The paper presents results of an investigation of the response of an incompressible fluid in a circular micropipe and a parallel-plate microchannel to a sudden time-independent pressure drop. Solutions of the problem were obtained analytically using the Laplace transform technique and numerically using the lattice Boltzmann method. The unsteady velocity profiles in the pipe and in the channel were obtained with the help of the infinite series solutions validated against numerical simulations. In line with the expectations, the flow asymptotically tends to the fully developed pattern, which is attained quicker for smaller Knudsen numbers. The solution enabled also obtaining relations to estimate the hydraulic resistance coefficient.

1.
M.
Calvert
and
J.
Baker
, “
Thermal conductivity and gaseous microscale transport
,”
J. Thermophys. Heat Transfer
12
,
138
145
(
1998
).
2.
S. A.
Schaaf
and
P. L.
Chambre
,
Flow of Rarefied Gases
(
Princeton University Press
,
Princeton
,
1961
).
3.
M.
Gad-el-Hak
, “
The fluid mechanics of microdevices—the Freeman scholar lecture
,”
ASME J. Fluids Eng.
121
,
5
-
33
(
1999
).
4.
S.
Stefanov
and
C.
Cercignani
, “
Monte Carlo simulation of the Taylor-Couette flow of a rarefied gas
,”
J. Fluid Mech.
256
,
199
-
213
(
1993
).
5.
A. A.
Avramenko
and
A. V.
Kuznetsov
, “
Instability of a slip flow in a curved channel formed by two concentric cylindrical surfaces
,”
Eur. J. Mech., B: Fluids
28
(
6
),
722
727
(
2009
).
6.
G.
Karniadakis
,
A.
Beskok
, and
N.
Aluru
,
Microflows and Nanoflows Fundamentals and Simulation
(
Springer
,
New York
,
2005
).
7.
B. J. N.
Wylie
, “
Application of two-dimensional cellular automaton lattice–gas models to the simulation of hydrodynamics
,” Ph.D. dissertation (
University of Edinburgh
, UK,
1990
).
8.
J. B.
Maxwell
,
Lattice Boltzmann Methods for Interfacial Wave Modelling
(
University of Edinburgh
,
1997
).
9.
C.
Cercignani
,
Theory and Applications of the Boltzmann Equation
(
Elsevier
,
1976
).
10.
X.
Shan
and
H.
Chen
, “
Lattice Boltzmann model for simulating flows with multiple phases and components
,”
Phys. Rev. E
47
,
1815
-
1819
(
1993
).
11.
X.
Shan
and
H.
Chen
, “
Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation
,”
Phys. Rev. E
49
,
2941
-
2948
(
1994
).
12.
X.
He
and
L.
Luo
, “
Theory of the lattice Boltzmann method: From the Boltzmann equation to the lattice Boltzmann equation
,”
Phys. Rev. E
56
,
6811
-
6817
(
1997
).
13.
Q.
Zou
and
X.
He
, “
On pressure and velocity boundary conditions for the lattice Boltzmann BGK model
,”
Phys. Fluids
9
(
6
),
1591
1596
(
1997
).
14.
A. I.
Tyrinov
,
A. A.
Avramenko
,
B. I.
Basok
, and
B. V.
Davydenko
, “
Modeling of flows in a microchannel based on the Boltzmann lattice equation
,”
J. Eng. Phys. Thermophys.
85
(
1
),
65
72
(
2012
).
15.
J.
Lihnaropoulos
and
D.
Valougeorgis
, “
Unsteady vacuum gas flow in cylindrical tubes
,”
Fusion Eng. Des.
86
(
9–11
),
2139
2142
(
2011
).
16.
P. L.
Bhatnagar
,
E. P.
Gross
, and
M.
Krook
, “
A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems
,”
Phys. Rev.
94
,
511
525
(
1954
).
17.
F.
Sharipov
and
I.
Graur
, “
General approach to transient flows of rarefied gases through long capillaries
,”
Vacuum
100
,
22
25
(
2014
).
18.
P.
Szymanski
, “
Quelques solutions exactes des équations de l’hydrodynamique du fluide visqueux dans le cas d’un tube cylindrique
,”
J. Math. Pures Appl.
97
(
11
),
67
107
(
1932
).
19.
W.
Müller
, “
Zum problem der anlaufströmung einer flüssigkeit im geraden rohr mit Kreisring- und Kreisquerschnitt
,”
ZAMM
16
,
227
-
238
(
1936
).
20.
W.
Gerbes
, “
Zur instationären, laminaren strömung einer inkompressiblen zähen flüssigkeit in kreiszylindrischen Rohren
,”
Z. Angew. Phys.
3
,
267
-
271
(
1951
).
21.
H.
Schlichting
and
K.
Gersten
,
Boundary Layer Theory
(
2000
).
22.
A. A.
Avramenko
and
A. V.
Kuznetsov
, “
Start-up flow in a channel or pipe occupied by a fluid-saturated porous medium
,”
J. Porous Media
12
(
4
),
361
367
(
2009
).
23.
T.
Indinger
and
I. V.
Shevchuk
, “
Transient laminar conjugate heat transfer of a rotating disk: Theory and numerical simulations
,”
Int. J. Heat Mass Transfer
47
(
14-16
),
3577
-
3581
(
2004
).
24.
I. V.
Shevchuk
, “
Unsteady conjugate laminar heat transfer of a rotating non-uniformly heated disk: Application to the transient experimental technique
,”
Int. J. Heat Mass Transfer
49
,
3530
3537
(
2006
).
25.
F.
Sharipov
, “
Application of the Cercignani-Lampis scattering kernel to calculations of rarefied gas flows
,”
Eur. J. Mech., B: Fluids
22
,
133
-
143
(
2003
).
26.
F.
Sharipov
and
V.
Seleznev
, “
Data on internal rarefied gas flows
,”
J. Phys. Chem. Ref. Data
27
,
657
-
706
(
1998
).
27.
A.
Agrawal
and
S. V.
Prabhu
, “
Deduction of slip coefficient in slip and transition regimes from existing cylindrical Couette flow data
,”
Exp. Therm. Fluid Sci.
32
,
991
-
996
(
2008
).
28.
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables
, edited by
M.
Abramowitz
and
I. A.
Stegun
(
Dover Publications
,
New York
,
1972
).
29.
B. R.
Munson
,
D. F.
Young
,
T. H.
Okiishi
, and
W. W.
Huebsch
,
Fundamentals of Fluid Mechanics
, 6th ed. (
John Wiley & Sons, Inc.
,
Hoboken, New Jersey
,
2009
).
30.
A.
Beskok
and
G. E.
Karniadakis
, “
Simulation of slip-flows in complex microgeometries
,”
ASME DSC
40
,
355
370
(
1992
).
You do not currently have access to this content.