Saffman-Taylor Instability of Hele-Shaw Cell
Chen Shu, Xizhi Cao, Enkhsanaa Sommers
Introduction Our group focuses on theoretically analyzing
two-flow Saffman Taylor instability in a Hele-Shaw cell with numerical methods.
A Hele-Shaw cell consists with two parallel plates are spaced infinitesimally thin. Less viscous flow injected from above or below into the more vicious fluid which is sitting between plates. The boundary condition of this flow is defined by surface tension and curvature.
In our case we will apply perturbation on Hele-Shaw cell and calculate expected number of fingers.
We compare numerical result with the experimental result
Fluid #1
Fluid #2
Derive Darcyโs law
Navier-Stokes equation:
(๐ขโ๐ป )๐ข=โ1๐๐ปP+ฮฝโ๐ป2๐ข+ +๐
๐๐ข๐๐ก
=0 (Since steady state uniform flow)
=0 gravity term is ignored since we are dealing with 2-D system
(1)
Derive Darcyโs law
The equation reduced to:Evaluate inertial term(IT) and viscous term(VT):
: h<<L we can ignore x and y axes
Reynold Number=๐๐๐๐๐ก๐๐๐ ๐ก๐๐๐๐ฃ๐๐ ๐๐๐ข๐ ๐ก๐๐๐
โช1 , h๐ก ๐๐๐๐๐๐๐ก๐๐๐ ๐ก๐๐๐๐๐ ๐๐๐๐๐๐๐ .
Derive Darcyโs law
Since, << 1 , so
0
Pressure change in z direction is small enough to neglect
(๐ขโ๐ป )๐ข=โ1๐๐ปP+ฮฝโ๐ป2๐ข
Derive Darcyโs law (4)
Boundary (5)
Pinner
Pouter
R2
R1
Laplace-Young BC:
General form:In our case:
R2 is big enough so that Pouter can be ignored
- surface tension
- curvature
Saffman-Taylor Instability Saffmen Taylor instability occurs when a less
viscous fluid is injected into a more viscous fluid.
It can also occur driven by gravity (without injection) if the interface is horizontal separating two fluids of different densities with the heavier fluid on the top.
In the rectangular configuration the system evolves until a single finger (the SaffmanโTaylor finger) forms.
In the radial configuration the pattern grows forming fingers by successive tip-splitting.
Exact solution to circular BC
At t=0 , ,
Pinner
Pouter
R2
R1
Linear stability analysis:
Perturbation: Pressure (R1) Velocity Curvature
Linear stability analysis:
= => () O(1): = O():
Linear stability analysis:
fits in Laplace equation so that .
Linear stability analysis:
Using the algebraic manipulation:
O (1): O ():
Left hand Side ๐ 1โur (๐ ,๐ )โจยฟR 1(1+ฮตฮท )ยฟ
is the growth rate of . If the perturbation is unstable (expand),
otherwise stable (shrink). In order to find the best perturbation, we need
to find the largest
Linear stability analysis:
Let
Linear stability analysis:
vs m
Relate Flow and
Paterson: )
How do we find number of fingers from the experiment result
Comparison between theoretical and experimental
result Width of Spacers
(mm)
Mass (g) Volume Inside
Syringe (mL)
Number of
Fingers (Measur
e)
Number of Fingers
(Theoretical)
Flow Rate
Relative Error
0.32 500 1.2 29 40 1.26E-07 0.275
0.32 1000 1.2 32 51 2.00E-07 0.37254902
0.32 1500 1.2 37 79 4.89E-07 0.53164557
0.32 2000 1.2 40 88 2.58E-06 0.545454545
0.52 200 1.2 18 19 1.20E-07 0.052631579
0.52 500 1.2 22 21 1.47E-07 0.047619048
0.52 1000 1.2 24 27 2.43E-07 0.111111111
0.52 1500 1.2 27 42 5.89E-07 0.357142857
0.83 500 1.2 11 8 1.04E-07 0.375
0.83 1000 1.2 12 8 1.30E-07 0.5
0.83 1500 1.2 14 11 2.46E-07 0.272727273
Result Analysis
Spacing 0.32mm has a large error.
Small spacing creates a lot of fingers and many of them are small which give us hard time to count
Spacing 0.52mm has the closest simulation result.
0.52mm minimize the errors come from experiment equipment
Spacing 0.83mm has a large error.
the theoretical value for number of fingers are small so a little error will create a large relative error, also the process is quick so itโs hard to tell the exact time
Error grows when mass gets bigger.
Flow rate grows in a really rapid way while the equation only handle it as
Comparison between theoretical and experimental result
Possible causes of error
Time error Counting error Error due to the Experiment equipment Our model is not accurate enough
References: References:
[1] Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series, Oxford University Press,
[3] Kondic, Lou (2014), Linear Stability Analysis of two phase Hele -Shaw Flow, Unpublished.
https://web.njit.edu/~kondic/capstone/2015/lin_stab.pdf
[4] Paterson, Lincoln (1981) Radial Fingering in a Hele Shaw Cell, Department of Engineering Physics, Research School of Physical Sciences, The Australian National University, < http://web.njit.edu/~kondic/capstone/2014/paterson_jfm_81.pdf>.
[4] http://en.wikipedia.org/wiki/Navier%E2%80%93Stokes_equations
[5] http://en.wikipedia.org/wiki/Surface_tension
[6] http://en.wikipedia.org/wiki/Young%E2%80%93Laplace_equation
[7] http://en.wikipedia.org/wiki/Curvature