Casas et al 2019a - Revision history

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 Understanding the physical mechanisms that govern the appearance of clustering is of extraordinary importance, not only due to its relevance to the assessment of the validity of the single-particle theory (which includes the MRE), but also in understanding the statistical distribution of particles in space, statistics on the flow properties being sampled (of relevance to the simulation of chemical reactions, for example) and the prediction of the collision rates (of relevance to the study of initiation of rain and snow, for example). Of direct relevance to the present discussion is the study by Aliseda et al.&nbsp;<span id='citeF-4'></span>[[#cite-4|[4]]], where the settling velocity of spheres was measured experimentally in a turbulent channel flow. A notable increase in the settling velocity was observed with respect to quiescent fluid conditions, especially for Stokes numbers on the order of unity. Furthermore, this velocity increased monotonically with the overall volume fraction, indicating the effect of three-way coupling. A phenomenological model based on the idea of particle clusters locally altering the average velocity seen by its constituent particles turned out to explain the observations very well. It is remarkable that the whole study was performed under volume fractions well below the limit given in Eq.~[[#eq-2.32|2.32]], which clearly demonstrates the limitations of the single-particle paradigm.
-The estimates provided in Section [[#2.2.4.2 Drag modification|2.2.4]] were derived under the assumption of an unbounded array of particles characterized by an average solid fraction under the hypothesis of having a homogeneous suspension. Therefore, the existence of significant inhomogeneities as those caused by preferential concentration in turbulent flows, invalidate the theory and the derived estimates.&nbsp;<span id='citeF-4'></span>[[#cite-4|[4]]] proposed a phenomenological model to explain the effect of preferential concentration in a turbulent suspension of settling particles. They observed that the average settling velocity of the particles belonging to a cluster was reasonably well predicted by Eq.~[[#eq-2.33|2.33]], with
+The estimates provided in Section [[#2.2.4.2 Drag modification|2.2.4]] were derived under the assumption of an unbounded array of particles characterized by an average solid fraction under the hypothesis of having a homogeneous suspension. Therefore, the existence of significant inhomogeneities as those caused by preferential concentration in turbulent flows, invalidate the theory and the derived estimates. Aliseda et al.&nbsp;<span id='citeF-4'></span>[[#cite-4|[4]]] proposed a phenomenological model to explain the effect of preferential concentration in a turbulent suspension of settling particles. They observed that the average settling velocity of the particles belonging to a cluster was reasonably well predicted by Eq.~[[#eq-2.33|2.33]], with
 <span id="eq-2.39"></span>

@@ Line 1,287: / Line 1,287: @@
 |}
-Which, as stated, recover the first order correction due to Oseen for the drag force at <math display="inline">S^* \rightarrow 0</math> (and thus <math display="inline">\alpha _p \rightarrow 0</math>), see Eq.~[[#eq-2.11|2.11]], and also the classic expression by&nbsp;<span id='citeF-49'></span>[[#cite-49|[49]]], at <math display="inline">\mathit{Re}_p \rightarrow 0</math>.
+Which, as stated, recover the first order correction due to Oseen for the drag force at <math display="inline">S^* \rightarrow 0</math> (and thus <math display="inline">\alpha _p \rightarrow 0</math>), see Eq.~[[#eq-2.11|2.11]], and also the classic expression by Brinkman&nbsp;<span id='citeF-49'></span>[[#cite-49|[49]]], at <math display="inline">\mathit{Re}_p \rightarrow 0</math>.
 Figure [[#img-5|5]] shows the magnitude of the correction coefficient <math display="inline">C_K</math> for a wide range of solid volume fractions, for different values of <math display="inline">\mathit{Re}_p</math>. Note that the correction to the drag is significant, even for very small Reynolds numbers if the solid volume fraction is greater than 1x10<sup>-3</sup>, in accordance with the rule of thumb of Eq.~[[#eq-2.32|2.32]], and it still appreciable past the 1x10<sup>-4</sup> mark. <div id='img-5'></div>

@@ Line 1,331: / Line 1,331: @@
 &nbsp;<span id='citeF-44'></span>[[#cite-44|[44]]] reviewed some of these formulations, recommending the one by Zaichik and Alipchenkov&nbsp;<span id='citeF-381'></span>[[#cite-381|[381]]] as the most comprehensive and robust over a wide range of <math display="inline">\mathit{St}</math> values (see below).
-Nonetheless, there remain a number of open questions and inconsistencies in the literature that need to be addressed. For instance, there is a significant consensus that the strength of clustering peaks at <math display="inline">\mathit{St} \sim 1</math> &nbsp;<span id='citeF-253'></span><span id='citeF-178'></span>[[#cite-253|[253,178]]], However &nbsp;<span id='citeF-328'></span>[[#cite-328|[328]]] concludes that the level of clustering, measured using a Voronoï tessellation of space, is most strongly related to the Taylor Reynolds number, less so to the average particle volume fraction and negligibly on the Stokes number. Moreover, Uhlmann and Chouippe&nbsp;<span id='citeF-349'></span>[[#cite-349|[349]]] has found that clustering scaling seems to depend on the mode in which it is measured.
+Nonetheless, there remain a number of open questions and inconsistencies in the literature that need to be addressed. For instance, there is a significant consensus that the strength of clustering peaks at <math display="inline">\mathit{St} \sim 1</math> &nbsp;<span id='citeF-253'></span><span id='citeF-178'></span>[[#cite-253|[253,178]]], However Sumbekova et al.&nbsp;<span id='citeF-328'></span>[[#cite-328|[328]]] concludes that the level of clustering, measured using a Voronoï tessellation of space, is most strongly related to the Taylor Reynolds number, less so to the average particle volume fraction and negligibly on the Stokes number. Moreover, Uhlmann and Chouippe&nbsp;<span id='citeF-349'></span>[[#cite-349|[349]]] has found that clustering scaling seems to depend on the mode in which it is measured.
 Understanding the physical mechanisms that govern the appearance of clustering is of extraordinary importance, not only due to its relevance to the assessment of the validity of the single-particle theory (which includes the MRE), but also in understanding the statistical distribution of particles in space, statistics on the flow properties being sampled (of relevance to the simulation of chemical reactions, for example) and the prediction of the collision rates (of relevance to the study of initiation of rain and snow, for example). Of direct relevance to the present discussion is the study by Aliseda et al.&nbsp;<span id='citeF-4'></span>[[#cite-4|[4]]], where the settling velocity of spheres was measured experimentally in a turbulent channel flow. A notable increase in the settling velocity was observed with respect to quiescent fluid conditions, especially for Stokes numbers on the order of unity. Furthermore, this velocity increased monotonically with the overall volume fraction, indicating the effect of three-way coupling. A phenomenological model based on the idea of particle clusters locally altering the average velocity seen by its constituent particles turned out to explain the observations very well. It is remarkable that the whole study was performed under volume fractions well below the limit given in Eq.~[[#eq-2.32|2.32]], which clearly demonstrates the limitations of the single-particle paradigm.

@@ Line 1,053: / Line 1,053: @@
 |}
-So that the next order correction will be significantly smaller than the classical Faxén correction up to the point when <math display="inline">\mathit{Re}_p</math> is not small any more (in the limit of quasi-steady Stokes flow it holds <math display="inline">\nabla ^4\boldsymbol{u} = 0</math> exactly). This means that in practice the small <math display="inline">\mathit{Re}</math> requirement will guarantee that the Faxén corrections will make a good enough job and thus the restriction Eq.~[[#eq-2.9|2.9]], that is, <math display="inline">a/L\ll 1</math> by itself does not appear to be restrictive, but rather indirectly through the violation of <math display="inline">\mathit{Re}_p \ll 1</math>. A similar argument can be made with the other forces bearing their corresponding Faxén correction. A significant study supporting this conclusion in the context of isotropic turbulence was provided by&nbsp;<span id='citeF-171'></span>[[#cite-171|[171]]], who tested the performance of the Faxén terms for a neutrally buoyant particle in direct numerical simulation (DNS) of isotropic turbulence. Their conclusion was that the first effects of the finite size of the particles were well captured by these terms up to <math display="inline">a \sim 4L</math> (where <math display="inline">L</math> is the Kolmogorov microscale). From then on, inertial (finite <math display="inline">\mathit{Re}_p</math>) effects quickly kick in.
+So that the next order correction will be significantly smaller than the classical Faxén correction up to the point when <math display="inline">\mathit{Re}_p</math> is not small any more (in the limit of quasi-steady Stokes flow it holds <math display="inline">\nabla ^4\boldsymbol{u} = 0</math> exactly). This means that in practice the small <math display="inline">\mathit{Re}</math> requirement will guarantee that the Faxén corrections will make a good enough job and thus the restriction Eq.~[[#eq-2.9|2.9]], that is, <math display="inline">a/L\ll 1</math> by itself does not appear to be restrictive, but rather indirectly through the violation of <math display="inline">\mathit{Re}_p \ll 1</math>. A similar argument can be made with the other forces bearing their corresponding Faxén correction. A significant study supporting this conclusion in the context of isotropic turbulence was provided by Homann and Bec&nbsp;<span id='citeF-171'></span>[[#cite-171|[171]]], who tested the performance of the Faxén terms for a neutrally buoyant particle in direct numerical simulation (DNS) of isotropic turbulence. Their conclusion was that the first effects of the finite size of the particles were well captured by these terms up to <math display="inline">a \sim 4L</math> (where <math display="inline">L</math> is the Kolmogorov microscale). From then on, inertial (finite <math display="inline">\mathit{Re}_p</math>) effects quickly kick in.
 ===2.2.3 Nonsphericity effects===

@@ Line 825: / Line 825: @@
 This way, the contribution from the outer region becomes a correction of the steady drag force, whose leading term, given by <math display="inline">3/8 \mathit{Re}_p</math>, could be used to provide the order of magnitude of this correction.
-In order to illustrate how it could be used in practice, let us assume that we wish to establish the upper limit to the particle Reynolds number to ensure that the correction is smaller than 5  as a convention to fix the range of validity of the MRE. Then the leading term in Eq.~[[#eq-2.11|2.11]] predicts this to happen for <math display="inline">\mathit{Re}_p \approx 1.4x10^{-1}</math>, and by this point the error in this expression is only <math display="inline">\approx 3x10^{-2}</math>, taking as a reference the empirical drag coefficient by Clift et al. (1978) (which has a root mean square error of <math display="inline">\approx 1x10^{-2}</math> with respect to the empirical data gathered by&nbsp;<span id='citeF-50'></span>[[#cite-50|[50]]] covering the range <math display="inline">0 < \mathit{Re}_p < 4x10^{3}</math>). This means that its range of validity is large enough to estimate the first-effects of inertia for error tolerances lower than 5%. In Fig. [[#img-2|2]] all these different approximations to the drag force are compared. <div id='img-2'></div>
+In order to illustrate how it could be used in practice, let us assume that we wish to establish the upper limit to the particle Reynolds number to ensure that the correction is smaller than 5  as a convention to fix the range of validity of the MRE. Then the leading term in Eq.~[[#eq-2.11|2.11]] predicts this to happen for <math display="inline">\mathit{Re}_p \approx 1.4x10^{-1}</math>, and by this point the error in this expression is only <math display="inline">\approx 3x10^{-2}</math>, taking as a reference the empirical drag coefficient by Clift et al. (1978) (which has a root mean square error of <math display="inline">\approx 1x10^{-2}</math> with respect to the empirical data gathered by Brown and Lawler&nbsp;<span id='citeF-50'></span>[[#cite-50|[50]]] covering the range <math display="inline">0 < \mathit{Re}_p < 4x10^{3}</math>). This means that its range of validity is large enough to estimate the first-effects of inertia for error tolerances lower than 5%. In Fig. [[#img-2|2]] all these different approximations to the drag force are compared. <div id='img-2'></div>
 {| class="floating_imageSCP" style="text-align: center; border: 1px solid #BBB; margin: 1em auto; width: 100%;max-width: 100%;"
 |-

← Older revision	Revision as of 11:05, 31 January 2019
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 The well known Stokes solution of the steady, low Reynolds number flow past a sphere (see Eq.~[[#eq-2.31|2.31]]) is obtained by completely neglecting fluid inertia. By applying the no-slip boundary conditions on the particle surface and the far-field velocity conditions at infinity and expanding the stream function in powers of <math display="inline">\mathit{Re}_p</math>, the hydrodynamic force on the particle can be calculated, to leading <math display="inline">\mathit{Re}_p</math> order, yielding the drag force of the MRE without the Faxén terms (this is the problem solved by Stokes in 1851). However, if one wishes to increase the order of this approximation following the same procedure, one soon realizes that there is no way to fulfil the far field boundary conditions in this case, since the higher order contributions to the perturbation caused by the particle do not vanish at infinity. This phenomenon is known as Whitehead's paradox&nbsp;<span id='citeF-248'></span>[[#cite-248|[248]]].
-Its resolution was possible due to ideas from Oseen, who noted that the assumption by which inertial terms are disregarded (under Eq.~[[#eq-2.8|2.8]]) is only valid near the particle, where viscous effects dominate. But, far from the particle (in particular at a distance such that <math display="inline">\mathit{Re}_p = \mathcal{O}(1)</math>, see&nbsp;<span id='citeF-284'></span>[[#cite-284|[284]]]), the approximation breaks down. This implies that there is an inherent inconsistency in requiring that the Stokes solution be valid in an infinite domain, and that it is necessary to consider inertia far from the particle in order to calculate the higher order corrections to the drag force. The final word on the issue was nonetheless given by&nbsp;<span id='citeF-284'></span>[[#cite-284|[284]]], who, using the technique of matched asymptotic expansions, re-derived Oseen's inertial, first-order correction to the steady drag force, corrected a flaw in Oseen's original reasoning and added an additional term to the expansion:
+Its resolution was possible due to ideas from Oseen, who noted that the assumption by which inertial terms are disregarded (under Eq.~[[#eq-2.8|2.8]]) is only valid near the particle, where viscous effects dominate. But, far from the particle (in particular at a distance such that <math display="inline">\mathit{Re}_p = \mathcal{O}(1)</math>, see&nbsp;<span id='citeF-284'></span>[[#cite-284|[284]]]), the approximation breaks down. This implies that there is an inherent inconsistency in requiring that the Stokes solution be valid in an infinite domain, and that it is necessary to consider inertia far from the particle in order to calculate the higher order corrections to the drag force. The final word on the issue was nonetheless given by Proudman and Pearson&nbsp;<span id='citeF-284'></span>[[#cite-284|[284]]], who, using the technique of matched asymptotic expansions, re-derived Oseen's inertial, first-order correction to the steady drag force, corrected a flaw in Oseen's original reasoning and added an additional term to the expansion:
 <span id="eq-2.11"></span>

@@ Line 589: / Line 589: @@
 The core of this work is contained in Chapters [[#2 The Maxey&#8211;Riley equation|2]] to [[#4 Forward and backward-coupled particulate flows|4]]. Of these, Chapter [[#2 The Maxey&#8211;Riley equation|2]] is the most theoretical in character. It is devoted to the analysis of the Maxey&#8211;Riley equation, as a fundamental model for the description of the motion of individual particles in a fluid, which at this point is assumed to be described by a known field. Section [[#2.2 Range of validity|2.2]] systematically explores the range of validity of the model with respect to several criteria, first in terms of the nondimensional values that appear in the equation itself and later in terms of additional variables involving a selection of simplifications introduced a priori in the development of the theory. In Section [[#2.3 Scaling analysis|2.3]] we apply a scaling analysis to the different terms in the model, providing estimates of their relative magnitude that may be applied in practice to simplify the basic model by neglecting the less important terms. Section [[#2.4 Summary|2.4]] contains a summary of the main results of the chapter, including Tables [[#table-2|2]], [[#table-4|4]] and [[#table-5|5]], that list the most important numerical estimates.
-Chapter [[#3 The numerical solution of the Maxey&#8211;Riley equation|3]] shifts from the analytic point of view of Chapter [[#2 The Maxey&#8211;Riley equation|2]] to the study of different numerical techniques to solve the Maxey&#8211;Riley equation, providing a detailed description of the algorithms involved. We emphasize the treatment of the history force term, whose numerical solution remains challenging, and that is often ignored for this reason. In Section [[#3.2 Overview of approaches for the treatment of the Boussinesq&#8211;Basset term|3.2]] we provide a state of the art concerning the numerical treatment of this term, comparing the accuracies of several methods of quadrature. Section [[#3.3 Improvements on the MAE|3.3]] contains the description of our method of choice for the quadrature, proposed by&nbsp;<span id='citeF-353'></span>[[#cite-353|[353]]], and of the extensions and improvements we have developed based on it. Section [[#3.4 Overall algorithm|3.4]] contains the description of the overall algorithm for the time integration of the equation of motion including the history term. In Section [[#3.5 The fractional calculus perspective|3.5]] we make a stop to connect the general theory of fractional calculus to the Maxey&#8211;Riley equation, which had only been very superficially sketched before. This section appears at this point to take advantage of the terminology and concepts introduced in the previous developments. Section [[#3.6 Performance of the methodology|3.6]] is devoted to a systematic analysis of the accuracy and efficiency of the numerical method of solution, applying it to a sequence of benchmarks of increasing complexity. We close the chapter with a summary of the most important results and developments.
+Chapter [[#3 The numerical solution of the Maxey&#8211;Riley equation|3]] shifts from the analytic point of view of Chapter [[#2 The Maxey&#8211;Riley equation|2]] to the study of different numerical techniques to solve the Maxey&#8211;Riley equation, providing a detailed description of the algorithms involved. We emphasize the treatment of the history force term, whose numerical solution remains challenging, and that is often ignored for this reason. In Section [[#3.2 Overview of approaches for the treatment of the Boussinesq&#8211;Basset term|3.2]] we provide a state of the art concerning the numerical treatment of this term, comparing the accuracies of several methods of quadrature. Section [[#3.3 Improvements on the MAE|3.3]] contains the description of our method of choice for the quadrature, proposed by Hinsberg et al.&nbsp;<span id='citeF-353'></span>[[#cite-353|[353]]], and of the extensions and improvements we have developed based on it. Section [[#3.4 Overall algorithm|3.4]] contains the description of the overall algorithm for the time integration of the equation of motion including the history term. In Section [[#3.5 The fractional calculus perspective|3.5]] we make a stop to connect the general theory of fractional calculus to the Maxey&#8211;Riley equation, which had only been very superficially sketched before. This section appears at this point to take advantage of the terminology and concepts introduced in the previous developments. Section [[#3.6 Performance of the methodology|3.6]] is devoted to a systematic analysis of the accuracy and efficiency of the numerical method of solution, applying it to a sequence of benchmarks of increasing complexity. We close the chapter with a summary of the most important results and developments.
 Chapter [[#4 Forward and backward-coupled particulate flows|4]] moves even further toward practice, being eminently applied in nature. We consider a series of applications representative to different families of industrial problems, describing several developments involved in their solution as we go along. Section [[#4.2 Beyond the MRE|4.2]] discusses how the equation of motion considered in the previous chapters can be modified using empirical relations to extend its range of applicability beyond the limits studied in Chapter [[#2 The Maxey&#8211;Riley equation|2]], for this will be necessary in the industrial applications that are considered later on. In Section [[#4.3 The continuous-phase problem|4.3]] we introduce the fluid phase as a problem to be solved for the first time, for which we make use of a well-established stabilized finite element formulation. We describe it in sufficient detail to clarify the terminology and set the stage for the generalizations introduced in Section [[#4.3 The continuous-phase problem|4.3]]. In Section [[#4.4 Derivative recovery|4.4]] we discuss the problem of derivative recovery, as a step necessary to obtain accurate estimates of the derivatives of the fluid field, once a solution is produced by the fluid solver. We give a brief state-of-the-art and compare several alternatives compatible to the finite element method. The description of the particles-fluid coupling is described very briefly in Section [[#4.5 Forward coupling|4.5]] where we basically bring together the different algorithmic parts involved. We quickly turn to the first application example, which is the subject of Section [[#4.6 Application example: T-junction bubble trapping|4.6]]. It consists on the numerical simulation of the phenomenon of air bubble trapping in T-shaped pipe junctions, which was only recently first studied by Vigolo et al.&nbsp;<span id='citeF-356'></span>[[#cite-356|[356]]]. This test exemplifies the use of the one-way coupled strategy, with no inter-particle interactions, representative of low-Reynolds number internal flows with low-density suspensions of small particles. We continue on to the study of the next test application in Section [[#4.7 Application example: Particle impact drilling|4.7]], where we study particle impact drilling, a technology used in the oil and gas industries to produce high penetration rate drilling systems. Here we again make use of a one-way coupled strategy for the fluid-particles coupling, although this time we consider the inter-particle contact as well. This is by far the most detailed of the examples provided and corresponds to consultancy work made during the Doctorat Industrial. Before moving on to the final example, the theory related to the backward-coupled flow must be introduced first, based on the theory of multicomponent flows (its rudiments can be found in Appendix [[#13 Multicomponent theory fundamentals|H]]) and the general finite element formulation introduced in Section [[#4.3 The continuous-phase problem|4.3]], which we specialize for this problem. Following this theoretical interlude, we present the last application example, a fluidized bed of Geldard-D particles in Section [[#4.9 Application example: fluidized bed|4.9]]. We close with a summary of the chapter in Section [[#4.10 Summary|4.10]].

@@ Line 587: / Line 587: @@
 ==1.3 Outline of this document==
-The core of this work is contained in Chapters 2 [[#2 The Maxey&#8211;Riley equation|2]] to [[#4 Forward and backward-coupled particulate flows|4]]. Of these, Chapter [[#2 The Maxey&#8211;Riley equation|2]] is the most theoretical in character. It is devoted to the analysis of the Maxey&#8211;Riley equation, as a fundamental model for the description of the motion of individual particles in a fluid, which at this point is assumed to be described by a known field. Section [[#2.2 Range of validity|2.2]] systematically explores the range of validity of the model with respect to several criteria, first in terms of the nondimensional values that appear in the equation itself and later in terms of additional variables involving a selection of simplifications introduced a priori in the development of the theory. In Section [[#2.3 Scaling analysis|2.3]] we apply a scaling analysis to the different terms in the model, providing estimates of their relative magnitude that may be applied in practice to simplify the basic model by neglecting the less important terms. Section [[#2.4 Summary|2.4]] contains a summary of the main results of the chapter, including Tables [[#table-2|2]], [[#table-4|4]] and [[#table-5|5]], that list the most important numerical estimates.
+The core of this work is contained in Chapters [[#2 The Maxey&#8211;Riley equation|2]] to [[#4 Forward and backward-coupled particulate flows|4]]. Of these, Chapter [[#2 The Maxey&#8211;Riley equation|2]] is the most theoretical in character. It is devoted to the analysis of the Maxey&#8211;Riley equation, as a fundamental model for the description of the motion of individual particles in a fluid, which at this point is assumed to be described by a known field. Section [[#2.2 Range of validity|2.2]] systematically explores the range of validity of the model with respect to several criteria, first in terms of the nondimensional values that appear in the equation itself and later in terms of additional variables involving a selection of simplifications introduced a priori in the development of the theory. In Section [[#2.3 Scaling analysis|2.3]] we apply a scaling analysis to the different terms in the model, providing estimates of their relative magnitude that may be applied in practice to simplify the basic model by neglecting the less important terms. Section [[#2.4 Summary|2.4]] contains a summary of the main results of the chapter, including Tables [[#table-2|2]], [[#table-4|4]] and [[#table-5|5]], that list the most important numerical estimates.
 Chapter [[#3 The numerical solution of the Maxey&#8211;Riley equation|3]] shifts from the analytic point of view of Chapter [[#2 The Maxey&#8211;Riley equation|2]] to the study of different numerical techniques to solve the Maxey&#8211;Riley equation, providing a detailed description of the algorithms involved. We emphasize the treatment of the history force term, whose numerical solution remains challenging, and that is often ignored for this reason. In Section [[#3.2 Overview of approaches for the treatment of the Boussinesq&#8211;Basset term|3.2]] we provide a state of the art concerning the numerical treatment of this term, comparing the accuracies of several methods of quadrature. Section [[#3.3 Improvements on the MAE|3.3]] contains the description of our method of choice for the quadrature, proposed by&nbsp;<span id='citeF-353'></span>[[#cite-353|[353]]], and of the extensions and improvements we have developed based on it. Section [[#3.4 Overall algorithm|3.4]] contains the description of the overall algorithm for the time integration of the equation of motion including the history term. In Section [[#3.5 The fractional calculus perspective|3.5]] we make a stop to connect the general theory of fractional calculus to the Maxey&#8211;Riley equation, which had only been very superficially sketched before. This section appears at this point to take advantage of the terminology and concepts introduced in the previous developments. Section [[#3.6 Performance of the methodology|3.6]] is devoted to a systematic analysis of the accuracy and efficiency of the numerical method of solution, applying it to a sequence of benchmarks of increasing complexity. We close the chapter with a summary of the most important results and developments.
@@ Line 597: / Line 597: @@
 We provide a series of appendices that carry substantial content. Our intent in doing so has been more to lighten the main text than to expand it with additional content. Perhaps the idea of ''modularity'', associated to the programming principle, has contaminated our writing style, but we hope it helps comprehension. For instance Appendix [[#6 DEM specifics|A]] contains most of the little text devoted to the discrete element method while Appendix [[#13 Multicomponent theory fundamentals|H]] is devoted to the description of the continuum theory related to the backward-fluid flow. The other appendices include formulas and numerical data of interest mainly to the interested reader willing to program the associated algorithms.
 Finally, the Nomenclature section provides a quick guide for the symbols used throughout. Only the symbols that are exclusively used in a single context, close to their definitions, have been left out of the list, to avoid overpopulating it.
 =2 The Maxey&#8211;Riley equation=

@@ Line 558: / Line 558: @@
 <li>To study the range of applicability of the Maxey&#8211;Riley equation as a model for the motion of the individual particles submerged in a fluid, improving the current knowledge on the subject and generating, where possible, practical estimates of direct application to numerical modelling. </li>
 <li>To study current alternatives for the numerical treatment of the history term in the equation of motion and compare them. </li>
-<li>To improve on the method of quadrature (the history term involves a time integral, as will be explained later) proposed by&nbsp;<span id='citeF-353'></span>[[#cite-353|[353]]] and provide a detailed study of its efficiency and accuracy, providing convincing evidence that it is not necessary to neglect this term to have an efficient numerical method. </li>
+<li>To improve on the method of quadrature (the history term involves a time integral, as will be explained later) proposed by van Hinsberg&nbsp;<span id='citeF-353'></span>[[#cite-353|[353]]] and provide a detailed study of its efficiency and accuracy, providing convincing evidence that it is not necessary to neglect this term to have an efficient numerical method. </li>
 <li>To report an account of relevant application examples of the proposed strategy with interest to the industry, as well as of the different technologies developed for their particular requirements. </li>
 <li>To generalize a stabilized finite element method and use it to discretize the backward-coupled flow equations. </li>