Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
STRUCTURAL
STABILITY
OF
MICROPOLAR
FLUID
FLOW
IN
A
CURVED
PIPE
WITH
DARCY–LAPWOOD–FORCHHEIMER–
BRINKMAN
MODEL
∗
A.
Prabal
†
M.
Devakar
‡
DOI 10.56082/annalsarscimath.2026.4.385
Abstract
In
this
article,
we
address
the
structural
stability
of
the
Darcy–
Lapwood–Forchheimer–Brinkman
model
for
a
micropolar
fluid
flowing
through
a
curved
pipe.
A
rigorous
analysis
is
carried
out
to
prove
the
continuous
dependence
of
solutions
of
the
model
on
several
physical
and
geometric
parameters
such
as
the
Reynolds
number,
curvature
ratio,
Forchheimer
coefficient,
and
micropolarity
parameter.
Keywords:
Darcy–Lapwood–Forchheimer–Brinkman
model,
micropo-
lar
fluid,
curved
pipe,
structural
stability,
continuous
dependence.
MSC:
35Q35,
76A05,
76S05,
35B30.
1
Introduction
and
mathematical
model
A
central
question
in
mathematical
modelling
is
whether
solutions
depend
continuously
on
variations
in
coefficients,
boundary
conditions,
or
the
gov-
∗
Accepted
for
publication
on
October
20,
2026
†
ds20mth001@students.vnit.ac.in
,
Department
of
Mathematics,
Visvesvaraya
Na-
tional
Institute
of
Technology
Nagpur,
440010
India
‡
m
devakar@mth.vnit.ac.in
,
Department
of
Mathematics,
Visvesvaraya
National
In-
stitute
of
Technology
Nagpur,
440010
India
385
Structural
stability
of
micropolar
fluid
flow
in
a
curved
pipe
386
erning
partial
differential
equations
themselves.
This
continuous
dependence
of
the
solutions
with
respect
to
the
model’s
structural
components
is
known
as
structural
stability.
For
real
life
problems,
variations
in
coefficients
of
the
governing
partial
differential
equations
correspond
to
changes
in
the
con-
stitutive
parameters,
therefore,
establishing
the
continuous
dependence
is
crucial
for
validating
the
mathematical
model’s
real-world
applicability.
A
comprehensive
treatment
of
structural
stability
and
related
continuous
de-
pendence
results
can
be
found
in
the
monographs
by
Ames
and
Straughan
[
1
]
and
Straughan
[
19
].
For
Newtonian
fluids,
structural
stability
of
the
flows
through
porous
media
with
the
Darcy
and
Brinkman
model
has
been
ex-
tensively
investigated
by
Morro
and
Straughan
[
13
],
Ames
and
Payne
[
2
],
and
Payne
and
Straughan
[
14
–
16
].
For
the
Brinkman–Forchheimer
model,
related
stability
results
of
Newtonian
fluid
flows
were
established
by
Payne
and
Straughan
[
17
],
C
¸elebi
et
al.
[
6
,
7
],
and
Liu
[
11
].
Lukaszewicz
[
12
]
estab-
lished
continuous
dependence
of
two-dimensional
micropolar
fluid
flows
on
the
microrotation
viscosity
coefficient.
To
the
best
knowledge
of
the
authors,
structural
stability
of
the
flows
in
curved
pipe
is
not
yet
explored.
There-
fore,
in
this
paper,
we
investigate
the
structural
stability
of
micropolar
fluid
flow
in
a
curved
pipe
with
Darcy-Lapwood-Forchheimer-Brinkman
model
on
the
pertinent
physical
and
geometric
parameters.
The
mathematical
model
of
the
flow
problem
under
consideration
is
presented
in
Subsection
1.1
.
A
priori
estimates
that
are
required
for
establishing
structural
stability
are
presented
in
Section
2
.
Main
theorems
establishing
the
structural
stability
of
the
flow
model
are
presented,
along
with
their
proofs,
in
Section
3
.
1.1
Mathematical
model
Figure
1:
Flow
Geometry.
We
consider
the
flow
of
a
micropolar
fluid
within
a
stationary
curved
pipe
of
arbitrary
cross-section
of
diameter
2
a
,
which
is
characterized
by
a
A.
Prabal,
M.
Devakar
387
constant
centerline
radius
R
>
a
.
A
typical
section
of
the
curved
pipe
is
shown
in
Figure
1
.
We
assume
that
the
pipe’s
cross-section,
Ω
⊂
R
2
,
is
a
bounded
domain
with
Lipschitz
boundary
∂
Ω.
The
flow
is
driven
by
a
constant
pressure
gradient
along
the
axis
of
the
pipe
and
it
is
assumed
to
be
axisymmetric.
Under
these
assumptions,
using
toroidal
coordinate
system
[
3
],
the
partial
differential
equations
governing
the
flow
of
micropolar
fluid
through
porous
medium
with
Darcy-Lapwood-Forchheimer-Brinkman
model
are
given
by
[
8
–
10
,
18
]
∇
·
(
H
u
)
=
0
,
(1)
ReH
(
u
·
∇
)
u
−
Reδu
2
3
ˆ
a
2
+
Reδu
2
u
3
ˆ
a
3
=
f
−
ReH
∇
p
+
ReG
ˆ
a
3
+
mH
∇
×
ν
+
mδν
3
ˆ
a
1
+
(1
+
m
)
∇
·
(
H
∇
u
)
−
(1
+
m
)
δ
2
H
u
2
ˆ
a
2
−
(1
+
m
)
δ
2
H
u
3
ˆ
a
3
−
σ
2
H
u
−
ReσC
F
H
|
u
|
u
,
(2)
ReHJ
(
u
·
∇
)
ν
−
ReJδu
3
ν
3
ˆ
a
2
+
ReJδu
3
ν
2
ˆ
a
3
=
l
−
2
mHν
ν
ν
+
mH
∇
×
u
+
mδu
3
ˆ
a
1
+
n
1
∇
·
(
H
∇
ν
ν
ν
)
−
n
1
δ
2
H
(
ν
2
ˆ
a
2
+
ν
3
ˆ
a
3
)
+(
n
2
+
n
3
)
H
∇
(
∇
·
ν
ν
)
+
(
n
2
+
n
3
)
δ
∇
ν
2
−
(
n
2
+
n
3
)
δ
2
H
ν
2
ˆ
a
2
(3)
in
Ω
,
where,
the
velocity
vector
u
=
(
u
1
,
u
2
,
u
3
)
and
the
microrotation
vector
ν
=
(
ν
1
,
ν
2
,
ν
3
)
are
independent
of
x
3
,
Re
=
ρUa/µ
is
the
Reynolds
num-
ber,
δ
=
a/R
∈
[0
,
1)
is
the
curvature
ratio,
σ
=
a/
√
K
is
the
porosity
parameter,
H
=
1
+
δx
2
is
the
scale
factor,
G
is
the
constant
axial
pres-
sure
gradient
so
that
the
total
pressure
is
p
(
x
1
,
x
2
)
−
G
x
3
,
m
=
k
µ
is
the
micropolarity
parameter,
J
is
the
nondimensional
gyration
parameter,
and
n
1
,
n
2
and
n
3
are
nondimensional
gyro-viscosity
parameters.
We
assume
that
min
{
n
1
,
n
2
+
n
3
}
>
0.
The
constants
µ
and
k
represents
the
usual
New-
tonian
viscosity
and
the
microrotation
viscosity.
ρ
is
the
density
of
the
fluid,
K
is
the
permeability
of
the
porous
medium
and
C
F
is
Forchheimer
coeffi-
cient.
The
functions
f
and
l
are
the
prescribed
dimensionless
body
force
and
dimensionless
bode
couple,
respectively,
U
is
the
characteristic
velocity
and
{
ˆ
a
1
,
ˆ
a
2
,
ˆ
a
3
}
is
the
orthonormal
basis
of
the
toroidal
coordinate
system
[
3
,
18
].
The
differential
operators
∇
=
∂
∂x
1
,
∂
∂x
2
,
0
and
∇·
=
∂
∂x
1
+
∂
∂x
2
.
Since
the
curved
pipe
is
stationary,
the
no-slip
and
the
hyperstick
bound-
ary
conditions
to
be
satisfied
are,
respectively,
u
i
=
0
and
ν
i
=
0
on
∂
Ω
for
i
=
1
,
2
,
3
.
(4)