1. Immersed Boundary Modeling of
Journal 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 the
immersed 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 Model
The 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 = LUρ
µ0
, De = λU
L , β =
µp
µ0
•Deformation tensor: d(u) = 1
2 ∇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, and
β ∈ (0, 1) denotes the fraction of the total viscosity that is viscoelastic.
Tether Forces and Maintaining Circular Shape
The F(X, t) expression in equation (2) is made up of two terms, FTi(Xi, t) and
Fsi(Xi, t). The force that connects the annulus points Xi(t) (in crimson) to target
points Zi(t) (in gray) is
FTi = −kTi (Xi(ξ, t) − Zi(ξ, t)) ,
where
Zi(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 a
discretized circle.
The force that connects the annulus points Xi(t) to every
other annulus point is
Fsil
= −
ksi
2
j=l
( Xil
− Xij
− ∆Rilj
)
(Xil
− Xij
)
Xil
− Xij
,
Visualizing Stress
Ellipses based on eigenvalues and eigenvectors of σ show
how a unit circle is stretched and compressed in the fluid
after one revolution of the inner annulus.
For a 2D simulation, σ can be represented as a matrix σ
where
σ =
σ11 σ12
σ21 σ22
(3)
Shown is the stress ellipses inside of the journal bearing
after one revolution of the inner annulus.
Mean Flux
The 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 the
inner 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 between
the two annuli.
Fluid: β = 1
2 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 Profiles
A 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 velocity
profiles 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) and
top (Right) gaps between the annuli. The model does not match the asymptotic
perfectly because the asymptotic is for a very small gap (c ≪ r2).
Fluid: β = 1
2, 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 annulus
in the lower gap.
•A viscoelastic response is observed more quickly for fluids as the De decreases.
Immersed Boundary Method
The Eulerian fixed grid communicates to the Lagragian moving grid using a
discretized forms of equations (1) and (2) with the help of a regularized delta
function.
Denoting a Lagrangian point by Xik
:
uk =
l,j
ul,jδh(xl,j − Xik
) h2
,
fl,j =
k
Fik
δh(xl,j − Xik
)∆sk.
These equations have δh(x) = φh(x)φh(y) where
φh(r) =



1
8h 3 − 2|r|
h + 1 + 4|r|
h − 4r2
h2 if |r| ∈ [0, h],
1
8h 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 at
a given time. They show the direction
a passive particle would travel provided
that the velocity field was fixed at a given
time and are represented as curves inside
the journal bearing.
Also shown are traces of the stress ma-
trices of equation (3). The level of inten-
sity increases with increasing time, and
can be seen to increase faster for lower
De, 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 of
Vancouver, 2008.
Contact Ryan Grove at qwpp@iup.edu