A classical Stokes’ second problem has been known for a long time and represents one of the few exact solutions of nonlinear Navier-Stokes equations. However, oscillatory flow in a semi-infinite domain of Newtonian fluid under harmonic boundary excitation only leads to fluid wind-milling back and forth in close wall vicinity. In this study, we are presenting the mathematical model and the numerical simulations of the Newtonian fluid and the shear-thinning non-Newtonian blood-mimicking fluid flow. Positive flow rates were obtained by periodic yet nonharmonic oscillatory motion of one or two infinite boundary flat walls. The oscillatory flows in semi-infinite or finite 2D geometry with sawtooth or periodic rectified-sine boundary conditions are presented. Rheological human blood models used were: Power-Law, Sisko, Carreau, and Herschel-Bulkley. A one-dimensional time-dependent nonlinear coupled conservative diffusion-type boundary layer equations for mass, linear momentum, and energy were solved using the finite-differences method with finite-volume discretization. It was possible to test the accuracy of the in-house developed computational programs with the few isothermal flow analytical solutions and with the celebrated classical Stokes’ first and second problems. Positive flow rates were achieved in various configurations and in absence of the adverse pressure gradients. Body forces, such as gravity, were neglected. The calculations utilizing in-phase sawtooth and rectified-sine wall excitations resulted in respectable net flow which stabilizes and becomes quasi-steady, starting from rest, after three to ten periods depending on the fluid rheology. It was assumed that rapid return stroke of the wall actuator resulted in total wall slip while forward wall motion existed with no-slip boundary condition. Shear “driving” and “driven” fluid regions were identified. The shear-thinning fluid rheology delivered many interesting results, such as pluglike flow. Constructive interference of diffusive penetration layers from multiple flat surfaces could be used as practical pumping mechanism in micro-scales.

References

References
1.
Lamb
,
H.
, 1945,
Hydrodynamics
,
6th ed.
,
Dover
,
New York
.
2.
Schlichting
,
H.
, 1979,
Boundary-Layer Theory
,
7th ed.
,
McGraw-Hill
,
Washington
.
3.
Batchelor
,
G. K.
, 2000,
An Introduction to Fluid Dynamics
,
Cambridge University
,
Cambridge, MA
.
4.
Carslaw
,
H. S.
, and
Jaeger
,
J. C.
, 1959,
Conduction of Heat in Solids
,
2nd ed.
,
Clarendon
,
Oxford, UK.
5.
Panton
,
R.
, 1968, “
The Transient for Stokes’s Oscillating Plate: A Solution in Terms of Tabulated Functions
,”
J. Fluid Mech.
,
31
(
4
), pp.
819
825
.
6.
Tokuda
,
N.
, 1968, “
On the Impulsive Motion of a Flat Plate in a Viscous Fluid
,”
J. Fluid Mech.
,
33
(
4
), pp.
657
672
.
7.
Zeng.
Y.
, and
Weinbaum
,
S.
, 1995, “
Stokes Problems for Moving Half-Planes
,”
J. Fluid Mech.
,
287
, pp.
59
74
.
8.
Erdogan
,
E. M.
, 2000, “
A Note on an Unsteady Flow of Viscous Fluid Due to an Oscillating Plane Wall
,”
Int. J. Non-Linear Mech.
,
35
, pp.
1
6
.
9.
Khaled
,
A. -R. A. A.
, and
Vafai
,
K.
, 2004, “
The Effect of the Slip Condition on Stokes and Couette Flows Due to an Oscillating Wall: Exact Solutions
,”
Int. J. Non-Linear Mech.
,
39
, pp.
795
809
.
10.
Muzychka
,
Y. S.
, and
Yovanovich
,
M. M.
, 2006, “
Unsteady Viscous Flows and Stokes’s First Problem
,”
Proc.
IMECE Paper No. 2006–14301, pp.
1
10
.
11.
Ai
,
L.
, and
Vafai
,
K.
, 2005, “
An Investigation of Stokes’ Second Problem for Non-Newtonian Fluids
,”
Numer. Heat Transfer, Part A
,
47
, pp.
955
980
.
12.
Ashraf
,
E. E.
, and
Mohyudinn
,
M. R.
, 2005, “
New Solutions of Stokes Problem for an Oscillating Plate Using Laplace Transform
,”
J. Appl. Sci. Environ. Mgt.
,
9
(
1
), pp.
51
55
.
13.
Daidzic
,
N. E.
, and
Hossain
,
Md. S.
, 2010, “
The Model of Micro-Fluidic Pump with Vibrating Boundaries
,”
14th AMME Conference
, AM-054 (MP-9), May 25–27,
Cairo, Egypt
.
14.
Patankar
,
S. V.
, 1980,
Numerical Heat Transfer and Fluid Flow
,
Hemisphere Publishing Corporation
,
Washington
.
15.
Press
,
W. H.
,
Vetterling
,
W. T.
,
Teukolsky
,
S. A.
, and
Flannery
,
B. P.
, 1992,
Numerical Recipes in FORTRAN: The Art of Scientific Computing,
2nd Eed.
,
Cambridge University
,
Cambridge, England
.
16.
Thomas
,
J. W.
, 1995,
Numerical Partial Differential Equations: Finite Difference Methods
,
Springer-Verlag
,
New York
.
17.
Tannehill
,
J. C.
,
Anderson
,
D. A.
, and
Pletcher
,
R. H.
, 1997,
Computational Fluid Mechanics and Heat Transfer
,
2nd ed.
,
Taylor & Francis
,
London
.
18.
Ferziger
,
J. H.
, and
Perić
,
M.
, 2002,
Computational Methods for Fluid Dynamic
,
3rd ed.
,
Springer-Verlag
,
Berlin
.
19.
Majumdar
,
P.
, 2005,
Computational Methods for Heat and Mass Transfer
,
Taylor & Francis
,
London
.
20.
Burmeister
,
L. C.
, 1993,
Convective Heat Transfer
,
2nd ed.
,
Wiley
,
New York
.
21.
Koplik
,
J.
,
Banavar
,
J. R.
, and
Willemsen
,
J. F.
, 1989, “
Molecular Dynamics of Fluid Flow at Solid Surfaces
,”
Phys. Fluid A
,
1
(
5
), pp.
781
794
.
22.
Koplik
,
J.
, and
Banavar
,
J. R.
, 1995, “
Continuum Deductions From Molecular Hydrodynamics
,”
Ann. Rev. Fluid Mech.
,
27
, pp.
257
292
.
23.
Thompson
,
P. A.
, and
Troian
,
S. M.
, 1997, “
A General Boundary Condition for Liquid Flow at Solid Surface
,”
Nature
,
389
, pp.
360
362
.
24.
Nguyen
,
N -T.
, and
Wereley
,
S. T.
, 2002,
Fundamentals and Applications of Microfluidics, MEMS Series
,
Artech House
,
Boston
.
25.
Yilmaz
,
F.
, and
Gundogdu
,
M. Y.
, 2008, “
A Critical Review on Blood Flow in Large Arteries; Relevance to Blood Rheology, Viscosity Models, and Physiologic Conditions
,”
Korea-Aust. Rheol. J.
,
20
(
4
), pp.
197
211
.
26.
Hsu
,
C. H.
,
Vu
,
H. H.
, and
Kang
,
Y. H.
, 2009, “
The Rheology of Blood Flow in a Branched Arterial System With Three-Dimensional Model: A Numerical Study
,”
J. Mech.
,
25
(
4
), pp.
N21
N24
.
27.
Lee
,
B. K.
,
Kwon
,
H. M.
,
Hong
,
B. K.
,
Park
,
B. E.
,
Suh
,
S -H.
,
Cho
,
M.-T.
,
Lee
C. S.
,
Kim
,
M. C.
,
Kim
,
C. J.
,
Yoo
,
S. S.
, and
Kim
,
H -S.
, 2001, “
Hemodynamic Effects on Atherosclerosis-Prone Coronary Artery: Wall Shear Stress/Rate Distribution and Impedance Angle in Coronary and Aortic Circulation
,”
Yonsei Med. J.
,
42
(
4
), pp.
375
383
.
28.
Kundu
,
P. K.
, and
Cohen
,
I. M.
, 2007,
Fluid Mechanics
,
4th ed.
,
Elsevier
,
New York
.
29.
Skelland
,
A. H. P.
, 1967,
Non-Newtonian Flow and Heat Transfer
,
Wiley
,
New York
.
30.
Briedis
,
D.
,
Moutrie
,
M. F.
, and
Balmer
,
R. T.
, 1980, “
A Study of the Shear Viscosity of Human Whole Saliva
,”
Rheol. Acta
,
19
, pp.
365
374
.
31.
Sankar
,
D. S.
, and
Hemalatha
,
K.
, 2007, “
A Non-Newtonian Fluid Flow Model for Blood Flow Through a Catheterized Artery – Steady Flow
,”
Elsevier’s Applied Mathematical Modeling
,
31
, pp.
1847
1864
.
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