49 
J. mt. area res., Vol. 8, 2023 

       

                Journal of Mountain Area Research 
            

 
 
 

MHD STAGNATION POINT FLOW WITH THERMAL RADIATION AND SLIP 

EFFECT OVER A LINEAR STRETCHING SHEET 
 

Ubaidullah Yashkun1,2, *, Khairy Zaimi1, Liaquat Ali Lund3 

 

1. Faculty of Applied Human and Sciences, Boundary Layer Research Group, Institute of Engineering Mathematics, 

Universiti Malaysia Perlis, Arau, Malaysia 

2. Department of Mathematics and Social Sciences, Sukkur IBA University, Sukkur, Pakistan 

3. School of Quantitative Sciences, Universiti Utara Malaysia, 06010 Sintok, Kedah, Malaysia 

 

ABSTRACT  
This research investigates the flow of stagnation point magnetohydrodynamic (MHD) and heat 

transfer along the stretched sheet in the existence of radiation and slip effects. With the help 

of similarity variables, the governing partial differential equations (PDEs) are transformed into 

ordinary differential equations (ODEs). The BVP4C technique in Matlab function has been used 

to simplify the governing ODEs. The numerical outcomes for temperature and velocity profiles, 

coefficient of skin friction and Nusselt Number have been achieved and matched with the 

findings in literature. The findings are compared to previously reported results. In addition, the 

impacts of numerous related parameters on the profiles of velocity and temperature are 

shown, and the results of every related parameter are presented using graphs. The velocity 

profile decreases as the magnetic force, suction, and permeability parameters rise. 

Keywords: Thermal radiation; slip effect; magnetohydrodynamic (MHD); suction; Prandtl 

number

*Corresponding author: (ubaidullah@iba-suk.edu.pk) 

1. INTRODUCTION 

In scientific literature, an extensive body of 

research has been done using stagnation 

points over the stretching sheet. It is widely 

used in a variety of engineering and 

manufacturing area, including insulating 

materials, polymer sheet production, glass 

drawing, production, displacement, and 

continuous casting, it is well-known in the 

researchers. With the use of comparable 

variables, Hiemenz [1] claimed that 

equations of Navier–Stokes may be 

translated into 3rd order ODEs, which 

became known as two-dimensional 

stagnation point flow. Sakiadis [2] gave 

mathematical solutions for boundary layer 

flow on the stretched surface in his paper. 

The exponentially stretched surface and 

the linearly stretched surface were further 

developed by Crane [3] in 1970. Chiam [4] 

investigated the flow of a stagnation point 

towards a bending plate. Several 

researchers, including Ishak et al. [5], have 

studied the MHD flow of stagnation point 

approaching a vertical stretched surface. 

Sadeghy et al. [6] examined flow of 

stagnation point of Maxwell fluid in a two-

dimensional. Attia [7] studied the viscous 

Full length article 

Vol. 8, 2023 

https://doi.org/10.53874/jmar.v8i0.175 

mailto:ubaidullah@iba-suk.edu.pk


Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

50 
J. mt. area res., Vol. 8, 2023 

fluid flow affecting on a porous stretched 

sheet for heat absorption in three 

dimensions using hydromagnetic flow of 

laminar and stagnation point. Recently, 

Ghasemi and Hatami [8] explored the 

stagnation point of nanofluid flow in 

conjunction with the effects of solar 

radiation for MHD nanofluid flow.  

As a result of recent advances in research, 

magnetohydrodynamics (MHD) may be 

thought of the mix of mechanics of fluids 

and electromagnetism, specifically the 

action of magnetic on an electrically 

conducting fluid, or both. Many studies 

have recently concentrated on the study 

of MHD flow because of its importance in a 

wide range in the field of Plasma Physics 

and engineering applications, including oil 

refinery, MHD power generators, the 

reactor of nuclear power, chemical 

reactions, magnetic mixers, and boundary 

layer control in aerodynamics [9-12]. Here, 

we will discuss several recent research that 

has concentrated on the MHD influence on 

heat transfer and flow concerns. These 

studies also focused on the flow and heat 

transfer of a viscous and incompressible 

fluid passing through a stretching surface 

along with magnetic field, among other 

fluids that are used in the industrial setting. 

According to Darzi et al. [13], for instance, 

they looked at the impact of heat transfer 

via the porous medium on the profile of 

velocity and temperature due to a thin 

liquid film flowing through a stretching 

sheet. 

Nandeppanavar et al. [14] used a porous 

stretched sheet and thermal radiation to 

investigate the MHD flow of stagnation 

point 2D via a porous medium. Abel and 

Nandeppanavar [15] investigated the 

impacts of non-uniform temperature 

sources and thermal radiation on MHD 

viscoelastic fluid movement and heat 

transmission past a stretched surface. 

Mustafa and Khan [16] explored the 2D 

steady next to the boundary, the mass 

transfer of Casson nano fluid, and the heat 

transmission of this liquid in existence of an 

exponential stretched surface.  Khalili et al. 

[17] quantitatively examined heat transfer 

of MHD flow of three distinct nanofluids 

induced through the shrunk plane inside 

porosity medium. Ghasemi et al. [18] 

simulated the MHD blood flow through 

porous vessels using both analytical and 

numerical methods. They classified blood 

like a 3rd grade fluid comprising 

nanoparticles, according to their 

classification. Asadi et al. [19] study the 

natural CuO-TiO2/water hybrid nanofluid at 

two infinite parallel vertical plates that are 

infinitely parallel to one another. They 

discovered that as volume percentage of 

nanoscale particles improves, width of 

momentum boundary increases while 

width of thermal layer decreases. 

Several practical applications relating to 

the boundary flow immerse in media of 

porous, including heat exchangers, 

geothermal engineering, oil industries, 

nuclear waste destruction, system of the 

grain storage, and others have drawn 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

51 
J. mt. area res., Vol. 8, 2023 

great interest to the research of boundary 

layer flow embedded in porous medium. 

Nield and Bejan [20] have highlighted the 

advantages of heat transfer flow across 

porous material and how it might be used. 

Several researchers, including Mohanty et 

al. [21], have studied approximate solutions 

for the transport of heat and mass of a 

micropolar fluid passing through porous 

media. Working on the MHD convective 

flow with reaction of chemical and 

radiation on porous medium [22]. Prasad et 

al. [23] explored the oscillatory reaction on 

viscoelastic fluid passing in porous media 

caused by a stretched sheet in a porous 

media. A study by Shamshuddin et al [24] 

investigated flow of MHD of the micropolar 

liquid using permeable medium over a 

moving surface that was incline. Many of 

the authors have recently completed 

simulations of porous medium interactions 

under a variety of flow conditions, which 

are detailed in the references [25-30]. 

This study expands the prior model of 

Agbaje et al. [31] to include the impact of 

MHD slip and radiation on heat transfer 

owing to flow of stagnation of nano-liquid 

across the stretching surface, which was 

previously neglected. Because of the linear 

similarity conversions, the conservation 

equalities are reduced to a dimensionless 

state. In order to simulate the nanofluid, an 

interfacial boundary condition is 

introduced, and an effective and reliable 

numerical approach, the BVP4C technique 

in Matlab function, according to the 

authors. Detailed research is conducted to 

determine how important parameters 

affect flow rate and energy output. The 

findings have been supported by results 

from prior investigations, which have been 

tabulated and found to have an excellent 

connection with the findings of the current 

study.  

2. PROBLEM FORMULATION 
In this study considered steady, MHD fluid 

flow in a porous form of boundary layer 

towards the stagnation point with effect of 

slip and thermal radiation long a stretched 

surface. The free stream velocity 𝑈∞(𝑥) = 𝑏𝑥 

and the stretching velocity 𝑈𝑤 (𝑥) = 𝑎𝑥 are 

along the flow direction. The constants  

𝑎 and 𝑏 areas 𝑎 > 0 and 𝑏 ≥ 0. The ambient 

temperature is 𝑇∞ and the mass flux velocity 

is 𝑉𝑤 (𝑥).  The flux velocity 𝑉𝑤 (𝑥) = −(𝑎𝜈)
1/2𝑆. 

The surface temperature of the stretching 

sheet is 𝑇𝑤 (𝑥) = 𝑇∞ + 𝑐𝑥
𝑛, where 𝑐 and 𝑛 are 

constant as 𝑐 > 0 at the heated surface. 

The proposed model is demonstrated using 

the following set of PDEs [31]. The physical 

model is presented in Figure 1. 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

52 
J. mt. area res., Vol. 8, 2023 

 

Figure 1. Geometry of Fluid Flow Model.

 

 

𝜕𝑢

𝜕𝑥
+

𝜕𝑣

𝜕𝑦
= 0,                       (1) 

𝑢
𝜕𝑢

𝜕𝑥
+ 𝑣

𝜕𝑣

𝜕𝑦
= 𝑈∞

𝑑𝑈∞
𝑑𝑥

+ 𝜈
𝜕2𝑢

𝜕𝑦2
+ 𝜎

𝐵0
2

𝜌
(𝑈∞ − 𝑢)

+
𝜈

𝐾
(𝑈∞ − 𝑢),                                      (2) 

𝑢
𝜕𝑇

𝜕𝑥
+ 𝑣

𝜕𝑇

𝜕𝑦
= 𝛼

𝜕2𝑇

𝜕𝑦2
+

𝑄

𝜌𝑐𝑝
(𝑇 − 𝑇∞) −

𝛼

𝐾

𝜕𝑞𝑟
𝜕𝑦

,     (3) 

with the boundary conditions 

𝑢 = 𝑈𝑤 (𝑥) + 𝐿 (
𝜕𝑢

𝜕𝑦
) , 𝑣 = 𝑉𝑤 (𝑥), 𝑇 = 𝑇𝑤 (𝑥),when 𝑦 = 0

𝑢 → 𝑈∞(𝑥), 𝑇 → 𝑇∞as𝑦 → ∞                                            (4) 

 

In the system of equations (1)-(4) 𝑢, 𝑣 are 

the velocity components along the axis of 

𝑥, 𝑦 respectively. The parameters are 

𝜌, 𝜈, 𝐾, 𝛼, 𝑐𝑝, 𝑄, 𝐿, 𝜎 and 𝑞𝑟  the fluid density, 

kinematic viscosity, porous medium 

permeability, thermal diffusivity, specific 

heat capacity, heat source/sink (𝑄 > 0, 𝑄 <

0), slip effect, electrical conductivity, and 

heat flux.  

PDEs (2-3) are transformed into ODEs using 

similarity transformation.  

 

𝑓(𝜂) =
𝜓

(𝑎𝜈)
1
2𝑥

, 𝜃(𝜂) =
𝑇 − 𝑇∞

𝑇𝑤 − 𝑇∞
, 𝜂 = (

𝑎

𝜈
)

1
2𝑦,      (5) 

 𝜓 is the stream function and its definition is 

as follows: 

𝑢 =
𝜕𝜓

𝜕𝑦
𝑎𝑛𝑑 𝑣 = −

𝜕𝜓

𝜕𝑥
                     (6) 

The velocity components 𝑢 and 𝑣 became. 

𝑢 = 𝑎𝑥𝑓′(𝜂), 𝑣 = −(𝑎𝜈)
1

2𝑓(𝜂)                (7) 

Where (′) represents differentiation w.r.t. 𝜂, 

𝛾 is the suction/injection parameter where 

𝛾 > 0 shows suction effect and 𝛾 < 0 

indicates the injection effect. The 

transformed ODEs are as   

𝑓‴ + 𝑓𝑓″ − 𝑓′2 + 𝐵2 + 𝑀(𝐵 − 𝑓′) − (𝛤 − 𝐵)𝑓′,     (8) 
1

𝑃𝑟
(1 +

4

3
𝑅) 𝜃″ + 𝐴𝜃 + 𝑓𝜃′ − 𝑛𝜃𝑓′                             (9) 

The boundary conditions are as 

𝑓(0) = 𝛾, 𝑓′ (0) = 1 + 𝛿𝑓″ (0), 𝜃(0) = 1,

𝑓′(∞) = 𝐵, 𝜃(∞) = 0
             (10) 

Where 𝐵 =
𝑏

𝑎
 is velocity ratio, 𝑀 =

𝜎𝐵0
2

𝑎𝜌
 is 

magnetic, 𝛤 =
𝜈

𝑎𝐾
 is the permeability 

 

 

  

 

  
 

Stagnation Point 

Porous Media 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

53 
J. mt. area res., Vol. 8, 2023 

parameter, 𝐴 =
𝑄

𝑎𝜌𝑐𝑝
 is the dimensionless 

heat generation/absorption coefficient, 

𝑅 =
4𝜎∗𝑇∞

3

𝐾𝐾𝑠
 is the thermal radiation and 𝛿 =

(
𝑏

𝜈
)1/2𝐿 is the slip parameter. 

The “substantial physical quantities of 

interest are the skin friction 𝐶𝑓 and Nusselt 

number 𝑁𝑢𝑥 are as 

𝐶𝑓 =
𝜏𝑤

𝜌𝑓 𝑈∞
2

, 𝑁𝑢𝑥                                                                             

=
𝑥𝑞𝑤

𝐾𝑓 (𝑇𝑤 − 𝑇∞)
,                                                                (11) 

where 𝜏𝑤   is the shear stress along the 

plate's surface and 𝑞𝑤  is the heat flow from 

the plate, the equations are as follows: 

𝜏𝑤 = 𝜇(
𝜕𝑢

𝜕𝑦
)𝑦=0, 𝑞𝑤

= −𝐾(
𝜕𝑇

𝜕𝑦
)𝑦=0 + (𝑞𝑟 )𝑦=0,                                                  (12) 

Putting Equation (5) into Equation (11) with 

(10), one can get 

𝐶𝑓 𝑅𝑒𝑥

1
2 = 𝑓″ (0),

𝑁𝑢𝑥

𝑅𝑒𝑥

1
2

= − (1 +
4

3
𝑅) 𝜃′(0),                                                   (13) 

  Where 𝑅𝑒𝑥 =
𝑈𝑤(𝑥)𝑥

𝜈
 is the Reynolds number. 

3. NUMERICAL PROCEDURE AND 

VALIDATION  
We used the Three Stage Labatto Three A 

Scheme in conjunction with the BVP4C 

technique to numerically solve the 

transformed ODEs (08-10) in order to find 

the solution. To make the problem simpler, 

all of the modified equations are expressed 

in the form of an initial value problem (IVP). 

In this case the initial values of 𝑓′′(0) and 

𝜃′(0) are not known. The BVP4C is used to 

determine the of 𝑓′′(0) and 𝜃′(0) by picking 

a specific value of 𝜂 → ∞ and a few starting 

guesses. The procedure of forecasting 

initial guesses is repeated until the desired 

value of the iteration is achieved. To turn 

Equations (08-10) into the first order ODEs, 

we must take into account the following 

variables: 

𝑓 = 𝑦1, 𝑓
′ = 𝑦2, 𝑓

′′ = 𝑦3, 𝜃 = 𝑦4, 𝜃
′ = 𝑦5            (14) 

𝑓′′′ = 𝑦2 ∗ 𝑦2 − 𝑦1 ∗ 𝑦3 − 𝐵
2 − 𝑀 ∗ (𝐵 − 𝑦2)

+ 𝛤𝑦2                                                    (15) 

𝜃′′ = (
3 ∗ 𝑃𝑟

3 + 4 ∗ 𝑅
) {𝑛 ∗ 𝑦4 ∗ 𝑦2 − 𝐴 ∗ 𝑦4 − 𝑦1

∗ 𝑦5}                                                     (16) 

Subject to boundary conditions 

𝑦1(0) = 𝛾, 𝑦2(0) = 1 + 𝛿 ∗ 𝛼1, 𝑦3(0) = 𝛼1
𝑦4(0) = 1, 𝑦5(0) = 𝛼2

                 (17) 

where 𝛼1 and 𝛼2 are assumed initial guesses 

which need to be determined. When 𝑀 =

𝛾 = 𝐴 = 𝑅 = 𝛿 = 0 and 𝑃𝑟 = 0.7, comparisons 

of coefficient of skin friction for various rates 

of 𝐵 with earlier published findings [31-33] 

are stated in Table 1 to assess the validity of 

the current numerical results. Also found is 

that an excellent link between the two 

variables has been established. Moreover, 

it is noted that a high degree of correlation 

is obtained. 

Table 1. Values of  𝑓′′(0) for some values of 𝐵 at 
𝑀 = 𝛾 = 𝐴 = 𝑅 = 𝛿 = 0 and 𝑃𝑟 = 0.7. 

B [31] [32] [33] Present 

results 

0.01 -0.9980 -0.9980 -0.99802 -0.998065 

0.02 -0.9958 - -0.99578 -0.995811 

0.05 -0.9876 - -0.98757 -0.987588 

0.10 -0.9694 -0.9694 -0.96938 -0.969387 

0.20 -0.9181 -0.9181 -0.91810 -0.918107 

0.50 -0.6673 -0.6673 -0.66732 -0.667263 

1.00 - - 0.00000 0 

2.00 - 2.0175 2.01750 2.0175027 

3.00 - 4.7292 4.72928 4.729282 

4.00 - - - 8.000429 

 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

54 
J. mt. area res., Vol. 8, 2023 

4. RESULTS AND DISCUSSIONS 
This section illustrates and discusses the 

effects of various relevant parameters such 

as velocity ratio parameter 𝐵, magnetic 

parameter 𝑀, permeability parameter 𝛤, 

heat generation/absorption coefficient 𝐴, 

thermal radiation 𝑅, Prandtl number 𝑃𝑟, 

and slip effect 𝛿 graphically on velocity 

and temperature profiles. In Figures 2–11, 

extensive numerical solutions are 

presented. The default values of the 

parameters considered as under, unless 

otherwise mentioned: 𝛾 = 1, 𝛤 = 0.1, 𝛿 =

0.1, 𝑛 = 0.2, 𝐴 = 𝐵 = 𝑀 = 0.1, and 𝑃𝑟 = 0.7. In 

Tables 2 and 3, coefficient of skin friction 

and heat transfer rate for different thermal 

radiation and velocity slip parameter 

values are listed. The effects of radiation 𝑅 

parameter on the temperature and 

velocity profiles are illustrated in Figures 2-3, 

respectively. A rise in the radiation 

parameter results cause to rise in the 

temperature of the fluid being heated. 

Essentially, the radiative energy enhances 

the impact of temperature and 

conduction at respective place on the 

surface, as a result, a vast amount of 

radiation energy is created, which raises 

the temperature of the fluid. On the other 

hand, there is no noticeable change in the 

profile of velocity as a horizontal surface 

has been considered. 

Table 2. Results of 𝑓′′(0) and −𝜃′(0) for different 
values of 𝑅 with 𝛾 = 1, 𝛤 = 0.1, 𝛿 = 0.1, 𝑛 = 0.2, 𝐴 =
𝐵 = 𝑀 = 0.1 and 𝑃𝑟 = 0.7. 

R 𝒇′′(𝟎) −𝜽′(𝟎) 

0 -1.338191965 0.945955902 

0.3 -1.338191965 0.705378882 

0.6 -1.338191965 0.574614931 

Table 3. Results of 𝑓′′(0) and −𝜃′(0) for different 

values of δ when = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝐴 = 𝐵 = 𝑀 =

0.1 and 𝑃𝑟 = 0.7. 

δ 𝒇′′(𝟎) −𝜽′(𝟎) 

0 -1.616258037 0.729409419 

0.1 -1.338191965 0.705378882 

0.3 -1.006890697 0.673515832 

 

 

Figure 2. Temperature profile at different values 

of R with 𝛾 = 1, 𝛤 = 0.1, 𝛿 = 0.1, 𝑛 = 0.2, 𝐴 = 𝐵 = 𝑀 =

0.1  

 

Figure 3. Velocity profile at different values of R 

with 𝛾 = 1, 𝛤 = 0.1, 𝛿 = 0.1, 𝑛 = 0.2, 𝐴 = 𝐵 = 𝑀 = 0.1 
and 𝑃𝑟 = 0.7. 

Figures 4-5 illustrate the influence of the 

velocity slip parameter δ on the profiles of 

temperature and velocity in the presence 

of the other parameters. When slip effects 

grow, it has been observed that the 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

55 
J. mt. area res., Vol. 8, 2023 

temperature increases. However, by 

gradually increasing the values of the 

velocity slip parameter, it is possible to 

achieve a steady reduction in the velocity 

profiles. The effect of the “Prandtl number 

Pr on the temperature profile is depicted 

and seen in the Figure 6. The Prandtl 

number is defined as the ratio of the 

momentum diffusivity to the thermal 

diffusivity of a given system. The increase in 

the values of Pr leads in a decrease in the 

temperature profile, which in turn results in 

a decrease in the thickness of the thermal 

boundary layer. definitely, a lesser thermal 

diffusivity causes a rise in the Prandtl 

number, which in turn aids in the reduction 

of the thickness of the thermal boundary 

layer. Because of this increase in the 𝑃𝑟, the 

flow at the boundary is pulled down, 

resulting in a change in the thickness of the 

thermal boundary layer. 

 

Figure 4. Temperature profile at different values 

of 𝛿 with 𝛾 = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝑅 = 0.3, 𝐴 = 𝐵 = 𝑀 =
0.1 and 𝑃𝑟 = 0.7. 

 

Figure 5. Velocity profile at different values of 𝛿 
with 𝛾 = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝑅 = 0.3, 𝐴 = 𝐵 = 𝑀 = 0.1 

and 𝑃𝑟 = 0.7. 

 

Figure 6. Temperature profile at different 𝑃𝑟 with 
𝛾 = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝐴 = 0.1, 𝑅 = 0.3, 𝐵 =

0.1 and 𝑀 = 0.1. 

The effects of the suction parameter 𝛾 on 

the fluid velocity and temperature profiles 

are illuminated in Figures 7-8, respectively. 

From this point on, we can see that the 

profile reduces with an increase in the 

value of 𝛾. The corresponding boundary 

layer thickness is also reduced, owing to 

the fact that the suction produces a 

reduction in the flow of the stream in a 

vertically descended manner. Also shown 

in Figure 8 is the relationship between the 

temperature and the suction/injection 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

56 
J. mt. area res., Vol. 8, 2023 

parameter 𝛾. It is obvious that increasing 𝛾 

decreases the temperature field. Because 

applying suction causes the number of fluid 

particles to be drawn into the wall, the 

temperature boundary layer drops.  

 

Figure 7. Velocity profile at different values of 𝛾 
with 𝛤 = 0.1, 𝑛 = 0.2, 𝑅 = 0.3, 𝐴 = 𝐵 = 𝑀 =

0.1and 𝑃𝑟 = 0.7. 

 

Figure 8. Temperature profile at different values 

of 𝛾 with 𝛤 = 0.1, 𝑛 = 0.2, 𝑅 = 0.3, 𝐴 = 𝐵 = 𝑀 =
0.1and 𝑃𝑟 = 0.7. 

 

The values of −𝜃′(0) in the solution for the 

variation of 𝑃𝑟 are shown in Figure 9, and it 

come to be positive for greater Prandtl 

number, indicating that heat moves from 

the heated surface to the ambient 

temperature of fluid. This fact also lends 

support to the previous findings of the 

current study, which show that 

temperature falls as the Prandtl number 

rises in significance. Furthermore, when the 

radiation parameter is expanded, the rate 

of heat transfer reduces; this fact 

demonstrates that heat flows from the 

ambient temperature in the direction of the 

heated surface. Slip effects, 

notwithstanding this, have the influence of 

slowing down the rate of heat transmission, 

as seen in Figure 10. The results of Figure 11 

similarly corroborate those of Figure 10, 

where the heat transfer rate decreases as 

the thermal radiation parameter is raised to 

greater levels. 

 

Figure 9. Temperature variation at different 

values of 𝑃𝑟 with 𝑅 at 𝛾 = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝐴 =
0.1, 𝐵 = 0.1 and 𝑀 = 0.1. 

 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

57 
J. mt. area res., Vol. 8, 2023 

Figure 10. Temperature variation at different 

values of 𝛿 with 𝑅 at 𝛾 = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝐴 =
0.1, 𝐵 = 0.1, 𝑀 = 0.1 and Pr=0.7. 

 

Figure 11.Temperature variation at different 

values of R with 𝛿 at 𝛾 = 1, 𝛤 = 0.1, 𝑛 = 0.2, 𝐴 =
0.1, 𝐵 = 0.1, 𝑀 = 0.1and 𝑃𝑟 = 0.7 

 

DECLARATIONS 

Funding: No funding was received for this 
study. 

Conflicts of interest/Competing interests: 
The authors declare no any conflict of 

interest/competing interests.  

Data availability: Not applicable 

Code availability: Not applicable 

Authors’ contributions:   

onceptualization, Ubaidullah Yashkun & Khairy 

Zaimi; Methodology, Ubaidullah Yashkun & 

Liaquat Ali Lund; Software, Ubaidullah Yashkun; 

Formal analysis, Ubaidullah Yashkun, Khairy 

Zaimi, Liaquat Ali Lund; Writing—original draft, 

Ubaidullah Yashkun & Liaquat Ali Lund; 

Supervision, Khairy Zaimi. All authors have read 

and agreed to the published version of the 

manuscript. 

 

CONCLUSION 
The aim of the current study was to offer a 

Three Stage Labatto Three a formula for 

addressing boundary value problems that 

included physical effects of various 

parameters. The equations of the 

electrically conducting incompressible 

fluid flow moving towards stagnation point 

through the stretched surface are 

approximated numerically in this study, and 

the results are presented in this paper. Flow 

modelling in porous media considers the 

impacts of heat source and sink, thermal 

radiation, velocity slip, magnetic field, and 

suction/injection parameters. Nonlinear 

ODEs are obtained from PDEs by the 

simulations of appropriate similarity 

transformations. Using the Three Stage 

Labatto Three A formula, the approximate 

solutions generated for the parameters: 

Nusselt number, skin friction coefficient, 

temperature profile, and velocity profile. 

The accuracy of the used method is 

validated by comparing current results to 

previously published literature. The 

following are the effects of relevant 

parameters: 

a. When the suction parameter was set to 

greater levels, the conductivity of the 

skin friction increased. 

b. The increasing value of the radiation 

parameter has ensured that the 

temperature profile along with the 

thickness of the thermal boundary layer 

have both raised. 

c. The amount of suction 𝑆 introduced into 

a steady fluid flow reduces the fluid flow 

velocity and temperature. 

d. The increasing Prandtl number has 

guaranteed that profile of temperature 

and the thickness of the thermal layer 

have both decreased.  

e. Because of a rise in the permeability 

parameter, the thickness of the 

boundary layer is increased. 



Ubaidullah et al., J. mt. area res. 08 (2023) 49-59 

58 
J. mt. area res., Vol. 8, 2023 

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Received: 05 Jan. 2023. Revised/Accepted: 26 Feb. 2023. 

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