Starting flow due to a suddenly applied pressure gradient in a circular tube containing two immiscible fluids is solved using eigenfunction expansions. The orthogonality of the eigenfunctions is developed for the first time for circular composite regions. The problem, which is pertinent to flow lubricated by a less viscous near-wall fluid, depends on the ratio of the radius of the core region to that of the tube, and the ratios of dynamic and kinematic viscosities of the two fluids. In general, a higher lubricating effect will lead to a longer time for the starting transient to die out. The time development of velocity profile and slip length are examined for the starting flows of whole blood enveloped by plasma and water enveloped by air in a circular duct. Owing to a sharp contrast in viscosity, the starting transient duration for water/air flow can be ten times longer than that of blood/plasma flow. Also, the slip length exhibits a singularity in the course of the start-up. For blood with a thin plasma skimming layer, the singularity occurs very early, and hence for the most part of the start-up, the slip length is nearly a constant. For water lubricated by air of finite thickness, the singularity may occur at a time that is comparable to the transient duration of the start-up, and hence, an unsteady slip length has to be considered in this case.

References

References
1.
Deen
,
W. M.
,
2012
,
Analysis of Transport Phenomena
,
2nd ed.
,
Oxford
,
New York
.
2.
Bird
,
R. B.
,
Stewart
,
W. E.
, and
Lightfoot
,
E. N.
,
2002
,
Transport Phenomena
,
2nd ed.
,
Wiley
,
New York
.
3.
Kapur
,
J. N.
, and
Shukla
,
J. B.
,
1964
, “
The Flow of Incompressible Immiscible Fluids Between Two Plates
,”
Appl. Sci. Res., Sec. A
,
13
(
1
), pp.
55
60
.
4.
Bhattacharyya
,
R. N.
,
1968
, “
Note on the Unsteady Flow of Two Incompressible Immiscible Fluids Between Two Plates
,”
Bull. Calcutta Math. Soc.
,
1
, pp.
129
136
.
5.
Wang
,
C. Y.
,
2011
, “
Two-Fluid Oscillatory Flow in a Channel
,”
Theor. Appl. Mech. Lett.
,
1
(
3
), p.
032007
.
6.
Wang
,
C. Y.
,
2017
, “
Starting Flow in a Channel With Two Immiscible Fluids
,”
ASME J. Fluids Eng.
,
139
(
12
), p.
124501
.
7.
Wu
,
Y. H.
,
Wiwatanapataphee
,
B.
, and
Hu
,
M.
,
2008
, “
Pressure-Driven Transient Flows of Newtonian Fluids Through Microtubes With Slip Boundary
,”
Phys. A: Stat. Mech. Appl
,
387
(
24
), pp.
5979
5990
.
8.
Wiwatanapataphee
,
B.
,
Wu
,
Y. H.
,
Hu
,
M.
, and
Chayantrakom
,
K.
,
2009
, “
A Study of Transient Flows of Newtonian Fluids Through Micro-Annuals With a Slip Boundary
,”
J. Phys. A: Math. Theor.
,
42
(
6
), p.
065206
.
9.
Matthews
,
M. T.
, and
Hastie
,
K. M.
,
2012
, “
An Analytical and Numerical Study of Unsteady Channel Flow With Slip
,”
Anziam J.
,
53
(
4
), pp.
321
336
.
10.
Crane
,
L. J.
, and
McVeigh
,
A. G.
,
2015
, “
Slip Flow along an Impulsively Started Cylinder
,”
Arch. Appl. Mech.
,
85
(
6
), pp.
831
836
.
11.
Avramenko
,
A. A.
,
Tyrinov
,
A. I.
, and
Shevchuk
,
I. V.
,
2015
, “
An Analytical and Numerical Study on the Start-Up Flow of Slightly Rarefied Gases in a Parallel-Plate Channel and a Pipe
,”
Phys. Fluids
,
27
(
4
), p.
042001
.
12.
Avramenko
,
A. A.
,
Tyrinov
,
A. I.
, and
Shevchuk
,
I. V.
,
2015
, “
Start-Up Slip Flow in a Microchannel With a Rectangular Cross Section
,”
Theor. Comput. Fluid Dyn.
,
29
(
5–6
), pp.
351
371
.
13.
Ng
,
C. O.
, and
Wang
,
C. Y.
,
2011
, “
Oscillatory Flow Through a Channel With Stick-Slip Walls: Complex Navier's Slip Length
,”
ASME J. Fluids Eng.
,
133
(
1
), p.
014502
.
14.
Ng
,
C. O.
,
2017
, “
Starting Flow in Channels With Boundary Slip
,”
Meccanica
,
52
(
1–2
), pp.
45
67
.
15.
Li
,
J.
, and
Renardy
,
Y.
,
2000
, “
Numerical Study of Flows of Two Immiscible Liquids at Low Reynolds Number
,”
SIAM Rev.
,
42
(
3
), pp.
417
439
.
16.
Busse
,
A.
,
Sandham
,
N. D.
,
McHale
,
G.
, and
Newton
,
M. I.
,
2013
, “
Change in Drag, Apparent Slip and Optimum Air Layer Thickness for Laminar Flow Over an Idealised Superhydrophobic Surface
,”
J. Fluid Mech.
,
727
, pp.
488
508
.
17.
Wang
,
C. Y.
, and
Bassingthwaighte
,
J. B.
,
2003
, “
Blood Flow in Small Curved Tubes
,”
ASME J. Biomech. Eng.
,
125
(
6
), pp.
910
913
.
18.
Szymanski
,
F.
,
1932
, “
Quelques Solutions Exactes Des Équations De L'hydrodynamique De Fluide Visqueux Dans Le Cas D'Un Tube Cylindrique
,”
J. Math. Pures Appl.
,
11
, pp.
67
107
.
19.
White
,
F. M.
,
2006
,
Viscous Fluid Flow
,
3rd ed.
,
McGraw-Hill
,
New York
.
20.
Magrab
,
E. B.
,
2014
,
An Engineer's Guide to Mathematica®
,
Wiley
,
New York
.
21.
Ethier
,
C. R.
, and
Simmons
,
C. A.
,
2007
,
Introductory Biomechanics: From Cells to Organisms
,
Cambridge University Press
,
Cambridge, UK
.
22.
Waite
,
L.
, and
Fine
,
J.
,
2007
,
Applied Biofluid Mechanics
,
McGraw-Hill
,
New York
.
23.
Vinogradova
,
O. I.
,
1995
, “
Drainage of a Thin Liquid Film Confined Between Hydrophobic Surfaces
,”
Langmuir
,
11
(
6
), pp.
2213
2220
.
You do not currently have access to this content.