Lateral leakage is encountered in many biological and chemical applications such as renal flows and filtration processes. In this paper, we report a comprehensive analytical and numerical method to examine pulsatile flow in a porous-walled tube, with leakage flow rate or permeation coefficient prescribed. In the first scenario, the analytical results have been obtained when the leakage flow rate is small as compared to the axial flow rate. Numerical simulations using ansysFluent were performed for cases where both the pulsatile Reynolds number based on the amplitude of the axial velocity and the leakage ratio (ratio of leakage velocity amplitude to the mean axial velocity amplitude) were varied. The comparison between the analytical and the numerical results indicates that the analytical solution for the axial velocity had an increasing deviation from the numerical results with the increasing pulsatile Reynolds number, or increasing leakage ratio. Interestingly, the analytical radial velocity almost overlapped with its numerical counterpart, for all considered cases. In the second scenario where a permeation coefficient for the leakage is prescribed, an analytical solution was obtained. Importantly, the solution in the second scenario suggests the criteria based on the wall permeability for the application of the analytical method developed herein.

References

References
1.
Wagner
,
R. C.
, and
Chen
,
S.-C.
,
1991
, “
Transcapillary TransportAQ5 of Solute by the Endothelial Vesicular System: Evidence From Thin Serial Section Analysis
,”
Microvasc. Res.
,
124
(
2
), pp.
139
150
.https://www.sciencedirect.com/science/article/abs/pii/002628629190082M
2.
Rector
,
F. C.
, Jr.
Van Giesen
,
G.
,
Kiil
,
F.
, and
Seldin
,
D. W.
,
1964
, “
Influence of Expansion of Extracellular Volume on Tubular Reabsorption of Sodium Independent of Changes in Glomerular Filtration Rate and Aldosterone Activity
,”
J. Clin. Invest.
,
43
(
3
), pp.
341
348
.
3.
Uchida
,
S.
, and
Aoki
,
H.
,
1977
, “
Unsteady Flows in a Semi-Infinite Contracting or Expanding Pipe
,”
J. Fluid Mech.
,
82
(
2
), pp.
371
387
.
4.
Afifi
,
N. A. S.
, and
Gad
,
N. S.
,
2003
, “
Interaction of Peristaltic Flow With Pulsatile Fluid Through a Porous Medium
,”
Appl. Math. Comput.
,
142
(
1
), pp.
167
176
.https://www.sciencedirect.com/science/article/abs/pii/S0096300302002916
5.
Ohki
,
M.
,
1980
, “
Unsteady Flow in a Porous, Elastic Circular Tube—Part 1: The Wall Contracting or Expanding in an Axial Direction
,”
Bull. JSME
,
23
(
179
), pp.
679
686
.
6.
Westerhoff
,
P.
,
Moon
,
H.
,
Minakata
,
D.
, and
Crittenden
,
J.
,
2009
, “
Oxidation of Organics in Retentates From Reverse Osmosis Wastewater Reuse Facilities
,”
Water Res.
,
43
(
16
), pp.
3992
3998
.
7.
Goerke
,
A. R.
,
Leung
,
J.
, and
Wickramasinghe
,
S. R.
,
2002
, “
Mass and Momentum Transfer in Blood Oxygenators
,”
Chem. Eng. Sci.
,
57
(
11
), pp.
2035
2046
.
8.
Sidnawi
,
B.
,
Chen
,
Z.
,
Sehgal
,
C.
,
Santhanam
,
S.
, and
Wu
,
Q.
,
2019
, “
On the Characterization of Blood Velocity in Arteries Using a Combined Analytical and Doppler Imaging Approach
,”
Phys. Rev. Fluids
,
4
(
5
), p.
053101
.
9.
Kozinski
,
A. A.
,
Schmidt
,
F. P.
, and
Lightfoot
,
E. N.
,
1970
, “
Velocity Profiles in Porous-Walled Ducts
,”
Ind. Eng. Chem. Fundam.
,
9
(
3
), pp.
502
505
.
10.
Tsangaris
,
S.
,
Kondaxakis
,
D.
, and
Vlachakis
,
N. W.
,
2007
, “
Exact Solution for Flow in a Porous Pipe With Unsteady Wall Suction and/or Injection
,”
Commun. Nonlinear Sci. Numer. Simul.
,
12
(
7
), pp.
1181
1189
.
11.
Chang
,
H. N.
,
Ha
,
J. S.
,
Park
,
J. K.
,
Kim
,
I. H.
, and
Shin
,
H. D.
,
1989
, “
Velocity Field of Pulsatile Flow in a Porous Tube
,”
J. Biomech.
,
22
(
11–12
), pp.
1257
1262
.
12.
Skalak
,
F. M.
, and
Wang
,
C.-Y.
,
1977
, “
Pulsatile Flow in a Tube With Wall Injection and Suction
,”
Appl. Sci. Res.
,
33
(
3–4
), pp.
269
307
.
13.
Majdalani
,
J.
,
Zhou
,
C.
, and
Dawson
,
C. A.
,
2002
, “
Two-Dimensional Viscous Flow Between Slowly Expanding or Contracting Walls With Weak Permeability
,”
J. Biomech.
,
35
(
10
), pp.
1399
1403
.
14.
Dauenhauer
,
E. C.
, and
Majdalani
,
J.
,
2003
, “
Exact Self-Similarity Solution of the Navier–Stokes Equations for a Porous Channel With Orthogonally Moving Walls
,”
Phys. Fluids
,
15
(
6
), p.
1485
.
15.
Rashidi
,
M. M.
,
Keimanesh
,
M.
, and
Rajvanshi
,
S. C.
,
2012
, “
Study of Pulsatile Flow in a Porous Annulus With the Homotopy Analysis Method
,”
Int. J. Numer. Methods Heat Fluid Flow
,
22
(
8
), pp.
971
989
.
16.
Malathy
,
T.
, and
Srinivas
,
S.
,
2008
, “
Pulsating Flow of a Hydromagnetic Fluid Between Permeable Beds
,”
Int. Commun. Heat Mass Transfer
,
35
(
5
), pp.
681
688
.
17.
Parsa
,
A. B.
,
Rashidi
,
M. M.
,
Anwar Bég
,
O.
, and
Sadri
,
S. M.
,
2013
, “
Corrigendum to “Semi-Computational Simulation of Magneto-Hemodynamic Flow in a Semi-Porous Channel Using Optimal Homotopy and Differential Transform Methods” [Computers in Biology and Medicine 43 (2013) 1142–1153]
,”
Comput. Biol. Med.
,
43
(
12
), p.
2311
.
18.
Ogulu
,
A.
, and
Amos
,
E.
,
2007
, “
Modeling Pulsatile Blood Flow Within a Homogeneous Porous Bed in the Presence of a Uniform Magnetic Field and Time-Dependent Suction
,”
Int. Commun. Heat Mass Transfer
,
34
(
8
), pp.
989
995
.
19.
Si
,
X. H.
,
Zheng
,
L. C.
,
Zhang
,
X. X.
, and
Chao
,
Y.
,
2011
, “
Homotopy Analysis Solutions for the Asymmetric Laminar Flow in a Porous Channel With Expanding or Contracting Walls
,”
Acta Mech. Sin.
,
27
(
2
), pp.
208
214
.
20.
Womersley
,
J. R.
,
1955
, “
Method for the Calculation of Velocity, Rate of Flow and Viscous Drag in Arteries When the Pressure Gradient Is Known
,”
J. Physiol.
,
127
(
3
), pp.
553
563
.
21.
Starling
,
E. H.
,
1896
, “
On the Absorption of Fluids From the Connective Tissue Spaces
,”
J. Physiol.
,
19
(
4
), pp.
312
326
.
22.
Hu
,
X.
, and
Weinbaum
,
S.
,
1999
, “
A New View of Starlings Hypothesis at the Microstructural Level
,”
Microvasc. Res.
,
58
(
3
), pp.
281
304
.
23.
Michel
,
C.
,
1997
, “
Starling: The Formulation of His Hypothesis of Microvascular Fluid Exchange and Its Significance After 100 Years
,”
Exp. Physiol.
,
82
(
1
), pp.
1
30
.
24.
Weinbaum
,
S.
,
1998
, “
1997 Whitaker Distinguished Lecture: Models to Solve Mysteries in Biomechanics at the Cellular Level; a New View of Fiber Matrix Layers
,”
Ann. Biomed. Eng.
,
26
(
4
), pp.
627
–6
43
.
25.
Nallapu
,
S.
, and
Radhakrishnamacharya
,
G.
,
2015
, “
Jeffrey Fluid Flow Through a Narrow Tubes in the Presence of a Magnetic Field
,”
Proc. Eng.
,
127
, pp.
185
192
.
26.
Jyothi
,
K. L.
,
Devaki
,
P.
, and
Sreenadh
,
S.
,
2013
, “
Pulsatile Flow of a Jeffrey Fluid in a Circular Tube Having Internal Porous Lining
,”
Int. J. Math. Arch.
,
4
(
5
), pp.
75
82
.http://www.ijma.info/index.php/ijma/article/view/2101
27.
El-Shahed
,
M.
,
2003
, “
Pulsatile Flow of Blood Through a Stenosed Porous Medium Under Periodic Body Acceleration
,”
Appl. Math. Comput.
,
138
(
2–3
), pp.
479
488
.https://www.sciencedirect.com/science/article/abs/pii/S0096300302001649
28.
Bhargava
,
R.
,
Takhar
,
H. S.
,
Rawat
,
S.
,
Bég
,
T. A.
, and
Bég
,
O. A.
,
2007
, “
Finite Element Solutions for Non-Newtonian Pulsatile Flow in a Non-Darcian Porous Medium Conduit
,”
Nonlinear Anal.: Modell. Control
,
12
(
3
), pp.
317
327
.www.researchgate.net/profile/O_Beg/publication/27399486_Finite_element_solutions_for_non-Newtonian_pulsatile_flow_in_a_non-Darcian_porous_medium_conduit/
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