Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
PERMANENT
SOLUTIONS
FOR
MHD
MODIFIED
STOKES
PROBLEMS
OF
SOME
MAXWELL
FLUIDS
WITH
POWER-LAW
DEPENDENCE
OF
VISCOSITY
ON
PRESSURE
∗
Constantin
Fetecau
†
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.15
Abstract
Modified
Stokes
problems
for
a
class
of
electrically
conducting
upper-
convected
incompressible
Maxwell
fluids
with
pressure-dependent
vis-
cosity
are
analytically
investigated
when
the
influence
of
magnetic
field
and
gravitational
acceleration
is
taken
in
consideration.
Simple
ana-
lytic
expressions
are
provided
for
the
dimensionless
permanent
veloc-
ity
and
shear
stress
fields.
They
satisfy
the
governing
equations
and
boundary
conditions
and
reduce
to
known
results
from
the
existing
literature
in
limiting
cases.
The
influence
of
physical
parameters
on
the
fluid
behavior
is
graphically
underlined
and
discussed.
Fluids
with
pressure-dependent
viscosity
flow
faster
than
ordinary
fluids
and
their
velocity
declines
in
the
presence
of
magnetic
field.
Permanent
shear
stress
for
the
modified
first
problem
of
Stokes
is
constant
on
whole
flow
domain
although
the
corresponding
velocity
is
function
of
spatial
variable.
Keywords:
MHD
modified
Stokes
problems,
Maxwell
fluids,
pressure-
dependent
viscosity.
MSC:
76A05.
∗
Accepted
for
publication
on
January
17,
2026
†
c
fetecau@yahoo.com
,
Academy
of
Romanian
Scientists,
3
Ilfov,
Bucharest
050044,
Romania
15
Permanent
solutions
for
MHD
modified
Stokes’
problems
16
1
Introduction
Generally,
in
the
existing
literature,
the
fluid
viscosity
is
considered
to
be
constant.
However,
the
experimentalists
clearly
showed
that
it
varies
with
pressure.
There
is
a
large
amount
of
works
regarding
this
problem.
For
the
literature
prior
to
1931
the
book
of
Bridgman
[
5
]
is
welcome.
Other
exper-
imental
studies
that
confirm
the
dependence
of
viscosity
on
pressure
have
been
developed,
for
instance,
by
Griest
et
al.
[
11
],
Johnson
and
Cameron
[
16
],
Johnson
and
Tevaarwerk
[
17
]
and
Bair
and
Winer
[
3
].
Some
experimental
investigations
were
done
by
Iqbal
and
Hasan
[
15
]
for
glycerin
and
Bair
and
Kottke
[
2
]
for
isopropanol
and
lubricants.
An
important
domain
where
the
influence
of
pressure
on
viscosity
cannot
be
ignored
is
the
elastohydrody-
namic
lubrication
[
25
]
where
the
fluid
viscosity
strongly
varies
with
pres-
sure.
A
comprehensive
review
concerning
this
problem
can
be
found
in
the
works
of
Binding
et
al.
[
4
]
and
Goubert
et
al.
[
10
].
However,
does
not
ex-
ist
a
pressure-viscosity
law
that
holds
for
all
types
of
fluids.
At
low
and
high
pressures,
for
instance,
the
linear
and
exponential
laws
are
adequate
to
describe
the
experimental
data
[
13
].
Here,
we
shall
consider
a
power-law
dependence
of
viscosity
on
pressure
and
the
gravitation
effect
will
be
taken
in
consideration.
In
the
same
time,
the
obtaining
of
exact
solutions
is
very
important
to
develop
a
research
domain
with
multiple
applications
in
lubrication,
micro
fluidics
and
geophysics,
crude
oil
and
fuel
oil
pumping,
polymer
and
food
processing
and
pharmaceutical
manufacturing
(see
for
instance
Hamrock
et
al.
[
12
]
and
Martinez-Boza
et
al.
[
19
]).
First
exact
steady
solutions
for
mo-
tions
of
Newtonian
fluids
with
pressure-dependent
viscosity
in
which
the
gravitation
is
taken
into
account
seem
to
be
those
of
Rajagopal
[
21
,
22
].
Velocity
fields
for
other
steady
motions
of
same
fluids
in
rectangular
ducts
were
also
determined
by
Akyildiz
and
Siginer
[
1
]
and
Housiadas
and
Gior-
giou
[
14
].
Exact
solutions
for
the
modified
Stokes’
problems
were
provided
by
Rajagopal
et
al.
[
23
],
Prusa
[
20
]
and
Fetecau
and
Agop
[
7
].
A
part
of
these
results
have
been
extended
to
Maxwell
fluids
with
pressure-dependent
viscosity
by
Fetecau
et
al.
[
9
]
and
Fetecau
and
Rauf
[
8
].
The
influence
of
magnetic
field
on
modified
Stokes
problems
for
Maxwell
fluids
with
expo-
nential
dependence
of
viscosity
on
pressure
has
been
recently
investigated
by
Fetecau
[
6
].
Interaction
between
a
moving
fluid
and
magnetic
field
has
multiple
applications
in
physics,
chemistry
and
engineering.
In
the
present
work
we
establish
simple
exact
expressions
for
the
di-
mensionless
velocity
and
shear
stress
fields
corresponding
to
the
modified
Stokes’
problems
for
incompressible
Maxwell
fluids
with
power-law
depen-
C.
Fetecau
17
dence
of
viscosity
on
pressure
of
index
two.
The
effects
of
magnetic
field
and
gravitational
acceleration
are
taken
into
consideration.
The
correctness
of
results
that
have
been
obtained
is
graphically
proved
comparing
with
known
results
from
the
literature.
Graphical
representations
are
used
to
show
the
oscillatory
demeanor
of
the
fluid
motion
in
the
case
the
second
problem
of
Stokes
and
the
influence
of
physical
parameters
on
the
fluid
velocity
in
the
case
of
the
first
problem
of
Stokes.
They
show
that
the
fluids
with
pressure-
dependent
viscosity
flow
faster
in
comparison
with
the
ordinary
fluids
and
the
fluid
velocity
declines
in
the
presence
of
a
magnetic
field.
2
Constitutive
and
governing
equations
Constitutive
equations
of
the
upper-convected
incompressible
Maxwell
fluids
with
pressure-dependent
viscosity
are
given
by
the
relations
[
18
]
T
=
−
p
I
+
S
,
S
+
λ
D
S
Dt
=
η
(
p
)(
L
+
L
T
)
,
(1)
where
T
and
S
are
stress
and
extra-stress
tensors,
respectively,
I
is
the
unit
tensor,
L
is
the
gradient
of
the
velocity
vector
w
,
p
is
the
Laplace
multiplier,
λ
is
the
relaxation
time,
D/Dt
denotes
the
time
upper-convected
derivative
and
η
(
p
)
is
the
fluid
viscosity.
For
simplicity,
like
Karra
et
al.
[
18
],
we
shall
refer
to
p
as
pressure.
In
the
following
we
shall
consider
the
case
when
η
(
p
)
=
µ
[
α
(
p
−
p
0
)
+
1]
2
.
(2)
Here,
µ
is
the
fluid
viscosity
at
the
reference
pressure
p
0
and
α
>
0
is
the
dimensional
pressure-viscosity
coefficient.
If
α
=
0,
the
equations
(1)
define
ordinary
upper-convected
incompressible
Maxwell
fluids.
If
λ
=
0,
these
equations
correspond
to
the
incompressible
Newtonian
fluids
with
pressure-
dependent
viscosity.
Let
us
assume
that
an
electrically
conducting
upper-convected
incom-
pressible
Maxwell
fluid
(ECUCIMF)
with
pressure-dependent
viscosity,
whose
constitutive
equations
are
given
by
the
relations
(1)
and
(2,
is
statio-
nary
between
two
infinite
horizontal
parallel
flat
plates.
After
the
initial
mo-
ment
t
=
0
the
lower
plate
begins
to
oscillate
along
its
plane
with
the
velocity
W
cos(
ωt
)
or
W
sin(
ωt
)
and
a
magnetic
field
of
constant
strength
B
acts
or-
thogonal
to
plates.
Here,
ω
is
the
oscillations
frequency
and
W
is
a
constant
velocity.
The
resulting
motions
correspond
to
the
magneto-hydrodynamic
(MHD)
modified
Stokes
second
problem.
Following
Rajagopal
[
21
],
we
are
looking
for
a
velocity
vector
w
and
a
pressure
p
of
the
form
w
=
w
(
z,
t
)
=
w
(
z,
t
)
e
y
,
p
=
p
(
z
)
,
(3)
Permanent
solutions
for
MHD
modified
Stokes’
problems
18
reported
to
a
fixed
Cartesian
coordinate
system
x,
y
and
z
in
which
e
y
is
the
unit
vector
along
the
y
-axis.
By
substituting
the
velocity
vector
w
(
z,
t
)
from
Eq.
(3)
in
the
second
relation
(1)
and
assuming
that
the
extra-stress
tensor
S
,
like
the
velocity
vector
w
is
also
a
function
of
z
and
t
,
it
results
that
its
non-null
component
τ
(
z,
t
)
=
S
yz
(
z,
t
)
satisfies
the
governing
equation
1
+
λ
∂
∂t
τ
(
z,
t
)
=
µ
[
α
(
p
−
p
0
)
+
1]
2
∂w
(
z,t
)
∂z
;
0
<
z
<
d,
t
>
0
,
(4)
where
d
is
the
distance
between
the
two
plates.
In
the
following
we
assume
that
the
fluid,
whose
magnetic
permeability
is
constant,
is
finitely
conducting.
In
addition,
the
induced
magnetic
field
is
negligible
in
comparison
with
the
applied
magnetic
field
and
there
is
no
surplus
electric
charge
distribution
inside.
In
these
conditions
the
balance
of
linear
momentum
reduces
to
the
relevant
relations
[
24
]
ρ
∂w
(
z,
t
)
∂t
=
∂τ
(
z,
t
)
∂z
−
σB
2
w
(
z,
t
)
,
dp
(
z
)
dz
+
ρg
=
0;
0
<
z
<
d,
t
>
0
.
(5)
In
these
relations
ρ
is
the
fluid
density,
σ
is
electrical
conductivity
and
g
is
the
gravitational
acceleration.
From
the
second
relation
(5)
it
results
that
p
(
z
)
=
ρg
(
d
−
z
)
+
p
0
where
p
0
=
p
(
d
)
.
(6)
The
fluid
pressure
at
the
lower
plate
is
p
(0)
=
p
0
+
ρgd
.
The
two
unknown
functions
w
(
z,
t
)
and
τ
(
z,
t
)
have
to
satisfy
the
next
initial
and
boundary
conditions:
w
(
z,
0)
=
0
,
τ
(
z,
0)
=
0;
0
≤
z
≤
d,
(7)
w
(0
,
t
)
=
W
cos(
ωt
)
or
W
sin(
ωt
)
,
w
(
d,
t
)
=
0;
t
>
0
.
(8)
Introducing
the
next
non-dimensional
variables,
functions
and
parameters
z
∗
=
1
d
z,
t
∗
=
ν
d
2
t,
w
∗
=
1
W
w,
τ
∗
=
d
µW
τ,
α
∗
=
αρgd,
ω
∗
=
d
2
ν
ω,
(9)
and
renouncing
to
the
star
notation,
one
finds
dimensionless
forms
of
the
governing
equations
1
+
We
∂
∂t
τ
(
z,
t
)
=
[
α
(1
−
z
)
+
1]
2
∂w
(
z,
t
)
∂z
;
0
<
z
<
1
,
t
>
0
,
(10)
∂w
(
z,
t
)
∂t
=
∂τ
(
z,
t
)
∂z
−
Mw
(
z,
t
);
0
<
z
<
1
,
t
>
0
.
(11)
C.
Fetecau
19
The
Weissenberg
number
We
and
the
magnetic
parameter
M
are
defined
by
the
relations
We
=
νλ
d
2
,
M
=
σB
2
µ
d
2
,
(12)
while
the
corresponding
initial
and
boundary
conditions
are
w
(
z,
0)
=
τ
(
z,
0)
=
0;
0
≤
z
≤
1
,
(13)
w
(0
,
t
)
=
cos(
ωt
)
or
sin(
ωt
)
,
w
(1
,
t
)
=
0;
t
>
0
.
(14)
Since
the
two
motions
induced
by
cosine
or
sine
oscillations
of
the
lower
plate
become
steady
or
permanent
in
time,
an
important
problem
for
the
experimental
researchers
is
to
know
the
time
to
reach
this
state.
In
order
to
determine
this
time
it
is
sufficient
to
know
the
permanent
(long
time
or
steady
state)
solutions
w
cp
(
z,
t
)
,
τ
cp
(
z,
t
),
or
w
sp
(
z,
t
),
τ
sp
(
z,
t
)
of
the
system
of
partial
differential
equations
(10,
(11)
with
the
boundary
conditions
(14).
These
solutions
are
independent
of
the
initial
conditions
(13).
Once
these
solutions
are
determined,
as
usual,
the
time
required
to
reach
steady
state
can
be
graphically
obtained.
It
is
the
time
after
which
the
diagrams
of
the
starting
solutions
w
c
(
z,
t
),
τ
c
(
z,
t
)
or
w
s
(
z,
t
),
τ
s
(
z,
t
)
(numerical
solutions)
overlap
those
of
the
corresponding
permanent
solutions.
This
is
the
reason
that,
in
the
following
section,
we
shall
provide
exact
expressions
for
the
permanent
solutions
only.
3
Analytic
expressions
for
the
permanent
solutions
3.1
Modified
Stokes
second
problem
In
order
to
determine
exact
solutions
for
the
system
of
partial
differential
equations
(10),
(11)
with
the
boundary
conditions
(14)
we
shall
use
the
complex
velocity
and
shear
stress
fields
w
com
(
z,
t
)
=
w
cp
(
z,
t
)
+
iw
sp
(
z,
t
)
,
τ
com
(
z,
t
)
=
τ
cp
(
z,
t
)
+
iτ
sp
(
z,
t
);
0
<
z
<
1
,
t
∈
R,
(15)
Permanent
solutions
for
MHD
modified
Stokes’
problems
20
where
i
is
the
imaginary
unit.
The
two
dimensionless
complex
entities
w
com
(
z,
t
)
and
τ
com
(
z,
t
)
have
to
satisfy
the
governing
equations
1
+
We
∂
∂t
∂w
com
(
z,
t
)
∂t
=
[
α
(1
−
z
)
+
1]
2
∂
2
w
com
(
z,
t
)
∂z
2
−
2
α
[
α
(1
−
z
)
+
1]
∂w
com
(
z,
t
)
∂z
−
M
1
+
We
∂
∂t
w
com
(
z,
t
);
0
<
z
<
1
,
t
∈
R,
(16)
1
+
We
∂
∂t
τ
com
(
z,
t
)
=
[
α
(1
−
z
)
+
1]
2
∂w
com
(
z,
t
)
∂t
;
(17)
0
<
z
<
1
,
t
∈
R,
with
the
boundary
conditions
w
com
(0
,
t
)
=
e
iωt
,
w
com
(1
,
t
)
=
0;
t
∈
R.
(18)
Equation
(16)
has
been
obtained
eliminating
τ
(
z,
t
)
between
Eqs.
(10)
and
(11).
Making
the
change
of
the
independent
variable
z
=
(1
+
α
−
e
r
)
/α,
(19)
the
equation
(16)
reduces
to
a
partial
differential
equation
with
constant
coefficients,
namely
1
+
We
∂
∂t
∂w
com
(
r,
t
)
∂t
+
M
1
+
We
∂
∂t
w
com
(
r,
t
)
=
α
2
∂
2
w
com
(
r,
t
)
∂r
2
+
∂w
com
(
r,
t
)
∂r
;
0
<
r
<
a,
t
∈
R.
(20)
The
corresponding
boundary
conditions
are
w
com
(0
,
t
)
=
0
,
w
com
(
a,
t
)
=
e
iωt
;
t
∈
R,
(21)
where
a
=
ln(1
+
α
).
The
linearity
of
the
partial
differential
equation
(20)
and
the
form
of
boundary
conditions
(21)
suggest
us
to
look
for
a
solution
of
the
form
w
com
(
r,
t
)
=
U
(
r
)
e
iωt
;
0
<
r
<
a,
t
∈
R.
(22)
C.
Fetecau
21
The
complex
function
U
(
·
)
that
has
to
be
solution
of
the
boundary
value
problem
d
2
U
(
r
)
dr
2
+
dU
(
r
)
dr
=
β
α
2
U
(
r
);
U
(0)
=
0
,
U
(1)
=
1
,
(23)
is
given
by
the
relation
U
(
r
)
=
e
r
1
r
−
e
r
2
r
e
r
1
a
−
e
r
2
a
;
r
1
,
2
=
−
α
±
α
2
+
4
β
2
α
,
(24)
where
β
=
(
M
+
iω
)(1
+
iωWe
).
Introducing
U
(
r
)
from
Eq.
(24)
in
(22)
one
finds
that
w
com
(
r,
t
)
=
e
r
1
r
−
e
r
2
r
e
r
1
a
−
e
r
2
a
e
iωt
;
0
<
r
<
a,
t
∈
R.
(25)
Now,
bearing
in
mind
the
notation
(15)
1
and
coming
back
to
the
initial
variables,
it
results
that
the
permanent
dimensionless
velocity
fields
w
cp
(
z,
t
)
and
w
sp
(
z,
t
)
corresponding
to
the
MHD
modified
second
problem
of
Stokes
for
ECUCIMFs
are
given
by
the
relations
w
cp
(
z,
t
)
=
Re
[
α
(1
−
z
)
+
1]
r
1
−
[
α
(1
−
z
)
+
1]
r
2
(
α
+
1)
r
1
−
(
α
+
1)
r
2
e
iωt
;
0
<
z
<
1
,
t
∈
R,
(26)
w
sp
(
z,
t
)
=
Im
[
α
(1
−
z
)
+
1]
r
1
−
[
α
(1
−
z
)
+
1]
r
2
(
α
+
1)
r
1
−
(
α
+
1)
r
2
e
iωt
;
0
<
z
<
1
,
t
∈
R.
(27)
The
corresponding
permanent
shear
stresses,
namely
τ
cp
(
z,
t
)
=
−
αRe
r
1
[
α
(1
−
z
)+1]
r
1
+1
−
r
2
[
α
(1
−
z
)+1]
r
2
+1
(
α
+1)
r
1
−
(
α
+1)
r
2
e
iωt
1+
iωWe
;
0
<
z
<
1
,
t
∈
R,
(28)
τ
sp
(
z,
t
)
=
−
αIm
r
1
[
α
(1
−
z
)+1]
r
1
+1
−
r
2
[
α
(1
−
z
)+1]
r
2
+1
(
α
+1)
r
1
−
(
α
+1)
r
2
e
iωt
1+
iωWe
;
0
<
z
<
1
,
t
∈
R,
(29)
have
been
determined
using
the
relations
(15),
(17)
and
(25).
Direct
compu-
tations
show
that
w
cp
(
z,
t
)
,
τ
cp
(
z,
t
)
and
w
sp
(
z,
t
)
,
τ
sp
(
z,
t
)
given
by
the
above
relations
satisfy
both
the
governing
equations
(10),
(11)
and
the
boundary
Permanent
solutions
for
MHD
modified
Stokes’
problems
22
conditions
(14).
Furthermore,
making
M
=
We
=
0
in
Eqs.
(26)
and
(27),
the
dimensionless
velocity
fields
(41)
and
(42)
obtained
by
Fetecau
and
Agop
[
7
]
for
incompressible
Newtonian
fluids
with
pressure-dependent
viscosity
are
recovered.
Now,
for
a
check
of
the
above
results,
let
us
remember
that
recently
Fetecau
[
6
]
has
determined
the
permanent
dimensionless
velocity
and
shear
stress
fields
corresponding
to
the
MHD
modified
Stokes
second
problem
for
ordinary
ECUCIMFs.
Their
expressions
are
given
by
the
relations
w
ocp
(
z,
t
)
=
Re
sinh
(1
−
z
)
(
M
+
iω
)(1+
iωWe
)
sinh
(
M
+
iω
)(1+
iωWe
)
e
iωt
;
0
<z<
1
,
t
∈
R,
(30)
w
osp
(
z,
t
)
=
Im
sinh
(1
−
z
)
(
M
+
iω
)(1+
iωWe
)
sinh
(
M
+
iω
)(1+
iωWe
)
e
iωt
;
0
<z<
1
,
t
∈
R,
(31)
τ
ocp
=
−
Re
cosh
(1
−
z
)
(
M
+
iω
)(1+
iωWe
)
sinh
(
M
+
iω
)(1+
iωWe
)
√
M
+
iω
√
1+
iωWe
e
iωt
;
0
<
z
<
1
,
t
∈
R,
(32)
τ
osp
=
−
Im
cosh
(1
−
z
)
(
M
+
iω
)(1+
iωWe
)
sinh
(
M
+
iω
)(1+
iωWe
)
√
M
+
iω
√
1+
iωWe
e
iωt
;
0
<
z
<
1
,
t
∈
R.
(33)
Figures
1
and
2
(see
below)
show
that
the
diagrams
of
w
cp
(
z,
t
)
,
w
sp
(
z,
t
)
and
τ
cp
(
z,
t
)
,
τ
sp
(
z,
t
)
tend
to
superpose
over
those
of
w
ocp
(
z,
t
)
,
w
osp
(
z,
t
)
and
τ
ocp
(
z,
t
)
,
τ
osp
(
z,
t
),
respectively,
when
the
pressure-viscosity
coefficient
α
→
0.
3.2
Modified
Stokes
first
problem
Let
us
suppose
that,
in
the
same
conditions
as
before,
the
lower
plate
begins
to
slide
in
its
plane
with
the
constant
velocity
W
after
the
initial
moment
t
=
0.
The
fluid
begins
to
move
and
its
motion
corresponds
to
the
MHD
modified
Stokes
first
problem
for
ECUCIMFs.
The
dimensionless
perma-
nent
velocity
and
shear
stress
fields
corresponding
to
this
problem
can
be
immediately
obtained
taking
ω
=
0
in
Eqs.
(26)
and
(28).
C.
Fetecau
23
Figure
1:
Convergence
of
the
dimensionless
permanent
velocities
w
cp
(
z,
t
)
and
w
sp
(
z,
t
)
to
w
ocp
(
z,
t
)
and
w
osp
(
z,
t
),
respectively,
at
the
moment
t
=
5
when
ω
=
π
3
,
M
=
0
.
8,
We
=
0
.
7
and
decreasing
values
of
the
parameter
α
.
Figure
2:
Convergence
of
the
dimensionless
permanent
velocities
τ
cp
(
z,
t
)
and
τ
sp
(
z,
t
)
to
τ
ocp
(
z,
t
)
and
τ
osp
(
z,
t
),
respectively,
at
the
moment
t
=
5
when
ω
=
π
3
,
M
=
0
.
8,
We
=
0
.
7
and
decreasing
values
of
the
parameter
α
.
Permanent
solutions
for
MHD
modified
Stokes’
problems
24
Their
expressions,
given
by
the
relations
w
Cp
(
z
)
=
[
α
(1
−
z
)+1]
q
1
−
[
α
(1
−
z
)+1]
q
2
(
α
+1)
q
1
−
(
α
+1)
q
2
;
0
<
z
<
1
,
(34)
τ
Cp
(
z
)
=
−
α
q
1
[
α
(1
−
z
)+1]
q
1
+1
−
q
2
[
α
(1
−
z
)+1]
q
2
+1
(
α
+1)
q
1
−
(
α
+1)
q
2
;
0
<
z
<
1
,
(35)
are
same
both
for
Newtonian
and
Maxwell
fluids.
This
is
possible
since,
for
such
motions,
the
governing
equations
are
identical
for
the
two
types
of
fluids.
In
the
last
two
relations
q
1
,
2
=
−
α
±
√
α
2
+
4
M
/
(2
α
).
In
the
absence
of
magnetic
field,
the
corresponding
permanent
dimensionless
solutions
w
Cp
(
z
)
=
(1
+
α
)(1
−
z
)
1
+
α
(1
−
z
)
,
τ
Cp
=
−
(
α
+
1)
,
(36)
have
been
determined
taking
the
limits
of
the
relations
(34)
and
(35)
when
M
→
0.
Of
course,
these
solutions
can
be
directly
obtained
solving
the
boundary
value
problem
corresponding
to
this
motion.
It
is
worth
to
men-
tion
the
fact
that
the
shear
stress
τ
Cp
,
unlike
the
fluid
velocity
that
is
a
function
of
the
spatial
variable
y
,
is
constant
on
the
entire
flow
domain.
Finally,
taking
α
=
0
in
Eqs.
(36)
the
classical
solutions
w
Cp
(
z
)
=
1
−
z,
τ
Cp
=
−
1
,
(37)
are
recovered.
4
Some
graphical
representations
and
discussions
In
the
previous
section,
analytical
expressions
of
the
permanent
dimension-
less
velocity
and
shear
stress
fields
corresponding
to
modified
Stokes
pro-
blems
for
some
ECUCIMFs
with
pressure-dependent
viscosity
have
been
derived
in
terms
of
Weissenberg
number
We
,
magnetic
parameter
M
and
the
pressure-viscosity
coefficient
α
.
The
obtained
results,
like
in
many
other
previous
works,
can
be
used
to
determine
the
required
time
to
reach
the
steady
or
permanent
state
for
the
respective
motions.
Here,
in
order
to
bring
to
light
some
physical
insights
of
the
obtained
results,
Figures
3
and
4
are
included
to
underline
the
oscillatory
behavior
of
the
fluid
motion
in
the
case
of
the
second
problem
of
Stokes
and
to
show
the
variations
of
the
fluid
velocity
with
respect
to
the
magnetic
field
M
and
the
pressure-viscosity
coefficient
α
.
In
Figure
3
,
in
which
the
variations
in
time
of
w
cp
(
z,
t
)
and
w
sp
(
z,
t
)
at
the
middle
of
the
channel
are
presented
at
three
values
of
the
Weissenberg
C.
Fetecau
25
Figure
3:
Time
variations
of
the
dimensionless
permanent
velocities
w
cp
(
z,
t
)
and
w
sp
(
z,
t
)
given
by
Eqs.
(26)
and
(27)
at
the
middle
of
channel
(
z
=
0
.
5)
when
ω
=
π
3
,
α
=
0
.
5,
M
=
0
.
8
at
three
values
of
Weissenberg
number
We
.
number
We
and
fixed
values
for
the
other
parameters,
it
is
clearly
visualized
the
oscillatory
behavior
of
the
two
motions
and
the
phase
difference
between
them.
In
the
same
time,
as
expected,
the
maximum
values
of
the
oscillations
amplitude
corresponding
to
the
two
motions
are
identical
at
same
values
of
physical
parameters
and
increase
for
increasing
values
of
the
Weissenberg
number
We
.
This
means
that
the
Maxwell
fluids
flow
faster
in
comparison
with
Newtonian
fluids.
Figure
4:
Profiles
of
the
dimensionless
permanent
velocity
w
Cp
(
z
)
given
by
Eq.
(34)
for
α
=
0
.
6
and
three
values
of
M
,
and
M
=
0
.
8
at
three
values
of
α
.
Figures
4
,
in
which
are
presented
profiles
of
the
dimensionless
perma-
Permanent
solutions
for
MHD
modified
Stokes’
problems
26
nent
velocity
field
w
Cp
(
z
)
at
decreasing
values
of
the
parameter
M
or
α
,
clearly
show
that
the
fluid
velocity
is
an
increasing
function
with
regard
to
the
pressure-viscosity
coefficient
α
and
declines
for
increasing
values
of
the
magnetic
parameter
M
.
Consequently,
the
fluids
with
pressure-dependent
viscosity
flow
faster
in
comparison
with
ordinary
fluids
and
their
velocity
diminishes
in
the
presence
of
a
magnetic
field.
5
Conclusions
In
this
work
the
modified
Stokes
problems
for
a
class
of
ECUCIMFs
with
power-law
dependence
of
viscosity
on
pressure
were
investigated
in
the
pre-
sence
of
a
constant
magnetic
field.
The
influence
of
gravitational
acceleration
was
also
taken
into
consideration
and
some
known
results
from
the
litera-
ture
have
been
recovered
as
limiting
cases
of
present
solutions.
Graphical
representations
have
been
used
to
validate
the
obtained
results
and
to
bring
to
light
some
characteristics
of
the
fluid
motion.
The
main
results
that
have
been
here
obtained
are:
-
Closed
form
expressions
have
been
established
for
the
dimensionless
permanent
velocity
and
shear
stress
fields
of
the
MHD
modified
Stokes
problems
for
a
class
of
ECUCIMFs
with
pressure-dependent
viscosity.
-
Obtained
solutions
satisfy
the
governing
equations
and
boundary
con-
ditions
and
reduce
to
known
results
from
the
existing
literature
in
limiting
cases.
-
Permanent
shear
stress
for
MHD
modified
Stokes’
first
problem
is
con-
stant
on
the
entire
flow
domain
although
the
corresponding
velocity
is
function
of
the
spatial
variable.
-
The
fluids
with
pressure-dependent
viscosity
flow
faster
than
ordinary
fluids
and
in
the
presence
of
a
magnetic
field
their
velocity
diminishes.
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