What is simmsus?

SIMMSUS is a research code, initially written in FORTRAN that simulates the motion of a system of interacting particles. These particles can be simulated in different scenarios and may interact through different physical mechanisms. The code was initially developed for studying the physics of magnetic spherical particles in suspensions in order to better understand the properties of magnetic fluids (ferrofluids).

History of the project

The code was initially developed for studying the physics of magnetic spherical particles in suspensions in order to better understand the properties of magnetic fluids (ferrofluids). Since the beggining of the development, the code was designed to consider the effects of Brownian motion, long range dipole-dipole and hydrodynamic interactions. The first physics simulated through SIMMSUS was the study of the alignment of the particles in the direction of an applied steady-state magnetic field to see wether the code was capable of capturing the behavior of the equilibrium magnetization predicted by theoretical asymptotic models available on the literature [1].

The comparison between the predictions of SIMMSUS and the asymptotic models served as a first validation of the numerical code. After this initial validation, we have implemented several additional features on the code and have used SIMMSUS to explore different physical situations regarding the behavior of magnetic fluids. Many of the studies conducted through SIMMSUS have been published in prestigious academic Journals, such as Physics of Fluids [2,3], Journal of Magnetism and Magnetic Materials [4,5,6], Powder Technology [7,8] and Mechanics Research Communications [9,10].

In its present version SIMMSUS is capable of simulating Brownian and non-Brownian suspensions and the user can turn on/off specific physical mechanisms in order to build a customized scenario. For example the user can mix magnetic particles with non-magnetic particles, turn on/off the gravitational field, apply an oscillatory or steady-state magnetic field over the particles, apply a simple or oscillatory shear over the particles, simulate different initial configurations (ordered, random or spherical distributions), simulate mono or polydisperse particles (with the same radius or with different radius) and the list goes on.

Regarding the application of time-dependent magnetic fields, we have conducted a rigorous study on the dynamical susceptibility response of ferrofluids using SIMMSUS and have validated the dynamical solution of the rotational motion of the dipole moments of the particles provided by the code by comparing the numerical solution with an asymptotic theoretical model [11]. The code seems to predict with excellent precision the dynamical behavior of the internal structure of ferrofluids.

General structure of the source-code

The code contains 8 files, from which 7 are necessary to produce the executable file (program) after the compilation and 1 is a configuration file. These are the following files:

• simmsus.f90
• input.f90
• main.f90
• subroutines.f90
• statistics.f90
• variables.f90
• makefile

In order to compile the source-code you must open a Linux terminal and run the make command. For the first time you should run the make command twice. In the first time the compiler will produce the modules from the files variables.f90 and subroutines.f90 and the second run will use these recently created modules to produce the executable file simmsus.ex.

Since the origins of its earlier development, SIMMSUS has been tested and compiled exclusively on Linux using the Intel Fortran Compiler and up to this version of the manual we have no idea if it compiles or runs using other compilers or operational systems.

In order to run a simulation, the user will need only 2 files: the executable file (simmsus.ex) and the input configuration file simconfig.dat. The user can drag these two files to a new folder to run a new set of numerical simulations defined on the configuration file simconfig.dat.

The configuration file is a text file with several questions that the user should answer in order to direct the path that SIMMSUS should cross in order to produce a specific set of simulations for the intended physics. We will talk about this file later. Figure bellow shows how the source-code files are related to each other.

The file simmsus.f90 basically calls other files from the source code structure in a straightfoward order. The first call is related to the reading of the simulation data. The file input.f90 reads the configuration file simconfig.dat and storage in logical, real and integer variables all the information necessary to run a set of simultaneous configurations. SIMMSUS may run a single simulation or perform several simultaneous numerical experiments varying the initial configuration of the particles and the set of random numbers used to emulate Brownian forces and torques. Therefore, the use may configure in the simconfig.dat file all the informations regarding all the simultaneous numerical experiments.

We will talk about the simconfig.dat file later on this README file. It is important to notice that the input.f90 file uses the variables module written in the variables.f90 file. We will also talk about this module later on this manual.

After reading the configuration file, simmsus.f90 calls the main.f90 file. This file calls the necessary subroutines in a specific order in order to run the simulations. Here, the dynamical variables are allocated in the computer memory, the initial condition is generated and the main loop occurs. Here the physics is processed and several output files are created along the simulation process. The main.f90 file uses the variables and subroutine modules.

The last one contains all the subroutines used on the code. After the processing stage, simmsus.f90 checks if the user has chosen to perform a statistical analysis of the calculated results and if so calls the statistics.f90 file, which also uses the variables module.

The makefile contains the compilation instructions. Here the name of the source code files, the compilation command (and consequently the compiler) is specified and any optional compilation flags are also defined. We will talk about the general logic and structure of each of these files in the next subsections of this README file.

simmsus.f90

This file simply calls the program subroutines in a specific order in order to execute a set of numerical simulations defined by the user in the configuration file (simconfig.dat). Besides calling these subroutine this file also counts the total simulation time and write a simple header that appears on the terminal when the user executes the solver.

input.f90

This subroutine reads the file simconfig.dat and storage the information in logical, integer and real varaibles. The variables presented in simconfig.dat must respect the formats defined in file input.f90. These formats are defined through the following commands:

505 FORMAT(1X,A40,1X,E11.4E2)

507 FORMAT(1X,A40,1X,I6)

508 FORMAT(1X,A40,L10)

Here the numbers 505, 507, 508 are simply shortcuts used to specify these formats. The format 505 defines a single keyboard space (1X) followed by 40 letters (A40) a single space (1X) and a real variables containing 11 digits expressed in scientific notation with 4 digits after the comma and 2 digits after the symbol E. Here is an example of the 505 format expressed in simconfig.dat:

VOLUME FRACTION OF PARTICLES...........: 1.0000E-02

The other formats, 507 and 508, are related to integer and logical variables respectively. New implementations should be added as options in the simconfig.dat file and altered on the input.f90 file also. The user is free to modify the structure of the menus on the configuration file simconfig.dat, but it is important to change the strucutre of the file input.f90 accordingly.

main.f90

The main.f90 file sets the order in which calculations are performed in order to achieve a specific set of numerical simulations. In this file the subroutines are sequentially called from the subroutines.f90 module. The heavy calculations are programmed in subroutines.f90. The sequence of procedures called by main.f90 are presented bellow.

1. Definition of constant internal numerical parameters;
2. Allocation of the matrices in the memory;
3. Definition of number formats for the output files;
4. Cleaning all important variables;
5. Determination of particle size distribution;
6. Calculation of the simulation box size;
7. Creation of output files;
8. Calculation of Green-functions and Lattice indexes structure;
9. Distribution of dipole moments;
10. Generation of the initial condition;
11. Calculation of field excitations;
12. Execution of the main simulation loop;
13. Deallocation of matrices from memory;

The main simulation loop is where the simulation occurs. This loop is composed by the following sequence:

1. Calculation of each force acting on each particle;
2. Solution of the linear momentum equation;
3. Evolution of the position of each particle;
4. Calculation of each torque acting on each particle;
5. Solution of the angular momentum equation;
6. Evolution of the orientation of each particle; This loop goes on until the total simulation time is reached. It is important to notice that position, velocity and dipole orientation are storaged in large matrices with the following indeces notation: $X(i,j,k)$,

where $i=[1,...,N_{rea}]$, $j = [1,...,N_{part}]$ and $k = [1,2,3]$. Here $N_{rea}$ denotes the number of simultaneous numerical experiments (realizations), $N_{part}$ represents the number of particles and $k$ the directions of 3D space. This way, $X(12,1234,2)$ represents the position of particle 1234 in the $12^{th}$ numerical experiment in the y direction. Since SIMMSUS also deals with the modeling of Brownian systems, in which a stochastic forcing is always present, the code was built to perform several simultaneous numerical experiments so a meaningful statistics could be obtained for a large number of independent numerical experiments. However, the user is also free to perfom a single numerical experiment. The number of numerical experiments as well as the number of particles in each simulation set is configured in the simconfig.dat file.

subroutines.f90

The inteligence of SIMMSUS is programmed in this module. Here all the heavy calculations are really performed. This file contains all the calculation routines implemented in the code, which are called by other files, such as main.f90 and statistics.f90. We won’t detail all the subroutines contained in this file here, since many of them are really simple and intuitive and all important details regarding SIMMSUS subroutines are commented in the code. However, some subroutines are more complex than others and we believe some discussion regarding some of them are important for the developer to understand some aspects of the logic of SIMMSUS.

statistics.f90

SIMMSUS has also a statistical analysis module implemented on file statistics.f90. This file is called whenever the user enables statistical analysis as true on file simconfig.dat. In order for SIMMSUS to perform this statistical analysis the user must enable record velocity on file as true on simconfig.dat. The statistical analysis module basically reads all the velocity components of all particles in all realizations from the output files named velocidade .plt and calculates several important statistical quantities based on ensemble averages of relevant physical variables. These calculations produce new output files containing the calculated variables, such as: average velocity of the system, Reynolds stress tensor of particle velocity fluctuations, time correlation function and the suspension's correlation time.

variables.f90

This file defines the global variables used in SIMMSUS. We ask for the developers to maintain an organized version of this file including a legend for each variable in the form of comments bellow the definition of the global variables of the problem.

makefile

This file is responsible for the source code compilation. Historically, throughout SIMMSUS development we have tested it using the Intel Fortran Compiler for Linux. Therefore, compilation directives for SIMMSUS in its present version automatically consider that the user is compiling SIMMSUS using Intel Fortran Compiler. In order to compile the code and produce an executable file the user must open a terminal in the folder where the files are located and type make and press enter twice. In the first time an error message will appear due to the inexistence of the module files, which are created after the user press make and type enter for the first time. In the second compilation attempt the error message will no long appear and the user will notice the creation of an executable file named simmsus.ex.

Physics and numerics

Periodic calculations

In order to understand the logic of SIMMSUS it is important to understand a few things. First of all, the calculations of forces, torques and particle’s velocities can be perfomed in a periodic or non-periodic structure. Here, we are not referring to the boundary conditions of the simulation cell, which are always periodic, but to the way in which these quantities are calculated. From the physics of multibody interacting systems it is known that long-range particle interactions have a certain decay. For example, the gravitational force between two masses decay with $1/r^2$, where $r$ is the distance between the center of the bodies.

A typical $1/r^n$ decay is said to be slow if $n \leq 3$. In this sense slow decays are felt from a great distance from the source which generates the disturbances in the surroundings. These slow decays lead to a famous convergence problem of the calculation of average properties of these systems, which demands a huge amount of particles to produce stable, convergent values of the transport properties of these suspensions. One way to solve this problem is to emulate an infinite suspension of particles using a periodic representation of the particulate system strucute called Lattice. A Lattice is a representation of a periodically replicated strucutre containing the central cell (the simulation box itself) and imaginary surrounding cells that mimics the configuration of the central cell. A visual representation of a Lattice strucutre is shown in the image bellow:

Using a Lattice representation it is possible to imagine what the black particle would see when looking to any direction: an infinite system with no beggining, no end and no walls. The use of a Lattice structure leads to a convergence of the system transport properties for long- range slow decays, but increases substantially the computational time, since now each particle on the central cell must interact with their neighbours in the real simulation box and with all the particles in the imaginary boxes surrounding the real one in the Lattice structure. Also, when we consider particle-particle interactions in a system with $N_{part}$ , interacting forces or torques are generally expressed in terms of a series in the form:

where $r_{ij}$ denotes the distance between particles $i$ and $j$ and $f(r_i, r_j)$ is a function of the system configuration at the instant of time in which the calculation is being performed. These functions differ from each kind of physical mechanism involved and are generally known as Green functions. These functions arive from physical principles and have to be modified using a sophisticated mathematical technique known as the Ewald [12] summation to be expressed in a Lattice structure. In SIMMSUS the user can define wether the periodic calculation is being performed for long-range dipolar interactions for both forces and torques. This definition occurs on the configuration file simconfig.dat. Generally, these periodic calculations are only necessary in non-dilute conditions. Usually for particle volume fractions above $\phi \geq 10$%, but the user should consult the reference [7] to understand the limits in which non-periodic calculations of magnetic forces and torques should produce precise results.

SIMMSUS also considers the effect of hydrodynamic interactions in Creeping flow, which are calculated using a mobility formulation where the particles velocities are directly linked with the non-hydrodynamic forces acting on each of them through equation

where $u_1$, $u_2$ ,..., $u_N$ denotes the velocities of particles 1, 2, . . . , N . The tensors $M_{ij}$ , for $i = 1,..., N$ and $j = 1,..., N$ , are second rank mobility tensors or square matrices that depend on the suspension configuration. These tensors couple the motion of particles given how the forces acting on a particle $j$ change the velocities of a particle $i$. Here, $f_i$ represents the sum of non-hydrodynamic forces acting on a particle i. Hence, when the user sets that hydrodynamic interactions must be computed in simconfig.dat, the calculations of the particle’s velocities are automatically performed in a period way. For more details regarding the mobility formulation used in SIMMSUS the reader should consult reference [2].

Whenever the user defines one or more of the following variables as true

ACCOUNT HYDRODYNAMIC INTERACTIONS

PERIODIC MAGNETIC TORQUES

PERIODIC MAGNETIC FORCES

SIMMSUS activates the following subroutines related to periodic computations:

• tabelagreen → computes the necessary Green functions and storages them in large pre-calculated tables;

• periodic_structure → creates the indexes of cells in the Lattice strucutre;

• periodic_interactions → computes all periodic interactions between the particles at a given time-step (this is the most expensive calculation performed by SIMMSUS);

It is important to mention that when the user enables account hydro- dynamic interactions in the file simconfig.dat SIMMSUS automatically considers a mobility formulation, which neglects the effect of particle inertia. Therefore, when considering hydrodynamic interactions we recommend the user to disable variable particle inertia in simconfig.dat.

Brownian motion and random numbers

In order to run the particle simulations random numbers must be generated. These random numbers are important to create random independent initial conditions and to compute Brownian forces and torques. Bellow we show a typical initial particle distribution that can be produced by SIMMSUS. Figure (a) shows an initial ordered distribution of particles while figure (b) illustrates a random initial distribution of particles.

Not only the initial particle distribution can be ordered or random, but also the initial dipole orientation, as shown in the picture bellow.

As we have mentioned, random numbers must also be generated in order to simulate the typical random walks produced by a particle in Brownian motion, as shown bellow:

The subroutine responsible for generating the sequences of random numbers used in a typical simulation is called randomica. This subroutine is defined in terms of the following arguments:

randomica(a,b,c,n,d)

where a,b,c,n,d denote

• a,b → the range of the random sequence;
• c → is a vector containing the random sequence generated;
• n → is the number of elements in vector c;
• d → is an integer used to produce a random seed for the random sequence that is being produced;

We have opted for implementing a customized subroutine for generating the random numbers used in our simulations instead of using the native random number FORTRAN subroutine. We have chosen this option due to several statistical tests performed in the beggining of the development of SIMMSUS used to validate the implementation of Brownian forces and torques. We have compared the ensemble average of the mean square displacement of a single Brownian particle as a function of time with Einstein’s theory and have found an excellent agreement using the implemented random number generation routine in SIMMSUS. For more details the user can consult reference [13].

Calculation of forces and torques

In SIMMSUS each type of force and torque is calculated using a specific subroutine. The following forces are considered in SIMMSUS:

• Long-range dipolar forces between the particles;
• Short-range repulsive forces between approaching particles;
• Contact forces for slightly overlapped particles;
• Brownian and gravitational forces;

With respect to the torques acting on the particles, SIMMSUS considers:

• Long-range dipolar torques between the particles;
• Field-dipole torques between the particles;
• Brownian torques;

Forces regarding field-dipole interaction are implemented in SIMMSUS through subroutine campo_externo, but are disabled in the present version of the code. It is known that the magnetic force acting on a dipole is proportional to the magnetic field gradient and since we have been using SIMMSUS to study problems where the simulation box is a continuum volume in which magnetic field gradients are very small we have been neglecting the force acting on a dipole due to an external magnetic field. Anyway, the user is free to alter this option in the source-code of the program whenever he/she wants.

The subroutines responsible for computing magnetic, repulsive, contact and gravitational forces are respectively

• forca_magnetica;
• respulsion;
• gravity;

Subroutine forca magnetica computes long-range dipolar forces between the particles in a non-periodic way. Subroutine repulsion computes both short-range repulsive and contact forces between the particles. Brownian forces and torques are both computed by the same subroutine, namely brow- nian. Magnetic torques between the particles are computed in a non-periodic way by subroutine torque magnetico. Periodic magnetic forces and torques are computed inside subroutine periodic interactions. For details regarding the expressions used to compute long-range periodic magnetic forces and torques the reader should consult reference [7].

Translational and rotational particle inertia

When hydrodynamic interactions are neglected, SIMMSUS gives two possi- bilities for computing the velocity of the particles: the first one considers particle inertia, while the second one neglects it. The physical principle behind the equation used to solve the velocities of the particles is Newton’s second law, for both the linear and angular velocity of each particle. In other words, the equations used for the solution of the motion of the particles are:

where $m$ and $I$ denotes respectively the mass and the inertia moment of a single particle, $v$ and $\omega$ represent the linear and angular velocities of the particle, $t$ denotes time and $F$ and $T$ represent respectively the forces and torques acting on a single particle. In the abscence of hydrodynamic interactions SIMMSUS considers a simple Stokes drag as the hydrodynamic force acting on each particle, this force is given in dimensional terms as $F_D = -6\pi \eta a$, where $\eta$ and $a$ represent the fluid’s viscosity and the radius of a single particle. It is important to keep in mind that the Stokes drag is only valid in Creeping-flow regime ($Re \ll 1$) and that is a basic premisse behind the mathematical equation of SIMMSUS. These equations are solved in SIMMSUS in their nondimensional version, given by

where the asterisks denotes nondimensional variables and St and Str denote respectively the Stokes number and its correspondent rotational version. These physical parameters are defined as:

where $U_s$ denotes the Stokes velocity of an isolated particle. These expressions are obtained considering typical velocities and timescales related to the Stokes velocity, which seems to be a good choise for sedimentation problem. Other possible scales generate different versions of the nondimensional equations solved by the code. SIMMSUS has three possible choices of nondimensional equations implemented and for more details the reader can refer to [13].

The important fact here is that the Stokes number provides a relation between the relaxation timescale of the particle with respect to a viscous dissipation timescale. For really small particles it is valid to assume $St \ll 1$, which leads us to neglect the effect of particle inertia. For inertialess (non-massive) particles in Creeping flow it is possible to obtain the velocity of a single particle by isolating $v$ from the expression of $F_D$ and balancing this hydrodynamic force with other non-hydrodynamic forces. This is the context assumed by SIMMSUS when the user sets variable particle inertia as false in the configuration file simconfig.dat. When neglecting particle inertia and hydrodynamic interactions we assume that hydrodynamic forces will balance non-hydrodynamic forces. Since the $F_D \sim v$, in this context $v$ is calculated in a dimensional form through:

where I is the second-rank identity tensor and $F_{NH}$ denotes the sum of non- hydrodynamic forces. These non-hydrodynamic forces are long-range dipolar forces, Brownian, gravitational, repulsive and contact forces. It is interesting to notice here that expression above could be written in a compact form as

where the self-mobility matrix $M_s$ is simply I/$(6\pi \eta a)$. It is interesting to mention that even though we could be interested in simulating the behavior of really small particles, which could be considered as inertialess particles, the consideration of a small effect of particle inertia could be interesting for numerical purposes, since a non-null Stokes number produces a certain response time of the particles with respect to the forces acting on them.

This small effect of particle inertia may be useful in order to reduce the noise provenient from Brownian motion and produce a cleaner numerical behavior. Previous tests have shown that this seems to be the case for the rotational movement of the particles. Therefore, the solution of the rotational motion of the particles in SIMMSUS always considers a small effect of particle inertia. A good value of the rotational Stokes number is 1.0E-01 which can be found on the beggining of file main.f90.

The subroutines responsible for solving the velocities of massive (with inertia) particles, their positions, angular velocities and evolving their dipole moments are respectively:

• resvel ;
• respos;
• resomega;
• evoldip;

The arguments in each of these subroutines are:

• resvel(a,b,c,d), where a is the velocity component of a given particle in a given numerical experiment, b denotes the numerical time-step, c represents the Stokes number of the particle and d is the sum of all forces acting on the particle in a given direction;

• respos(a,b,c), where a is the position in a given direction of a given particle in a given numerical experiment, b denotes the numerical time-step and c is the associated particle velocity;

• resomega(a,b,c,d), where a is the rotational velocity component of a given particle in a given numerical experiment, b denotes the numerical time-step, c represents the rotational Stokes number of the particle and d is the sum of all torques acting on the particle in a given direction;

• evoldip(a,b,c,d,e,f), where a,b,c are the particles dipole moments in each direction, d and e are the angular velocities of the particle in the directions of the dipole moments of the same diretions as the dipole moments b and c and f is the numerical time-step;

The evolution of the dipole moment of the particles is obtained through the solution of the simple kinematic equation:

where $\hat{d}$ is the direction of the particle’s dipole moment. Hence, SIMMSUS always assumes that the dipole moment of the particle is fixed on the particle.

Initial condition

SIMMSUS works with three different types of initial particle distributions: random, ordered or spherical aggregates. Random and ordered distributions have been shown in previous figures of this manual. The initial spherical aggregate condition is shown bellow:

The subroutines responsible for creating the initial condition are:

• particle distribution;
• box_size;
• distribui_dipolo;
• condicao_inicial;

Field excitations

In SIMMSUS the user can choose different ways of applying an external excitation on the multibody magnetic system of particles. When we refer here to external field excitations we are refering to an external applied magnetic field and/or an external applied shear rate on the system. The user can combine both kinds of excitations, which leads to very interesting dynamical responses. These excitations are calculated in the subroutines field excitations, rotating field and torque externo.

Each of these subroutines is responsible for a specific set of possible excitations as follows:

• field_excitations: is used for implementing a nonlinear applied magnetic field based on the solution of the Duffing’s harmonic oscillator, a double frequency excitation and a dynamical sweep of a single frequency excitation where the system periodically increases its excitation frequency. The last alternative is also implemented for the shear rate within this sub-routine;

• rotating_field: is used for implementing a rotating 2D magnetic field;

• torque_externo: is used to calculate the magnetic torques on the particles arising from an external field. This subroutine is used for all the 1D applied fields, which include a steady-state and a simple oscillatory field;

Bellow we list the equations responsible for each of the implemented field excitations on SIMMSUS.

• Oscillatory field $\rightarrow$ $H(t) = H_0 \sin(\omega t) \hat{e}_z$
• Double period oscillatory field $\rightarrow$ $H(t) = H_0 [\sin(\omega_1 t) + \sin(\omega_2 t)] \hat{e}_z$
• Rotating field $\rightarrow$ $H(t) = H_0 [\sin(\omega t)\hat{e}_y + \cos(\omega t)\hat{e}_z]$
• Nonlinear Duffing oscillator $\rightarrow$ $H(t) = f(t) \hat{e}_z$, where $f(t)$ is the solution of the following nonlinear ordinary differential equation (ODE):

Variables $\omega$, $\omega_1$ ,$\omega_2$, $C_1$, $C_2$, $C_3$ and $C_4$ are all defined by the user in the configuration file simconfig.dat.

Bellow we show the excitation signals for an oscillatory (a), double period oscillation (b) and nonlinear Duffing field excitation for different values of parameter $C_1$ (c) and (d).

Finally, the next figure illustrates the excitation signal in the case of a dynamical sweep of the fields frequency, which can also be enabled and configured by the user in simconfig.dat file.

Code validation

SIMMSUS has been tested in different physical scenarios and using different database for validation purposes. Here we show some quantitative simulation results documented in previous publications that shows the capacity of the code to capture relevant aspects on the physics of magnetic and non-magnetic suspensions.

Bellow we show the average sedimentation velocity of non-Brownian and non-magnetic particles subjected to hydrodynamic interactions. While figure (a) considers a random initial condition, figure (b) considers an ordered array, whose initial condition is show in the next figure.

In the context of Brownian, magnetic suspensions, we present the following plots. In these plots we show the equilbrium magnetization of a suspension subjected to a steady state magnetic field (a) and the temperature derivative of a suspension of magnetic Brownian particles subjected to an oscillatory field with respect to the frequency of the field (b). In both plots the symbols denote numerical results obtained by SIMMSUS and the lines represent consolidated asymptotic theories. For more details regarding these plots we recommend the reader to consult the references displayed in this README file.

Gallery

References

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[2] Gontijo, R. G., and F. R. Cunha. "Numerical simulations of magnetic suspensions with hydrodynamic and dipole-dipole magnetic interactions." Physics of fluids 29.6 (2017). DOI:10.1063/1.4986083.

[3] Guimarães, A. B., F. R. Cunha, and R. G. Gontijo. "The influence of hydrodynamic effects on the complex susceptibility response of magnetic fluids undergoing oscillatory fields: New insights for magnetic hyperthermia." Physics of Fluids 32.1 (2020). DOI:10.1063/1.5128411.

[4] de Carvalho, Douglas Daniel, and Rafael Gabler Gontijo. "Reconstructing a continuous magnetization field based on local vorticity cells, CFD and Langevin dynamics: A new numerical scheme." Journal of Magnetism and Magnetic Materials 514 (2020): 167135.DOI:10.1016/j.jmmm.2020.167135.

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[6] Gontijo, Rafael Gabler, and Andrey Barbosa Guimarães. "Langevin dynamic simulations of magnetic hyperthermia in rotating fields." Journal of Magnetism and Magnetic Materials 565 (2023): 170171. DOI:10.1016/j.jmmm.2022.170171.

[7] Gontijo, R. G., and F. R. Cunha. "Dynamic numerical simulations of magnetically interacting suspensions in creeping flow." Powder technology 279 (2015): 146-165. DOI:10.1016/j.powtec.2015.03.033.

[8] Gontijo, R. G., S. Malvar, and F. R. Cunha. "Magnetic particulate suspensions from the perspective of a dynamical system." Powder technology 297 (2016): 165-182. DOI:10.1016/j.powtec.2016.04.010.

[9] Gontijo, R. G., and S. Malvar. "Microstructural transition in an ordered set of magnetic spheres immersed in a carrier liquid." Mechanics Research Communications 83 (2017): 12-17. DOI:10.1016/j.mechrescom.2017.03.001.

[10] Gontijo, R. G. "Heat transfer increase for a laminar pipe flow of a magnetic fluid subjected to constant heat flux: An initial theoretical approach." Mechanics Research Communications 91 (2018): 27-32. DOI:10.1016/j.mechrescom.2018.05.005.

[11] Berkov, D. V., L. Yu Iskakova, and A. Yu Zubarev. "Theoretical study of the magnetization dynamics of nondilute ferrofluids." Physical Review E—Statistical, Nonlinear, and Soft Matter Physics 79.2 (2009): 021407. DOI:10.1103/PhysRevE.79.021407.

[12] Ewald, P. "Die Berechnung optischer und elektrostatischer Gitterpotentiale". Annalen der Physik. 369 (3): 253–287. DOI:10.1002/andp.19213690304.

[13] Gontijo, R. G. "Micromechanics and microhydrodynamics of magnetic suspensions (in Portuguese)". PhD Thesis, Graduate Program in Mechanical Sciences, University of Brasília, DOI:10.13140/RG.2.1.1181.2563.