only on the source/sink intensity, but also on the friction between obstacles and plates, as well as on the particle interaction.
The considered problem is motivated by applications encountered in industrial settings. Foremost is the often noted example of injection moulding, where a high viscosity fluid is pumped through a strategically placed hole into a mould of prescribed shape. During the injection process air must be allowed to escape from the mould, and as such vents must be properly placed around the mould at the locations which will be the last to be filled by the fluid. In the simple case this reduces to studying the free moving boundary of a HeleShaw cell containing stationary obstacles (see [4] for a more extensive description).
Meanwhile the field of microfluidics presents a more recent domain in which the modeling of multiple moving inclusions within the HeleShaw cell may prove useful. An interesting example was given in [5], where a HeleShaw cell with fine patterns etched into the channel roof was utilized to control the movement of water drops within a fluid flow. Such a system has potential benefits when performing parallel experiments in molecular and cell biology, where using twodimensional arrays enables the investigator to, in effect, obtain simplified data sets for multiple experiments simultaneously.
Additionally, the method utilized in this paper to represent inelastic collisions means that the model may be used to examine problems where particles coalesce (or ’clump’), such as the study of biological systems where agglutination occurs.
Finally it is the hope of the authors that this model may later assist in the developing of a numerical tool capable of simulating the mass transport mechanism for the fluid flow in a narrow channel. This type of problem is most notably encountered in hydraulic fracturing technologies, with respect to the slurry flow inside the fracture (see, e.g., [6, 7] and references therein).
Following [8] we reduce our problem to a mixed boundary value problem for the Laplace equation in a multiply connected domain. Supposing existence of moving particles in the flow we have to add extra equations describing this movement (also accounting for the different friction properties of the particles). Its geometric solution (parametrization of the moving front and trajectories of the moving particles) is sought from equations given in terms of the Green’s function for the above mixed boundary value problem. Our main theoretical tool is an asymptotic analysis of the model based on the approximation of the Green’s function.
The properties of Green’s functions for various differential operators subject to different boundary conditions are of great importance in finding the solution to the above problems, and thus for many applications (see, e.g. [9, 10]). There are only a few known results for the exact representation of Green’s functions which correspond to the simplest boundary conditions and simple geometry (see, e.g. [11, 12]). Therefore, elaborating on novel asymptotic and approximate methods becomes crucial to the theory and application of Green’s functions. The use of asymptotic methods for approximation of Green’s functions goes back to the classical paper by Hadamard [13], where the method of regular perturbation was performed. Recently, V. Maz’ya, A. Movchan and M. Nieves developed the methods of asymptotic approximation and obtained several uniform asymptotic formulas for Green’s functions related to different boundary value problems for a number of differential operators in the case of singular perturbations of the domains. In [14] a number of uniform asymptotic approximations of the Green’s function for mixed and Neumann problems in domains with small holes/inclusions were obtained. A review of Maz’ya–Movchan–Nieves results on asymptotic expansion of the Green’s function for elliptic problems in perforated domains is presented in [15]. Special attention is paid to Green’s kernel for multiply connected domains in \({\mathbb {R}}^3\). For domains in \({\mathbb {R}}^2\) with several small inclusions a uniform asymptotic formula for the Green’s function in transmission problems of antiplane shear was given in [16]. An extended exposition of methods and results for asymptotic approximation of Green’s functions is presented in [17]. We use these results in the study of the HeleShaw problem in domains with several small inclusions.
This article presents an essential extension of the method proposed in [18] (for a single fixed obstacle in the HeleShaw flow) and in [19] (where an individual particle is supposed to move in the flow). Our first aim here is to show that the developed method continues to obtain comparably good results for several (fixed or immobile) inclusions. We note that the density, size and shape of the obstacles do not constrain our asymptotic analysis (though the computations in this paper are only dealing with circular objects; this choice has the advantage of allowing us to present the discovered features in a more straightforward manner). The more novel challenge is for us to take into account the various interactions (elastic, fully nonelastic and intermediate) between particles. We perform numerical simulations involving different scenarios of the obstacles’ behavior, as well as conducting simulations with differing numbers of inclusions (up to 540).
Our paper is organized as follows. The problem formulation is given in Sect. 2, with the final set of equations relative to our model collected under the title Problem HS _{ M }.
A modification of the Maz’ya–Movchan–Nieves formula for the Green’s function of a mixed boundary value problem for the Laplace equation is presented in Sect. 3. It accounts for the geometrical and physical assumptions of the considered model. The components of the proposed uniform asymptotic formula (3) are additionally presented in this chapter, while further discussion of their nature and possible representations is relegated to the supplementary material.
The final system of equations is presented in Sect. 4.1, and a computational model is developed to solve the approximate system in Sect. 4.2.
In Sect. 5 the accuracy of the computational model is examined, followed by the providing of numerical simulations involving different scenarios for the obstacles’ movement, examining the effect of particle inclusion on the fluid flow.
We conclude in Sect. 6 with the discussion of the obtained results highlighting the different features of the flow in presence of moving obstacles.
2 Problem formulation
We consider a twodimensional potential flow of viscous incompressible fluid in the HeleShaw cell caused by a source/sink placed at the origin 0 with finite number of rigid obstacles in the flow, which is supposed to be very slow. Each obstacle is moving without rotation due to the pressure in the flow. Their movement can be faster or slower depending on specific friction between an obstacle and the plates of the HeleShaw cell.
Let us denote by \(\varOmega (t)\ni 0\) an open simply connected domain encircled by the free boundary \(\varGamma (t) = \partial \varOmega (t)\) at each time instant t ≥ 0. We study the flow at (in principle unknown) time interval \(I = [0, T]\) which could be rather small in the case of the sink (extraction of the fluid).
The obstacles are represented by fixed shape closed subdomains of \(\varOmega (t)\). To avoid technical difficulties, we accept a circular shape of the the obstacles of the radius \(\varepsilon _k\) and denote the (moving) center of obstacles by \({\mathbf{z}}_k(t)\), i.e. \(F_k(t) := \left\{ {\mathbf{z}}\in {\mathbb {R}}^2: {\mathbf{z}}  {\mathbf{z}}_k(t) \le \varepsilon _k\right\} , k = 1, \ldots , M\). The domain occupied by the fluid is denoted \(\varOmega _M(t) = \varOmega (t)\setminus \bigcup \limits _{k=1}^{M} F_k\).
 1.Particle distribution:

The initial distribution of the particles/obstacles must not place them too close to the boundary, with the minimum distance determined by the radius of the particle. Stated explicitly: \(dist \{\delta F_k(0),\delta \varOmega (0)\}=b_k > \varepsilon _k\).

Similarly the obstacles/particles must not come too close to the source/sink, with the radius of the particle determining the minimum: \(dist \{\delta F_k(0), {\mathbf{0}} \} = q_k > \varepsilon _k\).

The inclusions must not overlap with each other, this can be stated as \(\bigcap \limits _{k=1}^{M} F_k = \emptyset \). More explicitly we have, for all \(k,l=1,\ldots ,M, k\ne l, t\in I\): \(d_{k,l}=  z_k(t)z_l(t)  > \varepsilon _k + \varepsilon _l\)

 2.
Domain size: The characteristic size of the initial domain \(\varOmega (0)\) is of order \(10^0\), with bounds on the minimum and maximum initial distance as follows: \(0<c\le \min dist\{0,\varOmega (0)\} \le \max dist\{0,\varOmega (0)\}\le 1\), where \(c<b_k+2\varepsilon _k\).
Now we are at the position to formulate our problem (cf. [2, 8]) which is a generalization of the models considred in [18, 19].
Problem HS _{ M } Find 2M + 3 unknown realvalued functions
 (1)
\({\mathbf{w}}(s, t)\in \varGamma (t)\) for all \((s, t)\in \partial \, {\mathbb {U}} \times I\);
 (2)
\({\mathbf{w}}(\cdot , t) : \partial \, {\mathbb {U}} \rightarrow \varGamma (t)\) is a \({\mathcal {C}}^2\)diffeomorphism for each fixed \(t\in I\);
 (3)
\({\mathbf{w}}^{(0)}(s) = {\mathbf{w}}(s, 0)\) is a given \({\mathcal {C}}^2\)diffeomorphism of the unit circle \(\partial \, {\mathbb {U}}\), which describes the boundary \(\varGamma (0)\) of initial domain \(\varOmega _{M}(0)\);
 (4)\({\mathcal {G}}({\mathbf{z}}; \zeta ; t)\) is Green’s function of the operator\(\triangle \) subject to the mixed boundary value problem, i.e. for each fixed \(t\in I\)$$\begin{aligned}&\Delta {\mathcal {G}}({\mathbf{z}}; \zeta ; t) + \delta ({\mathbf{z}}  \zeta ) = 0, \; {\mathbf{z}}, \zeta \in \varOmega _{M}(t);\\&{\mathcal {G}}({\mathbf{z}}; \zeta ; t) = 0, \; z\in \varGamma (t), \zeta \in \varOmega _{M}(t);\\&\frac{\partial {\mathcal {G}}({\mathbf{z}}; \zeta ; t)}{\partial n_{\mathbf{z}}} = 0, \; z\in \partial F_k(t), \zeta \in \varOmega _{N}(t),\quad k=1,\ldots , M; \end{aligned}$$
 (5)
\(\partial _{t}\, {\mathbf{w}}(s, t) =  \frac{Q h^2}{12 \mu }\cdot \nabla {\mathcal {G}}({\mathbf{w}}(s, t); O; t)\) for all \((s, t)\in \partial \, {\mathbb {U}} \times I\);
 (6)
\(\begin{aligned}\frac{d^2 {\mathbf{z}}_k(t)}{d t^2} + \frac{\kappa _k \pi \varepsilon ^2_k}{m_k} \frac{d {\mathbf{z}}_k(t)}{d t}&= \frac{Q \varepsilon _k}{m_k} \int \limits _{0}^{2 \pi } {\mathcal {G}}(z_{k,1}(t) + \varepsilon _k \cos \theta , z_{k,2}(t)\\&\quad + \varepsilon _k \sin \theta ; \zeta ; t) \cdot {\mathbf{n}}^{(k)}(\theta ) d \theta ;\end{aligned}\)
 (7)
\({\mathbf{z}}_k(0)={\mathbf{z}}_k^{(0)}, {\mathbf{z}}_k^{\prime }(0)={\mathbf{z}}_k^{(1)}\).
Here h is the width of the HeleShaw cell, μ is viscosity coefficient of fluid in the cell, Q is the strength of the source/sink, \(\kappa _k\) are the friction coefficients for the contact of kth obstacle and the plates of the cell, \(m_k = \pi \varepsilon ^2_k \rho _k\) is the mass of the kth obstacle, \({\mathbf{n}}^{(k)}(\theta )\) is the internal normal vector on the boundary of kth obstacle, \(k=1, \ldots , M\).
The function \({\mathbf{w}}(s, t) = (w_1(s, t), w_2(s, t))\) determines the parametrization of the unknown free boundary \(\partial \varOmega (t)\). Meanwhile the movement of each obstacle can be described in terms of the location of its variable center \({\mathbf{z}}_k(t)\), which is permissible as the inclusions are moving as rotation free rigid bodies. The assumption of nonrotation is valid as the small size and circular shape of each particle mean that the pressure function around its boundary can be assumed constant, while the term for friction with the fluid will be negligible compared to that between the particle and the plates of the cell. It is worth noting that the lubricative force between particles is not modeled in this paper, as it is not required to test the accuracy and limitations of the proposed system, however this can be added at a later time without requiring modification of the underlying analytical formulation.
Existence of the solution to the above problem can be shown in a way similar to that for existence of the solution for the flow in the HeleShaw cell with air bubbles in the flow (see, e.g. [21] and references therein).
The aim of our study is to get an approximate solution to the problem HS _{ M }, and to create a numerical system based upon it which describes the different behaviour of small obstacles in the HeleShaw flow.
3 Uniform representation of Green’s function
The method of uniform asymptotic approximation of the Green’s function related to different boundary value problems for a number of differential operators in singularly and regularly perturbed domains was created and developed by V. Maz’ya, A. Movchan and M. Nieves and is summarized in the recent book [17]. This method has additionally been used successfully to examine the effect of an individual obstacle or particle in the HeleShaw cell [18, 19]. In our analysis of Problem HS _{ M } slight modifications to the asymptotic formula for \(N_{\varepsilon }\) [16, (7.1)] are required, however as the method remains similar it won’t be repeated here in full. The full details of the formulation and method are explained more fully in the supplementary material.
Here \({\mathcal {N}}^{(k)}\) are solutions to the modified Neumann problems in the exterior of \(cl\, \omega _{\varepsilon }^{(k)}, {{\mathcal {D}}}^{(k)}\) are dipole vectors corresponding to inclusion \(\omega _{\varepsilon }^{(k)}\) and H corresponds to the regular part of the Green’s function.
The uniform estimate for the remainder in the above mentioned [16, Thm 7.1] is proved in \(L_{\infty }\)norm. It remains valid in our case for any multiply connected domain \({\varOmega _{M}(t)}\) with sufficiently small interval I of time variable t.
In addition to the formulae for these functions solving Problem HS _{ M } requires both the derivatives and integrals to be calculated. These calculations are again placed in the Complimentary material, and the results are expressed in Sect. 4.1.
4 Computational algorithm
With the asymptotic relationships required now derived, it is possible to articulate the approximate solution to Problem HS _{ M } and create a numerical model capable of performing simulations for multiple particles within the HeleShaw cell. Additionally, the results of these simulations can be used to more carefully examine the effect of initial particle distribution on both the evolution of the fluid boundary and the dynamic behaviour of the particles themselves.
4.1 Final system of differential equations
4.2 Description of the scheme
In all simulations, as the asymptotic approximation reduces in accuracy near the source/sink, or near to the fluid boundary, three conditions are imposed to prevent the results being adversely affected. The first two conditions are that, for any particle k, there must always be a minimum distance of \(\varepsilon _k\) to the source/sink and to the fluid boundary. Additionally it will be required that the minimum radial distance between the source/sink and the fluid boundary always remains larger than 0.1.
5 Numerical examples and discussions
5.1 Computational accuracy
The second benchmark is built in exactly the same way on the assumption that the reference solution, \({\tilde{\varvec{w}}}(s,t)\), describes a circular shape. Its evolution in time, defined by the radius R(t), can be easily determined from the fluid balance.
For the second benchmark example, the circular domain, we obtained similar trends (for this reason we do not illustrate them), but the solution accuracy was slightly better.
5.2 Collision strategies and particle interactions
In any fluid containing multiple particles and an explicit boundary there exist a multitude of forces which aren’t directly accounted for within the present model. Rotational forces, such as the Saffman and Magnus forces, needn’t be considered here as the model assumes that the particles are nonrotating. Further effects such as the virtual mass and Basset forces on particle movement are likely to be small compared to the friction term from their contact with the cell wall (see [27] for more information on related forces).
The primary situation in which forces arise for which additional algorithms are required is the case of particle collision. In principle use of the Green’s function should prevent particle overlap, however as an asymptotic approximation is used this possibility isn’t eliminated within the final system of equations directly. Complicating matters further is the fact that, for particle collisions within a fluid, the lubricative force prevents both perfectly elastic and inelastic collisions, and it also decelerates particles near to the boundary (see [28]). Determining the exact effect of this force on the final dynamics of the system however is beyond the scope of this paper.
To compensate for this two separate steps are taken. The first is to increase the friction coefficient of particles which become close to the boundary, which can be easily achieved and prevents any objects leaving the fluid domain. The second is that, in the case of particle collisions, both the perfectly elastic and inelastic cases will be modeled. While this will not produce the most accurate representation of the dynamics of particle interaction within the fluid these two cases provide the opposing ends of the spectrum, and as such if both can be modeled then the more accurate case involving the lubricative force can be added later through the use of additional algorithms. Further these cases are very computationally efficient, and will ensure the model does not become overly cumbersome.
In the conducted tests, the presence of a small number of particle collisions lead to only a minimal increase in the computation time, and the decrease in accuracy was negligible compared to that induced by other parameters. Additionally it is clear from results concerning the relative boundary deformation between cases (Fig. 8) that the difference between perfectly elastic and inelastic collisions is relatively small compared with the total change in boundary deformation. The fact that the collision type has only a minor influence on the deformation of the free boundary only improves the case for these efficient approximations over the more intensive option of computing the lubricative force.
5.3 Particle position and parameters
With the abilities of the model established, we can now use it to examine the effect of particle inclusion on the fluid flow within the HeleShaw cell. The flow can be best understood as traveling in the shortest possible straight line to the boundary (sink) from the source (boundary). Particles placed in the path of this line will redirect the flow around it. Therefore positioning objects near to the source/sink will disrupt the largest portion of the domain, as the proportion of the flow disturbed is greatest, and the fluid on the opposite side of the source/sink will extend/drain faster to compensate. Conversely obstacles near the boundary will produce very localized effects, although these effects will be far larger in magnitude.
As a result of the fluid flow behaving in this manner the shape of the domain, size of the inclusions and movement of the particles over time will play a crucial role in determining the effect of particle placement on the fluid boundary evolution. Particles with a low (or zero) initial velocity, or a sufficiently high friction coefficient, will remain almost stationary relative to the movement of the fluid. In such situations the large magnitude localized effects are often only present over short time periods. Similarly a small inclusion size will only disrupt a small portion of the fluid flow, and as such the overall effects will be minimal.
The localized effects for particles near the boundary, and wider domain effects for particles near the source, are both clearly present in Fig. 10a. Additionally it can be seen in Fig. 10b that these effects compound as the number of particles increases. These results perfectly match the previous description in terms of the fluid flowing between the source and the boundary. Further the results are well within the expected level of accuracy for this problem variant (see Fig. 4, \(\varepsilon =10^{1}\)).
5.4 Simulations with many particles
While there is no theoretical maximum for the number of particles which can be simulated the computation time will obviously become a limiting factor. In practice the particle numbers which can be reasonably computed is far more dependent on the distance between each particle and; adjacent particles, the source/sink and the boundary. As a result, with proper initial distribution, simulations involving hundreds of particles can be easily completed within a reasonable computation time (typically 1.5–4.5 h).
It is clear from Fig. 12 that the deformation of the boundary in the case with many particles is far more complicated than those previously examined, with systems containing particles with a similar distribution and identical volume fractions resulting in notably different changes to the fluid boundary. That the systems with particles in an ordered distribution had a more periodic and predictable effect on the boundary deformation is not surprising, however the fact that simulations involving large numbers of particles lead to a far smoother boundary deformation in both cases, and can easily be conducted using this model, provides a method by which internal effects can be more readily studied.
6 Outline and discussion
An asymptotic approximation of the fluid flow within a HeleShaw cell containing multiple freemoving particles has been been obtained, which models both the movement of the fluid boundary and the inclusions. A model based on this system of equations has been created in a MatLab environment, which is capable of dealing with situations involving hundreds of particles and any potential collisions between them. The accuracy of the final model was assessed for various numbers and sizes of inclusions.
A brief examination into the effect of the particle inclusion on the fluid flow, primarily through the resulting boundary distortion, was conducted. The key finding was the dual nature of the effect of initial particle distribution on the fluid boundary, with particles close to the boundary causing very localized effects, while those near to the source/sink affect the wider domain.
We now have a credible tool with which to simulate the fluid flow in a narrow channel subject to the presence of multiple different inclusions and obstacles. This has clear applicability, namely with regards to investigating the effective properties of the fluid when it contains a large number of particles.
Footnotes
 1.
It is worth noting that the constants \(c, b_k, d_{k,l}\) and \(q_k\) aren’t dependent on any individual \(\varepsilon _k\).
 2.
Unknown magnitudes \({\mathbf{w}}, {\mathcal {G}}, {\mathbf{z}}_0\) depend on time t from a rightsided neighborhood I of t = 0. In fact, for our problem we need to determine the value of \({\mathcal {G}}({\mathbf{z}}; \zeta ; t)\) only at the point \(\zeta = O\), but we keep the extra variable \(\zeta \) for computational reasons.
 3.
 4.
An analytical solution in the case of a single circular inclusion is forthcoming (J.S. Marshall: Analytical solutions for HeleShaw moving boundary flows in the presence of a circular cylinder), but there are currently no published solutions for circular inclusions with which to compare.
Notes
Acknowledgments
D.P, S.R, and G.M. gratefully acknowledge the support of the European Union Seventh Framework Marie Curie Programme PARM2 (Project Reference: PIAPGA2012284544PARM2), and M.W. acknowledges the European Union FP7 project INTERCER2 (Reference: PIAPGA2011286110INTERCER2). The authors are grateful to Dr. Michael Nieves for fruitful discussion on the asymptotic approximation of Green’s function.
Supplementary material
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