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Emergence of nonlinear dynamics in 3D cavity

mixing of linear fluids

Roger E. Khayat

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Abstract

The emergence of nonlinear dynamics in cavity mixing is examined using theBEM. The formulation is implemented for the simulation of 3D transientcreeping flows of Newtonian and linear viscoelastic fluids of the Oldroyd-B type.A boundary-only formulation in the time domain is proposed for viscoelasticflows. It is found that, for simple cavity flow induced by a rotating vane atconstant angular velocity, the traction at the vane tip and cavity face exhibitsnonlinear periodic dynamical behavior with time for fluids obeying linearconstitutive equations.

1 Introduction

Mixing flows constitute a class of the moving-boundary type, whichremain little explored because of their geometric complexity. The presentstudy examines 3D cavity mixing, where the flow domain separates twoboundaries: the outer boundary (cavity), which is stationary, and theinner boundary (rotating vane), which moves and induces the flow. Inaddition to the presence of a moving boundary, the flow involved isunsteady. The flow becomes periodic after initial transients have died outfollowing the early stages of the process. Periodic behavior resultsobviously from the periodicity in geometry and regularity in stirringmotion. For moving-boundary problems the implementation of

Transactions on Modelling and Simulation vol 20, © 1998 WIT Press, www.witpress.com, ISSN 1743-355X

472 Boundary Elements

conventional volume methods, such as the finite-element method (FEM),can often be extremely costly given the requirement for domainremeshing at each time step of the procedure, especially for complex 3Dflow. Remeshing or mesh refinement is not required when a boundary-integral approach, such as the boundary element method (BEM), is used.The BEM offers the obvious advantage when dealing with moving-domain problems as it necessitates only the discretization of theboundary and not that of the inner volume domain. The 3D problem istherefore reduced to computing the flow field on the 2D boundary. Thisis the case for the present flow inside a cavity induced by the movementof the vane; only the (inner) cavity wall together with the (outer) surfaceof the vane need to be discretized.

Although the fluids investigated obey a linear constitutive equation,the flow exhibits a rich sequence of nonlinear dynamical behavior. Thefluids examined are creeping Newtonian and linear viscoelastic of theOldroyd-B type [1]. Similar nonlinear dynamics is usually expected inthe presence of nonlinear effects such as fluid inertia or elasticity (whichleads to the well-known rod-climbing Weissenberg phenomenon).

Unlike many existing BEM formulations for linear viscoelastic

problems in the frequency domain [2,3], the boundary-integral equations

in the current study are derived and solved in the time domain. Thederivation of the boundary-integral equation for viscoelastic flow isbased on the Laplace transform of the flow variables. The association ofthe integral transform of the viscoelastic solution with that associatedwith the Newtonian flow problem is similar to the correspondenceprinciple for linear viscoelastic solids or the elastico-viscoelastic analogy

[3]-

2 Governing equations for viscoelastic flow

linear momentum equations are then given by Consider a fluid occupying

a 3D region, Q(t) that may change with time, t. The fluid is assumed to

be incompressible Newtonian or viscoelastic. Inertia and body forces areassumed negligible. The conservation of mass and

V-u(x,t) = 0, V-a(x,t) = 0, xe fi(t) u T(t) (1)

where V is the gradient operator, x the position vector, u(x, t) the

velocity vector and o(x, t) the total stress tensor. It is convenient to


Boundary Elements 473

express o(x, t) as a sum of Newtonian and elastic contributions, suchthat:

G(x, t) = - p(x, t)I + \iT(x, t)+ T(x, t), (2)

where p(x, t) is the hydrostatic pressure, T(X, t) is the elastic part of the

stress tensor, \JL is the viscosity of the fluid, T(x, t) = Vu(x, t) + [Vu(x, t)]*

is the rate-of-strain tensor, and I is the identity tensor. A suitable

constitutive equation for the excess stress tensor T(x, t) must be selected,where

) = |ir(x,t) + T(x,t). (3)

Although the flow is expected to be significantly influenced by theconstitutive model, the choice of a suitable model is not critical in thepresent study. The study's major objective is to investigate the influenceof fluid elasticity on an already complex flow behavior as it arises forNewtonian fluids alone. It is then more prudent to adopt as simple aviscoelastic constitutive equation as possible. Moreover, the assumptionof linear constitutive behavior makes the approach not adequate tohandle highly nonlinear viscoelastic phenomena. Thus, although largestrains are present in the flows examined here, only small strain rates areassumed to be involved, making the usually important nonlinearities inthe constitutive equation rather negligible. Such nonlinearities typicallystem from convective and upper-convective terms, the dependence ofviscosity and relaxation time on shear and elongation rates. In this study,the constitutive equation for T corresponds to an Oldroyd-B fluid [1]:

%l x, t) + T(x, t) = |Li[r(x, t) + %2 Ax, t)]x e Q(t) u T(t), (4)

where and/^ (0 < < /^ ) are the relaxation and retardation times

of the fluid. Note that a dot means partial differentiation with respect totime. Despite its simplicity, the Oldroyd-B model is known to beadequate to describe the rheology of polymer solutions [1]. Typically, thesolution is composed of a polymer solute in a Newtonian solvent, with

viscosity |ip and jLi , respectively. In this case, the governing equations for

u(x, t), p(x, t) and T(X, t) follow from eqs. (l)-(4), and may be writtenhere as:

V-u(x,t) = 0, 5



V • T(X, t) + mV Vx, t) - Vp(x, t) = 0 ,

x e Q(t) u T(t), (6)

,t). (7)

Note that, in this case, the retardation time is related to the relaxationtime and the polymer-to-solvent viscosity ratio:

(8)

In the limit \JL$ —> 0, eqs. (6) and (7) reduce to the equations

corresponding to Maxwell flow or polymer melt. The Newtonian limit is

recovered if, further, Xi —> 0.

The geometry of the cavity mixer considered in this study is shownin fig. 1. It is emphasized that the current formulation and solutionmethod can handle other more complex flows with moving

boundary(ies). Typically, the velocity is prescribed on the moving part of

the boundary, Fj (t), and the velocity or the traction is imposed on the

remaining stationary part Fg. In this case, F(t) = Fg 4r"F (t) is the

boundary surrounding £2(t), with Fg being stationary and F (t)

undergoing rigid-body rotation. The flow is thus induced by the

movement of F (t) and/or by the imposed velocity or traction on Fg. In

the present work, the velocity is always prescribed on Fm(t), so that

u(x, t) = um(t), x E Fm(t), (9a)

and either the velocity or the traction is imposed on the stationary part:

u(x, t) = ug(t) or t(x, t) = ts(t), x E Fg , (9b)

where the traction is defined as t(x, t) = a(x, t)-n(x, t), n(x, t) being the

unit normal vector at the boundary, pointing away from O(t). As to the



initial conditions, the fluid is assumed to be at rest initially; the fluid is ina stress free state::

u(x, t = 0) s 0, t(x, t = 0) = 0. x e Q(t = 0) u F(t = 0). (10)

The assumption of initial equilibrium may seem incompatible with theassumption that the acceleration term in the momentum equation isnegligible. This is certainly true if the initial jump in the boundarycondition(s) is significant. However, since the viscosity of the fluid andthe imposed velocity are typically low, the assumption of negligibleacceleration, even initially, may still be valid. Conditions (10) greatlysimplify the solution procedure for the viscoelastic flow as will be seenbelow.

3 Generalized boundary-integral equation for

an Oldroyd-B fluid

The first step in deriving the integral equation consists of taking theLaplace transform of the governing eqs. (5)-(7). Since the fluid isincompressible, the transformed continuity and momentum equationsretain the same form in the frequency domain:

V-u(x,s) = 0, (11)

V T(X, s) + V u(x, s) - Vp(x, s) = 0 , xeQuT, (12)

where a bar over the velocity or stress variable designates Laplacetransformation. An expression for the transformed excess stress is alsoobtained from eq. (7) in terms of the transformed rate-of-strain tensor,

F(X,S), which is mathematically equivalent to Newton's law of

viscosity:

(13)

If eq. (13) is inserted in eq. (12), and eq. (11) is used, then themomentum equation in the frequency domain takes the same form as forStokes flow:



^ _ _ fXoS + I

Note that an equivalent viscosity, p, is now obtained, which is a

function of the Laplace parameter s. The problem is solved in the

frequency domain similarly to the flow of a Newtonian fluid. In this case,the acting force for the fundamental solution starts to act initially at time

t = 0. In other words, the force in real time is given by F8(x - y)S(t),

where 8 is the Dirac delta function. Substitution of the fundamental

singular solution into the Reciprocal theorem, using (1), dropping the

force vector, and interchanging the label x and y, lead to the following

integral equation in the frequency domain:

,s) • J(x I y)d - n(y) • [u(y,s) • K(x

= c(x)u(x,s),(15)

where c(x) is equal to one for x belonging to the interior of Q, and for a

point on the boundary F, its value depends on the jump in the value of

the first integral on the boundary as the boundary is crossed. Thus c = 1/2if the boundary is Lyapunov smooth, which requires that a local tangentto the boundary exists everywhere. This assumption, however, is notvalid in the vicinity of sharp corners, cusps or edges. In such cases, aseparate treatment is needed.

The inverse Laplace transform of eq. (15) is taken to obtain thedesired integral equation in the time domain. It is not difficult to see thatthe Laplace inversion of eq. (15) leads to the following time-dependentintegral equation:

J -v,,., p ^

r(t)

n(y , t) •

-jot

r(t)

= c(x,t)| X 2 + u ( x , t ) , xe'

(16)



for the class of problems envisaged in the present study.

The major difficulty in dealing with the solution of eq. (16) is the

explicit presence of the stress tensor rather than the traction vector.Equation (16) is valid for the general transient viscoelastic flow withmoving boundary, where the normal to the boundary changes with time.A direct relation between velocity and traction is not possible unless thenormal vector is constant with time. If the domain occupied by the fluid

is fixed T(t) = T, then the traction may be directly related to the velocityas for Stokes flow. In this case, eq. (16) reduces to:

J /#

Tr / I v n—t / \ i t/ui/v,t/K(x I yjJT = c(x) A, ——— + ufx,i

' dt

(17)

r.

Note that the flow field may still be time dependent although the domaindoes not change with time.

4 Numerical implementation and solution

procedure

Consider again cavity mixing, which is typically illustrated in fig. 1. It isassumed that the velocity is always imposed on the stationary and

moving boundaries, Fg and F (t), respectively. In this case F(t) = F^ u

TmW- On r%, stick conditions apply so that the velocity is zero and eq.

(9b) leads to u%(t) = 0, V t. But if u%(t) vanishes, so does its (time)

derivative. Similarly, both u (t) and its derivative are specified on F (t).

Since the velocity is completely specified on the boundaries, then eq.

(16) will be solved to give the stress or traction on T(t). It is convenientto introduce generalized traction and velocity as:

+ orx, urx, u = AJ ^ <

(18)

so that eq. (16) reduces to:



JP(y, t) - J(x I y)dFy - Jn(y ,t) - U (y, t) • K(x I y)dFy

(19)

Equation (19) is a Fredholm integral of the second kind, with P(x, t)being the unknown. However, the traction at the boundary is yet to bedetermined since it is the variable of main interest in the present study.

Once P(x, t) is determined from eq. (19), the time derivative of thestress tensor must be approximated by finite difference. If the Euler

scheme is used and higher-order terms in the time increment, At, are

neglected, then P(x, t) can be written as:

P(y , 0 = ( — + * V(y, t) - — P(y, t - At) + 0( At) (20)i^At , At

This discretization, however, does not yet give t(x, t) at the boundary,

which is the quantity of main interest here, unless a(x, t - At) is known.

The stress at the previous time step can be determined from an additionalintegral equation that can be derived from eq. (19) similarly to Stokesflow. However, a further approximation may be used, which greatlysimplifies the numerical computation, if the normal vector at the currenttime is expanded around the previous time step. If only the leading term

in the Taylor expansion of the normal vector is kept, such that n(y, t) ~

n(y, t - At) + 0(At), then eq. (20) is reduced, leading to the expression for

the current traction:

X] H- At

The current formulation and its computer implementation areintended for typical mixing problems. The geometry involved is usuallycomplex and three-dimensional. Typically, a mixing process involves themotion of part of the boundary, such as in lid-driven cavity flow, or thepresence of a second moving boundary such as a rotor or a stirrer (fig. 1 ).The discretization of the vane does not pose any difficulty. Thekinematics of each of the discretization elements is known (imposed)given the boundary is undergoing rigid-body motion. In the present work,



the motion of the moving boundary is always known so the velocity willbe prescribed on Fj (t).

Equation (19) is solved using constant elements. The boundary isdiscretized into a finite number of constant triangular elements. The

velocity and traction are thus assumed to be constant over each surfaceelement and equal to the values at the centroid. In this case, since thereare no corner nodes on which the unknown variables are evaluated, thevalue of c(x, t) = 1/2. The resulting discretized equations represent a setof linear algebraic equations in the traction at the boundary once theintegrals are evaluated over each element.

5 Transient rotating flow and nonlinear

dynamics

Consider now the 3D flow inside the rotating mixer shown in fig. 1. The

cubic cavity is of unit side: - 0.5 < x < 0.5, - 0.5 < y < 0.5, - 0.5 < z < 0.5,

with the origin coinciding with the center of the cube. The flow isinduced by the rotation of a flat rectangular rotor (vane) initially

occupying the region - 0.1 < x < 0.1, - 0.4 < y < 0.4, - 0.4 < z < 0.4 as

illustrated in the figure. At t = 0, the rotor is set in counterclockwisemotion (around the vertical y-axis) at an angular speed of one revolutionper second. Since the fluid responds immediately to the motion of therotor, the flow field is the same for both Newtonian and (linear)

viscoelastic fluids. For this reason, viscoelasticity only influences theevolution of stress or traction at the rotor and inner cavity wall, whichwill now be examined under conditions of complete stick at theboundaries.

The evolution of the traction at the rotor tip, at the point locatedinitially at (0., 0., 0.4), is monitored over a period of two revolutions.

Both Newtonian fluids (ki = 0 s) and viscoelastic fluids of the Maxwell

type with A,i = 0.1, 0.2 and 0.3 s are examined. The evolutions of thetraction component t% is shown in fig. 2. For a Newtonian fluid, incontrast to viscoelastic fluids, there is a small but nonzero t% value at t =0 (fig. 2) equal (in magnitude) to the slope u%,z(0., 0., 0.4). The periodic

behavior in t% is clearly depicted from fig. 2. For ^i > 0 s, t% increases

from zero, and eventually settles in turn into a similar periodic signalwith the same frequency as the Newtonian signal. There is a phase shiftrelative to the Newtonian signal that increases as the level of fluidelasticity increases. The amplitude of the stress decreases as the fluid



becomes more elastic. The figure also shows that any modulation or

nonlinear behavior in the time signature tends to disappear as k,

increases.Figure 2 reflects clearly a nonlinear behavior in the evolution of

the traction. This nonlinear character is further evidenced when the long-term phase portrait is examined after transient effects die out. This isdepicted from fig. 3, which shows the trajectories in the (t%, t%) plane forNewtonian and viscoelastic fluids. The phase trajectories clearlyillustrate the periodic nature of the force signal with a decrease in

amplitude as X\ increases. More importantly, the nonlinear character of

the signal decreases gradually for the viscoelastic fluids, until the signal

becomes essentially linear for X\ = 0.3 s. The initial growth of the

viscoelastic traction, and its approaching the Newtonian limit with aphase shift, are reminiscent of problems in sudden inception of steady-state and oscillatory plane and rotating Couette flows [1].

6 Conclusion

In this study, the applicability of the boundary-element method to mixingis demonstrated for Newtonian and viscoelastic fluids. The mixingprocess is typically unsteady and involves the presence of a movingboundary (rotor or stirrer). The rotating flow is investigated as inducedby the action of a vane inside a rectangular cavity. The influence ofviscoelasticity is clearly illustrated in the case of sudden inception in themotion of the vane. The evolution of the tractions at the rotor tip revealscomplex dynamics and transient behavior. While the traction for aNewtonian fluid settles into periodic motion right from the beginning, thetraction for a viscoelastic fluid is found to undergo a transient evolutionbefore it reaches periodic behavior. The viscoelastic signal exhibitsgenerally a phase shift and has an amplitude that decreases as the level offluid elasticity increases. While the constitutive equations used (forcreeping Newtonian and Maxwell fluids) are linear, the response of thetraction is highly nonlinear.

References

[1] R. B. Bird, R. C. Armstrong, and O. Hassager. Dynamics ofPolymeric Liquids (vol. 1, John Wiley & Sons, N. Y., 1987).



[2] T. Kusama and Y. Mitsui, Boundary element method applied to linearviscoelastic analysis, App. Math Modelling 6 (1982) 285.

[3] R. M. Christensen, Theory of Viscoelasticity, 2nd edt., Pergamon,1982.

STATIONARY CAVITY

Fig. 1. Schematic view of mixing flow inside a cavity. Rotating vane in acubic cavity of unit side. Figure illustrates coordinate system and initialposition of the vane.



800

600 -

400 -

-200 -

-400 -

-600 -

-800

Fig. 2. Evolution of t% at the tip of the rotor, at the point initiallylocated at (0., 0., 0.4). The long-term signal period is 1 s correspondingto a full revolution of the rotor. Newtonian fluid ( ), Maxwell fluidwith Aj =0.1 s (—-), 0.2 s (- - -) and 0.3 s (••»).

800

600 -

400 -

200 -

-200 -

-400

-600

-800 -400 -200 200

Fig. 3. Long-term phase-plane behavior of the traction components t%and t% at the tip of the rotor, at the point initially located at (0., 0., 0.4).


y q/ - wit press...creeping flows of newtonian and linear viscoelastic fluids of the oldroyd-b type....

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