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 Darcy’s law

Navier-Stokes equation:

(𝑢∗𝛻 )𝑢=−1𝜌𝛻P+ν∗𝛻2𝑢+ +𝑔

𝜕𝑢𝜕𝑡

=0 (Since steady state uniform flow)

=0 gravity term is ignored since we are dealing with 2-D system

(1)

Derive Darcy’s law

The equation reduced to:Evaluate inertial term(IT) and viscous term(VT):

: h<<L we can ignore x and y axes

Reynold Number=𝑖𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑡𝑒𝑟𝑚𝑣𝑖𝑠𝑐𝑜𝑢𝑠𝑡𝑒𝑟𝑚

≪1 , h𝑡 𝑒𝑛𝑖𝑛𝑒𝑟𝑡𝑖𝑎𝑙 𝑡𝑒𝑟𝑚𝑖𝑠𝑖𝑔𝑛𝑜𝑟𝑒𝑑 .

Derive Darcy’s law

Since, << 1 , so

0

Pressure change in z direction is small enough to neglect

(𝑢∗𝛻 )𝑢=−1𝜌𝛻P+ν∗𝛻2𝑢

Derive Darcy’s 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 Saffman–Taylor 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 𝑅1∗ur (𝑟 ,𝜃 )∨¿R 1(1+εη )¿

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 it’s 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