Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder
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Copyright © 2023 by American Scientific Publishers Journal of Nanofluids
All rights reserved. Vol. 12, pp. 29–35, 2023
Printed in the United States of America (www.aspbs.com/jon)
Statistical Analysis of Non-Newtonian Couple
Stress Fluid Induced in Stretching Cylinder
Hiranmoy Mondal1, ∗ , Subhabrata Dey2 , Archita Biswas2 , Sruti Gupta2 , and Sukhendu Samajdar3
1
Department of Applied Mathematics, Maulana Abul Kalam Azad University of Technology, 700064, West Bengal, India
2
Department of Applied Statistics, Maulana Abul Kalam Azad University of Technology, 700064, West Bengal, India
3
Department of Materials Science & Technology, Maulana Abul Kalam Azad University of Technology, 700064,
West Bengal, India
The paper provides the impact of suction and injection on convection laminar incompressible couple stress fluid
flow and magnetic field using spectral quasi linearization methods as the major novelty of our work. This work is
to addresed heat transfer is an important process in many engineering, industrial, residential, and commercial
buildings. Thus, this study aims to analyze the effect of MHD and non-Newtonian couple stress fluid runs over
a permeable stretched cylinder. The leading formulation is transmuted into ordinary differential equations via
similarity functions. The coupled equations with non-linearly terms are resolved numerically through utilization
of MATLAB code for spectal quasi linearization methods (SQLM). Convergence regions for solutions are dis-
cussed. Graphical results illustrating the impacts of various emerging parameters are presented in discussion.
ARTICLE
The statistical declaration and probable error for skin friction and Nusselt number are numerically computed
and discussed through Tables. From obtained outcomes it is concluded that magnitude of skin friction increases
at the cylindrical surface for higher values of couple stress parameter and Reynolds number. Nusselt number
IP: 5.10.31.151
or heat transfer rate also enhances On:
at the surface of Fri, 27 Sep
cylinder 2024
in the 01:21:32of Reynolds number.
presence
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KEYWORDS: Couple Stress Fluid, Stretching Cylinder, SQLM.
1. INTRODUCTION presence of porous medium passing across a cone were
In Newtonian theory of fluids, fluid has been regarded as studied by Ahmad et al.3
continuous material ignoring the fact that fluid particle’s Of late Hadjesfandiari et al.4 have formulated an inno-
size or micro structural property affects the flow features vative reliable couple stress premises in which the diffi-
of fluid. However, in practical field results may differ from culties illustrated by Stokes can be eliminated. This novel
above mentioned assumption. Because blood flow or poly- theory provides an excellentsource for elementary stud-
mer extrusions or some lubricants, applications of colloidal ies as well as hugenumber of fluid mechanics applica-
suspensions designate that structural characteristic of con- tions. Wang5 explores the features of fluid passing over
tinuum at microscopic level is needed. Nanofluids have stretchable cylinder. Same was carry forwarded by Ishak
attracted the attention of several scientists owing to the and Najar.6 Numerical treatment of couple stress liquid
important applications in the technology sector. The heat over infinite vertically placed cylinder was reported by
transformation of convection liquids like ethylene glycol, Rani et al.7 A power-law fluid run over stretching surface
kerosene, water as well as oil can be used in numerous was addressed by Jalil et al.8 Couple stress flow between
engineering equipments, for example devices of the elec- permeable contracting or expanding path was investigated
trons and heat transfer. by Khan et al.9 Flow of magnetised couple stress liquid
Verma et al.1 numerically discussed the effects of Soret over oscillatory stretched surface was communicated by
and Dufour with thermal radiation on MHD flow around Ali et al.10
a vertical cone. The 2-dimensional MHD nanofluid flow The progress in discovering the model of couple stress
passing over a Plate or cone were discussed by Ahmad fluid that can contribute to enhancing the flow properties
et al.2 The investigation of MHD micropolar fluid in the always be the main focus. Among the available additional
extension on the fluid flow problem, the MHD effects
∗
are among the applicable elements should be deliberated.
Author to whom correspondence should be addressed.
Email: hiranmoymondal@yahoo.co.in Slip flow of radiating couple stress liquid over stretching
Received: 4 November 2021 surface was explored by Ref. [11]. Hayat and Ahmad12
Accepted: 19 January 2022 scrutinize peristaltic couple stress flow inside revolving
J. Nanofluids 2023, Vol. 12, No. 1 2169-432X/2023/12/029/007 doi:10.1166/jon.2023.1905 29Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder Mondal et al.
non-uniform channel. Literatures connecting these issues field having strength B0 is applied along radial direction.
are presented in Refs. [13–15]. The surface of the cylinder is subjected to the temperature
Theory of physics confirms us about the subsistence Tw and the ambient fluid temperature is T . The induced
of two kinds of convection, first natural convection and magnetic field effects are considered to be negligible due
second forced convection. When natural along with force to the fact magnetic Reynolds number has been considered
convection acts jointly to transport heat, then mixed con- negligibly small.
vection originates. In this circumstance forces arising from We maintain our study with the hypothesis that there
pressure and buoyancy perform together. Mixed convec- is no chemically reactive species, no slips take place, all
tive flow of couple stress liquid through parallel path was body forces along with viscous dissipation and joule heat-
demonstrated by Srinivasacharya and Kaladhar.16 Ojjela ing is ignored. Based on the above assumption the govern-
and Kumar17 studied the unsteady chemically reactive ing equations are as follows (Asad et al.41 ):
18
flow between parallel surfaces. Umavathi et al. analysed u w w
the flow considering heat source or sink. Fluid blessed + + =0 (1)
z r r
with couple stress runs over oscillatory stretched surface
was examined by Khan et al.19 Literatures introduced in
u u f 1 u ¯
Refs. [20–23]. depict more about such flows. A great num- u +w = r − 4u
ber of reports have been stimulated by the major func- z r f r r r f
tion of hall and ion slip effect on heat and mass transfer B02 u
MHD flow with different types of fluid model are analyzed + g T T − T − (2)
f
24–40
numerous thermal system. 2
Being encouraged by the aforementioned literatures, in T T T 1 T
u +w = f + (3)
this article we have disclosed the scenario of couple stress z r r 2 r r
ARTICLE
liquid crossing over a stretched cylinder. We have presup- where u and w are the velocity components of the fluid
posed the flow to be mixed convective and radiating in in the directions z and r respectively, T is the nanofluid
character. Prime equations have been framed in its non- temperature, ¯ stands for couple stress viscosity coeffi-
dimensional structure. Then solution is being sketched out cient, T authenticates thermal expansion, f is the den-
IP: 5.10.31.151 On: Fri, 27 Sep 2024 01:21:32
using novel SQLM mechanism. Parametric effects retained sity of the
Copyright: American Scientific fluid, f is the fluid dynamic viscosity, f =
Publishers
velocity, temperature and noteworthy exergy discussion. f /cp f represents thermal diffusivity, f denotes the
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Now the upcoming section enlightens mathematical for- thermal conductivity for nanofluid, cp f denotes the spe-
mulation of the problem. cific heat of nanofluid,
Also after boundary layer approximation we obtain
2. MATHEMATICAL FORMULATION 4 u 2 3 u 1 2 u 1 u
Let us consider the steady two dimensional laminar cou- 4u = + − + (4)
r 4 r r 3 r 2 r 2 r 3 r
ple stress fluid flow caused by a stretched cylinder with The requisite boundary conditions are as follows:
radius a as depicted in Figure 1. We presume r-axis along
the radial direction while z-axis has been taken parallel U0 z T
u= Uw = w = w0 −k = hf Tf −Tw at r =a
to the axis of the cylinder. The stretching velocity is of l y
the form Uw = U0 z/l where U0 > 0 and l corresponds u→ 0 T → T w →0 as r → (5)
to the characteristic length. Uniform transverse magnetic
Also it should be noted that couple stress vanishes out-
side the boundary layer, and then we also have
u 2 u
→ 0 and → 0 as r → (6)
r r 2
Invoking the following dimensionless relations
⎫
−a U0 f ⎪
f ⎪
U0 z
u= f w= ⎪
⎪
l r l ⎬
(7)
r 2 − a2 U0 T − T ⎪
⎪
= = ⎪
⎪
2a f l Tw − T ⎭
One can have the transformed form of Eqs. (6)–(7) as
2f + 1 + 2 f − Re 8 2 f + 8 1 + 2 f iv
+1 + 22 f v + Gr + ff − f − Mf = 0
2
Fig. 1. Schematic of the problem. (8)
30 J. Nanofluids, 12, 29–35, 2023Mondal et al. Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder
1 + 2 + 2 + Pr f = 0 (9) Table II. Numerical values of covariance and correlation coefficient for
Nusselt number.
Also the boundary conditions (8) and (9) take the
shape as Parameters Covariance Correlation
Re −08018976 −09906756
f 0 = 1 f 0 = −fw −006990366 −09808269
0385935 09989474
0 = −Bi1 − 0 at =0 Gr 05826931 −04389585
f → 0 f → 0 f → 0 fw −04087654 −09921106
→0 as → (10)
Now invoking (10) into (20), we get the requisite expres-
It is to be noted that fw < 00 corresponds to suction, fw >
00 indicates injection and fw = 00 signifies impermeable sion for reduced skin friction and reduced Nusselt number
surface. The non-dimensional appearances of the relevant as follows:
parameters are Cfr = Cf Re1/2
z = f 0 (13)
⎫
¯ ⎪ N ur = Nu Re−1/2 = − 0 (14)
= Couple stress parameter = ⎪
⎪
z
⎪
⎪
f a 2
⎪
⎪ where Rez = Uw z/lf is the local Reynold’s number.
1/2 ⎪
⎪
⎪
⎪
f l ⎪
⎪
= Curvature parameter = ⎪
⎪
U0 a2 ⎪
⎪ 4. STATISTICAL APPROACH
⎪
⎪
f ⎪
⎪
Pr = Prandtl number = ⎪
⎪
Here the maximum and minimum of the parameters of
⎪
⎪
⎪
⎪
Nusselt Number are shown in the required Table I along
ARTICLE
f
⎪
⎪
Ul ⎪
⎬
with the mean and median values of those parameters as
Re = Reynolds number = 0 (11) obtained.
f ⎪
⎪
⎪ Now, as we know for Skewness, from the formula that
⎪
⎪
g T TIP: − T l 2
5.10.31.151 ⎪
⎪ On: Fri, if
27 (Q3-Q2)
Sep 2024is01:21:32
greater than (Q2-Q1) then it is positively
Gr = Grashoff number = w
⎪
⎪
U0 z Copyright:
2 ⎪
American
⎪ Scientific
skewed, Publishers
if less than then negatively skewed and if equals
⎪
⎪
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⎪
⎪ symmetric. So, applying this rule we can say that
lB02 ⎪
⎪
M = Magnetic parameter = ⎪
⎪ parameters Re, R, Gr are positively skewed, parameters ,
f U0 ⎪
⎪
⎪
⎪ are negatively skewed and only parameter fw is sym-
⎪
⎪
⎪ metric in nature.
l ⎪⎪
⎪
fw = suction/injection parameter = w0 ⎪ And, for Kurtosis as we know if it’s value is greater than
U0 f ⎭ 3, then it is leptokurtic, less than 3 then platykurtic and if
equals 3 then mesokurtic. So, now as per results obtained
from the table all the parameters i.e., Re, , , R, Gr, fw
3. PHYSICAL QUANTITIES are platykurtic in nature.
The physical quantities of the stream profile are skin The Table II finds the Numerical values of Covariance
friction and Nusselt number. They are characterized as and Correlation coefficient for Nusselt Number. Table II
follows: shows that the value of corellation are in the range.
w zqw ⎫
Cf = Nu = ⎪
⎪
1/2f Uw2 f T w −T
⎬ 5. CALCULATION
u T ⎪
⎪ The probable error is the value which is added or sub-
where w = f and qw = − f ⎭ tracted from the correlation coefficient to obtain the upper
r r=a r r=a
(12) limit and the lower limit respectively, within which the
Table I. Numerical values of first quartile (Q1), median (Q2), third Quartile (Q3), maximum and minimum of the parameters, mean, skewness and
kurtosis of the parameters.
Parameters Minimum Maximum Mean Median (Q2) Q1 Q3 Skewness Kurtosis
Re 10 6.0 2.857 20 20 35 09268158 2.778368
05 2.5 1.271 12 05 1850 03099562 1.588231
01 2.5 1.029 10 02 16 04606427 1.760768
Gr 05 4.0 1.657 12 07 225 08542307 2.534069
fw −05 0.5 0.0666 01 −015 035 −03270641 1.840772
J. Nanofluids, 12, 29–35, 2023 31Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder Mondal et al.
value of correlation coefficient expectedly lies. The prob- Table VI. Numerical values of probable error and r/P E r for skin
able error of the correlation coefficient can be obtained friction coefficient.
by applying
√ the following formula: PEr = 067451 − Parameters Probable error (P.E(r)) r/P.E (r)
r 2 / n, where r signifies the correlation coefficient and
n marks the number of observation. The correlation coef- Re 0.061621170 14297567
0.073708481 11610044
ficient is not remarkable if the value of r is less than PE 0.070808655 −12172095
this discloses that there is no correlation between the vari- Gr 0.121282543 6167696
ables. The correlation is said to be evident when the value fw 0.040559394 22767125
of r is 6 times more than the PE and insignificant when r
is less than PE(r).
Here, we observe from the Table II that parameters Re,
, fw have a fairly strong negative relationship with the
6. STATISTICAL RULE Nusselt Number and parameter Y has a fairly extremely
The values of r/PE(r) are dispensed in the Tables III strong positive relationship with the Nusselt Number.
and IV for Nusselt Number. From this Table it is obvi- While from the Table II.I, we observe that parameters Y ,
ous that no values have fulfilled the relation, r/PE(r) > 6, R have a fairly strong negative relationship with the Skin
which specifies that the correlation coefficient is statisti- friction coefficient and parameters Re, , Gr, fw have a
cally insignificant for those all parameters. fairly extremely strong positive relationship with the Skin
The values of r/PE(r) are dispensed in the Tables V and friction coefficient.
VI for Skin friction coefficient. From this Table we can As a consequence we come to an end that some corre-
see that some values have fulfilled the relation, r/PE(r) > lation coefficients are tremendous and the parameters are
6, which specifies that the correlation coefficient is sta- greatly interconnected to the physical attributes.
tistically significant for those all parameters (Re, , fw),
ARTICLE
while for the other parameters the correlation coefficient
is insignificant. 7. NUMERICAL SOLUTIONS USING
In case of perfect correlation that is r = 1, we get the SPECTRAL QUASI-LINEARIZATION
perfect significant positive correlation and if r = −1, METHODS (SQLM)
IP: 5.10.31.151 On:weFri, 27 Sep 2024 01:21:32
Copyright: American Scientific
get the perfect significant negative correlation. The numerical implemented to solve the modelled dif-
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Delivered byferential
Ingenta equations. Here we approach spectral quasi
linearization (SQLM) to achieve numerical outcomes
Table III. Numerical values of probable error for Nusselt number.
of coupled nonlinear equations together with boundary
Parameters Probable error (P.E(r)) condition.
Re 001045792
Let us consider fr , r be the solutions of equations at
00005793901 r th
stage of iteration and fr+1 , r+1 at r + 1th stage.
00001320485 Now employing SQLM scheme to the equations along
Gr 02223052 with boundary condition, we acquire the following itera-
Fw 0004327766 tive systems:
v
a0 r fr+1 + a1 r fr+1
iv
+ a2 r fr+1 + a3 r fr+1 + a4 r fr+1
Table IV. Numerical values of r/P Er for Nusselt number.
+a5 r fr+1 + a6 r r+1 = Rf (15)
Parameters r/P.E(r)
b0 r r+1 + b1 r r+1 + b2 r fr+1 = R (16)
Re −93.78798
Subject to,
−7571.16
⎫
Gr −1.974576 fr+1 0 = 1 fr+1 0 = −fw ⎪
⎪
Fw −229.2431 0
⎬
r+1 = −Bi 1 − r+1 0 fr+1 = 1 (17)
⎪
⎪
⎭
fr+1 = 0 fr+1 = 0 r+1 = 0
Table V. Numerical values of covariance and correlation coefficient for
skin friction coefficient. The coefficients in (20)–(21) are as follows:
Parameters Covariance Correlation
⎫
a0 r = − Re1+22 a1 r = −8 Re1+2 ⎪
⎪
⎪
⎪
Re 01248104 08810328 ⎪
⎬
a2 r = 1+2−8 Re 2 a3 r = 2 +fr
007633721 08557587
−01759238 −08618897 a4 r = −M −2fr a5 r = fr a6 r = Gr ⎪
⎪
⎪
⎪
Gr 06351431 07480339 ⎪
⎭
fw 01848454 09234208 b0 r = 1+2 b1 r = Prfr b2 r = Prr
(18)
32 J. Nanofluids, 12, 29–35, 2023Mondal et al. Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder
The Chebyshev polynomial has been employed with 8. RESULTS AND DISCUSSION
Gauss-Lobatto points defined by The flow of couple stress on a stretching cylinder is inves-
tigated numerically by considering magnetic effect and
i Biot number. By selecting appropriate similarity variables,
xi = cos i = 0 1 2 N −1 ≤ xi ≤ 1 (19)
N the equations that reflect the stated flow are transformed
to ordinary differential equations. A numerical scheme is
where N symbolizes the number of collocation points.
used to give a clear knowledge of the behaviour of flow
The whole coordination (20)–(21) is worked out inside
fields, which have been followed for the graphical frame
the region [0, L] insteadof [0, ); where, L being a
work. The accuracy of our couple stress model we have
large number, corresponds the boundary clause at infin-
examined the values of − 0 for different values of
ity and L must be a larger number. Thus, the region [0, Prandtl number and listed in Table VII. Then the numerical
L] changed to [−1, 1] via linear transformation defined
data have been compared with Ishak et al.,6 when others
by, = L x + 1/2. The key feature of spectral colloca- values as = R = Gr = M = = fw = 00. We observed
tion scheme is to launch a system of differentiation matrix that values are in good accord.
to approximate the derivative of unknown variables at the The effect of dimensionless parameters on involved pro-
collocation points as a matrix product: files is studied using graphs in this section. The impact
N of couple stress parameter ( = 05, 1.2, 1.5, 2.5, 3)
dFr
= Djk f k = DFm j = 0 1 2 (20) on f is exposed in Figure 2. The velocity profiles
d k=0 decreases as the value of couple stress parameter rises due
to higher magnetic field which cause a resistance to flow
where D = 2D/L and F = f 0 , f 1 , and hence velocity decays. The increasing values of couple
f 2 f N T is the vector formation of the functions. stress parameter upsurges the temperature profile. Here,
ARTICLE
Derivatives of higher order are classified as power of the radius of the cylinder increases as the increases.
D as: Figure 3 demonstrates the variation of temperature profile
Fr p = Dp Fr (21) for varied . The plot explains that the increasing values
IP: 5.10.31.151 On: Fri, of Seprises
27 the01:21:32
2024 temperature profile. Temperature rises for
where p denotes the order of derivatives. higher Prandtl number.
Copyright: American Scientific Publishers
Now the matrix appearance of spectral collocation Delivered by Ingenta
scheme containing the differentiation of anonymous func-
tions are as follows: Table VII. Comparison of − 0 for various values of Pr.
Pr Ishak et al.6 Present work
A1 1 f + A1 2 = Rf (22)
0.2 0.1691 0.169187852
2.0 0.9114 0.911423120
A2 1 f + A2 2 = R (23) 7.0 1.8954 1.895443213
70.0 6.4622 6.462205674
Here,
⎫
A1 1 = diag a0 r D 5 +diag a1 r D 4 +diag a2 r D 3 ⎪
⎪
⎬
+diag a3 r D 2 +diag a4 r D +diag a5 r I
⎪
⎪
⎭
A1 2 = diag a6 r I
(24)
2
A2 1 = diag b0 r D + diag b1 r D
(25)
A2 2 = diag b2 r I
Rf = f
2
(26)
R = − Pr f (27)
where diag and I are the diagonal and identity matrices of
order N + 1 × N + 1. Now the entire systems can be
framed as:
A11 A12 Fr+1 Rf
= (28)
A21 A22 r+1 R Fig. 2. Impact of on velocity profile.
J. Nanofluids, 12, 29–35, 2023 33Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder Mondal et al.
Fig. 3. Impact of on temperature profile.
Fig. 6. Impact of Bi on velocity profile.
ARTICLE
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Fig. 4. Impact of fw on velocity profile.
Fig. 7. Impact of Bi on temperature profile.
Consequently frictional characteristics between fluid
layers enhance and aids fluid velocity to run slow as
depicted in Figure 4. Besides the reverse features is According to Figure 5 the temperature is raising through-
observed for injection parameter. The range of suction and out the suction procedure compared to impermeable one.
injection parameter fw is −0.5, −0.2, 0.0, 0.2, 0.5. Inside Figures 6 and 7 displays the impact of Biot number
the cylindrical parametric effect is undoubtedly distinct. (Bi = 0.2, 0.5, 1.5, 2, 2.5) over velocity and temperature
profiles. Here velocity and temperature enhances for both
the cases.
9. CONCLUSION
In this research article we formulated and analyzed a
model of flow of a couple stress fluid induced in stretch-
ing cylinder. The flow and temperature of the couple stress
parameters were shown to have an effect on the fluid on
the cylinder. Some of the notable effects are:
(a) Increasing the couple stress parameter decreases
velocity and increases temperature profile throughout the
stretching cylinder.
(b) The perfect significant correlation for the different of
couple stress, Reynolds numbers, magnetic and suction
Fig. 5. Impact of fw on temperature profile. parameters exists in case of skin friction.
34 J. Nanofluids, 12, 29–35, 2023Mondal et al. Statistical Analysis of Non-Newtonian Couple Stress Fluid Induced in Stretching Cylinder
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ARTICLE
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Copyright: American Scientific Publishers
dria Engineering
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