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Viscoelastic Fluids Mathematical Models of Viscoelastic Flows Numerical Methods of Viscoelastic Flow Simulations Preliminary Results

Mathematical Models and Computational Methodsfor Viscoelastic Fluids

Olabanji Y. ShonibareAdvisors: Prof. Kathleen Feigl & Prof. Franz Tanner

February 16, 2015

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Contents

1 Viscoelastic Fluids

Introduction

Flow Behaviour of Viscoelastic Fluids Vs Newtonian Fluids

2 Mathematical Models of Viscoelastic Flows

Conservation Equation of Continuum MechanicsRheology and Constitutive Equations

Memory Fluids

3 Numerical Methods of Viscoelastic Flow Simulations

Governing EquationsDiscretization of the Transport Equation

Iterative Solution Algorithm

4 Preliminary Results

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Introduction

Viscoelastic Fluids

They are fluids that possess both viscous and “elastic” properties. Theyexhibit flow phenomena that cannot be explained by the Newtonian

Viscous law e.g. Weissenberg effect . Hence, they are also referred tonon-Newtonian fluids .

Examples of non-Newtonian fluids are Polymeric fluids used to makeplastic articles, dough used to make bread and pasta, biological fluidssuch as synovial fluids found in joints and blood.

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The Weissenberg Effect

(N) Centrifugal forces cause fluid to move away from the rod. (V)

Centripetal forces causes fluid to move towards the center

Vi l i Fl id M h i l M d l f Vi l i Fl N i l M h d f Vi l i Fl Si l i P li i R l

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The Tubeless Siphon

(N) Siphoning stops when tube is lifted out of container. (V) Fluidcontinues to flow up and through the siphon.

Viscoelastic Fl ids Mathematical Models of Viscoelastic Flo s N merical Methods of Viscoelastic Flo Sim lations Preliminar Res lts

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Die Swell

Die swell is caused by relaxation of extended polymer coils as the stress in a

polymeric liquid reduces from high, flow producing stresses present within

the die to the low stresses associated with the extruded stream moving

through ambient air.

Source: R.B. Bird, R.C. Armstrong, O. Hassager, ”Dynamics of Polymeric Liquids, Fluid Mechanics,” John Wiley and Sons, vol. 1, New

York, 1977.


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Dimensionless Groups

For Newtonian fluids, the Reynolds number (Re ) is the most importantdimensionless group while for viscoelastic fluids, the Deborah number(De ) is the most important dimensionless group.

The Deborah number is defined as

De = λ

t flow

where λ is the characteristic time of the material and t flow is the time

interval during which a typical fluid element experiences a significantsequence of kinematic event. When De ≥ De crit , elastic effects becomesrelevant, where De crit depends on the flow problem. Typically, De crit = 1.


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Contents


Introduction




Memory Fluids


Governing Equations

Discretization of the Transport Equation




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Conservation Equation of Continuum Mechanics

Conservation Equation

Two physical laws that govern the Isothermal definition of matter are thelaw of Conservation of Mass and the law of Conservation of LinearMomentum.The Law of Conservation of Mass:

∂ρ

∂ t + ∇ · (ρv) = 0

The Law of Conservation of Momentum:

∂ρv

∂ t + ∇ · (ρv v)

= −∇ ·Π + ρg (1)

where Π = τ + p I and τ is called the extra stress tensor . It containscontributions to stress that results from fluid deformation.


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Rheology and Constitutive Equations

Constitutive Equation

A Constitutive equation is an equation that expresses the molecularstresses generated in the flow in terms of kinetic variables such asvelocities, derivatives of velocities and strain.


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Rheology and Constitutive Equations

Newtonian Fluids

The Constitutive Equation for an Incompressible Newtonian Fluid is givenby

τ = −µγ̇ (2)

where γ̇ , the rate of strain tensor is given by

γ̇ = ∇v + (∇v)T


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Memory Fluids

Memory Fluids

For Generalized Newtonian Constitutive Equation,

τ (t ) = −η(γ̇ )γ̇ (t ).

Since γ̇ (t ) represents only the instantaneous deformation, there can beno effect of the history of the deformation on the stress in these models.

To construct a Constitutive equation with memory, we must includeterms that involve expressions such as γ̇ (t − t o ), the value of γ̇ at a time

t o seconds in the past.


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Memory Fluids

Maxwell Model

The Maxwell fluid Constitutive Model (differential form),

τ + λ∂ τ

∂ t = −ηo γ̇


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Memory Fluids

Maxwell Model (Integral form)

To calculate stress, we solve the Maxwell model for stress tensor toobtain the Maxwell Model (in integral form)

τ (t ) = −

t

−∞

ηo

λ e −

(t −t )λ

γ̇ (t )dt (3)

This model calculates the stress at the time of interest t as the integralover events at past times t . Hence, it is not only a function of the

instantaneous shear-rate tensor but also a function of the history of theshear rate tensor.


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Memory Fluids

What Next?

During Modeling of any process, it is always prudent to begin with thesimplest models (i.e. linear models like the previous ones) and to move tomore complex, non-linear equations only if the linear equations areinadequate.


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Memory Fluids

Upper Convected Jeffreys (Oldroyd-B) Fluids

τ + λ1∇

τ = −ηo

γ̇ + λ2

∇

γ̇

The model above is no longer linear because of the introduction of nonlinear terms in the velocity gradient ∇v by the convected derivative

terms.A major drawback with this model and other quasi-linear models is thatthey are unable to model both shear-thinning and non-zero secondnormal stress differences in shear.

The upper convected derivative of an arbitrary tensor, A, is defined as :

∇

A ≡ D A

Dt − (∇v)T · A − A · ∇v


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Memory Fluids

Oldroyd 8-Constant model

Oldroyd proposed a constitutive equation by adding to the UpperConvected Maxwell model all possible terms that are linear in either τ orγ̇ and at most quadratic in γ̇ . The resulting equation is the Oldroyd8-Constant Model

τ + λ1∇

τ + 1

2(λ1 − µ1)(γ̇ · τ + τ · γ̇ ) +

1

2µo (tr τ )γ̇ +

1

2ν 1(τ : γ̇ )I

= −ηo

γ̇ + λ2

∇

γ̇ + (λ2 − µ2)(γ̇ : γ̇ ) + 1

2ν 2(γ̇ : γ̇ )I

(4)

where ηo , λ1 and λ2 are the zero shear viscosity, relaxation time andretardation time, respectively, from the Maxwell and Jeffreys models.The other five constants, µ0, µ1, µ2, ν 1, ν 2 are associated with theadditional nonlinear terms.


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Memory Fluids

Giesekus Equation

This includes terms that are second order in stress, which are absent inOldroyd 8-Constant model,

τ + λ∇

τ + αλ

ηo

τ · τ = −ηo γ̇

This model has three parameters: the zero shear viscosity, ηo , therelaxation time, λ and α, which is a parameter that relates to the

anisotropy of the drag encountered by flowing polymer segments.


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Memory Fluids

Phan-Thien-Tanner (PTT) Fluid

The Phan-Thien-Tanner Model is given by

(f (tr (τ p ))) τ p + λ∇

τ p = ηp (∇v + (∇v)T )

f (tr (τ p )) =

1 + ληp

trτ p

, linear form

exp

ληp

trτ p

, exponential form


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Contents


Introduction




Memory Fluids


Governing Equations





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Governing Equations

Governing Equations

The governing equations are the Mass and Momentum balance

∇ · v = 0

ρ

∂ v

∂ t + ∇ · (v v)

= −∇p + ∇ · τ


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Governing Equations

Governing Equations

where τ = τ s + τ p .

The Newtonian law holds for the solvent contribution

τ s = ηs (∇v + (∇v)T

For the polymeric contribution, τ p , we consider the Linear PTT equation

1 + λ

ηp

tr (τ p ) τ p + λ∇

τ p = ηp [∇v + (∇v)T ]


G

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Governing Equations

High Weissenberg Number Problem (HWNP)

Figure: A planar contraction-expansion flow cell geometry

Early attempts to predict experimental observation in this (benchmark)geometry were largely frustrated by the presence of singularities at there-entrant corners, which causes breakdown of numerical codes atsignificant values of Weissenberg number, a situation which is well knownto rheologists as the High Weissenberg Number Problem (HWNP).


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Governing Equations

Stabilization technique (DEVSS)

The DEVSS technique involves the addition of an elliptic contribution tothe momentum equation, which is solved implicitly, and then explicitlysubtracting its counterpart in a source term:

ρ∂ v

∂ t + ∇ · (v v)

= −∇p + ∇ · τ

Using τ = τ s + τ p , τ s = ηs (∇v + (∇v)T ) and DEVSS technique, weobtain

ρ

∂ v∂ t + ∇ · (v v)

− (ηs + φ)∇ · (∇v) = −∇p + ∇ · τ p − φ∇ · (∇v)

where φ is a positive number usually taken to be ηp .


Governing Equations

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Governing Equations

Stabilization technique (DEVSS)

Thus, the method may be written as follows

ρ

∂ v

∂ t + ∇ · (v v)

−∇ · (ηo ∇v) = −∇p + ∇ ·Σp

1 +

λ

ηp tr (τ p )

τ p + λ

∇

τ p = ηp [∇v + (∇v)T ]

where the term ∇ ·Σp is given by ∇ · τ p −∇ · (ηp ∇v)

table

algorithm



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Transport Equation

∂ v∂ t

+ ∇ · (v v)

= 1ρ

[−∇p + ∇ · τ ]

Figure: Computational cell for finite volume method



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Temporal term

The temporal term is integrated over a control volume and evaluatedusing a second-order accurate scheme such as backward difference

scheme:

d

dt

V

vdV ≈ 3(vP V )n+1 − 4(vP V )n + (vP V )n−1

2t



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Diffusion term

V

∇ · (ν ∇v)dV =

S

(ν ∇v) · d S =

i ν i ∇vf ,i · Si

In one dimension, the face gradient term (∇v)f · S can be evaluatedbetween the center of the cell of interest, P and center of a neighboringcell, N using

(∇v )f · S ≈ v N − v P

|d| |S|

where d is the length vector between the cell centers P and N.



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p q

Gradient term

V

∇pdV = S

pd S ≈i

p f ,i Si

The face pressure field, p f is evaluated using central differencing.



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Divergernt term

The stress term in the momentum equation is treated as a source termand it is discretized as follows

V

∇ · τ dV =

S

τ · d S ≈i

τ f ,i · Si

The cell face stress, τ f , is evaluated using central differencing.



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Convection term

V

∇ · (v v)dV = S

(v v) · d S ≈i

vf ,i vf ,i · Si =i

F vf ,i

where F = vf · S is the velocity flux across the face f .



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Interpolation of velocity field, vf

Central Differencing Scheme (CDS)

vf = f x vP + (1 − f x )vN ,

f x = |fN |/|PN |. |fN | is the distance between f and cell center N , and|PN | is the distance between cell centers P and N .

Second-order accurate but unbounded



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Interpolation of velocity field, vf

Upwind Differencing Scheme (UDS)

vf =vP , F > 0

vN , F < 0

First-order accurate, but highly stable

highly diffusive particularly when the flow direction is skewed relative

to grid lines



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Minmod Scheme (Roe (1985))

It is a TVD scheme, which could be seen as a second-orderextension of Upwind-biased discretization schemes.

In 1-d, the general form of the face value, v f within a discretizationscheme for convective flux may be written as

v f = v P + 1

2ψ(r ) (v N − v P )

where r = v P − v S

v N − v P



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Minmod Scheme (Roe (1985))

for the UD Scheme, ψ(r ) = 0for the CD Scheme, ψ(r ) = 1

for the LUD Scheme, ψ(r ) = r

for the Minmod Scheme, ψ(r ) = min(r , 1) , r > 0

0 , r ≤ 0



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The solution strategy involves a modification to the PISO (PressureImplicit with Splitting of Operators) algorithm by explicitly introducingelastic stress unknowns in the momentum equation, which requires someadditional steps.

In order to apply Rhie-Chow interpolation, the momentum equation isexpressed as follows

∂ v

∂ t + ∇ · (v v) −∇ · (ν ∇v) −

1

ρ∇ ·Σp = −

1

ρ∇p

where ν = ηo

ρ and ∇ ·Σp is given by ∇ · τ p −∇ · (ηp ∇v).



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1 For a given velocity field, vni , the constitutive equation is solved

implicitly for the elastic stress component τ ∗

p ,i , and Σ∗

p ,i iscomputed. governing equations

2 Using the stress field, Σ∗

p ,i , the discretized momentum equation issolved without the contribution of the pressure gradient term toobtain the velocity components, v∗i - This represents the predictor

step for velocity.3 Using the predicted velocity v∗i , the pressure correction equation is

solved for p ∗, the predictor step for pressure.4 Correct the velocity field v∗i , by solving the velocity field v∗∗i using

the estimated pressure field p ∗.

5 Using the corrected velocities, v∗∗

i , the discretized constitutiveequation is solved again to obtain the corrected stress field, τ ∗∗p ,i .

6 Repeat steps 1-5 using v∗∗i , p ∗, and τ ∗∗

p ,i as improved estimates for

the solution at the new time step values vn+1i , p n+1, and τ

n+1p ,i until

all corrections are negligibly small.

7 March to the next time step.



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p ,i , and Σ∗












n+1p ,i until





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p ,i , and Σ∗












n+1p ,i until





I i S l i Al i h

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p ,i , and Σ∗












n+1p ,i until





I i S l i Al i h

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p ,i , and Σ∗












n+1p ,i until





It ti S l ti Al ith

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p ,i , and Σ∗












n+1p ,i until




C t t

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Contents


Introduction




Memory Fluids


Governing Equations

Discretization of the Transport EquationIterative Solution Algorithm



Test Geometry

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Test Geometry

A planar abrupt contraction with contraction ratio, H /h, of 3.97 : 1 (H = 0.0127, h = 0.0032) was chosen as test problem because of the availability of literature data for validation of developed code.

Figure: 4:1 Planar contraction geometry


Flow Properties and Model Parameters

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Flow Properties and Model Parameters

Q [cm3.s −1] U inlet [cm.s −1] γ̇ [s −1] ρ [Kg .m−3] Re De

252 3.875 48.4 803.87 0.56 1.45

Model λ [s ] ηp [Pa.s ] ηs [Pa.s ]LPTTS 0.25 0.03 1.422 0.002

constitutive equation


Mesh Properties

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Mesh Properties

Figure: Computational mesh

Number of CVs x min/h y min/h

20700 0.0065 0.017


Color Plot of U

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Color Plot of U


Streamlines

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Streamlines


Color Plot of τ

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Color Plot of τ xx


Velocity profile along centerline for Wi = 1 45

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Velocity profile along centerline for Wi = 1.45

0 20 40 60 80 100 120

0.4

0.6

0.8

1.0

1.2

1.4

xh

u U


Velocity profiles along centerline for

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y p gWi = 0.5,Wi = 1.0,Wi = 1.45,Wi = 5.0

Wi0.5

Wi1.0

Wi1.45

Wi5.0

0 20 40 60 80 100 120

0.0

0.5

1.0

1.5

xh

u U

80.0 80.5 81.0 81.5 82.0 82.5 83.01.40

1.42

1.44

1.46

1.48

1.50


Velocity profiles on vertical lines in the downstream section

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Velocity profiles on vertical lines in the downstream section

Wi0.5

Wi1.0

Wi1.45

Wi5.0

0.0 0.2 0.4 0.6 0.8 1.0

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

xh

u U


Conclusion

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Conclusion

The governing equations are discretized using a Collocated finite volumemethod, which has been implemented in the OpenFOAM library:

The convection term of the governing equations is treated usinghigh resolution schemes (e.g. TVD schemes) which provides better

numerical stability and accuracy for hyperbolic PDEs.Further stability was achieved through stress-splitting techniquessuch as DEVSS method, which enhances the elliptic character of thegoverning equations.

The iterative solution strategy is based on the PISO

predictor-corrector algorithm and Rhie-Chow interpolation scheme,which has been modified for Viscoelastic flow calculations.


Future Work

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Numerical Simulation of Segmented Two-phase Flows inMicrochannels using Volume of Fluid/ Level Set Method.


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Thank You!

Questions, please.


Acknowledgements

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Acknowledgements

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Special thanks to the following professors: Dr. Allan Struthers, Dr.

Alexander Labovsky, Dr. Jiguang Sun, Dr. Tamara Olson, Dr.

Zhengfu Xu, Dr. Iosif Pinelis, Dr. Franz Tanner, Dr. Kathleen

Feigl.The great lectures i have been opportuned to take with you havecontributed immensely to this work and would also serve as an impetusfor future projects.The constructive advices given by Prof. Mark Gockenbach has been

invaluable.

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