Abstract

Despite the significant and ongoing interest in Green's functions from scientists, engineers, and mathematicians, the area remains underdeveloped with respect to understanding problems from laminar fluid flow and magnetohydrodynamics (MHD) in porous media. The purpose of this paper is to partially address this gap by constructing a new and explicit representation of the Green's function for a boundary value problem that is derived from laminar flow in channels with porous walls in the presence of a transverse magnetic field. We discuss some interesting consequences of our constructed Green's function, including: the establishment of an equivalent integral equation; and the generation of new information regarding solutions to our boundary value problem. We discover that, for any given transverse magnetic field, our laminar flow problem has a unique solution in a particular location provided the Reynolds number is sufficiently small, and that the solution may be approximated by Picard iterations.

References

1.
Green
,
G.
,
1828
,
An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism
,
T. Wheelhouse
,
Nottingham, UK
.
2.
Burkhardt
,
H.
,
1894
, “
Sur les fonctions de Green relatives a un domaine d'une dimension
,”
Bull. Math. Soc. France
,
22
, pp.
71
75
.10.24033/bsmf.484
3.
Bôcher
,
M.
,
1901
, “
Green's Function in Space of One Dimension
,”
Bull. Amer. Math. Soc.
,
7
(
7
), pp.
297
299
.10.1090/S0002-9904-1901-00802-6
4.
Troy
,
W.
,
Dutta
,
M.
, and
Stroscio
,
M.
,
2021
, “
Green's Function Solutions of One- and Two-Dimensional Dual-Phase-Lag Laser Heating Problems in Nano/Microstructures
,”
ASME J. Heat Mass Transfer-Trans.
,
143
(
11
), p.
112502
.10.1115/1.4051882
5.
Vick
,
B.
,
Mahan
,
J. R.
,
Yarahmadi
,
M.
, and
Priestley
,
K. J.
,
2023
, “
Second-Law Considerations in Monte Carlo Ray-Trace and Discrete Green's Function Analysis of Coupled Radiation and Conduction Heat Transfer
,”
ASME J. Heat Mass Transfer-Trans. ASME
,
145
(
8
), p.
082801
.10.1115/1.4062174
6.
Blackledge
,
J. M.
,
2005
, “
Woodhead Publishing Series in Electronic and Optical Materials
,”
Chapter 5 - Green Functions, Digital Image Processing
,
Horwood Publishing
,
Chichester, West Sussex
.
7.
Greenberg
,
M. D.
,
1971
,
Application of Green's Functions in Science and Engineering
,
Prentice Hall
,
Englewood Cliffs, NJ
.
8.
Dieudonné
,
J.
,
1969
, “
Le Point de Vue du Mathématicien Concernant la Place du Calcul Dans la Mathématique D'aujourd'hui [the Mathematician's View on the Position of Calculating in Today's Mathematics]
,”
Nico
,
2
, pp.
2
16
.
9.
Myint-U
,
T.
, and
Debnath
,
L.
,
2007
,
Linear Partial Differential Equations for Scientists and Engineers
,
Birkh'́auser Boston
,
Boston, MA
.
10.
Kythe
,
P. K.
,
2011
,
Green's Functions and Linear Differential Equations: Theory, Applications, and Computation
,
CRC Press
,
New York
.
11.
Cabada
,
A.
,
2014
,
Green's Functions in the Theory of Ordinary Differential Equations
,
Springer
,
New York
.
12.
Bellman
,
R.
, and
George
,
A.
,
1985
,
Green's Functions for Partial Differential Equations
,
Springer
,
Dordrecht
, The Netherlands, pp.
243
247
.
13.
Terrill
,
R. M.
,
1964
, “
Laminar Flow in a Uniformly Porous Channel
,”
Aeronaut. Q.
,
15
(
3
), pp.
299
310
.10.1017/S0001925900010908
14.
Proudman
,
I.
,
1960
, “
An Example of Steady Laminar Flow at Large Reynolds Number
,”
J. Fluid Mech.
,
9
(
4
), pp.
593
602
.10.1017/S002211206000133X
15.
Terrill
,
R. M.
,
1965
, “
Laminar Flow in a Uniformly Porous Channel With Large Injection
,”
Aeronaut. Q.
,
16
(
4
), pp.
323
332
.10.1017/S0001925900003565
16.
Suryaprakasarao
,
U.
,
1961
, “
Laminar Flow in Channels With Porous Walls in the Presence of a Transverse Magnetic Field
,”
Appl. Sci. Res., Sect. B
,
9
(
4–5
), pp.
374
382
.10.1007/BF02921819
17.
Terrill
,
R. M.
, and
Shrestha
,
G. M.
,
1964
, “
Laminar Flow Through Channels With Porous Walls and With an Applied Transverse Magnetic Field
,”
Appl. Sci. Res., Sect. B
,
11
(
1–2
), pp.
134
144
.10.1007/BF02922219
18.
Shrestha
,
G. M.
,
1967
, “
Singular Perturbation Problems of Laminar Flow in a Uniformly Porous Channel in the Presence of a Transverse Magnetic Field
,”
Q. J. Mech. Appl. Math.
,
20
(
2
), pp.
233
246
.10.1093/qjmam/20.2.233
19.
Cole
,
K.
,
Beck
,
J.
,
Haji-Sheikh
,
A.
, and
Litkouhi
,
B.
,
2010
,
Heat Conduction Using Greens Functions
,
Taylor & Francis
,
Boca Raton, FL
.
20.
Tisdell
,
C. C.
,
2023
, “
Improved Perturbation Solution for Viscous Flow in a Dilating–Contracting Permeable Channel With Velocity Slip
,”
Phys. Fluids
,
35
(
6
), p.
061708
.10.1063/5.0159711
21.
Tisdell
,
C. C.
,
2023
, “
Improved Perturbation Solution for Two-Dimensional Viscous Flow Between Expanding or Contracting Permeable Walls
,”
J. Biomech.
,
155
, p.
111642
.10.1016/j.jbiomech.2023.111642
22.
Eberhard
,
Z.
,
1985
,
Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems
,
Springer
,
New York
.
23.
Almuthaybiri
,
S. S.
, and
Tisdell
,
C. C.
,
2022
, “
Laminar Flow in Channels With Porous Walls: Advancing the Existence, Uniqueness and Approximation of Solutions Via Fixed Point Approaches
,”
J. Fixed Point Theory Appl.
,
24
(
3
), p.
55
.10.1007/s11784-022-00971-8
24.
Hao
,
M. L.
, and
Tisdell
,
C. C.
,
2023
, “
When is the Porous, Laminar Flow Problem With Slip Condition Well Posed? and Where Does the Solution Lie?
,”
Transp. Porous Media
,
147
(
2
), pp.
281
303
.10.1007/s11242-023-01907-7
25.
Stakgold
,
I.
, and
Holst
,
M. J.
,
2011
,
Green's Functions and Boundary Value Problems
,
Wiley
,
Hoboken, NJ
.
26.
Agarwal
,
R. P.
, and
O'Regan
,
D.
,
2008
,
Ordinary and Partial Differential Equations: With Special Functions, Fourier Series, and Boundary Value Problems
,
Springer Science & Business Media
,
New York
.
27.
Hartmann
,
J.
,
1937
, “
Magnetohydrodynamic Flow Within a Parallel Plate Channel in the Presence of an Applied Cross Magnetic Field
,”
Kgl Danske Vidensk Selskal Math-Fys Medd
,
15
, pp.
6
9
.
28.
Tisdell
,
C. C.
,
2019
, “
On Picard's Iteration Method to Solve Differential Equations and a Pedagogical Space for Otherness
,”
Int. J. Math. Educ. Sci. Technol.
,
50
(
5
), pp.
788
799
.10.1080/0020739X.2018.1507051
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