Immersed Boundary Modeling ofJournal Bearings in a Viscoelastic Fluid

R.R. GROVE AND J.C. CHRISPELL

DEPARTMENT OF MATHEMATICS, INDIANA UNIVERSITY OF PENNSYLVANIA, INDIANA, PA 15705

ABSTRACT: The flow created by two eccentric rotating cylinders immersed in both Newtonian and viscoelastic fluids is simulated with a model created using theimmersed boundary method. A model of this form allows for the transient behavior of the flow in the bearing to be studied as it transitions from rest to a steady state.

Viscoelastic Fluid ModelThe non-dimensional Navier-Stokes/Oldroyd-B equations that model an incom-

pressible, viscoelastic fluid (with stress diffusion term ν) are:

σ +De

(

∂σ

∂t+ u · ∇σ − (∇u)σ − σ(∇u)T − ν∆σ

)

− 2βd(u) = 0 in Ω

Re

(

∂u

∂t+ u · ∇u

)

+∇p− 2(1− β)∇ · d(u)−∇ · σ = f in Ω

∇ · u = 0 in Ω

∂X(ξ, t)

∂t=

∫

Ω

u(x, t)δ(x−X(ξ, t)) dx (1)

f(x, t) =

∫

Γ

F(X, t)δ(x−X(ξ, t)) dξ (2)

•Nondimensional parameters: Re = LUρµ0

, De = λUL, β =

µp

µ0

•Deformation tensor: d(u) = 12

(

∇u + (∇u)T)

•The fluid velocity, pressure and extra stress are denoted by u, p, and σ respec-tively. The fluid Deborah and Reynolds number are written as De and Re, andβ ∈ (0, 1) denotes the fraction of the total viscosity that is viscoelastic.

Tether Forces and Maintaining Circular ShapeThe F(X, t) expression in equation (2) is made up of two terms, FT i(Xi, t) and

Fsi(Xi, t). The force that connects the annulus points Xi(t) (in crimson) to targetpoints Zi(t) (in gray) is

FT i = −kT i (Xi(ξ, t)− Zi(ξ, t)) ,

whereZi(ξ, t) =

(ri cos(ωit + ξ) + xi, ri sin(ωit + ξ) + yi∣

∣i ∈ 1, 2

Hookean springs with stiffness ksi and rest length ∆Rilj

are used to connect every point to every other point on adiscretized circle.

The force that connects the annulus points Xi(t) to everyother annulus point is

Fsil = −ksi

2

∑

j 6=l

(∥

∥Xil −Xij −∆Rilj

∥

∥)(Xil −Xij)∥

∥Xil −Xij

∥

∥

,

Visualizing StressEllipses based on eigenvalues and eigenvectors of σ show

how a unit circle is stretched and compressed in the fluidafter one revolution of the inner annulus.

For a 2D simulation, σ can be represented as a matrix σ

where

σ =

[

σ11 σ12σ21 σ22

]

(3)

Shown is the stress ellipses inside of the journal bearingafter one revolution of the inner annulus.

Mean FluxThe mean flux for this model is the average amount of fluid moving through one

of the gaps between the annuli over a time interval τ , and is calculated by using

Φi(t) =1

τ

∫ t0+τ

t0

φidt,

where φi across the bottom and top gap of the annuli, respectively, is:

φ2 =

∫ α1

α2

uxdy and φ1 =

∫ α3

α4

uxdy.

τ = 0.5 was selected so that the mean flux throughout the first revolution of theinner annulus could be analyzed with Re = 10 (Top-Left), Re = 50 (Top-Right),Re = 100 (Bottom-Left), and Re = 200 (Bottom-Right) at the bottom gap betweenthe two annuli.

Fluid: β = 12 and De = 0.0, 0.1, 0.5, 1.0, and 2.0.

•The larger the Re, the longer it takes each simulation to converge to steady state.•For Re = 10, 50 and 100 it can be seen that the bearing reaches steady state within

a single revolution of the inner annulus for the simulations with De = 0.0 and 0.1.•The higher the De, the longer it takes each simulation to converge to steady state.

Velocity ProfilesA velocity profile in this model is a collection of the measures of velocity in the x-

direction at any point in the top or bottom gap between the two annuli. The velocityprofiles of the Newtonian model is compared to the known asymptotic (Left)

u1 = ω2r2

(

1−Y

B

)

along with the velocity profiles of a viscoelastic model at the bottom (Middle) andtop (Right) gaps between the annuli. The model does not match the asymptoticperfectly because the asymptotic is for a very small gap (c ≪ r2).

Fluid: β = 12, Re = 50, and De = 0.0, 0.1, 0.5, 1.0, and 2.0.

•The larger De simulations appear to have a larger velocity between the annulusin the lower gap.

•A viscoelastic response is observed more quickly for fluids as the De decreases.

Immersed Boundary MethodThe Eulerian fixed grid communicates to the Lagragian moving grid using a

discretized forms of equations (1) and (2) with the help of a regularized deltafunction.

Denoting a Lagrangian point by Xik:

uk =∑

l,j

ul,jδh(xl,j −Xik)h2,

fl,j =∑

k

Fikδh(xl,j −Xik)∆sk.

These equations have δh(x) = φh(x)φh(y) where

φh(r) =

18h

(

3− 2|r|h+√

1 + 4|r|h− 4r2

h2

)

if |r| ∈ [0, h],

18h

(

5− 2|r|h−

√

−7 + 12|r|h

− 4r2

h2

)

if |r| ∈ [h, 2h],

0 otherwise.

Velocity Streamlines and Stress Tracers

Shown are streamlines which are tan-gent to the velocity vector of the flow ata given time. They show the directiona passive particle would travel providedthat the velocity field was fixed at a giventime and are represented as curves insidethe journal bearing.

Also shown are traces of the stress ma-trices of equation (3). The level of inten-sity increases with increasing time, andcan be seen to increase faster for lowerDe, since the effects of viscoelasticity oc-cur faster at lower De.

Conclusion/Future Research•Viscoelasticity changes the dynamics of the flow.•As De increases, it takes longer for the flow inside the journal bearing to reach

steady state.• Ideas for future research include making a larger inner annulus, the addition

of gravity into the model, and having the inner annulus not fixed in space.•Future research will include longer simulation times and more emphasis on

the re-circulation of the fluid.

[1]R. G. Larson, The structure and rheology of complex fluids. Oxford university press, 1999.

[2]C. S. Peskin, The immersed boundary method. Acta numerica, 11.0 (2002): 479-517.

[3]K. Liu. Viscoelastic flows within eccentric rotating cylinders. Master’s thesis, University ofVancouver, 2008.

Contact Ryan Grove at qwpp@iup.edu