Format Template


 

 Vol. 5, No. 2 | July – December 2021 
 

 

 

SJCMS | P-ISSN: 2520-0755| E-ISSN: 2522-3003 | Vol. 5 No. 2 July – December 2021 

1 

Mathematical Analysis of Magnetized Rotating 

Nanofluid Flow Over nonlinear shrinking surface: 

Duality and Stability

Sumera Dero1, Ghulam Hyder Talpur2, Abbas Ali Ghoto3, Shokat Ali1 

Abstract: 

In this study, the magnetohydrodynamic (MHD) effect on the boundary layer rotating flow 

of a nanofluid is investigated for the multiple branches case. The main focus of current research 

is to examine flow characteristics on a nonlinear permeable shrinking sheet. Moreover, the 

governing partial differential equations (PDEs) of the problem considered are reduced into 

coupled nonlinear ordinary differential equations (ODEs) with the appropriate similarity 

transformation.  Numerical results based on the plotted graphs are gotten by solving ODEs with 

help of the three-stage Labatto IIIA method in bvp4c solver in MATLAB. To confirm numerical 

outcomes, current results have been compared with previously available outcomes and found in 

good agreement. Skin friction coefficients, Nusselt and Sherwood numbers, velocity profiles, 

temperature profiles, and concentration profiles are examined. The results show that dual (no) 

branches exist in certain ranges of the suction parameter i.e., S≥Sc (S<Sc). Further, profiles of 
velocity decrease for rising values of Hartmann number in the upper branch, while a reverse 

trend has been noticed in the lower branch. Profiles of temperature and concentration enhance 

for the increasing values of thermophoresis in both branches. stability analysis of the branches 

is also done and noticed that the upper branch is a stable branch from both branches. Finally, it 

is noted that the stable branch has symmetrical behavior with regard to the parameter of rotation. 

Keywords: 3D flow; nanofluid; Rotating shrinking surface; Dual Branches; Stability analysis. 

1. Introduction  

Recently, scholars are implicated in the 
analysis of rotational flows within stretching 
and shrinking boundary layer problems 
because of their widespread use in the system 
of rotor-stator, food processing, spinning 
devices, the architecture of gas turbines, disk 
cleaners, and many others. Wang [1] examined 
the flow of rotating fluid through the stretching 
sheet where momentum boundary layer 
thickness was observed to decrease as the 
parameter of the rotational impact increased. 
Takhar et al. [2] considered a rotating fluid 

 
1Institute of Mathematics and Computer Science, University of Sindh, Jamshoro, Pakistan. 
2Department of Statistics, University of Sindh, Jamshoro 76080, Pakistan. 
3Department of Mathematics and Statistics, Quaid-e-Awam University, Pakistan. 

Corresponding Author: sumera.dero@usindh.edu.pk 

flow on the stretching surface with the 
characteristics of a magnetic number. Shafique 
et al. [3] investigated the rotating effect in the 
Maxwell fluid by considering binary chemical 
reactions and energy activation characteristics. 
They have found that the hydrodynamic 
boundary layer thins when rotation 
parameter λ is incremented. Oscillatory 
behavior in both x- and y-components of 
velocity is observed when rotation 
parameter λ is sufficiently large. Rashad [4] 
used mathematical modeling to check the 
effect of the non-steady MHD flow of rotating 
fluid. Recently, Ullah et al. [5] found that 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 5 No. 2 July – Dec 2021 

2 

temperature and concentration were increasing 
functions of the porosity and the Forchheimer 
parameters during the examination of the 
rotating flow of the nanofluid. In addition, 
Hayat et al. [6] extended the rotational flow 
model to examine the characteristics of 
homogeneous-heterogeneous nanofluid 
reactions. Lund et al., [7] reviewed the 3D flow 
of rotating nano-fluid on an exponential plane 
and discovered that the solution was not unique 
when the value of the rotating parameter was 
less than 0.1. Some important effects of the 
various physical parameter on the rotating flow 
can be found in [7-12]. 

The influence of MHD has attracted a lot 
of attention from researchers because of its 
extensive range of uses in various fields of 
science. It was introduced by Hannes Alfvén 
(1908-1995) who was a famous Swedish 
physicist. In 1970, he received the Physics 
Nobel prize for his pioneering work in MHD 
and major applications in numerous portions of 
plasma physics. In general, the presence of the 
magnetic field and the electrically conductive 
flow of the fluid give rise to the induced 
electrical current. More interest in MHD flow 
began in 1930 when Hartmann invented an 
electromagnetic pump. After that, many 
researchers considered MHD in their studies 
such as William [13-14], Eastman et al. [15], 
and Hossain [16] due to its extensive uses.  
Moreover, magnetic fields have an impact on 
numerous artificial and natural flows, which 
are essential elements used in some industries 
like pumping, heating, and levitating metals in 
the core of the earth. For example, solar flares 
and sunspots are generated by the solar 
magnetic field [17-18]. For medical and other 
applications, the ideal properties of the 
finished product are determined by drawing 
these strips or filaments into an electrically 
conductive liquid under the influence of a 
magnetic field. Hsiao [19] investigated 
numerical solutions for MHD two-dimension 
steady flow of boundary layer in a micropolar 
nanofluid. The author considered 
Buongiorno’s model [20] with a viscous 
dissipation effect on a linear non-permeable 
stretching surface. The linear non-permeable 
stretching surface was used because only one 
solution was considered. Further, it was 

discovered that an upsurge in the magnetic 
field decreases in the magnitudes of Nusselt 
number and velocities. Recently, Dero et al. 
[21] used the MHD effect on nanofluid of 
micropolar where the effect of the thermal 
radiation had been studied. The shooting 
method was then adopted to solve the resultant 
ODEs and triple solutions were obtained. 
Some important effects of the MHD on fluid 
flow can be seen in these articles [22-28]. 

In view of the development of new 
technologies over the last few decades, the use 
of convective fluids for heat transfer, such as 
oil, ethylene glycol, and water minerals, has 
increased significantly. These kinds of fluids 
are an essential part of numerous industrial 
sectors including air-conditioning, 
transportation, and power generation [29]. It 
seems that these convection fluids could not 
meet the requirements of the rate of heat 
transfer and cooling. In this regard, different 
fluid upgrade procedures have been applied as 
there is a necessity to make novel kinds of fluid 
that are extra viable in relation to heat transfer 
act to meet the increasing demands of modern 
technology and innovation in miniaturization 
and process intensification of equipment [30]. 
Keeping in mind the final goal to attain such, it 
has recently been anticipated to mix 
insignificant amounts of nanometers from 10 
nm to 50 nm of nanoparticles in convectional 
fluids, subsequent in nanofluids [31-33]. 
Studies have shown that the fraction of particle 
volume, that is the concentration of volumetric 
of the nanoparticle in nanofluid is related to the 
nanofluid thermal conductivity [34-35]. As 
associated with convectional fluid, the results 
of experiments on nanofluid noted that the 
expressively of thermal conductivity expanded 
with little changes in nanoparticle volume 
fractions. The nanofluids’ thermal 
conductivity with the base fluid of water 
containing nanoparticles 𝑇𝑖𝑂2 (27 nm), 𝑆𝑖𝑂2 
(12 nm), and 𝐴𝑙2𝑂2 (13 nm) have been 
measured by Masuda et al. [36]. Abareshi et al. 
[37] and Das et al. [38] suggested that there is 
a significant rise in temperature with an 
increase in thermal conductivity. CuO (28.6 
nm)/water and 𝐴𝑙2𝑂2 (38.4 nm)/water 
nanofluids at different temperatures 
fluctuating from 21°C to 51°C were studied by 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 5 No. 2 July – Dec 2021 

3 

Das et al. [39]. Theoretically, due to an 
increase in Brownian motion along with 
nanoparticles and nanofluid's bulk temperature 
T, it is expected that more energy can be 
exchanged from one region to the next as time 
increased. 

The flow of boundary layer idea on an 
incessant stretched sheet along constant 
velocity was initially presented by Sakiadis 
[40-41]. Ever since, frequent research on 
boundary layer flow through a stretched sheet 
had been carried out because of its extensive 
applications in industries such as the 
production of glass fiber, hot rolling, paper 
production, and polymer sheets extrusion [42]. 
Crane [43] was the first who consider a fluid 
flow problem on a stretched surface and 
presented a solution in an analytical form. In 
recent years, Awaludin et al. [44] investigated 
heat transfer of a stagnation point flow with the 
effect of heat sink/source on a stretched/shrunk 
parameter.  They observed that dual solutions 
existed indefinite ranges of 
stretching/shrinking parameters. Recently, 
Raza et al. [45] studied the three-dimensional 
boundary layer flow of a rotating nanofluid in 
the presence of suction/injection. The 
numerical results revealed two solutions 
existed which depended on the values of 
magnetic, stretching/shrinking, porosity, and 
suction parameters.  

Based on the literature review conducted 
and published, there is no such study in which 
a rotational nanofluid model with Brownian 
motion and thermophoresis effect on the 
stretching surface has been considered for 
multiple branches with their stability analysis. 
Due to this research gap, this study is 
conducted numerically as the nanofluid 
rotational flow has important applications in 
industry, engineering, and so on. In light of this 
fact, the Buongiorno model was considered to 
be preparing a nanofluid model in the hope that 
our findings will provide valuable help and 
reduce the cost of experiments. 

2. Mathematical Formulation 

The steady three-dimensional flow of 
nanofluid along with heat transfer has been 

considered over the shrinking surface as 
presented in Figure 1. Sheet at 𝑧 = 0 in the 
direction of 𝑥-axis i.e, 𝑢𝑤(𝑥) = −𝑐𝑥

𝑛 and 
velocity of mass flux is 𝑤𝑤(𝑥) =

𝑆√𝑐𝜗𝑥(𝑛−1) 2⁄ . Moreover, the temperature at 
the wall (ambient) is 𝑇𝑤 (𝑇∞). The sheet is 
supposed to rotate with velocity 𝛺0 about the 
𝑧-axis that is vertical to the sheet. A uniform 
field of external magnetic 𝐵 = 𝐵0 has been 
used to act with the z-axis. Taking into 
account the momentum, temperature, and 
concentration boundary layers, the flow of 
nanofluid can be presented in the form of 
PDEs as follows [45]: 

 

Figure 1. Physical model. 

𝜕𝑢

𝜕𝑥
= −(

𝜕𝑤

𝜕𝑧
+
𝜕𝑣

𝜕𝑦
)                                       (1) 

𝑢
𝜕𝑢

𝜕𝑥
− 2𝛺0𝑣 = 𝜗

𝜕2𝑢

𝜕𝑧2
−
𝜎𝐵2𝑢

𝜌
− 𝑣

𝜕𝑢

𝜕𝑦

− 𝑤
𝜕𝑢

𝜕𝑧
                             (2) 

𝑢
𝜕𝑣

𝜕𝑥
+ 2𝛺0𝑢 = 𝜗

𝜕2𝑣

𝜕𝑧2
−
𝜎𝐵2𝑣

𝜌
− 𝑣

𝜕𝑣

𝜕𝑦

− 𝑤
𝜕𝑣

𝜕𝑧
                             (3) 

𝑢
𝜕𝑇

𝜕𝑥
− 𝛼

𝜕2𝑇

𝜕𝑧2
= −𝑣

𝜕𝑇

𝜕𝑦
− 𝑤

𝜕𝑇

𝜕𝑧
+ 

                  𝜏1  [𝐷𝐵  
𝜕𝐶

𝜕𝑧
 
𝜕𝑇

𝜕𝑧
 + 

𝐷𝑇
𝑇∞
 (
𝜕𝑇

𝜕𝑧
)
2

]   (4) 

𝑢
𝜕𝐶

𝜕𝑥
+ 𝑣

𝜕𝐶

𝜕𝑦
+ 𝑤

𝜕𝐶

𝜕𝑧
= 𝐷𝐵  

𝜕2𝐶

𝜕𝑧2
 + 

                       
𝐷𝑇
𝑇∞
 
𝜕2𝑇

𝜕𝑧2
                                     (5) 

The related boundary conditions (BCs) (1-
5) are 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 5 No. 2 July – Dec 2021 

4 

{

𝑣 = 𝑤𝑤(𝑥) ,𝑢 = 𝑢𝑤(𝑥),𝑣 = 0,at 𝑧 = 0 
𝑇 = 𝑇𝑤,𝐶 = 𝐶𝑤  at 𝑧 = 0

𝑢 →  0,𝑣 →  0,𝑇 → 𝑇∞,as 𝑧 →  ∞
𝐶 → 𝐶∞ as 𝑧 →  ∞

(6) 

Note that respective velocities in 
directions of 𝑥,𝑦, and 𝑧-axes are 𝑢, 𝑣, and 𝑤. 
Further, 𝜗,𝜎,𝛼,𝑇,𝜏1,𝐷𝐵,𝐷𝑇, and 𝐶 are the 
corresponding kinematics viscosity, electrical 
conductivity, thermal diffusivity, temperature, 
the ratio between heat capacitances of the 
nanoparticles and base fluid, Brownian 
diffusion, thermophoresis diffusion, and 
nanoparticle fraction or concentration of 
nanofluid.  

Lund et al. [46]’s similarity variables are 
employed as follows: 

{
 
 
 
 
 
 

 
 
 
 
 
 
 𝑢 = 𝑐𝑥𝑛𝑓′(𝜂),𝑣 = 𝑐𝑥𝑛𝑔(𝜂) 

𝑤 = −√
𝑐𝜗(𝑛 + 1)

2
𝑥(𝑛−1) 2⁄

[𝑓 +
𝑛 − 1

𝑛 + 1
𝜂𝑓′]

𝜂 = 𝑧√
𝑐(𝑛 + 1)

2𝜗
𝑥(𝑛−1) 2⁄

∅(𝜂) =
(𝐶−𝐶∞)

(𝐶𝑤−𝐶∞)
⁄

𝜃(𝜂) =
(𝑇 − 𝑇∞)

(𝑇𝑤−𝑇∞)
⁄ ,

                   (7) 

where prime shows the derivative with 
respect to 𝜂 and 𝑐 is a positive constant. By 
putting Equation (7) in (2-5) leads to  

𝑓′′′ + 𝑓𝑓′′ −
2𝑛

𝑛 + 1
𝑓′2 +

4𝛺

𝑛 + 1
𝑔

−
2𝑛

𝑛 + 1
𝑀𝑓′ = 0            (8) 

𝑔′′ + 𝑓𝑔′ −
2𝑛

𝑛 + 1
𝑓′𝑔 −

4𝛺

𝑛 + 1
𝑓′

−
2𝑛

𝑛 + 1
𝑀𝑔 = 0             (9) 

1

𝑃𝑟
𝜃′′ + 𝜃′𝑓 + 𝑁𝑏∅′𝜃′ + 𝑁𝑡(𝜃′)2

= 0                                 (10)  

∅′′ + 𝑆𝑐𝑓∅′ +
𝑁𝑡

𝑁𝑏
𝜃′′ = 0                          (11) 

Along with BCs 

{
  
 

  
 
𝑓′(0) = −1,𝜃(0) = 1,∅(0) = 1

𝑓(0) = −𝑆√
2

𝑛 + 1

𝑔(𝜂) →  0,𝑓′(𝜂) →  0 𝑎𝑠 𝜂 → ∞

𝜃(𝜂) → 0,∅(𝜂) → 0  𝑎𝑠 𝜂 → ∞

        (12) 

Here 𝛺 =
𝛺0

𝑐
 is rotation parameter, 𝑃𝑟 =

𝜗𝑓

𝛼𝑓
 is Prandtl, 𝑁𝑏 =

𝜏1𝐷𝐵(𝐶𝑤−𝐶∞)

𝜗
 is Brownian 

motion parameter, 𝑁𝑡 =
𝜏1𝐷𝑇(𝑇𝑤−𝑇∞)

𝜗𝑇∞
  is 

thermophoresis parameter, and 𝑆 is the 
injection parameter (𝑆 > 0) and suction 
parameter (𝑆 < 0). 

The skin friction coefficients, local 
Nusselt, and Sherwood numbers can be 
defined as 

{
 
 
 
 

 
 
 
 𝐶𝑓𝑥 =

𝜇

𝜌𝑢𝑤
2
(
𝜕𝑢

𝜕𝑧
)|𝑧 = 0

𝐶𝑓𝑦 =
𝜇

𝜌𝑣𝑤
2
(
𝜕𝑣

𝜕𝑧
)|𝑧 = 0

𝑁𝑢𝑥 = −
𝑥

(𝑇𝑤−𝑇∞)
(
𝜕𝑇

𝜕𝑧
)|𝑧 = 0,

𝑆ℎ𝑥 = −
𝑥

(𝐶𝑤−𝐶∞)
(
𝜕𝐶

𝜕𝑧
)|𝑧 = 0

            (13) 

Putting Equation (7) in Equation (13) 
gives 

{
 
 
 
 
 

 
 
 
 
 √𝑅𝑒𝑥𝐶𝑓𝑥 =

𝑛 + 1

2
𝑓′′(0)

√𝑅𝑒𝑦𝐶𝑓𝑦 =
𝑛 + 1

2
𝑔′(0)

√
1

𝑅𝑒𝑥
𝑁𝑢𝑥 = −𝜃

′(0),

√
1

𝑅𝑒𝑥
𝑆ℎ𝑥 = −∅

′(0)

                          (14) 

where 𝑅𝑒𝑥 =
𝑥𝑢𝑤

𝜗
 and 𝑅𝑒𝑦 =

𝑦𝑣𝑤

𝜗
 are the 

local Reynold numbers. 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

Sukkur IBA Journal of Computing and Mathematical Science - SJCMS | Vol. 5 No. 2 July – Dec 2021 

5 

3. Temporal Stability Analysis 

In the previous section, dual solutions of 
Equations (8-11) with BCs (12) are noted; 
these branches are important as BCs (12) are 
also fulfilled in the lower branch. It is critical, 
however, to obtain a solution that can maintain 
its stability when subjected to minor 
disturbances. The stability of two solutions is 
so evaluated in order to identify a branch that 
is perfectly appropriate to the actual natural 
situation. The first step for stability is to 
transform Eqs (2-5) to the unsteady form as per 
the stability criteria as follows [47]:  

𝜕𝑢

𝜕𝑡
+ 𝑢

𝜕𝑢

𝜕𝑥
− 2𝛺0𝑣 = 𝜗

𝜕2𝑢

𝜕𝑧2
−
𝜎𝐵2𝑢

𝜌
− 

                           𝑣
𝜕𝑢

𝜕𝑦
− 𝑤

𝜕𝑢

𝜕𝑧
                       (15) 

𝜕𝑣

𝜕𝑡
+ 𝑢

𝜕𝑣

𝜕𝑥
+ 2𝛺0𝑢 = 𝜗

𝜕2𝑣

𝜕𝑧2
−
𝜎𝐵2𝑣

𝜌
− 

                  𝑣
𝜕𝑣

𝜕𝑦
− 𝑤

𝜕𝑣

𝜕𝑧
                                (16) 

𝜕𝑇

𝜕𝑡
+ 𝑢

𝜕𝑇

𝜕𝑥
− 𝛼

𝜕2𝑇

𝜕𝑧2
= −𝑣

𝜕𝑇

𝜕𝑦
− 𝑤

𝜕𝑇

𝜕𝑧
+ 

      𝜏1  [𝐷𝐵  
𝜕𝐶

𝜕𝑧
 
𝜕𝑇

𝜕𝑧
 + 

𝐷𝑇
𝑇∞
 (
𝜕𝑇

𝜕𝑧
)
2

]            (17) 

𝜕𝐶

𝜕𝑡
+ 𝑢

𝜕𝐶

𝜕𝑥
+ 𝑣

𝜕𝐶

𝜕𝑦
+ 𝑤

𝜕𝐶

𝜕𝑧
= 𝐷𝐵  

𝜕2𝐶

𝜕𝑧2
 + 

                               
𝐷𝑇
𝑇∞
 
𝜕2𝑇

𝜕𝑧2
                         (18) 

where 𝑡 indicates the time. As a result, a 
new variable, 𝜏 = 𝑐𝑥𝑛−1𝑡, is established. 
Equation (7) is articulated as follows: 

{
 
 
 
 
 
 

 
 
 
 
 
 
 𝑢 = 𝑐𝑥𝑛𝑓′(𝜂,𝜏),𝑣 = 𝑐𝑥𝑛𝑔(𝜂,𝜏) 

𝑤 = −√
𝑐𝜗(𝑛 + 1)

2
𝑥(𝑛−1) 2⁄

[𝑓 +
𝑛 − 1

𝑛 + 1
𝜂𝑓′] ,𝜏 = 𝑐𝑥𝑛−1𝑡

𝜂 = 𝑧√
𝑐(𝑛 + 1)

2𝜗
𝑥(𝑛−1) 2⁄

∅(𝜂,𝜏) =
(𝐶−𝐶∞)

(𝐶𝑤−𝐶∞)
⁄

𝜃(𝜂,𝜏) =
(𝑇 − 𝑇∞)

(𝑇𝑤−𝑇∞)
⁄ ,

         (19) 

Substituting Equation (19) in Equations 
(15-18) leads to 

𝑓𝜂𝜂𝜂 + 𝑓𝑓𝜂𝜂 −
2𝑛

𝑛 + 1
𝑓𝜂
2
+

4𝛺

𝑛 + 1
𝑔 − 

2𝑛

𝑛 + 1
𝑀𝑓𝜂 −

2𝑛

𝑛 + 1
𝑓𝜏𝜂 = 0                        (20) 

𝑔𝜂𝜂 + 𝑓𝑔𝜂 −
2𝑛

𝑛 + 1
𝑓𝜂𝑔 −

4𝛺

𝑛 + 1
𝑓𝜂 − 

2𝑛

𝑛 + 1
𝑀𝑔 −

2𝑛

𝑛 + 1
𝑔𝜏 = 0                          (21) 

1

𝑃𝑟
𝜃𝜂𝜂 + 𝑓𝜃𝜂 + 𝑁𝑏∅𝜂𝜃𝜂 + 𝑁𝑡(𝜃𝜂)

2
− 

𝜃𝜏 = 0                                                             (22) 

∅𝜂𝜂 + 𝑆𝑐𝑓∅𝜂 +
𝑁𝑡

𝑁𝑏
𝜃𝜂𝜂 − 𝑆𝑐∅𝜏 = 0        (23) 

Along with BCs 

{
  
 

  
 
𝑓′(0,𝜏) = −1,𝜃(0,𝜏) = 1,∅(0,𝜏) = 1

𝑓(0,𝜏) = −𝑆√
2

𝑛 + 1

𝑔(𝜂,𝜏) →  0,𝑓′(𝜂,𝜏) →  0 𝑎𝑠 𝜂 → ∞

𝜃(𝜂,𝜏) → 0,∅(𝜂,𝜏) → 0  𝑎𝑠 𝜂 → ∞

(24) 

Now, obtain the solutions of steady flow 
from (8-11) as 𝑓(𝜂) = 𝑓0(𝜂),𝑔(𝜂) = 𝑔0(𝜂), 
𝜃(𝜂) = 𝜃0(𝜂),and ∅(𝜂) = ∅0(𝜂), it is 
assumed  

{
 

 
𝑓(𝜂,𝜏) = 𝑓0(𝜂) + 𝑒

−𝜀𝜏𝐹(𝜂,𝜏)

𝑔(𝜂,𝜏) = 𝑔0(𝜂) + 𝑒
−𝜀𝜏𝐺(𝜂,𝜏)

𝜃(𝜂,𝜏) = 𝜃0(𝜂) + 𝑒
−𝜀𝜏𝐻(𝜂,𝜏)

∅(𝜂,𝜏) = ∅0(𝜂) + 𝑒
−𝜀𝜏𝐽(𝜂,𝜏)

               (25) 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

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6 

where the unidentified eigenvalue is 𝜀 
where its value needs to fix a stable branch. In 
addition,  𝐹(𝜂,𝜏),𝐺(𝜂,𝜏), 𝐻(𝜂,𝜏), and 𝐽(𝜂,𝜏) 
all their derivatives are assumed small relative 
to 𝑓0(𝜂),𝑔0(𝜂), 𝜃0(𝜂), and ∅0(𝜂). Now, 
substituting the correlation (25) in Equations 
(20-24), we get the following resultant 
Linearized Eigenvalue Problem (LEVP) 
system as follows: 

𝐹0
′′′ + 𝑓0𝐹0

′′ + 𝐹0𝑓0
′′ −

4𝑛

𝑛 + 1
𝑓0
′𝐹0
′ +

4𝛺

𝑛 + 1
𝐺0 

2𝑛

𝑛 + 1
𝑀𝐹0

′ +
2𝑛

𝑛 + 1
𝜀𝐹0

′ = 0                      (26) 

𝐺0
′′ + 𝑔0

′𝐹0 + 𝐺0
′𝑓0  −

2𝑛

𝑛 + 1
(𝑓0

′𝐺0 + 𝐹0
′𝑔0) − 

4𝛺

𝑛 + 1
𝐹0
′  +

2𝑛

𝑛 + 1
𝜀𝐺0 = 0                         (27) 

1

𝑃𝑟
𝐻0
′′ + 𝜃0

′𝐹0 + 𝐻0
′𝑓0 + 𝑁𝑏(∅0

′ 𝐻0
′ + 𝐽0

′𝜃0
′) + 

2𝑁𝑡𝜃0
′𝐻0

′ + 𝜀𝐻0 = 0                                    (28) 

𝐽0
′′ + 𝑆𝑐∅0

′ 𝐹0 + 𝐽0
′𝑓0 +

𝑁𝑡

𝑁𝑏
𝐻0
′′ + 

𝑆𝑐𝜀𝐽0 = 0                                                       (29) 

subject to BCs 

{
 

 
𝐹0(0) = 0,𝐹0

′(0) = 0,𝐺0(0) = 0,

𝐻0(0) = 0,𝐽0(0) = 0

𝐹0
′(𝜂) →  0,𝐺0(𝜂) →  0 𝑎𝑠 𝜂 → ∞

 𝐻0(𝜂) → 0,𝐽0(𝜂) → 0 as 𝜂 → ∞

       (30) 

All feasible eigenvalues would be acquired 
(𝜀) by solving the LEVP system. In Equation 
(30), a relaxed boundary condition requires 
being used to achieve the sequence of 
eigenvalues. The boundary condition 𝐹0

′(𝜂) →
 0 as 𝜂 → ∞ is now restrained to 𝐹0

′′(0). 

4. Results and Discussions 

Non-linear Equations (8-11) subject to BCs 
(10) has been numerically solved with bvp4c 
solver in MATLAB. We have compared the 

values of √𝑅𝑒𝐶𝑓𝑥 and √𝑅𝑒𝐶𝑓𝑦 with the results 
of Zaimi et al. [48] over the stretching surface 
(i.e, 𝑓′(0) = 1) in Table 1. From these results, 
we notice that the numerical outcomes signify 
good a correlation with the earlier findings. 
Henceforth, the code of MATLAB can be 

employed with full conviction to investigate 
the problem under discussion. The effect of 
numerous physical parameters such as 
magnetic number (0 ≤ 𝑀 < 0.5), rotation 
parameter (Ω ≤ 0.04), positive number (2 ≤
𝑛 ≤ 3), Brownian motion parameter (0.1 ≤
𝑁𝑏 ≤ 0.5), thermophoresis parameter (0.1 ≤
𝑁𝑡 ≤ 0.5), and suction parameter (𝑆 ≥ 3.5) 
are conversed and illustrated in figures. 

Table 1. Values of √𝑅𝑒𝐶𝑓𝑥 and √𝑅𝑒𝐶𝑓𝑦 are 
compared when  𝑓′(0) = 1 = 𝑛 and 𝑀 =

𝑆 = 0. 
  √𝑅𝑒𝐶𝑓𝑥  √𝑅𝑒𝐶𝑓𝑦  

Ω [48] Presen
t 
results 

 [48] Prese
nt 
results 

0 -1.00 -1.000  0.0000 0.000
0 

0.5 –
1.13
84 

-
1.1384 

 –
0.5128 

-
0.512
8 

1 –
1.32
50 

-
1.3250 

 –
0.8371 

-
0.837
1 

2 –
1.65
23 

-
1.6523 

 –
1.2873 

-
1.287
3 

3 –
1.92
89 

-
1.9289 

 –
1.6248 

-
1.624
8 

4 –
2.17
16 

-
2.1716 

 –
1.9054 

-
1.905
4 

5 –
2.39
01 

-
2.3901 

 –
2.1506 

-
2.150
6 

 

The existence of multiple solutions enables 
one to explore those parameters lead to the 
existence of two branches. The reduced skin 
friction variants 𝑓′′(0),𝑔′(0), heat transfer 
−𝜃′(0), and −∅′(0)  are shown in Figures 2-5 
for various values of 𝑛. Moreover, 𝑆𝑐 =
−2.40392,−2.6249,−2.8285 is the 



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7 

equivalent critic of 𝑛 = 2,2.5,3 where 𝑆𝑐 is the 
critical point where all solutions exist at 𝑆 =
𝑆𝑐. Dual branches are noted as 𝑆 ≥ 𝑆𝑐 and 
when 𝑆 < 𝑆𝑐 there is no solution. The 
estimation of boundary layers beyond such 
critical values is no longer justified. Reduced 
skin friction (𝑓′′(0)) reduces when 𝑛 is 
increased in the upper branch. Further, 𝑔′(0)  
decreases when 𝑛 increases in the lower 
branch. On the other hand, 𝑔′(0) rises when 
values of 𝑛 are increased in the upper branch. 
In addition, the behavior of 𝑔′(0) and 𝑆 are 
inversely proportional in the lower branch. 
Nature of reduced heat transfer (−𝜃′(0)) can 
be seen in Figure 4 in which −𝜃′(0) enhances 
in both branches when the effect of 𝑆 reduces, 
while the opposition movement has been 
examined in both branches for the rising values 
of 𝑛. Similarly, the effects of suction and 
positive constant were drawn in Figure 5 in 
order to examine their effects on the 
nanoparticle fraction of nanofluid. As 
previously noticed in Figure 4, the same 
behavior is noted. 

 

Figure 2. 𝑓′′(0) for numerous values of 𝑆 and 
𝑛. 

 

Figure 3. 𝑔′(0) for numerous values of 𝑆 and 
𝑛. 

 

Figure 4. −𝜃′(0) for numerous values of 𝑆 
and 𝑛. 

 

Figure 5. −∅′(0) for numerous values of 𝑆 
and 𝑛. 

Figures 6-9 allude to the impacts of rising 
magnitudes of 𝑀 on profiles of velocity 
𝑓′(𝜂),𝑔(𝜂), temperature profiles 𝜃(𝜂), and 
concentration profiles ∅(𝜂). Figures 6 and 9 
show that 𝑓′(𝜂) and 𝑔(𝜂) decline for the rising 
magnitudes of 𝑀 in the upper solution, but the 
opposite movement is noticed in the lower 
solution. It is apparent from these estimates 
that for significant values of M, the thickness 
of momentum boundary layers of 𝑓′(𝜂) and 
𝑔(𝜂) are decreased in the stable branch. 
Physically, the decreasing behavior is due to 
the magnetic field effect on the nanofluid 
experience of the force induced by the 
electrical current. This electrically conductive 
nanofluid interacts with a transverse magnetic 
field that induces the Lorentz forces. The 
Lorentz’s force reduces the velocity flow and 
thus reduces the thickness of layer. The 
temperature and concentration of nanofluid 
enhance in both branches when the magnetic 
effect increases (see Figures 8-9). 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

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8 

 

Figure 6. 𝑓′(𝜂) for different values of 𝑀. 

 

Figure 7. 𝑔(𝜂) for different values of 𝑀. 

 

 

 

The space is intentionally left blank to adjust 
the fingers. 

 

Figure 8. 𝜃(𝜂) for different values of 𝑀. 

 

Figure 9. ∅(𝜂) for different values of 𝑀. 

Figures 10-11 display the effect of 
increasing values of 𝑁𝑡 on the dimensionless 
temperature profiles 𝜃(𝜂) and concentration 
profiles ∅(𝜂), respectively. These figures 
show that 𝜃(𝜂) and ∅(𝜂) increase for the 
increasing values of 𝑁𝑡 in both branches. 
These increments in the thickness of boundary 
layers are due to the fact that the higher effect 
of 𝑁𝑡 supports molecules of nanoparticles and 
fluid to transfer heat to the next layer and 
therefore temperature and concentration 
increase.  



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Duality and Stability (pp. 01 - 13) 

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9 

 

Figure 10. 𝜃(𝜂) for different values of 𝑁𝑡. 

 

Figure 11. ∅(𝜂) for different values of 𝑁𝑡. 

 

Figures 12-13 are prepared to see the 
variations in temperature profiles 𝜃(𝜂) and 
concentration profiles ∅(𝜂) for various 
magnitudes of 𝑁𝑏, respectively. Figure 12 
indicates that 𝜃(𝜂) rises for the increasing 
values of 𝑁𝑏 in both branches. This situation 
is true because, in the fluid flow process, it is 
possible to surge in the rate of heat transfer in 
the presence of the thermophoresis effect. The 
concentration of boundary layers, on the other 
hand, decreases in thickness in both branches. 
The consequence of Brownian motion can be 
described as the nanoparticles extending in the 

entire fluid and thus decrease in the 
concentration profiles. 

 

Figure 12. 𝜃(𝜂) for different values of 𝑁𝑏. 

 

Figure 13. ∅(𝜂) for different values of 𝑁𝑏. 

Figure 14 illustrates that 𝜃(𝜂) decreases for 
increasing values of 𝑃𝑟 in both branches. The 
development of nanofluid, a blend of center 
fluid and nanoparticles, depends on values of 
𝑃𝑟. The rising 𝑃𝑟 values enhance the base fluid 
viscosity, resulting in a decline in the thickness 
of the thermal boundary layer and thus a 
decrease in heat transfer for the higher 𝑃𝑟 
values. This is because the extremely viscous 
nanofluid results in poor conductivities of 
thermal that influence the phenomenon of 



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10 

conduction to reduce the thickness of thermal 
boundary layer. In the extremely viscous 
nanofluid, the motions of dispersed 
nanoparticles are often more impacted because 
of lower nanoparticle forces between one 
another. 

 

 

Figure 14. 𝜃(𝜂) for different values of 𝑃𝑟. 

 

Figure 15 is plotted for ∅(𝜂) to analyze the 
effects of 𝑆𝑐. It is observed that ∅(𝜂) decreases 
for the increasing values of 𝑆𝑐 in both 
branches. This decrease in ∅(𝜂) for large 
values of 𝑆𝑐 is justified due to the fact that 𝑆𝑐 
is directly proportional to the kinematic 
viscosity of the nanofluid. The increasing 
values of 𝑆𝑐 increase the viscosity of the 
nanofluid which results in a decrease in ∅(𝜂). 
Finally, it is noted that the stable branch has 
symmetrical behavior with regard to the 
parameter of rotation (See Figure 16). It can be 
easily concluded from Figure 16 that the 
symmetrical branches belong to this fluid 
model. 

 

Figure 15. ∅(𝜂) for different values of 𝑆𝑐. 

 

Figure 16. 𝑔(𝜂) for different values of Ω. 

Governing Equations (26-29) have been 
resolved by employing the bvp4c function. 
The results of the smallest eigenvalues are 
given in Table 2. The governing system 
provides an infinite range of eigenvalues. The 
smallest negative eigenvalues; 𝜀 < 0 implies 
that the flow has an initial disruption 
development that may disrupt the flow and, 
ultimately, induce unstable flow. Besides that, 
the smallest positive eigenvalues; 𝜀 > 0 
specifies that an initial decay of disturbance 
occurs in the flow, are showing the stable flow. 



Sumera Dero (et al.), Mathematical Analysis of Magnetized Rotating Nanofluid Flow Over nonlinear shrinking surface: 
Duality and Stability (pp. 01 - 13) 

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11 

Table 2. Values of 𝜀 for 𝑆 and 𝑛 where 𝑀 =
0.1 = 𝑁𝑡,𝑁𝑏 = 0.3,𝑆𝑐 = 𝑃𝑟 = 1,𝛺 = 0.04. 

𝑛 𝑆  𝜀 

  Upper 
branch 

Lower 
branch 

2 -2.403 0.0001 -0.0001 

 -2.6 0.1271 -0.0478 

 -3 0.5973 -0.8696 

2.5 -2.625 0.0005 -0.0007 

 -2.8 0.3857 -0.5945 

 -3 0.9643 -1.0642 

3 -2.83 0.0002 -0.0009 

 -3 0.4585 -0.3946 

 -3.2 1.0962 -0.9738 

 

5. Conclusion 

In this study, we investigate MHD 
nanofluid 3D flow through a non-linear 
shrinking sheet for the heat transfer 
performance with multiple branches and 
stability analysis characteristics. The 
numerical analysis is conducted by applying 
the three-stage Labatto IIIA method in bvp4c 
solver to study the multiple branches of the 
problem with the stability analysis of the 
branches. Fluid suction/injection is found to 
have a major effect on the distribution of 
velocity, temperature, and concentration, 
which transitively influences the presence of 
multiple branches within the boundary layer. 
The main findings of the current study are 

1. For 𝑆 ≥ 𝑆𝑐, there are two branches of the 
nanofluid problem solution, namely a lower 
and an upper branch. It is noticed that the lower 
one is not a physically suitable branch.  

2. Upper branch is a stable branch from both 
branches. 

3. An increase in the thermophoresis 
parameter advances nanofluid temperature 
along with concentration profiles.  

4. Brownian motion parameter reduces the 
nanofluid concentration. 

5. Increasing Hartmann number causes the 
reduction of nanofluid velocity uniformly due 
to the presence of Lorentz force. 

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