Nonlinear flow response of soft hair beds …...Supplementary Information Jos´e Alvarado, Jean Comtet, and A. E. Hosoi Massachusetts Institute of Technology Emmanuel de Langre Ecole

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Supplementary Information

Jose Alvarado, Jean Comtet, and A. E. HosoiMassachusetts Institute of Technology

Emmanuel de LangreEcole Polytechnique

(Dated: April 25, 2017)

Nonlinear flow response of soft hair beds

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

SUPPLEMENTARY INFORMATIONDOI: 10.1038/NPHYS4225

NATURE PHYSICS | www.nature.com/naturephysics 1

http://dx.doi.org/10.1038/nphys4225

http://www.nature.com/naturephysics

2

PERTURBATION ANALYSIS OF THE EQUATION OF EQUILIBRIUM

In this section, we present a perturbation analysis of the equation of equilibrium (Main Text, Eq. 2)

0 =d2θ(s)

ds2

(1− L

∫ 1

0

cos θ(s�)ds�)+ v cos θ(s), (1)

with s = s/L, L = L/H, v = 4ηL2vEφa2H , and s� a dummy integration variable. We have boundary conditions:

θ(s = 0) = θ0 θ′(s = 1) = 0.

We seek a solution θ(s) in powers of the small parameter v

θ(s) =∑n

vnθ(n)(s),

where the θ(n) are functions to be determined. Inserting this solution into the equation of motion yields the followingequation and solution, to zeroth order:

0 = θ(0)′′(s)

(1− L

∫ 1

0

cos θ(0)(s�)ds�

)

with θ(0)(s = 0) = θ0 and θ′(0)(s = 1) = 0

↓ �

0 = θ(0)′′(s)

θ(0)(s) = θ0.

with ′ denoting differentiation with respect to s and where we assume (�) that the tips of undeformed hairs (v = 0)

do not contact the opposing surface (L∫ 1

0cos θ(0)(s

�)ds� < 1). Inserting again yields, to first order:

0 = vθ′′(1)

(1− L cos θ0

)+ v cos θ0

with θ(1)(s = 0) = 0 and θ′(1)(s = 1) = 0

θ(1)(s) =cos θ0

1− L cos θ0

(s− s2

2

). (2)

Inserting again yields, to second order:

0 = v2θ′′(2)

(1− L cos θ0

)+ v2θ′′(1)L

∫ 1

0

θ1 sin θ0ds� − v2θ(1) sin θ0

with θ(2)(s = 0) = 0 and θ′(2)(s = 1) = 0

θ(2)(s) =cos θ0 sin θ0(1− L cos θ0

)3

[s

(−1

3

)+ s2

(L cos θ0

6

)+ s3

(1− L cos θ0

6

)+ s4

(−1− L cos θ0

24

)].

Henceforth we shall consider the special case θ0 = 0, corresponding to straight hairs cantilevered perpendicularlyto the anchoring surface. In this case, θ(2) vanishes. We therefore have, to second order:

θ(s) ≈ v

1− L

(s− s2

2

).






3

This parabolic hair profile is similar to the textbook problem of a cantilevered beam with an end load [1]. However,

the prefactor v1−L

can only be found by perturbation analysis. The resulting dimensionless hair height h is given by:

h[θ(s)] =

∫ 1

0

cos θ(s�)ds� ≈ 1− 1

15

v2

(1− L)2.

Finally, the rescaled impedance Z is given by:

Z = h1− L

1− Lh≈ 1− 1

15

v2

(1− L)3.

CORRECTING FOR HAIR TAPER

The hairs that result from our experimental process (Main Text, Methods) do not have a uniform thickness profile.Rather, they are wider at the base (∼ 400µm) and thinner at the tip (∼ 250µm). Most of the taper only occurs ina layer ∼ 1mm from the hair base (cf. Main Text, 1b). The origin for this taper is characteristic of the laser-cuttingprocess, which delivers more power to the side of the working material where the laser is in focus. Our model assumesthat hairs have a uniform thickness profile, given by a circular cross section with constant radius a. Rather thanmodifying our model to account for a tapered profile, we approximate tapered hairs, which have radius profile a(s)(Fig. 6a), by uniform hairs with constant effective radius ae and effective length Le (Fig. 6b). In this section, wecalculate the position (d, h) of a tapered hair tip (Fig. 6c), as well as the position (de, he) of a uniform hair tip(Fig. 6d), both as a function of the forcing parameter to second order. By setting d = de and h = he, we solve for Le

and ae.First, we re-express the forcing parameter v for the more general case where the hair radius a(s) = a0a(s) may vary

along the hair contour, where a0 is an arbitrary reference length scale and a is a dimensionless parameter characterizingthe taper profile:

v =4ηvL2

EφHa2=

4ηvL2δ2

EHa4tapered−−−−−→hair

4ηvL2δ2

EHa(s)4=

4ηvL2δ2

EHa40

1

a(s)4≡ v0a(s)

−4.

Thus, the dimensionless forcing parameter v can be expressed as a product of a dimensionless reference v0 and thedimensionless profile a(s). We can now perform a perturbation analysis similar to that for Eq. 1, but in powers of thesmall parameter v0 for the equation:

0 =d2θ(s)

ds2

(1− L

∫ 1

0

cos θ(s�)ds�)+ v0a(s)

−4 cos θ(s).

Solving to zeroth order gives, as before:

θ(0)(s) = θ0 = 0.

To first order, we have:

0 = v0θ′′(1)(s)(1− L) + v0a(s)

θ(1)(s) =1

1− L

∫ s

0

∫ 1

s�a(s��)−4ds��ds�,

where s� and s�� are dummy integration variables. This solution reduces to Eq. 2 for uniform hairs with a(s) = a =const. Solving to second order yields, as before, for straight hairs:

θ(2)(s) = θ0 = 0.






4

Therefore, the complete solution reads, to second order in v0:

θ(s) =v0

1− L

∫ s

0

∫ 1

s�a(s��)−4ds��ds�.

Re-expressing this solution in dimensional form yields:

θ(s) =4ηvδ2

E

1

H − L

∫ s

0

∫ L

s�a(s��)−4ds��ds�.

We use this solution to calculate the locus of points (x(v), y(v)) that the hair tip sweeps in response to fluid stressesfrom the velocity v:

x(v) =

∫ L

0

sin θ(s)ds ≈∫ L

0

θ(s)ds =4ηvδ2

E

1

H − L

∫ L

0

∫ s

0

∫ L

s�a(s��)−4ds��ds�ds

≡ 4ηvδ2

E

1

H − Lν(1)

y(v) =

∫ L

0

cos θ(s)ds ≈∫ L

0

(1− 1

2θ(s)2

)ds = L− 1

2

(4ηvδ2

E

1

H − L

)2 ∫ L

0

[∫ s

0

∫ L

s�a(s��)−4ds��ds�

]2

ds

≡ L− 1

2

(4ηvδ2

E

1

H − L

)2

ν(3).

The quantities ν(1) and ν(3) relate the taper profile a(s) to the locus of points (x(v), y(v)). They have units ofinverse length and inverse length cubed, respectively. Hairs with identical loci (d, h) have identical ν(1) and ν(3), aslong as one can assume Le ≈ L. Hence, our problem of equating d = de and L = Le has been reduced to equatingν(1) and ν(3) for the tapered and uniform case.For the tapered case, we perform image analysis (see below) to measure a discrete array of hair radius values [ai]

Ni=1

from micrographs of hair samples (Fig. 6e). Hence, ν(1) and ν(3) can be approximated by sums:

ν(1)tapered−−−−−→hair

N∑k=1

k∑j=1

N∑i=j

ai−4∆s3

ν(3)tapered−−−−−→hair

N∑k=1

k∑j=1

N∑i=j

ai−4

2

∆s5,

where ∆s is the distance between pixels. Meanwhile for the uniform case, ν(1) and ν(3) can be calculated directly:

ν(1)uniform−−−−−→hair

ae−4

∫ Le

0

∫ s

0

∫ Le

s�ds��ds�ds =

1

3

Le3

ae4

ν(3)uniform−−−−−→hair

ae−8

∫ Le

0

(∫ s

0

∫ Le

s�ds��ds�

)2

ds =2

15

Le5

ae8.

Rearranging:

Le =6

5

ν(1)2

ν(3)

ae =

(2332

53ν(1)

5

ν(3)3

)1/4

Inserting computed values of ν(1) and ν(3) for the discrete case into the solution above yields the desired effectivelength Le and effective radius ae. As shown in Fig. 7, the effective quantities Le and ae attain reasonable values whichmimic the response of tapered hairs.






5

IMAGE ANALYSIS FOR EXTRACTING HAIR PROFILES

In this section, we describe the image-analysis techniques used to obtain the discretized hair profiles from theprevious section. All analysis was performed with a custom-written Mathematica script (Fig. 7).

1. We manually draw a line which connects the base of the hair to the tip of the hair and record the location ofthe endpoints. This line represents the backbone of the hair.

2. We rotate the image such that the backbone line is horizontal.

3. We crop the image to a region of interest (ROI). The ROI’s width is given by the backbone line’s width. TheROI’s height is such that the hair is contained but does not include features from neighboring hairs.

4. We binarize the image such that white pixels correspond to the hair and black pixels correspond to the back-ground.

5. We sum the number of white pixels N for each column i of pixels along the backbone. Multiplying by the pixelsize ∆s yields the hair thickness profile 2ai = N ∆s as a function of the backbone contour s = ∆s i.

6. Given ai as a function of i, we determine the quantities ν(1) and ν(3), and thus effective hair length Le andthickness 2a.

FLOW PENETRATION INTO THE HAIR LAYER

In this section, we check the validity of our assumption in the main text that flow does not penetrate into thehair bed. In order to account for flow penetration in the hair layer, we solve the Stokes equation in the fluid bulk(h < z < H) to recover the fluid velocity field u = u(z)ex, and we solve the Brinkman equation in the hair-bed layer(0 < z < h) to recover the pore-averaged fluid velocity field U = U(z)ex (Fig. 8a,b). At the interface z = h we assume

continuity of the field u(z = h) = U(z = h) = Ub as well as continuity of fluid shear stress du(z=h)dz = dU(z=h)

dz = τb.The latter condition assumes that momentum is not transferred directly into the hairs via a point force applied atthe tip, which we assume in the main text. We first solve for the velocity field u in the bulk:

η∇2u−∇p = 0.

Assuming ∇p = 0, we have:

d2u

dz2= 0. (3)

With the following boundary conditions:

u(z = h) = Ub u(z = H) = v,

the solution to Eq. 3 is given by

u(z) =Ub(H − z) + v(z − h)

H − hh < z < H. (4)

Now we consider pore-averaged flow U through the hair bed with the Brinkman equation:

∇p = −η

kU+ ηe∇2U,

with η the fluid viscosity, ηe an effective viscosity, and k the permeability. Assuming ∇p = 0, η = ηe, and k = ∆2

with ∆ the effective pore size (in our case, ∆ = δ − d, with δ the spacing between hairs and d the effective hairdiameter), we have:






6

0 = − U

∆2+

d2U

dz2. (5)

With the following boundary conditions:

U(z = h) = Ub U(z = 0) = 0,

the solution to Eq. 5 is given by

U(z) = Ubeh−z∆

e2z/∆ − 1

e2h/∆ − 10 < z < h. (6)

In order to solve for Ub in Eqs. 4 and 6, we apply continuity of shear stress at the interface:

v − Ub

H − h=

du(z = h)

dz= τb =

dU(z = h)

dz=

2Ube2h/∆

∆(e2h/∆ − 1)− Ub

∆.

Solving for Ub yields

Ub = v∆(e2h/∆ − 1)

(e2h/∆ + 1)(H − h) + ∆(e2h/∆ − 1).

We see that penetration of flow, characterized by Ub is controlled in part by the dimensionless parameter ∆/h. InFigure 8c we plot the resulting continuity solutions for the velocity field in the bulk u together with the pore-averaged

flow field U . We show profiles for different values of the area packing fraction φ ≈ d2

δ2 (color) as well as the undeformedhair height h (panels) corresponding to experimental conditions in the main text. Upon inspection of these plots, itappears that flow does not significantly penetrate the hair bed for φ = 0.30 (dark blue) and 0.10 (blue). Meanwhilefor φ = 0.03 (light blue), the flow penetrates significantly into the hair layer.

In order to determine the effect of hair packing fraction φ on the resulting stresses in the system, we express thepore size ∆ as a function of φ: ∆ = d

√1/φ− 1. Next, we compute the shear stress applied on the upper plate:

τ = ηv − Ub

H − h= ηv

e2h/∆ + 1

(e2h/∆ + 1)(H − h) + ∆(e2h/∆ − 1).

In the limit ∆ → 0, we recover the form of shear stress assumed in the main text:

τ∆→0 =ηv

H − h.

In the limit of undeformed hairs, for which h is independent of flow penetration, the impedance ratio ZZ∆→0

measuresthe effect of flow penetration at finite ∆ on the impedance Z of the system:

Z

Z∆→0=

(e2h/∆ + 1)(H − h)

(e2h/∆ + 1)(H − h) + ∆(e2h/∆ − 1).

We plot the impedance ratio ZZ∆→0

as a function of pore size ∆ in Fig. 8d for different values of hair height used inexperiment. We find that for the higher hair packing fractions (φ = 0.3 and φ = 0.1), the impedance ratio is within therange 0.75–0.95. Therefore, the effect of flow penetration due to a finite pore size introduces a correction factor whichis of order one for these experimental conditions. This is particularly interesting from the perspective of biologicalhair surfaces, which can have relatively high packing fractions— up to φ = 0.55 for the case of the glycocalyx [2].For the lower packing fraction (φ = 0.03) tested in experiment, the impedance ratio is within the range 0.55–0.75.






7

Therefore, this value of φ represents a critical value, below which flow penetration in the hair bed starts to have anon-negligible effect on the measured impedance in our experiments.Note that, unlike the model in the main text, the model presented here does not undergo a singularity in the limit

of sparse hairs (φ → 0, ∆ → ∞). We have:

Z∆→∞ =η

H.

That is, the hairs do not affect the system, and we recover a Couette flow profile across the entire gap. In the limith → H, the singularity present in the main text is also regularized, as the impendance can be expressed as

Zh→H = ηe2h/∆ + 1

∆e2h/∆ − 1.

[1] B. Audoly and Y. Pomeau, Elasticity and geometry: from hair curls to the non-linear response of shells (Oxford UniversityPress, 2010).

[2] S. Weinbaum, J. M. Tarbell, and E. R. Damiano, Annual Review of Biomedical Engineering 9, 121 (2007).






8

b

a

FIG. 1. Schematic representing the Taylor-Couette geometry with inner radius Ri and outer radius Ro = Ri + H; anddeformable hairs with length L. a: Top view. b: Side view.






9

a

c

b

-2 -1 0 1 2

-1.5

-1.0

-0.5

0.0

log10v

log10Z˜

-2 -1 0 1 2 3

-1.5

-1.0

-0.5

0.0

log10v˜

log10Z˜

-2 -1 0 1 2

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

log10v

dlog10Z˜/dlog10v

L

0

0.2

0.4

0.6

0.8

1.0

FIG. 2. The rescaled velocity v = v(1− L cos θ0

)−3/2

reduces Z to univariate form. a: Plot of Z as a function of the

dimensionless velocity v = πηL2vEφa2H

. Color corresponds to L (legend, bottom right). b: Plot of the scaling exponent d log Zd log v

,

showing that all Z(v) curves scale with v−1/2 in the limit of high v. c: Transforming the abscissa of panel (a) with the rescaledvelocity v collapses curves.






10

0.1

0.1

0.3

0.5

0.7

0.9

0.3 1 3 10 30 100

FIG. 3. The dimensionless velocity v roughly preserves hair contour shape. Hair contours are numerically computed (Methods)

and displayed for varying L and v.

a b

1.3

1.6

1.9

2.2

2.5

2.8

3.1weak

asymmetry

10 20 30 40 50 60 70 800.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

/ °

rectification

weak

asymmetry

FIG. 4. Asymmetry of angled hair beds in the reconfiguration regime. a: Schematic depicting the degree of asymmetry for thethree response regimes, owing to hair contours for flow with (left) and against (right) the grain. b: Contour plot of the peak

value of Z−/Z+ as a function of the dimensionless hair length L and the anchoring angle θ0. Symbols correspond to the datashown in the main text, Fig 4f.






11

0.1 0.5 1 5 10 50 1000.2

0.5

1

2

v˜

Z˜model 0 [°]

40

30

20

10

0

FIG. 5. Rescaled impedance Z as a function of rescaled velocity v. Color corresponds to anchoring angle (legend, lower left).

Hair dimensions satisfy L cos θ0 = 0.47. Solid lines correspond to flow against the grain (Z−), dashed lines with the grain (Z+).






12

Tapered Hair Uniform Haira b

c d

e f

FIG. 6. Accounting for tapered radius profile of hairs. a: Schematic of a tapered hair profile a as a function of the contourcoordinate s. b: Schematic of a uniform hair profile with constant effective thickness ae and length Le. c,d: The hair tip,characterized by (d, h), trace an arc (dotted line) as a function of v. e: Tapered hair profile is given by an image, which yieldsa discretized array ai. f : Uniform hair profile is given by ∞ for s < L− Le, and ae for L− Le < s < L.






13

a

c

d

e

f

b

200 400 600 800

100

200

300

400

FIG. 7. Image analysis algorithm to extract hair-thickness profile. a: Image of hair specimen. Red line denotes manually drawnline connecting hair base to hair tip. b: Rotated image. Red box denotes region of interest. c: Cropped region of interest.d: Thresholded image. e: Discretized hair-thickness profile 2ai as a function of the contour parameter s = ∆s i. f : Image oftapered hair with effective uniform hair profile (green area) overlaid.






14

vz

velocity field

(u or U)

h

H

0

Brinkman eq.

Stokes eq.

Ub

v

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

0 1 2 3 4 5 6 7

0.0

0.2

0.4

0.6

0.8

1.0

a

c

d

b

0.0 0.5 1.0 1.5 2.0

0.4

0.5

0.6

0.7

0.8

0.9

1.0

FIG. 8. Modeling flow penetration into the hair-bed layer. a: Schematic depicting a hairy surface (bottom) opposing a smoothsurface (top) moving with a velocity v. b: Schematic of the velocity flow field in the fluid bulk and hair-bed layer, whichare determined by solving the Stokes equation and Brinkman equation, respectively. c: Plots of the continuity solutions fordifferent hair heights h (panel) and hair packing fraction φ (color, see legend, left). d: Plots of the impedance ratio Z

Z∆→0as a

function of pore size ∆ for different hair heights h (color, see legend right).






Documents

Nonlinear flow response of soft hair beds …...Supplementary Information Jos´e Alvarado, Jean Comtet, and A. E. Hosoi Massachusetts Institute of Technology Emmanuel de Langre Ecole