Simulating the dispensing of complex rheological fluids on arbitrary geometriesusing the immersed boundary method

Simon Ingelsten1, Johan Göhl1, Andreas Mark1 and Fredrik Edelvik1

1 Fraunhofer-Chalmers Research Centre for Industrial Mathematics, 412 88 Gothenburg, Sweden

ABSTRACTA numerical framework to simulate dispensingof fluids with different types of complex rheol-ogy is presented. The framework is based on anincompressible flow solver that uses immersedboundary methods and an automatically gen-erated octree grid. It therefore supports simu-lation with arbitrary geometry and of movingapplication with various injection models withminimum setup time. The framework also sup-ports rheology models spanning from purelyviscous models to more advanced viscoelasticstress models.

Three examples of applications are given,namely for seam sealing in automotive bodies,swirled adhesive extrusion and wire fed addi-tive manufacturing of metal. The wide range ofapplications shows the potential of the frame-work and indicates that many more possible ap-plications are conceivable.

INTRODUCTIONProcesses that include dispensing of fluids withcomplex rheology are common in various typesof industrial applications, e.g. seam sealing ap-plication in automotive bodies, adhesive extru-sion and additive manufacturing. As the de-mand for tools to simulate and predict the out-come of the processes increases, so does theneed for robust and efficient numerical frame-works and models that accurately describe thespecific features of such flows. In this papersuch a framework is presented.

The simulation framework is based onIPS IBOFlow R�, an incompressible finite vol-ume flow solver developed at the Fraunhofer-Chalmers Research Centre for Industrial Math-ematics in Gothenborg, Sweden. The solver

has previously been used to simulate conjugateheat transfer10, fluid-structure interaction12 andtwo-phase flows with shear thinning fluids forseam sealing11, 13, 15 and adhesive extrusion14.

For the flows in consideration, four com-mon main properties can be identified,

1. Two-phase flow of the applied material andthe surrounding air

2. Moving application along a prescribed path

3. Arbitrary substrate geometry

4. Non-Newtonian material rheology

All these properties require careful modellingin order to correctly account for them in simu-lations.

Since non-Newtonian fluid flow is consid-ered, a subject of importance is of course therheology model. For flow of viscoelastic flu-ids it may be sufficient to use a purely vis-cous model such as the Carreau model7, es-pecially if the flow is dominated by inertia.For flows dominated by elasticity, however,the accuracy of such models are often insuf-ficient as they lack the ability to store elasticenergy. For such flows the proper approachis to instead use a constitutive model that de-scribes the transient evolution of the viscoelas-tic stress tensor. A wide range of viscoelasticconstitutive models can be found in the litera-ture, ranging from simpler linear models suchas the Upper-Convected Maxwell (UCM) andOldroyd-B models2 and more physically accu-rate nonlinear models such as the Phan ThienTanner (PTT) model1 and the Finitely Extensi-ble Nonlinear Elasticity (FENE) models3. Themain improvement of the nonlinear models

1

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compared to the linear models is that they pre-vent unbounded normal stresses by limiting theextension of the polymers in the viscoelasticfluid.

Another type of rheology that is consideredin this paper is temperature-dependent viscos-ity. This is typically relevant for e.g. liquidmetal flow.

The aim of this work is to present the build-ing blocks of a numerical framework for simu-lating the flows discussed above and highlightsome example applications. The rest of the pa-per is structured as follows. First the govern-ing equations are defined, then the numericalmethod is described. Finally three example ap-plications are shown and some conclusions aredrawn. The examples are application of seamsealing in automotive bodies, swirled extrusionof adhesive and wire fed additive manufactur-ing of metal.

GOVERNING EQUATIONSThe non-Newtonian fluid flow is described bethe incompressible momentum and continuityequations

r ∂u∂ t

+ru ·—u = �—p+— ·s + f, (1)

— ·u = 0, (2)

where r is fluid density, u velocity, p pressure,t the deviatoric stress and f a body force. Thedeviatoric stress is divided into a purely viscouspart and a viscoelastic stress as

t = 2µS+ tv, (3)

where µ is viscosity, S the rate of strain

S =12�—u+(—u)T� , (4)

and tv the viscoelastic stress.The non-Newtonian rheology models are

divided into two main categories. The first con-sists of purely viscous fluids, for which tv ⌘ 0

and µ > 0 is a function of the local state of theflow. Two such models are considered, one be-ing the Carreau model, reading7

µ(g) = µ• +(µ0 � µ•)�1+(x g)2� n�1

2 , (5)

where g = |S| is the local shear rate, µ• andµ0 the viscosities at zero and infinite shear rate,respectively, x is a characteristic time and n apower law index. The other viscous model con-sidered is the temperature dependent viscosityfor liquid metals and alloys calculated as9

µ(T ) = µT0 exp

✓� EA

RT

◆, (6)

where T is the local temperature, µT0 a viscos-

ity constant, EA the activation energy and R theideal gas constant. In this case the temperatureis described by the heat transport equation.

∂T∂ t

+u ·—T = — · ( kT

rcp—T )+ST , (7)

where kT is thermal conductivity, cp heat ca-pacity and ST a source term.

The second main type of rheology modelconsists of the viscoelastic models, for whichµ � 0 is constant and tv is described by a con-stitutive equation. In this work the linear formof the Phan-Thien-Tanner (PTT) model is used,reading

l5t v +

✓1+

elh

Tr(tv)

◆tv = 2hS, (8)

where Tr(tv) denotes the trace of tv and5t v is

the upper convected derivative

5t =

dtdt

�—uT · t � t ·—u, (9)

where d/dt denotes the material time deriva-tive.

The two-phase flow of the non-Newtonianfluid and the surrounding air is modelled withthe Volume of Fluid (VOF) method. A single

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set of momentum and continuity equations isthen used for the whole domain and the respec-tive phases are described by the local volumefraction a 2 [0,1]. In areas where only air ispresent a = 1 and in areas where only the non-Newtonian fluid is present a = 0. A locationwhere 0 < a < 1 lies on the interface betweenthe phases. Local properties in the flow, e.g.density or viscosity, are calculated as

f = afair +(1�a)fNN, (10)

where fair is the property of the air and fNN thatof the non-Newtonian fluid. The evolution of ais described by the advection equation

∂a∂ t

+u ·—a = 0. (11)

NUMERICAL METHODThe simulations are performed withIPS IBOFlow R�, which is an incompress-ible flow solver. The key feature of the solveris the use of immersed boundary methods incombination with an octree grid that is auto-matically generated and dynamically refinednear objects and interfaces. Finite volumediscretization of (1), (2), (7) and (11) is carriedout on the octree grid, and the SIMPLECalgorithm is used to couple pressure and mo-mentum. In each time step the following stepsare performed:

1. Calculate non-Newtonian stresses (µ or t)

2. Solve (1) and (2) iteratively for u and p.

3. Solve (11) for a

4. Solve additional equations (e.g. for T )

Boundary conditions from interior objectsin the computational domain are treated usingthe mirroring immersed boundary method6, 8.The velocity field is then implicitly mirroredacross the boundary surface, such that theresulting velocity at the surface satisfies theboundary condition for the converged solution.

There is therefore no need for a boundary fit-ted grid. The combination with the automati-cally generated octree grid is therefore highlyefficient for simulation of flow with arbitraryboundary geometry and moving free surfaces.

Extra care needs to be taken for the dis-cretization of (11) for a . Since the equationincludes no natural diffusion, it is importantto minimize the introduction of numerical dif-fusion in order to maintain a sharp interfacebetween the phases. The compact CICSAMscheme4 is therefore used.

An injection model is used to describe theapplication of material with a nozzle movingalong a predefined path. The injection step canbe summarized in two main parts:

1. Refine octree grid in the neighbourhood ofinjection.

2. Identify injection cells and add material bychanging a and setting the velocity.

The velocity in the injection cells is set to

uinj = uflow +uapp, (12)

where uflow is the injection velocity based onthe material flow rate and uapp the velocity ofthe applicator. Due to possible geometry dis-crepancies between the real nozzle shape andthe injection cells, a correction step is used tocontrol the inlet flow. In each time step the to-tal volume of injected material in the domain iscalculated. A flow rate correction is then calcu-lated as

Vcorr = max✓

xsVtot �Vtot,nom

Dt,xlVnom

◆, (13)

where xs 2 (0,1] is introduced to give smoothflow variations and xl 2 [0,1) enforces a lowerlimit relative to the nominal flow rate. Vtot isthe calculated volume, Vtot,nom the nominal to-tal volume, Vnom the nominal flow rate and Dtthe fluid time step. The volume flow rate forthe injection is then calculated as

V = xr(Vnom +Vcorr)+(1�xr)Vold, (14)

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where xr 2 (0,1) is a relaxation factor and Voldis the flow rate used in the previous time step.

A geometrical model is used to determinethe injection cells. The models depend on thetype of dispensing application, but can be di-vided in two main categories:

• Direct injection.

• Projected injection.

In the direct injection approach the mate-rial is injected at the nozzle opening and thegeometrical shape of the nozzle directly deter-mines the injection cells. In Figure 1 an exam-ple is shown of the injection cells for a circu-lar adhesive nozzle. The octree discretizationwith refinements are also shown along with thebead, which is visualized with the contour sur-face a = 1/2.

Figure 1. Injection cells (solid cubes) for acircular nozzle model.

The projected injection approach relies onthe assumption that the flow of material fromthe nozzle to the target surface has a largestokes number, such that the effect of the sur-rounding air is negligible. The simulation ofthis part of the application may then be replacedby a projection model. Injection is then insteadperformed at the target surface, which signif-icantly reduces the computational effort. Formore details, the reader is referred to13.

The non-Newtonian stresses are calculatedfrom the flow field. For the purely viscousmodels the local viscosity is calculated di-rectly from the velocity field using (5) or fromthe temperature field using (6), depending onwhich model is being used.

For the viscoelastic fluids, the constitu-tive equation must be solved and the resultingstresses must be coupled to the fluid momen-tum equation. For this, a novel Lagrangian-Eulerian method is used, in which the vis-coelastic stresses are solved and stored in a La-grangian node set that is convected by the fluidand the stresses are interpolated to the Eulerianfluid grid using radial basis functions (RBF).The approach is motivated by the lack of nat-ural diffusion in most viscoelastic constitutiveequations. The Lagrangian frame also providesa natural description to transport the stresses re-sulting from the deformation history of fluid el-ements. An illustration of the concept can beseen in Figure 2.

Figure 2. Sketch of stresses convected in aLagrangian node.

In each node the updated position and stressare obtained by solving the system of ordinarydifferential equations

⇢x = utv = G(tv,—u)

, (15)

where ˙(•) denotes time derivative and the righthand side G(tv,—u) follows directly from (8).The local properties needed for solving thesystem, i.e. u and —u, are interpolated fromthe octree grid to the current position of thenode. When (15) is solved and the updatedstresses are available in the Lagrangian nodes,the stresses are interpolated to the cell centersof the Eulerian fluid grid using RBF. An inter-polant f (x) of a function f (x) that is known ina finite set of points is then calculated as the

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weighted sum5

f (x) =N

Âi=1

wif(|x�xi|), (16)

where xi are the points where f is known, withe corresponding weights, N the number ofpoints and f(r) the RBF. The weights are deter-mined by solving the linear system implied bythe constraint that the interpolant should be ex-act where f is known, i.e. f (xi) = f (xi) 8i =1, . . . ,N. This gives

Aw = f, (17)

where

Ai j = f(|xi �x j|), (18)

w = [wi · · ·wN ]T , (19)

f = [ f (xi) · · · f (xN)]T . (20)

When the components of tv are known inthe cell centers they are integrated over the cellsusing Gauss’s divergence theorem as

Z

cell— · tdV = Â

fA f n f · t f , (21)

where the sum is taken over the cell faces andA f , n f and tv, f are the area, normal vector andstress tensor at the respective faces.

RESULTSIn this section, three example applications arepresented. The examples cover the rheologymodels and injection model approaches de-scribed above.

The first example considers flat bead seamsealing, which is applied in automotive bod-ies to cover holes and cavities to prevent corro-sion. The rheology of the shear thinning sealant

is described with the Carreau model with pa-rameters µ0 = 264.76Pas, µ• = 3.096Pas, x =0.4796s and n = 0.45359. The parametersare determined by fitting the viscosity resultingfrom (5) to data from a rheometer shear sweep.The projected injection approach is used to de-scribe the input of material into the computa-tional domain. The flow exiting the sealingnozzle consists of a thin bead, where the widthis a function of the distance from the nozzleand the flow rate. The injection model is con-structed by measuring the width at varying dis-tance for a range of flow rates relevant for theprocess. The data is used in the simulations toreconstruct the bead and predict the impact atthe target surface with ray tracing. In Figure3 an experimental bead and the correspondingreconstructed bead from a simulation is shown.

Figure 3. Experiment (left) and reconstructionin simulation (right) of the sealing bead in the

air.A set of beads in the floor of a Scania truck

cab are simulated. The corresponding realbeads are 3D-scanned to enable comparison be-tween the scanned and the simulated beads, re-spectively. In Figure 4 an overview is shown ofthe scanned truck floor and sealing beads. Theflow rates used to apply the beads vary between15 ml/s and 35 ml/s and the movement speed ofthe applicator is around 0.3 m/s to 0.5 m/s

In Figure 5 the simulated and scannedbeads, respectively, are shown in an area of thetruck floor. The bead passes a corner and in-cludes an area with a thick sealant layer. Highflow rates and thick layers of sealant typicallyincreases the computational complexity of theproblem. It is observed, however, that the sim-ulation gives an accurate prediction of the realbead in this case.

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Figure 4. Scanned sealing beads in truck floor.

Figure 5. Simulated (left) and scanned (right)sealing bead in the Scania truck.

In Figure 6 a similar visual comparisonbetween the simulated and scanned beads isshown in an area where the bead is applied ona vertical wall. This causes the sealant to flowdownwards along the wall before finally stick-ing. The simulation captures the geometricalfeatures resulting from this scenario.

Figure 6. Simulated (left) and scanned (right)sealing bead in the Scania truck.

A detailed comparison of the bead crosssection is performed in a relevant area for thebeads. The positions are defined in Figure 7. InFigure 8 the comparisons in the cross sectionsare shown. The results show that the simula-tions are in good agreement with the scanneddata in the studied areas.

The second example is of extrusion of ad-

Figure 7. Cross sections (green planes) usedfor detailed comparison of the sealing beads.

Figure 8. Detailed comparison of thesimulated and scanned beads in the cross

sections defined in Figure 7.

hesive material with a swirled nozzle. In thistype of application a very small circular noz-zle rotates with high speed, forming a thread-like spiral shaped bead in the air. The flowexhibits a significant degree of elasticity, andthe viscoelastic PTT model is therefore usedto model the adhesive rheology. The parame-ters l = 0.0821s, h = 3065Pas and e = 0.5and µ = 60Pas are determined using rheome-ter data from a shear sweep and an oscillatingfrequency sweep.

A simulation is performed with the flow rate1 ml/s and a rotation speed of 15000 rpm. Thenozzle diameter is 0.6 mm and the radius ofthe rotation 1 mm. The movement speed of thenozzle is zero, in order to focus on the effectsof the swirl technique. In Figure 9 snap shotsare shown for a time series in the simulation.The spiral pattern of the flow in the air causedby the stresses in the material is clearly visible.

The third and final example is of wire fedadditive manufacturing of metals, a process thatis used to build or repair structures with com-plex geometry. A metal wire is continuouslyfed to the focus spot of a laser beam, caus-

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Figure 9. Snapshots of E-swirl simulation at2 ms (top left), 50 ms (top right), 100 ms(bottom left) and 150 ms (bottom right).

ing the material to melt. The liquid metal thenflows onto the substrate surface and eventuallysolidifies as it cools down.

The direct injection approach with a cir-cular inflow is used for this application. Theviscosity is temperature dependent and iscalculated from (6) with parameters µT

0 =1.665mPas and EA = 48.586kJ/mol. The heatequation (7) is solved in the whole domain witha conjugate heat transfer solver for the cou-pled heat transfer between the solid, the liquidmetal and the surrounding air. The solidifica-tion of the material is modelled by convertingfluid cells to solid cells if the temperature fallsbelow the solidification temperature.

A simulation is carried out of a straight sin-gle layer. The inlet temperature is 5000 K andthe diameter of the inlet is 1.14 mm. The move-ment speed is 14 mm/s and the flow rate is0.34 ml/s. In Figure 10 the bead is shown attime 50 ms and at 250 ms, respecively.

The temperature field is significant for therheology of the liquid metal. The tempera-ture history can also provide input to the mi-cro structure of the final product. In Figure 11the temperature field at 250 ms is shown in thesubstrate and the bead, which has been clippedto make the result inside the bead visible. The

Figure 10. Simulation of wire fed additivemanufacturing with bead (contour of a = 1/2)and injection cells (solid cubes) at time 50 ms

(left) and 250 ms (right).

resulting viscosity in the bead is also shown.As can be expected, the temperature is veryhigh near the injection area but decreases fur-ther away. At a certain distance from the injec-tion the temperature is sufficiently low for themetal to solidify. Following from the tempera-ture field, the viscosity is low near the injectionbut is significantly higher in the cooler parts.

Figure 11. Simulated temperature (top) andviscosity (bottom) for wire fed additive

manufacturing. Solidified cells representedwith black grid.

CONCLUSIONSA numerical framework to simulate varioustypes of dispensing of non-Newtonian fluidshas been presented. The framework supports awide range of applications with injection mod-

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els for different types of nozzles and a vari-ety of rheology models with different levels ofcomplexity.

Three different example applications havebeen demonstrated, covering both direct andprojected injection models as well as purelyviscous rheology models and time dependentviscoelastic stress models. The wide range ofthe applications included indicates that manydifferent applications could be simulated usingthe proposed approach. Future work thereforeincludes expanding the scope of the frameworkby adding new injection and rheology models.

ACKNOWLEDGEMENTSThis work has been supported in part by theSwedish Governmental Agency for InnovationSystems, VINNOVA, through the FFI Sustain-able Production Technology program, and theCompentence Centre for Additive Manufactur-ing - Metal, CAM2. The support is gratefullyacknowledged. Scanned geometries and exper-imental data for the sealing simulations wereprovided by Scania trucks.

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