Universidade de Brasília
Departamento de Matemática
Prof. Dr. Yuri Dumaresq Sobral
General aspects

1  General interests

I am interested in the application of advanced mathematical tools, such as numerical analysis, nonlinear dynamical systems, integrable PDEs, wave propagation and perturbation methods, specially to problems in fluid dynamics, but not limited to this subject. More specifically, my primary research interests are particulate flows, ferrohydrodynamics and granular materials. I have been working on problems that interconnect these three areas, and I have applied both analytical and numerical methods to accomplish my goals.
More specifically, I am interested in fluidised beds, and I have devoted my efforts in studying their stability and the difficult problem of the formation of bubbles. I focus on theoretical studies to predict the general behaviour of instabilities and most of my results have been complemented with different kinds of numerical simulations. I am also interested in flows of magnetic particles and how magnetic interactions among particles affect the stability fluidised beds, in one side, and on the stability of the suspension (related to aggregation), on the other side. My interest in flows of magnetic particles has led me to work on general ferrohydrodynamics, and I have worked on simple problems related to the rheological characterisation of magnetic fluids, on oil recovery and on free boundary problems of magnetic fluids.
As the result of fruitful collaborations with colleagues, I have expanded my research interests. With colleagues in France and in the UK, I have started to look at granular materials problems, more specifically on the modelling of dissipation in dense flows of grains. With colleagues from Brazil, I have started to look at some problems of elliptic PDEs, more specifically on the numerical implementation of the mountain pass algorithm applied to find solutions of nonlinear equations on some special manifolds of functions. These adventures in new fields of research are very enriching and I shall mention a few words about this in the following sections. Finally, I am very much interested in problems that arise from modelling, that generally give birth to challenging, and potentially interesting, problems in differential equations, fluid dynamics and applied mathematics in general.
This statement contains descriptions of the lines of research I have been following during the last few years, and also short descriptions of the directions where I hope this research will lead in the near future.

2  Particulate flows

2.1  Fluidised beds: Modelling and Stability Analysis

Fluidised beds are systems in which a bed of solid particles is suspended by an upward fluid flow. When the flow is strong enough, the particles become fully mobile and occupy a large region of the reservoir where the flow takes place. The particles are then said to be fluidised. The name fluidised bed is due to the fact that the particles in this condition can be stirred and poured as a fluid.
One of the most intriguing aspects of fluidisation is its unstable behaviour. In most cases of practical applications, it is very difficult to obtain a stable fluidisation, in which large gradients of concentration of particles on the scale of the fluidisation reservoir are not present. In fact, large regions free of particles, usually called bubbles, are seen to propagate along the bed and are responsible for significant changes of the dynamics of the flow.
The research in fluidised beds is a very complicated subject, as there is still a lot of uncertainty on governing equations and there are still no set of equations that are widely accepted. The most accepted ones are obtained by considering both the particles and the fluid as continua, which leads to the following equations [1]:
fluid







∂ϵ

∂t
+ ∇·(ϵu ) = 0
   

∂t
(ϵρf u ) + ∇·(ϵρfu u ) = ∇·(ϵTf ) + ϵρf gf

particles







∂ϕ

∂t
+ ∇·(ϕv) = 0
   

∂t
(ϕρp v ) + ∇·(ϕρpv v ) = ∇·(ϕTp ) + ϕρp g + f
The important quantities in these equations the velocities of the fluid u and the particulate phase v and the local particle concentration ϕ and local porosity ϵ.
In [1,2,3,4], I have performed linear stability analyses to characterise the unstable behaviour of one-dimensional fluidised beds and link it to physical parameters and mechanisms. It is in fact the inertia of the particles that induce the growth of the instabilities, winning over any possible stabilisation mechanism. However, when the amplitude of the instabilities grows, we observed, inertia is no longer dominant and it is dissipation on the particulate phase that sets the basic properties of the instabilities. The results depended on the choice for the stress tensors Tf and Tp for both phases and on the model of the fluid-particle interaction force f. In [1,3,4], I discuss several models for the particle pressure and particle viscosities several constitutive models, as well as for the fluid particle interaction force, which deserved special attention in [5], where we studied the case of an isolated sedimenting sphere aiming to find justifications for our choices of f.
In my recent article [6], I discuss a toy model for the 2D fluidised bed, in which the driving force instability is the stratification of particles originated by one-dimensional instabilities. We performed a stability analysis and computer simulations of this toy model and observed the formation of upward propagating regions empty of particles, as a result of the tilt-and-slide mechanism that occurs when the stratification of the flow is disturbed horizontally, as observed in fig.(1). However, it is not clear yet whether such regions will evolve to a bubble in fluidised beds, and I am now working on an improvement of the model that I sketched in [1] to try and establish a connection between the two problems.
Concentration
Figure 1: Concentration profiles of a stratified fluid disturbed horizontally. Extracted from [6].
There are indications that the problem of bubble formation in fluidised beds might be linked to fluctuations of particle concentration and velocity. In order to have access to these quantities more accurately and in a less model-dependent way than the continuum formulation, I am now, together with collaborators from École Normale Supérieure de Lyon, in France , implementing a hybrid algorithm that simulates the motion of the particles explicitly, via a molecular dynamics algorithm that solves Newton's third law for each individual particle, for prescribe models of contact forces, and the equations of kinematics for a particle, coupled with a finite differences solution of the averaged equations for the fluid phase. In this equation, particle concentration is no longer an unknown, but given from the molecular dynamics solution of the motion of the particles. A snapshot of a standard simulation of a fluidised bed of glass particles in water is shown in fig.(2). More details of this project can be found in the PhD thesis of a student I helped co-supervising on this topic in Lyon [7]. The proposed formulation for this problem also allows us to study the rheology of the particulate phase and, eventually, propose models for both particle pressure and viscosity, and we want to venture on that area in the near future.
Concentration
Figure 2: Snapshot of simulation of a fluidised bed via the hybrid method that my colleagues and I are working on. The colour map represents the particle concentration field calculated from the distribution of particles and the arrows the velocity of the fluid.

2.2  Granular Materials

Granular materials are fascinating: they have at the same time properties that are typical of fluids and others that are typical of solids. There has been major progress in the understanding of such material with the advances of molecular dynamics simulations. I am investigating, with my collaborators from École Normale Supérrieure de Lyon, in France, and at Trinity Hall, Cambridge, in the UK, the problem of a rotating drum that moves down an incline [8], see fig.(3). Despite the simplicity of this problem, we have been able to identify several regimes of motion and the underlying physical criteria that differentiate them. We have also been able to perform a detailed study of the energy dissipation by collision of grains, and made some effort in trying to link it with an equivalent viscosity.
Drum
Figure 3: The problem of the granular cylinder down an incline and some results showing that, for a filling ratio $\phi$ and inclination $\alpha$, we can find a steady motion downhill.
The simplicity of the molecular dynamics algorithm applied to granular materials makes it a perfect topic for undergrads to perform simple research projects. In fact, I am investigating, with two undergraduate students, the problem of crater formation and stability in granular beds and also the problem of segregation in heterogeneous media with another undergraduate student.

3  Modelling and Characterisation of Magnetic Fluids

During my work related to the stability of fluidised beds, I got involved in fluidised beds of magnetic particles and how one could stabilise such flows by applying a magnetic field [9,10,11]. It turns out that the stabilisation of one-dimensional instabilities in magnetic fluidised beds does occur, but it depends on the angle between the direction of fluidisation direction and the direction of the applied field. Being again a fluidised beds problem, there are many uncertainties related to the closure models of the governing equations, mainly related to the Maxwell stress tensor, that accounts for the magnetic interaction, and also to the magnetisation of the particulate phase. There is still a lot that can be done in this area, for instance a numerical simulation of the one and two-dimensional instabilities and investigate how the magnetic field inhibits the growth of instabilities and also the gravitational instability. Furthermore, in order to study in more detail these problems, I want to apply the hybrid methodology to this problem and for that I'll need to implement a scheme to calculate pairwise magnetic interactions of the particles.
On that matter, I am now working on suspensions of magnetic particles that interact not only magnetically, but also hydrodynamically. With my collaborators from the University of Brasília, we have studied the problem of two magnetic particles interacting in a sedimentation problem, recently published in [12]. Our main interest was to determine how one can predict the rate of formation of irreversible aggregates in such suspensions and characterise, in terms of the parameters of the flow, the stability of a dilute suspension of magnetic particles. We have observed that there are intense fluctuations of the dipoles of the particles as two particles approach each other, and this can cause both diffusion in the suspension and aggregation, depending on the intense are the magnetic interactions.
Somewhere during my research on magnetic fluidised beds, I got involved problems of real magnetic fluids, frequently called ferrofluids. These are suspensions of nanometric magnetic particles dispersed in a carrier fluid, in general synthetic oils. When a magnetic field is applied, these particles tend to align themselves in the direction of the field and the fluid becomes magnetically responsive. Ferrohydrodynamics is a fascinating subject and there is still a lot to be done on that area. For instance, we are now investigating the shape of the meniscus of a magnetic fluid in the presence of a magnetic field [13]. The shape y(x) of the meniscus can be modelled by a correction to Young-Laplace's equation, in order to account for the magnetisation of the fluid, given by:
y" = Bo(y+D)−Bomχ1(1+χ1)
1+ 2

D
y+ 1

D
y2
.
We have found many analytical solutions of this equation in different regimes of the parameters Bo and Bom, and we also performed a numerical study to evaluate the shape in very nonlinear cases. An example of our results is shown in fig.(4).
Menisco
Figure 4: The shape of a the meniscus of a magnetic fluid in the presence of a magnetic field. Note the change of the average level of the surface, as well as its very nonlinear shape.
My collaborators from the Universidade de Brasília and I are now, together with a colleague from the University of California Santa Barbara, just starting to devote our attention to the problem more general problems of inter-facial dynamics of magnetic fluids, more specifically drops and emulsions of magnetic fluids. We want to perform numerical simulations of the response of magnetic drops in sedimentation and shear flows to characterise the rheology of dilute emulsions.
In addition, we are now studying the asymmetry of the stress tensors of magnetic fluids [14], given that they are polar and that there are internal torques originated by magnetic interactions. There is not much done on the literature about the spin viscosity ηr that appears in the equation for the conservation of angular momentum ω,
ρJ
∂ω

∂t
+u·∇ω
r2ω+μ0M×H.
We are now investigating a way to measure experimentally this quantity, based on standard rheological experiments in a rheometer.
Finally, some time ago, I have also investigated the usage of magnetic particles to clean up potentially harmful oil spills [15]. In this work, we found scaling laws that predict how efficiently big regions of oil, over which magnetic nanoparticles would have been dropped, would be attracted by magnetic fields [15,16]. Then, using perturbation methods and numerical simulations of the full governing equations [15,17,18], we showed how the pipe flow pattern of magnetic fluid and water would look like.

4  Other interests

Numerical implementation of the Mountain Pass Theorem

Together with collaborators from the Universidade de Brasília, I am now working on the numerical implementation of the Mountain Pass Theorem to find weak solutions of elliptic problem as critical points of an associated functional [19]. Solutions of the problem




−∇2 u
= g(u)
in
Ω ⊂ RN
u
= 0
on
∂Ω,
  with   I(u)= 1

2



 

|∇u|2dx−
x

0 
f(t) dt
dx
are found on the mini-max levels of the functional I(u). We are now implementing a new algorithm that, through successive appropriate projections, searches for the mini-max in a special Pohozaev manifold that contains all the critical points of I(u), thus optimising the search process.

Modelling voice production

Together with collaborators from the Universidade de Brasília, I am now involved in a multinational research project (Brazil, France, Argentina and Chile) concerning the modelling of voice production. To that extent, we use a lumped model of the vocal folds, to which we write a set of coupled nonlinear second-order ODEs that describe. I am interested in using a stochastic forcing on these equations to study its bifurcation properties, and its links to diseases of the vocal folds.

References

[1]
Sobral, Y. D., Instabilities in fluidised beds, PhD Thesis, University of Cambridge, 2008.
[2]
Sobral, Y. D., Cunha, F. R. A linear stability analysis of a homogeneous fluidized bed. Tendencies in Computational and Applied Mathematics, v. 3, n. 2, p. 197-206, 2002.
[3]
Sobral, Y. D. Hydrodynamic and magnetic stability of fluidised beds, MSc Dissertation, Universidade de Brasília, 2004.
[4]
Sobral, Y. D., Hinch, E. J. Characterisation of one-dimensional linear and nonlinear concentration waves in fluidised beds. In preparation, 2012.
[5]
Sobral, Y. D., Oliveira, T. F., Cunha, F. R. On the unsteady forces during the motion of a sedimenting particle. Powder Technology, v. 178, p. 129-141, 2007.
[6]
Sobral, Y. D., Hinch, E. J. Gravitational overturning in stratified particulate flows. SIAM Journal on Applied Mathematics, Vol. 71, No. 6, p. 2151-2167, 2011.
[7]
Grenard, V. Structuration and fluidification of carbon black gels. PhD Thesis, École Normale Supérieure de Lyon, 2011.
[8]
Sobral, Y. D., Taberlet, N., Tokieda, T. Energy dissipation in a granular drum in an incline. In preparation, 2012.
[9]
Sobral, Y. D., Cunha, F. R. A stability analysis of a magnetic fluidized bed. Journal of Magnetism and Magnetic Materials, v. 258, p. 464-467, 2003.
[10]
Sobral, Y. D., Cunha, F. R. Wave hierarchy of concentration waves in magnetic fluidized beds. Journal of Magnetism and Magnetic Materials, v. 289, p. 111-114, 2005.
[11]
Sobral, Y. D., Cunha, F. R. Stabilization of concentration waves in fluidized beds of magnetic particles. Submitted, 2012.
[12]
Cunha, F. R., Gontijo R. G., Sobral, Y. D. Symmetry breaking of particle trajectories due to magnetic interactions in a dilute suspension. Journal of Magnetism and Magnetic Materials, v. 326, p. 240-250, 2013.
[13]
Cunha, F. R., Gontijo R. G., Sobral, Y. D. Magnetic field effects on the Shape of a Magnetic Fluid Meniscus. In preparation, 2012.
[14]
Cunha, F. R., Gontijo R. G., Sobral, Y. D. On the spin viscosity of asymmetrical magnetic fluids. In preparation, 2012.
[15]
Cunha, F. R., Sobral, Y. D. Characterization of the physical parameters in a process of magnetic separation and pressure driven flows of a magnetic fluid in a cylindrical tube. Physica A, v. 343C, p. 36-64, 2004.
[16]
Sobral, Y. D., Cunha, F. R. Drift velocity and stretching of polarized drops in magnetic fields. Journal of Magnetism and Magnetic Materials, v. 289, p. 318-320, 2005.
[17]
Cunha, F. R., Sobral, Y. D. Asymptotic solution for pressure driven flows of magnetic fluids in pipes. Journal of Magnetism and Magnetic Materials, v. 289, p. 314-317, 2005.
[18]
Ramos, D. M., Cunha, F. R., Sobral, Y. D., Rodrigues, J. L. A. F. Numerical Simulation of Magnetic Fluids in Laminar Pipe Flows. Journal of Magnetism and Magnetic Materials, v. 289, p. 238-241, 2005.
[19]
Maia L. A., Sobral, Y. D., Ruviaro R. Mountain Pass Algorithm via Pohozaev Manifold. In preparation, 2012.



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