Modelling of a free-surface ferrofluid flow

Transcription

1 Modelling of a free-surface ferrofluid flow M. Habera a,, J. Hron a a Mathematical Institute, Faculty of Mathematics and Physics, Charles University in Prague, Soolovsa 83, Prague Abstract The Cauchy s stress tensor of a ferrofluid exposed to an external magnetic field is subject to additional magnetic terms. For a linearly magnetizable medium, the terms results in interfacial magnetic force acting on the ferrofluid boundaries. This force changes the characteristics of many free-surface ferrofluid phenomena. The aim of this wor is to implement this force into the incompressible Navier-Stoes equations and propose a numerical method to solve them. The interface of ferrofluid is traced with the use of the characteristic level-set method and additional reinitialization step assures conservation of its volume. Incompressible Navier-Stoes equations are formulated for a divergence-free velocity fields while discrete interfacial forces are treated with continuous surface force model. Velocity-pressure coupling is implemented via the projection method. To predict the magnetic force effect quantitatively, Maxwell s equations for magnetostatics are solved in each time step. Finite element method is utilized for the spatial discretization. At the end of the wor, equilibrium droplet shape are compared to nown experimental results. 1. Introduction Free surface fluid flows and processes involved in a fluid behavior fascinated scientists since the very beginning of the scientific history. Problems as a breaup of a liquid jet, droplet formation and merging, rising bubbles etc. still lacs deeper understanding because of a complex and nonlinear equations governing such phenomena. In addition, they play a role in many industrial processes: fuel injection, fibre spinning, in-jet printing, etc. All these phenomena become even more attractive in terms of ferrohydrodynamics. Ferrofluid reacts to a magnetic field and changes its shape and rheology. This wor attempts to present a physical model and numerical method succesful in capturing multiphasefreesurface) ferrofluid flow with special focus on the interface development. 2. Theory and methods 2.1. Balance of momentum for ferrofluid Balance of momentum for continuous medium in Eulerian setting reads ρu) t + [u ] ρu) = T + f, 1) with ut, x) : I Ω R d being Eulerian velocity, a dimension d, T a Cauchy s stress tensor and ft, x) : I Ω R d volume density forces. Corresponding author addresses: habera@arlin.mff.cuni.cz M. Habera), hron@arlin.mff.cuni.cz J. Hron) The continuous medium Ω := Ω air t) Ω ferr t), t I = [0, T ], Ω C 0,1, in this wor is composed of two immiscible phases - surrounding fluid, e.g. air, Ω air t) and ferrofluid, Ω ferr t). So called constitutive relation must be specified. Here, we adopt several assumptionssimplifications) on the behaviour and properties of the medium, namely 1. the Ω fluid is incompressible, newtonian, the stress tensor without magnetic field T n = ˆT n D) = mi + 2ηD, with D := ) 1 2 u + u T the rate of deformation tensor, m the Lagrange multiplier pressure ) enforcing the incompressibility constraint, η dynamic viscosity, 2. the ferrofluid Ω ferr is linearly magnetizable, isotropic and homogenous, 3. there are no macroscopic electric currents and ferrofluid is non-conductive, we wor in field of magnetostatics, 4. the only effect of magnetic field on ferrofluid is additional magnetic stress tensor T m = µ 2 H 2 I + µh H. 2) With the assumptions above the total stress tensor T = ˆTD, H) = T n + T m. The surface tension effect must be included in multiphase flows. We chose so called continuous surface force, where surface forces are approximated with volume body forces concentrated on the interface. This allows us to write f = f s + f g, f s = σκδn, f g = ρge z, Preprint submitted to Elsevier July 14, 2016

2 where σ is surface tension coefficient, κ curvature, δ an infinitely smooth representative of the Dirac delta distribution concentrated on the interface, n outward normal to the interface, density ρ and gravitational acceleration g. Moreover, according to [8] there exists a tensor T s, such that 2.2. Magnetostatics f s = T s, T s = σδ I n n). As assumed above, the total stress tensor is function of magnetic field intensity H, it must be found a priori, before the momentum balance is solved. Magnetostatic equations and constitutive relation for linearly magnetizable medium are B = 0, H = 0, B = µh. 3) 2.3. Diffused interface, level-set function So called characteristic level-set function is constructed to trac the domains Ω air t), Ω ferr t) and their boundaries It is a regularized characteristic function of Ω ferr t), such that { C 0 x Ω air, distx, Ω air ) > /2, Ω), = 1 x Ω ferr, distx, Ω ferr ) > /2, and χ Ωferr L2 Ω) 0 as 0+. The epsilon parameter is the thicness of the diffused interface. With the aid of this function all discontinuous piecewise constant physical quantities are approximated, therefore ρ = ρ ferr ρ1 air, η = η ferr η1 air, µ = µ ferr µ1 air, 4) for density, dynamic viscosity and absolute permeability respectively. Outer normal to the implicitly represented interface is approximated as n Advection Since the evolution of boundaries Ω ferr t) and Ω air t) is the essence of this method, one must advect/transport the level-set function in the velocity field u. It is achieved solving t + u = 0. 5) This distorts the profile of level-set function and deteriorates it s property as regularized characteristic function of the ferrofluid phase. Level-set function is therefore reinitialized similarly to [5], [7], [6] τ + [ 1 )nt 0, x)] =, 6) note the change in the term on right-hand-side. tae full laplacian instead of normal derivative. We Numerical methodology Generally, evolution equations are here discretized in time with finite differences FDM). Finite element FEM) method is used to discretize in spatial variable. In the sense of FDM we denote a discrete solution at -th time level with upper index so u x) ut, x), etc. Because Cauchy s stress tensor is function of magnetic field, we must see H before the momentum balance is solved. For the Ω domain is simply connected the potential character of magnetic field intensity allows us to define H =: ξ. Wea formulation for the magnetostatics problem 3) is: find ξ 1 V := H0 Ω) such that for all s V µ 1 s ξ 1 n ds + µ 1 ξ 1, s ) = 0 7) Ω with u, v) := Ω uvdx being the standard L2 Ω) scalar product. The term ξ 1 n = H 1 n is substitued with boundary conditions. The balance of momentum 1) contains in each time step two unnown quantities, velocity u and pressure m enforcing the incompressibility constraint u = 0. To overcome this difficulty Chorin s projection method is used [9], [5], [7]. First find tentative u V := H 1 0 Ω; R d ), such that for all v V ρ u u 1), v ) ρ 1 u u, v ) m 1, v ) + η u + u Re ) T ), v ) + ) I 1 We 1 2 ), v ) 1 µ 1 Mg 2 H 2 ) 1, v + µ 1 H H, v ) = 0 8) Mg where Re, We are Reynolds and Weber numbers. The dimensionless number Mg := ρ ref u 2 ref µ ref is often called Magnetic Reynolds number. Href 2 Then, in so called pressure correction step find m V that for all q V u, q ) + m ref x ref u ref ρ ref m m 1 ) ρ 1 ), q = 0. Finally, correct the velocity to have zero divergence by finding u V that for all v V u u, v ) m ref m m 1 ) ) + x ref u ref ρ ref ρ 1, v = 0. 10) We use implicit Euler schema for the advection and reinitialization equation. Dimensionless wea formulation 9)

3 for the advection problem 5) is: find V := H 1 0 Ω) such that for all ϕ V 1 1, ϕ ) u ref x, u ϕ ) = 0. 11) ref Dimensionless wea formulation for the reinitialization 6): find n V such that for all ϕ V n τ n 1, ϕ ) n 1 n ) ), ϕ + n, ϕ) = 0. 12) In the Figure 1 we give a simple overview of the method used. Wea formulations and mesh generation are implemented into FEniCS [3]. Magnetostatics, 7) Balance of momentum, 8), 9), 10) 3. Results Advection, 11) Reinitialization, 12) next time step, 1 Figure 1: Schema of developed numerical method. p m h Figure 2: Convergence of pressure inside a droplet in equilibrium. y/cm x/cm Simple model validations Here we give a simple validation of the developed model and methodology in trivial static cases. In all examples the dimension d = 2. First, consider an ideally circular ferrofluid droplet with radius r = 2 mm placed in the center of square computational domain Ω. From well nown Young-Laplace equation there is a jump in pressures, p = σ R, for droplet with radius R and surface tension σ. Magnetic field is not assumed. We depicted in Figure 2 a ratio of the theoretical pressure difference and the incompressibility constraint p jump, m, computed from our model. This ratio is evaluated for several values of mesh cell diameter, h. As a mesh is more refined and h 0 + we have p m 1. Another simple example is a magnetic field intensity H in a domain with ferrofluid droplet. Let have a droplet with radius r = 1 mm, permeability of the surrounding phase be the permeability of free space, µ air = 4π 10 7 N A 2 and susceptibility of the droplet be χ = 1. Boundary conditions are set in 7) so that H n = 1 A m 1 at top and bottom boundaries. Note, that due to applied magnetic field, the magnetic intensity increases at the top and bottom tips of the droplet. The magnetic stress tensor 2) yields to the force which elongate the droplet. 3 Figure 3: Magnetic field intensity H in A m 1 with ferrofluid droplet present Equillibrium ferrofluid droplet shape Let have ferrofluid droplet of initial radius r = 1 mm and set the boundary conditions H n = 12 A m 1. The droplet elongates [10], [13] due to the effect of magnetic field. Developed model could capture this behaviour. In equillibrium state the level-set function of ferrofluid phase is depicted in Figure 4 and magnetic field intensity in Figure 5. Parameters are inspired by [10] where experimental results are presented. Their experiment is compared to our model in Figure Ferrofluid dripping Ferrofluid dripping is experimentally studied in [1] and [2]. It is noted, that magnetic field substantially changes equilibrium of droplet formation and formed droplet and evolution dynamics are affected by magnetic field. Here we give a simple numerical example of such effect. Pendant droplet profile is depicted in Figure 6.

4 Figure 6: Pendant droplet profile. Magnetic field H n = 4 A m 1 and χ = 1 in the left picture, while no magnetic field is present in the right picture. Profiles are taen at time 33 ms after simulation started. Figure 4: Equillibrium droplet shape. Boundary conditions H n = 12 A m 1 and χ = Surface tension is set to σ = 13.5 mn m 1. Viscosity and density do not play role in equillibrium state, since u 0. Experimental result from [10] is included in actual dimensions in dotted rectangle. y/cm x/cm time evolution is done solving advection equation. To preserve volume of ferrofluid level-set function is reinitialized. In the last part we give wea formulations of derived equations and show some simple numerical experiments. The possible extensions of this wor could be the implementation of non-newtonian constitutive laws, with viscosity depending on magnetic field, etc. [4] Another numerical experiments are needed to verify the validity of the model. Dynamics of droplet pinch-off process and influence of magnetic field on it are now experimentally studied [1], [2], so simulations of this phenomena are be welcomed. Forces due to surface tension and magnetic field are written in terms of tensors, wea formulation maes it possible to per-partes derivatives and less regularity is needed for physical quantities. Several wors with finite-volume approach were validated in recent years [10], [11], [12], [13] on the other hand, this is the first time finite-elements are used, according to authors best nowledge. Figure 5: Magnetic field for equillibrium droplet shape. Boundary conditions H n = 12 A m 1 and χ = Surface tension is set to σ = 13.5 mn m Conclusions The main goal of this wor was to formulate the equations of ferrohydrodynamics and to put together advanced mathematical techniques, so simple free-surface ferrofluid problems could be attaced. In the beginning a ferrofluid is treated in terms of continuum mechanics and several assumptions lead us to the balance law form. Continuous surface force method is utilized for surface tension force. The implicit description of the interface is adopted and 4

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