low reynolds number fluid-structure interactions: modeling...

Introduction Perturbative calculation Computational modeling Extension: Thick plates Summary

Low Reynolds Number Fluid-Structure Interactions:Modeling the Response of Soft Microfluidic Devices

Ivan C. Christov & Tanmay C. Shidhorehttp://tmnt-lab.org/

School of Mechanical EngineeringPurdue University

Minisymposium on Analysis, Control and Inverse Theory of Flows, Material Structures,Acoustics, and Their Interactions

SIAM Conference on Analysis of Partial Differential Equations

Baltimore, Maryland

December 10, 2017

Christov & Shidhore (Purdue) Modeling Low Reynolds # FSIs SIAM PD17 1 / 14

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Lab on a chip

“The ability to perform laboratory operations on small scales

using miniaturized (lab-on-a-chip) devices has many benefits.

Designing and fabricating such systems is extremely challenging,

but physicists and engineers are beginning to construct highly

integrated and compact labs on chips with exciting functionality

...” – Nature 442 (2006)

But there’s much to learn ...

– Wall Street Journal (Oct. 26, 2015)




Fabrication and response of microfluidic devicesTypical lab-on-a-chip device fabrication via replica molding of PDMS.(Sollier, et al., Lab Chip, 2011)

cured polymer is still sticky having some difficulties to crosslink

(E ¼ 1.1 MPa). These data confirm that changing the cross-

linking ratio for PDMS is not a reliable solution towards more

rigid polymer prototyping materials.

As discussed by Inglis,27 recipes for harder PDMS have been

recently developed as alternatives to classical Sylgard 184

PDMS. For example, the h-PDMS by IBM Zurich has been

reported to be 4.5 times stiffer than standard PDMS40,41 but

Inglis reported difficulty in release of these polymers from the

molds.27 Here we tested another PDMS, also from Dow Corning

with an expected higher stiffness (biomedical grade PDMS,

Silastic 7-4860). Unfortunately, this Silastic PDMS is very

viscous, which greatly complicates its processing, and above all

still was measured to have a tensile elastic modulus similar to

1 : 10 PDMS (2.4 versus 2.5 MPa) (Fig. 3.A).

The other prototyping polymers investigated are much stiffer

than PDMS. The Young’s modulus measured for TPE is

"1.2 GPa, similar to thermoplastics like PC (2 GPa) and three

orders of magnitude higher than 1 : 10 PDMS. This is important

since design performance for TPE chips should be replicated

when later transferring to mass producible thermoplastics,

reducing the need to re-optimize designs late in the commer-

cialization process. When cured as indicated in our process flow,

NOA81 exhibits a tensile modulus of 325 MPa, which is lower

than the 1 GPa reported by Bartolo et al.34 but still two orders of

magnitude higher than PDMS. As explained previously, such

a difference is likely due to the reduced curing time that was

required for successful device assembly and sealing. Finally, the

tensile modulus measured for PUMA is 91 MPa, which is lower

than TPE and NOA81. Notably, PUMA was observed to have

different properties in its stress-strain curve than TPE and

NOA81, with a short elastic behavior, for extension only from

0 to 2 mm, followed by a large plastic behavior, where PUMA

pieces undergo irreversible extension up to 60 mm.

Deformation under flow

The low tensile modulus of PDMS offers several advantages for

microfluidic applications as (i) it helps in demolding without

Fig. 2 Protocols for fabrication of PDMS, PUMA, TPE and NOA chips. Our primary test device was a straight and rectangular channel (5 cm long,

60 mm wide, 52 mm deep). A restriction (1000 mm long, 30 mm large) was added 4 cm downstream the inlet in order to validate the replication fidelity for

more complex structures. (i) PDMSmicrofluidic devices were fabricated using standard replica molding processes,26 with a cross linker to polymer ratio

of 1 : 10, and maintaining 2 h as the curing time for repeatable mechanical measurements. As PUMA, NOA and TPE adhere to SU8 and make

demolding impossible, a PDMS master mold with the same polarity as the silicon master was produced as indicated by Kuo et al.,31 using octade-

cyltricholosilane (Sigma-Aldrich, 0.2% v/v in pure ethanol) to passivate the surface. (ii) The PUMA chips were prepared as previously described by

Kuo et al.,32 especially using 1 min of UV exposure for PUMA channels, 5 min for the PUMA-coated glass slide, and additional 5 mins to complete the

bonding after the conformal sealing. (iii) TPE devices were fabricated following the protocol published by Fiorini et al.,28–30 with UV exposure for 70 s in

our conditions (l ¼ 364 nm, 400 W, Dymax Model 2000 Flood). (iv) NOA chips were made using a protocol adapted from Bartolo et al.34 and W€agliet al.,35 especially using 2 s of UV exposure for NOA channels, 3 s for the NOA-coated glass slide, and an additional 3 s to complete the bonding after the

conformal sealing.

3756 | Lab Chip, 2011, 11, 3752–3765 This journal is ª The Royal Society of Chemistry 2011

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Resulting microchannels are soft. (Gervais, et al., Lab Chip, 2006; Ozsun, et al., JFM, 2013)

images were acquired, filtered and rendered using Imaris 4.2

image analysis software (Bitplane Inc.). Cross-section contours

were obtained using built-in algorithms from the Matlab image

processing toolbox (Mathworks Inc., Natick, MA).

Results and discussion

Numerical simulations

In order to understand the nature of coupled fluid–structure

interactions, simulations focus on predicting the extent of the

channel deformation as well as the resulting pressure distribu-

tion inside the device. The models also reveal the extent of the

flow perturbation due to the channel deformation (Fig. 3).

These results (and eqn (12) describing the average fluid

velocity U(z) at a position z down the channel) show that the

average fluid velocity increases towards the channel outlet as

the channel cross-section decreases, while the volumetric flow

rate remains constant. This acceleration of the flow in a

deforming channel is a potential source of error in predicting

quantities such as the shear stress in a PDMS device for cell

attachment assays.

Confocal imaging and displacement measurements

Channel deformation was verified directly using confocal

microscopy. 3D rendered images of the channels capture the

shape of the channels under various flow conditions (Fig. 4).

The experimental data reveal the much higher deformation

predicted by the geometry change at the inlets of the device

(Fig. 5 and 6, first 3 data points). The rapid tapering of the

channel (Fig. 2) effectively produces an unconstrained

displacement (stress-free) boundary condition at the edges.

Both simulation and experimental data show that this effect

dies away within a few widths inside the channel. This entrance

effect is not taken into account in the simplified model

provided in eqn (10). However, since in most applications,

channels are much longer than wide, neglecting it is usually a

reasonable approximation.

By using the raw data, measurements of the maximum

displacement (located at mid channel width) were performed

and compared to the numerical simulations and scaling model

(eqn (10)). Two types of measurements were performed in both

250 mm and 500 mm wide channels: (i) A flow rate was specified

and the maximum displacement measured for various points

along the channel axis (Fig. 5 and 6). (ii) An axial position was

Fig. 2 Schematics of the microfluidic channel used (channel to scale).

Deflection is measured in the channel’s narrower region (250 mm or

500 mm). A series of triangular ticks (20 mm wide) are patterned at the

channel wall every 500 mm for accurate positioning of confocal images.

Fig. 3 3D simulation of the velocity (color) and pressure (grayscale)

profiles under an imposed pressure drop of 1 bar. The channel

geometry represented is 25 mm 6 500 mm 6 1 cm (Re = 15). Young’s

modulus is assumed to be of E = 1 MPa. The decreasing cross-sectional

area along the channel axis is responsible for the change in the fully

developed flow profile and for the acceleration of the flow.

Fig. 4 3D rendering of the entrance of two microfluidic channels of

the same material and geometry (250 mm wide, 26 mm thick, 1 cm long,

E = 2.2 MPa). Top channel: 300 mL min21 imposed flow rate (Re =

13). The tapering of the channel can be observed by comparing the

cross-sections in the first and second segments. Long dash lines:

topographic displacement curves. Bottom channel: 1 mL min21

imposed flow rate (no displacement reference measurement).

Fig. 5 Maximum displacement vs. axial position (E = 2.2 MPa). m:

26 mm 6 250 mm channel at a flow rate of 300 mL min21. &: 30 mm 6500 mm channel at a flow rate of 800 mL min21. 6: Numerical

simulation under the same material and flow conditions.

504 | Lab Chip, 2006, 6, 500–507 This journal is ! The Royal Society of Chemistry 2006

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Non-invasive pressure measurements

(a) (b)

xz

y

05

1015

–0.5

00.5

z (m

m)

x (mm)0 5 10 15

–20

0

20

40

3.6 kPa

100 Pa

1.6 kPa

50 kPa

20 kPa

10 kPa

Chip

Bottom wall

ShimChannelzy

(d)

(e) (f)

(c)

xy

100

101

102

102 103 104 105 103 104 105

100

101

102

FIGURE 1. (a) A one-dimensional channel with a deformable top wall. (b) The two-dimensionaldeflection ⇣2d(x, z) of S1 (t = 200 nm, 2h0 = 175 µm) under hydrostatic pressure of p = 2.3 kPaas measured optically. The inset shows a cross-sectional view of the channel. Clamps hold thechip (with the thin membrane at the centre) and the bottom wall together. The two walls areseparated by a precision shim; an O-ring (black ovals) seals the channel. (c) Cross-sectionalline scans from the image in (b) showing the parabolic z-profile of the deformable wall underpressure. (d) The one-dimensional (average) wall deflection ⇣(x) at several different pressuresfor the same sample. The indicated values are gauge pressure values. These are the localconstitutive curves. (e) The peak deflection ⇣p of the PDMS deformable walls as a function ofpressure p. (f ) ⇣p of the SiN deformable walls as a function of p; the dashed lines are the p1/3

asymptotes. Note that the ⇣p and the p axes do not cover the same ranges in (e) and (f ). Errorbars are smaller than the symbols.

pressure distribution in a planar channel flow and validate our measurements againstthe analytic approximation in (1.2) for a long channel. Our method does not dependupon the particulars of the local constitutive relation ⇣(x) = ⇣(p(x)). In other words, itremains independent of the nature of the wall response, providing accurate results forbuckled walls and elastically stretching walls alike.

734 R1-3

For a rigid conduit, q = R−1h ∆p (∼ Poiseuille).

What if flow and deformation are coupled? q = f (∆p)?




Problem formulation: Static response

h0

x = 0

z = 0

u(x, z)

ℓy = 0

x

y

z

q

w

x = -w/2 x = w/2

For (h0/`)Re � 1, theNavier–Stokes equations forthe fluid become:

Stokes

{0 = −∇p + µ∇2v ,∇ · v = 0,

BCs

{v = 0 at y = 0, h(x , z).

h(x , z) = h0 + u(x , z).

For the top wall: Kirchoff–Love equation for bending of thin plates,[Et3

12(1− ν2)

]︸︷︷︸

≡B

∇2‖∇

2‖u = p, u =

∂u

∂x= 0 at x = ∓w

2; z = 0, `.

All other walls are rigid.




AsymptoticsPerturbation expansion, {V ,P,U} ∼

∑i

∑j ε

iδj{V ,P,U}ij ,assuming 0 < h0/`︸︷︷︸

≡ε

� h0/w︸︷︷︸≡δ

� 1. [Capital letters = dimensionless.]

At the leading order, we get the usual lubrication approximation:

VZ ,00(X ,Y ,Z ) = −1

2

dP00

dZ[H(X ,Z )− Y ]Y .

Then, the flow rate follows

Q = − 1

12

dP00

dZ

∫ +1/2

−1/2H(X ,Z )3 dX .

Still need to know H(X ,Z ) to finish the calculation ... at leadingorder, plate eq. reduces to beam eq. in the streamwise cross-section(H = 1 + βU):

∂4U00

∂X 4= P00(Z ), β :=

uc

h0=

h30∆p

Bδ4= O(1).




Flow rate–pressure drop relationPutting all the pieces together:

Q =P(Z )

12(1− Z )

[1 +

β̃

20P(Z )

��β̃2

630P(Z )2 +

��

��β̃3

48, 048P(Z )3

]where β̃ = β/24 and ∆P = P(0).

Deformation (β ↑) increases Q for a fixed ∆P, decreases P overall:

0.2 0.4 0.6 0.8�

1

2

3

4

5

(�) Δ�

increasing β

0.2 0.4 0.6 0.8 1.0�

2

4

6

8

10

12

(�) �(�)

increasing β

β= �

β= �

β= �

β= �

Figure: (a) ∆P–Q, different β̃; (b) P(Z ), Q = 1, different β̃.




Explicit expression for the velocity fieldEven for β = O(1), keeping only terms up to P2 is a good approx.,

P(Z ) = −10

β̃

[1−

√1 +

12

5β̃Q(1− Z )

].

Hence,

VZ (X ,Y ,Z ) = 6Q

[1 +

12

5β̃Q(1− Z )

]−1/2

×{

1 + β̃P(Z )(X + 1

2

)2 (X − 1

2

)2 − Y}Y .

-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

X

Y

(a)

-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

X

Y

(b)

-0.4 -0.2 0.0 0.2 0.4

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

X

Y

(c)

0.32

0.64

0.96

1.28




Two-way fluid–structure interaction simulations

Set up simulations to benchmarkagainst (Ozsun, et al., JFM, 2013):

I Case 1: `×w = 15.5 mm× 1.7 mm,h0 = 244µm, t0 = 200µm(t0/w ' 0.12);E = (1.6± 0.2) MPa, νs = 0.499.

I Case 2: `×w = 15.5 mm× 1.7 mm,

h0 = 155µm, t0 = 605µm

(t0/w ' 0.35);

E = (1.6± 0.2) MPa, νs = 0.499.

FSI simulations usingANSYS/Fluent:

I Partitioned approach w/ “systemcoupling” module.

I BCs as discussed before; iterate tosteady state.

I Large-deformations are considered;

grid-independence was verified.

X

Y

Z

t

h

w

ANSYS®

Fluent

ANSYS®

Mechanical

Fluid Stresses

Incremental

Displacement

System

Coupling




Top-wall deformation: FSI simulations vs. thin-plate theoryThe leading-order deformation shape shape is self-similar (in Z ) :

∂4U

∂X 4= P(Z ) ⇒ U(X ,Z )

P(Z )=

1

24

(X − 1

2

)2 (X + 1

2

)2.

FSI simulations of the “S4” case from (Ozsun, et al., JFM, 2013) verify this:

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.001

0.002

0.003

0.004

0.005

U/P

(Z)

z = 2000μm

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.001

0.002

0.003

0.004

0.005

U/P

(Z)

z = 5000μm

50mL/min40mL/min30mL/min20mL/min10mL/minChristov et al.

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.001

0.002

0.003

0.004

0.005

U/P

(Z)

z = 8000μm

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.001

0.002

0.003

0.004

0.005

U/P

(Z)

z = 12000μm

Case 1: t0/w ' 0.12(≈ a thin plate).




Flow rate–pressure drop: Experiment, numerics & theory

•s: “S4” data (Ozsun, et al., JFM, 2013); ×s: our FSI simulation.

Dashed line – rectangular conduit (∼Poiseuille), q = h30w∆p/(12µ`).

Solid curve – (new) perturbative theory:

q =h3

0w ∆p

12µ`

[1 +

1

480

w4

Bh0∆p +

1

362, 880

(w4

Bh0

)2

(∆p)2 +1

664, 215, 552

(w4

Bh0

)3

(∆p)3

].




Mindlin’s thick-plate-bending theory

Governing equations (Mindlin, ASME J. Appl. Mech., 1951) in terms ofdeformation u, rotation of the normal φ, stress resultants Q and M.

Equilibrium conditions (perturbative approach ⇒ keep only ∂/∂x):

∂Q∂x

+ p(z) = 0,

∂M∂x−Q = 0.

Constitutive relations are

M = B∂φ

∂x,

Q = κGt

(∂u

∂x+ φ

),

where κ is a “shear correction factor,” and G = E/[2(1 + νs)] iselastic shear modulus.




Top-wall deformation: Simulations vs. thick-plate theoryFollowing (Mindlin, ASME J. Appl. Mech., 1951) thick-plate theory, we getanother leading-order self-similar (in Z ) deformation shape:

U(X ,Z )

P(Z )=

1

24

(1

4− X 2

)[2(t/w)2

κ(1− νs)+

(1

4− X 2

)].

FSI simulations of the “S5” case from (Ozsun, et al., JFM, 2013) verify this:

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.002

0.004

0.006

0.008

0.010

U/P

(Z)

z = 2000μm

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.002

0.004

0.006

0.008

0.010

U/P

(Z)

z = 5000μm

50mL/min40mL/min30mL/min20mL/min10mL/minThick-plate, κ=1.0

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.002

0.004

0.006

0.008

0.010

U/P

(Z)

z = 8000μm

−0.4 −0.2 0.0 0.2 0.4X

0.000

0.002

0.004

0.006

0.008

0.010

U/P

(Z)

z = 12000μm

Case 2: t0/w ' 0.35(6≈ a thin plate).




Flow rate–pressure drop: Experiment, numerics & theory

0 10 20 30 40 50q (mL/min)

0

5

10

15

20

25

30

35

Δp (kPa)

Experimental data (Ozsun et al.)Simulation dataRigid channelThick-plate (κ=1.0)Fit, Gervais et al.

Case 2: “S5” data (Ozsun, et al., JFM, 2013) vs. our simulations & theory.

Solid curve – (new) perturbative theory for thick plates:

q =h3

0w ∆p

12µ`

{1 +

[1

480+ f1(t/w)

]w4

Bh0∆p +

[1

362, 880+ f2(t/w)

](w4

Bh0

)2

(∆p)2

+

[1

664, 215, 552+ f3(t/w)

](w4

Bh0

)3

(∆p)3

}Christov & Shidhore (Purdue) Modeling Low Reynolds # FSIs SIAM PD17 13 / 14



SummarySystematically derived flow rate–pressure drop relations for adeformable shallow microchannel, explicitly showing 3 dependencies:

q ≈ h30w∆p

12µ`

[1 + F

(∆p

E,h0

w,t

w

)]

Refs: I.C.C., V. Cognet, T.C. Shidhore, H.A. Stone, “Flow rate–pressure drop relation

for deformable shallow microfluidic channels,” in rev. for J. Fluid Mech., arXiv:1712.02687

T.C. Shidhore, I.C.C., “Static response of deformable microchannels: A comparative

modelling study,” in rev. for J. Phys.: Condens. Mat., arXiv:1709.03002.

Thank you for your attention!


PURDUE UNIVERSITY COLLEGE OF ENGINEERING BRAND MANUAL page 31

THE SIGNATUREAlways use the correct artwork for the schools of Engineering signatures. The left side is always the university name as independent usage or the name of the division level for lockup usage with Frutiger Black font. The right side can be changed to the name of the College or divisions with Frutiger Light font.

VISUAL IDENTITY

LOGO USAGELOGO WITHTHE SCHOOL NAME

Supported by NSF grant CBET-1705637.


http://arxiv.org/abs/1712.02687

http://arxiv.org/abs/1709.03002

http://engineering.purdue.edu/ME

http://www.nsf.gov/

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