We examine the fluid mechanics of viscous flow through filters consisting of perforated thin plates. We classify the effects that contribute to the hydraulic resistance of the filter. Classical analyses assume a single pore size and account only for filter thickness. We extend these results to obtain an analytical formula for the pressure drop across the microfilter versus the flow rate that accounts for the non-uniform distribution of pore sizes, the hydrodynamic interactions between the pores given their layout pattern, and wall slip. Further, we discuss inertial effects and their order of scaling.

1.
L. A.
Spielman
, “
Particle capture from low-speed laminar flows
,”
Annu. Rev. Fluid Mech.
9
,
297
319
(
1977
).
2.
H.
Wyss
,
D.
Blair
,
J.
Morris
,
H. A.
Stone
, and
D.
Weitz
, “
Mechanism for clogging of microchannels
,”
Phys. Rev. E
74
,
061402
(
2006
).
3.
J.
Lin
,
D.
Bourrier
,
M.
Dilhan
, and
P.
Duru
, “
Particle deposition onto a microsieve
,”
Phys. Fluids
21
,
073301
(
2009
).
4.
M. T.
Tyree
,
M. H.
Zimmermann
, and
M. H.
Zimmermann
,
Xylem Structure and the Ascent of Sap
(
Springer
,
New York
,
2002
).
5.
K. H.
Jensen
,
D. L.
Mullendore
,
N. M.
Holbrook
,
T.
Bohr
,
M.
Knoblauch
, and
H.
Bruus
, “
Modeling the hydrodynamics of phloem sieve plates
,”
Front. Plant Sci.
3
,
151
(
2012
).
6.
S.
Gravelle
,
L.
Joly
,
F.
Detcheverry
,
C.
Ybert
,
C.
Cottin-Bizonne
, and
L.
Bocquet
, “
Optimizing water permeability through the hourglass shape of aquaporins
,”
PNAS
110
,
16367
16372
(
2013
).
7.
X.
Yang
,
J. M.
Yang
,
Y.-C.
Tai
, and
C.-M.
Ho
, “
Micromachined membrane particle filters
,”
Sens. Actuators
73
,
184
191
(
1999
).
8.
D. J.
Tritton
,
Physical Fluid Dynamics
, 2nd ed. (
Oxford University Press
,
New York
,
1988
).
9.
I.
Ahmed
and
A.
Beskok
, “
Rarefaction, compressibility, and viscous heating in gas microfilters
,”
J. Thermophys. Heat Transfer
16
,
161
170
(
2002
).
10.
H. Y.
Yang
,
Z. J.
Han
,
S. F.
Yu
,
K. L.
Pey
,
K.
Ostrikov
, and
R.
Karnik
, “
Carbon nanotube membranes with ultrahigh specific adsorption capacity for water desalination and purification
,”
Nat. Commun.
4
,
2220
(
2013
).
11.
D. L.
Mullendore
,
C. W.
Windt
,
H.
Van As
, and
M.
Knoblauch
, “
Sieve tube geometry in relation to phloem flow
,”
Plant Cell
22
,
579
593
(
2010
).
12.
M. M.
Couette
, “
Etudes sur le frottement des liquides
,”
Ann. Chim. Phys.
21
,
433
510
(
1890
).
13.
R. A.
Sampson
, “
On Stokes's current function
,”
Philos. Trans. R. Soc. London, Ser. A
182
,
449
518
(
1891
).
14.
R.
Roscoe
, “
The flow of viscous fluids round plane obstacles
,”
Philos. Mag.
40
,
338
351
(
1949
).
15.
H. L.
Weissberg
, “
End correction for slow viscous flow through long tubes
,”
Phys. Fluids
5
,
1033
(
1962
).
16.
Z.
Dagan
,
S.
Weinbaum
, and
R.
Pfeffer
, “
An infinite-series solution for the creeping motion through an orifice of finite length
,”
J. Fluid Mech.
115
,
505
523
(
1982
).
17.
H.
Hasimoto
, “
On the flow of a viscous fluid past a thin screen at small Reynolds numbers
,”
J. Phys. Soc. Jpn.
13
,
633
639
(
1958
).
18.
K.-K.
Tio
and
S. S.
Sadhal
, “
Boundary conditions for Stokes flows near a porous membrane
,”
Appl. Sci. Res.
52
,
1
20
(
1994
).
19.
C. Y.
Wang
, “
Stokes flow through a thin screen with patterned holes
,”
AIChE J.
40
,
419
423
(
1994
).
20.
J.
Happel
and
H.
Brenner
,
Low Reynolds Number Hydrodynamics
(
Springer
,
Hingham, MA, USA
,
1983
).
21.
V. K.
Rohatgi
,
An Introduction to Probability Theory and Mathematical Statistics
(
Wiley
,
New York
,
1976
).
22.
S. A.
Schaaf
and
P. L.
Chambre
,
Flow of Rarefied Gases
(
Princeton University Press
,
1961
).
23.
E.
Lauga
and
H. A.
Stone
, “
Effective slip in pressure-driven Stokes flow
,”
J. Fluid Mech.
489
,
55
77
(
2003
).
24.
J. C.
Maxwell
, “
On stresses in rarified gases arising from inequalities of temperature
,”
Philos. Trans. R. Soc. London
170
,
231
256
(
1879
).
25.
A. B.
Basset
,
A Treatise on Hydrodynamics with Numerous Examples
(
Dover
,
New York
,
1961
).
26.
A. K.
Sreekanth
, “
Slip flow through long circular tubes
,” in
Rarefied Gas Dynamics
, edited by
L.
Trilling
and
H. Y.
Wachman
(
Academic Press
,
New York
,
1968
), pp.
667
680
.
27.
F. C.
Johansen
, “
Flow through pipe orifices at low Reynolds numbers
,”
Proc. R. Soc. A
126
,
231
245
(
1930
).
28.
O. G.
Tietjens
and
L.
Prandtl
,
Applied Hydro- and Aeromechanics
(
Dover
,
New York
,
1957
).
29.
W. N.
Bond
, “
Viscosity determination by means of orifices and short tubes
,”
Proc. Phys. Soc. London
34
,
139
(
1921
).
30.
S.
Goldstein
,
Modern Developments in Fluid Dynamics
(
Dover
,
New York
,
1965
).
31.
L. G.
Leal
,
Laminar Flow and Convective Transport Processes
(
Butterworth-Heinemann
,
Boston
,
1992
).
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