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\begin{document}
\title{Hydromagnetic Stagnation--Point Flow of Casson Fluid towards a
Convectively Heated Stretching/Shrinking Sheet}
\author[1]{Winifred Mutuku}%
\author[2]{Onyekachukwu Oyem}%
\affil[1]{Kenyatta University}%
\affil[2]{Islamic University in Uganda}%
\vspace{-1em}
\date{\today}
\begingroup
\let\center\flushleft
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\maketitle
\endgroup
\selectlanguage{english}
\begin{abstract}
This study investigates the hydromagnetic stagnation point flow to
Casson fluid towards a vertically stretching sheet with convective
surface boundary condition. Casson fluid model is used to characterize
the non-Newtonian fluid behavior. Using similarity transformation, the
governing nonlinear partial differential equations are transformed into
coupled nonlinear ordinary differential equations which are solved
numerically by employing Runge-Kutta Fehlberg integration scheme with
shooting technique. Graphical results showing the effects of various
thermophysical parameters on the velocity and temperature profiles are
presented and discussed quantitatively.%
\end{abstract}%
\sloppy
\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\emph{ARTICLE} &\tabularnewline
\bottomrule
\end{longtable}
\textbf{Hydromagnetic Stagnation--Point Flow of Casson Fluid towards a
Convectively Heated Stretching/Shrinking Sheet}
\textbf{Winifred N. Mutuku\textsuperscript{1} \textbar{} Anselm O.
Oyem*\textsuperscript{2}}\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\textsuperscript{1}Department of Mathematics and Actuarial Science,
Kenyatta University, Nairobi, Kenya \textsuperscript{2}Department of
Mathematics and Statistics, Islamic University in Uganda, Uganda
\textbf{Correspondence} *Corresponding Author name, Corresponding
address. Email: anselmoyemfulokoja@gmail.com \textbf{Funding
Information} Project Name Grant/Award Number: XXXXXXXX &
\textbf{Abstract} This study investigates the hydromagnetic stagnation
point flow to Casson fluid towards a vertically stretching sheet with
convective surface boundary condition. Casson fluid model is used to
characterize the non-Newtonian fluid behavior. Using similarity
transformation, the governing nonlinear partial differential equations
are transformed into coupled nonlinear ordinary differential equations
which are solved numerically by employing Runge-Kutta Fehlberg
integration scheme with shooting technique. Graphical results showing
the effects of various thermophysical parameters on the velocity and
temperature profiles are presented and discussed quantitatively.
\textbf{KEYWORDS} Hydromagnetic, Stagnation point flow, Casson fluid,
Convective heating.\tabularnewline
\bottomrule
\end{longtable}
\section*{Introduction}
{\label{introduction}}
Hydromagnetic boundary layer flow of an electrically conducting viscous
incompressible fluid with a convective surface boundary condition is
frequently encountered in many biological, industrial and technological
applications such as extrusion of plastics in the manufacture of Rayon
and Nylon, MHD generators, the cooling of nuclear reactors, geothermal
energy extraction, purification of crude oil, textile industry, polymer
technology, metallurgy, drag reduction in aerodynamics, MHD blood flow
meters among others. Since the pioneering work on MHD boundary layer
flow by Sakiadis \(\mathbf{[1]}\), numerous aspects of steady and
unsteady boundary layer flow of convectional fluids as well as
nanofluids have been investigated by various authors\(\mathbf{[2-6]}\).
Stagnation-point flow, describing fluid motion over a continuously
stretching surface in the presence of electromagnetic fields are
significant in many engineering processes with various industrial
applications such as extraction of polymer sheet, paper production,
metallurgy, polymer processing, glass blowing, glass-fibre production,
plastic films drawing, filaments drawn through a quiescent electrically
conducting fluid subject to a magnetic field and the purification of
molten metals from non-metallic inclusions. The quality of the final
product depends to a great extent on the rate of cooling at the
stretching surface, thus for superior products the heat transfer should
be controlled. Since the pioneering work in this area by
Crane\(\mathbf{[7]}\), who investigated steady boundary layer flow of
a viscous fluid over a linearly stretching plate, many aspects of this
problem have been investigated by other authors\(\mathbf{[8-}\mathbf{12}\mathbf{]}\).
Non-Newtonian fluid flows are encountered in chemical, material and
industrial processing engineering. In particular, such materials are
involved in geophysics, oil reservoir engineering, bioengineering,
chemical and nuclear industries, polymer solution, cosmetic processes,
paper production, design of thrust bearings and radial diffusers etc.
These fluids exhibit a non-linear relationship between shear and the
rate of strain which deviate significantly from the Newtonian fluids
(Navier-Stokes) model making it difficult to express these properties in
a single constitutive equation. Owing to the complexity of these fluids,
there is not a single constitutive equation which exhibits all their
properties. Thus, various models have been used for non-Newtonian
fluids, with their constitutive equations varying greatly in
complexity\(\mathbf{[13-16]}\). The different types of non-Newton fluids
are viscoelastic fluid, couple stress fluid, micropolar fluid, power-law
flow, Casson fluid, among other types.
Casson fluid behaves like an elastic solid, with a yield shear stress
existing in the constitutive equation. It is a is a shear thinning
liquid which is assumed to have an infinite viscosity at zero rate of
shear, a yield stress below which no flow occurs, and a zero viscosity
at an infinite rate of shear. This implies that if a shear stress
greater than yield stress is applied; it starts to move whereas if a
shear stress less than the yield stress is applied to the fluid, it
behaves like a solid. Examples of Casson fluids are fluids such as
yoghurt, molten chocolate, cosmetics, nail polish, tomato puree, jelly,
honey, soup, concentrated fruit juices, human blood. The first model for
such fluids was formulated by Casson \(\mathbf{[17]}\)whose objective
was to investigate the flow behaviour of pigment oil suspensions of the
printing ink type. Thereafter, several researchers studied Casson fluid
pertaining to different flow situations\(\mathbf{[18-}\mathbf{2}\mathbf{6}\mathbf{]}\).
Medikare\emph{et al} . \(\mathbf{[22]}\) investigated MHD
stagnation-point flow of a Casson fluid over a nonlinearly stretching
sheet with viscous dissipation.
Owing to the numerous application of non-Newtonian fluids in industrial
processes and the literature gap in hydromagnetic Casson fluid flow has
given a strong motivation to understand their behaviour in several
transport processes. The current study extends the work of
Medikare\emph{et al} . \(\mathbf{[22]}\) by incorporating buoyancy
force and considering a convective boundary layer in the numerical
analysis of the hydromagnetic stagnation-point flow of a steady,
incompressible Casson fluid towards a shrinking/stretching sheet.
\section*{governing Equations}
{\label{governing-equations}}
Consider a steady, incompressible two-dimensional stagnation-point flow
of an electrically conducting Casson fluid towards a vertically
stretching sheet at \(y=0\) with the flow confined in the
region\(y>0\). Along the stretching surface in the
\(x\)-axis, two equal and opposite forces are being applied
with a uniform magnetic field strength\(B_{0}\) applied
perpendicular to the surface. The induced magnetic field is neglected
and the ambient fluid is moved with a velocity\(U_{\infty}\left(x\right)=ax\). The
rheological equation of state for an isotropic and incompressible flow
of a Casson fluid\(\mathbf{[18,26]}\) is given by
\begin{equation}
\tau_{\text{ij}}=\left\{\begin{matrix}\left(\mu_{B}+\frac{P_{y}}{\sqrt{2\pi}}\right)2e_{\text{ij}}\ ,\ \ \ \ \pi>\pi_{c}\\
\left(\mu_{B}+\frac{P_{y}}{\sqrt{2\pi_{c}}}\right)2e_{\text{ij}}\ ,\ \ \ \ \pi>\pi_{c}\\
\end{matrix}\right.\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (1)\nonumber \\
\end{equation}
where, \(\pi\) is the product of the component of deformation
rate with itself, \(\pi=e_{\text{ij}}e_{\text{ij}}\), \(e_{\text{ij}}\) are
the\(\left(i,\ j\right)^{\text{th}}\) component of the deformation rate and
\(\pi_{c}\) is a critical value of this product based on the
non-Newtonian model, \(\mu_{B}\) is plastic dynamic velocity of
the non-Newtonian fluid and \(P_{y}\) is the yield stress of
the fluid.
The MHD boundary layer equations for the steady stagnation-point flow
are given by\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\frac{\partial u}{\partial x}+\frac{\partial v}{\partial y}=0\) & (2)\tabularnewline
\midrule
\endhead
\(u\frac{\partial u}{\partial x}+v\frac{\partial u}{\partial y}=U_{\infty}\frac{dU_{\infty}}{\text{dx}}+\ \upsilon\left(1+\frac{1}{\beta}\right)\frac{\partial^{2}u}{\partial y^{2}}+\beta g\left(T-T_{\infty}\right)-\frac{\sigma B_{0}^{2}\left(x\right)}{\rho}\left(u-U_{\infty}\right)\) & (3)\tabularnewline
\(u\frac{\partial T}{\partial x}+v\frac{\partial T}{\partial y}=\alpha\frac{\partial^{2}T}{\partial y^{2}}+\selectlanguage{greek}\selectlanguage{english}\frac{\text{μα}}{k}\left(1+\frac{1}{\beta}\right)\left(\frac{\partial u}{\partial y}\right)^{2}+\selectlanguage{greek}\frac{\text{ασ}B_{0}^{2}\left(x\right)}\selectlanguage{english}{k}\left(u-U_{\infty}\right)^{2}.\) & (4)\tabularnewline
\bottomrule
\end{longtable}
The boundary conditions at the sheet surface and free stream is given
as:\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(u\left(x,0\right)=U_{w}\left(x\right),\text{\ \ \ }v\left(x,0\right)=0,\ \text{\ \ }-k_{f}\frac{\partial T}{\partial y}\left(x,0\right)=h_{f}\left[T_{f}-T\left(x,0\right)\right]\text{\ \ }\text{\ \ }\text{at}\ \ y=0\ \) & (5)\tabularnewline
\midrule
\endhead
\(u\left(x,\infty\right)\rightarrow U_{\infty}\left(x\right),\ \ \ T\left(x,\ \infty\right)\rightarrow T_{\infty}\text{\ \ \ }\text{as\ \ \ }\ \ y\rightarrow\infty\) & (6)\tabularnewline
\bottomrule
\end{longtable}
where \(u\), \(v\) are the velocity components
in \(x,y\) direction respectively, \(\rho\) is the
viscosity,\(\beta=\mu_{B}\sqrt{2\pi_{c}}/P_{y}\) is the non-Newtonian or Casson parameter,
\(U_{w}=bx\) is the shrinking/stretching velocity for the sheet
with \(b\) being the shrinking/stretching constant,
\(b<0\)corresponds to shrinking, \(b>0\)
corresponds to stretching and\(U_{\infty}=\text{ax}\) is straining velocity of
the stagnation point flow with \(a(>0)\) being straining
constant.
The stream functions \(u=\partial\psi/\partial y\) and\(v=-\partial\psi/\partial x\)
satisfies the continuity Eq. (2). In order to simplify the mathematical
analysis of the problem, we introduce the following similarity variables\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(\eta=\left(a\backslash\upsilon_{f}\right)^{\frac{1}{2}}y\ ;\text{\ \ }\psi=\left(a\upsilon_{f}\right)^{\frac{1}{2}}\text{xf}\left(\eta\right);\ \ \selectlanguage{greek}\text{\ θ}\selectlanguage{english}\left(\eta\right)=\frac{T-T_{\infty}}{T_{f}-T_{\infty}}\text{\ .}\) & (7)\tabularnewline
\bottomrule
\end{longtable}
Using Eq. (7), Eqs. (3 -- 6) are transformed to a set of coupled
non-linear ordinary differential equations
\begin{equation}
\left(1+\frac{1}{\beta}\right)\frac{d^{3}f}{d\eta^{3}}+f\frac{d^{2}f}{d\eta^{2}}+\left(\selectlanguage{greek}\frac{\text{df}}{\text{dη}}\selectlanguage{english}\right)^{2}+Gr\theta-M\left(\selectlanguage{greek}\frac{\text{df}}{\text{dη}}-\selectlanguage{english}1\right)+1=0\ \ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(8)\nonumber \\
\end{equation}\begin{equation}
\frac{d^{2}\theta}{d\eta^{2}}+Prf\selectlanguage{greek}\selectlanguage{english}\frac{\text{dθ}}{\text{dη}}\selectlanguage{greek}\selectlanguage{english}+\left(1+\frac{1}{\beta}\right)\text{PrEc}\left(\frac{d^{2}f}{d\eta^{2}}\right)^{2}+PrEcM\left(\selectlanguage{greek}\frac{\text{df}}{\text{dη}}-\selectlanguage{english}1\right)^{2}=0\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(9)\nonumber \\
\end{equation}
and the boundary conditions are
\begin{equation}
\begin{matrix}f\left(0\right)=0,\text{\ \ }\ \selectlanguage{greek}\frac{\text{df}}{\text{dη}}\selectlanguage{english}\left(0\right)=\lambda,\text{\ \ \ }\selectlanguage{greek}\selectlanguage{english}\frac{\text{dθ}}{\text{dη}}\selectlanguage{greek}\selectlanguage{english}\left(0\right)=Bi\left[\theta\left(0\right)-1\right]\\
\selectlanguage{greek}\frac{\text{df}}{\text{dη}}\selectlanguage{english}\left(\infty\right)=1,\text{\ \ }\ \selectlanguage{greek}\selectlanguage{english}\frac{\text{dθ}}{\text{dη}}\selectlanguage{greek}\selectlanguage{english}\left(\infty\right)=0\ .\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\\
\end{matrix}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(10)\nonumber \\
\end{equation}
From the above Eqs. (8), (9) and (10), prime denote differentiation with
respect to \(\eta\), \(\lambda\) is the velocity ratio
parameter,Gr is Grashof number\emph{,}
\(M\) is Magnetic field parameter, \(\Pr\) is
Prandtl number, Ec is Eckert number
andBi is the Biot number respectively defined as
follows;
\begin{equation}
M=\selectlanguage{greek}\frac{\sigma B_{0}^{2}}{\text{ρa}}\selectlanguage{english};\ Gr=\selectlanguage{greek}\frac{\text{gβ}\left(T_{w}-T_{\infty}\right)}\selectlanguage{english}{U_{\infty}a};\ Pr=\frac{\upsilon}{\alpha};\ Ec=\frac{U_{\infty}^{2}}{c_{p}\left(T_{w}-T_{\infty}\right)};\ Bi=\frac{h}{k}\sqrt{\frac{\upsilon}{a}};\ \lambda=\frac{b}{a}\ .\text{\ \ \ \ \ \ \ }(11)\nonumber \\
\end{equation}
The physical quantities of practical interest are the skin friction
coefficient \(C_{f}\) and the local Nusselt number
\(Nu_{x}\) defined as
\begin{equation}
C_{f}=\frac{\tau_{w}}{\rho U_{w}^{2}},\ \ Nu_{x}=\frac{xq_{w}}{\alpha\left(T_{w}-T_{\infty}\right)}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (12)\nonumber \\
\end{equation}
where \(\tau_{w}\) is the shear stress or skin friction along the
stretching sheet and \(q_{w}\) is the heat flux from the sheet
and defined thus
\begin{equation}
C_{f}\left(Re_{x}\right)^{\frac{1}{2}}=\left(1+\frac{1}{\beta}\right)\frac{d^{2}f}{d\eta^{2}}\left(0\right),\text{\ \ }\ \frac{Nu_{x}}{\left(Re_{x}\right)^{\frac{1}{2}}}=-\selectlanguage{greek}\selectlanguage{english}\frac{\text{dθ}}{\text{dη}}\selectlanguage{greek}\selectlanguage{english}\left(0\right),\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ (13)\nonumber \\
\end{equation}
where \(Re_{x}=U_{w}x/\upsilon\) is the local Reynolds number.
\section*{numerical method}
{\label{numerical-method}}
The set of coupled ordinary differential equations Eq. (8) and Eq. (9)
with boundary conditions Eq. (10) are computed numerically using
shooting method with Runge-Kutta Fehlberg integration scheme. This
method involves transforming the dimensionless coupled nonlinear
differential equations into a set of first order differential equations
after which, the fourth order Runge-Kutta Fehlberg integration scheme is
employed until the given boundary conditions are satisfied.
Thus, we define the new variables as;\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
\(x_{1}=f,\ \ x_{2}=f^{{}^{\prime}},\ \ x_{3\ =\ }f^{{}^{\prime\prime}},\ \ x_{4}=\ \theta,\ \text{\ \ \ \ \ \ \ }x_{5}=\ \theta^{{}^{\prime}}.\) & \((14)\)\tabularnewline
\bottomrule
\end{longtable}
Eqs. (8) -- (10) are then reduced to the following system
\begin{equation}
x_{3}^{{}^{\prime}}=\ \left(\frac{1}{1+\ \beta}\right)\left[-x_{1}x_{3}-\ {x_{2}}^{2}-\text{Gr}x_{4}+M\left(x_{2}-1\right)-1\right]\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\ \ \ \ \ \ \ \ \ \ \ \ (15)\nonumber \\
\end{equation}\begin{equation}
x_{5}^{{}^{\prime}}=\ -\Pr x_{1}x_{5}-\left(1+\ \frac{1}{\beta}\right)\text{PrEc}{x_{3}}^{2}\ -\text{MPrEc}\left(x_{2}-1\right)^{2}\text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(16)\nonumber \\
\end{equation}
subject to the following initial conditions,
\begin{equation}
x_{1}\left(0\right)=0,\ \ x_{2}\left(0\right)=\lambda,\ \ x_{5}\left(0\right)=\ \selectlanguage{greek}\text{βi}\selectlanguage{english}\left[\theta\left(0\right)-1\right],\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }(17)\nonumber \\
\end{equation}\begin{equation}
x_{2}\left(0\right)=s_{1},\ \ x_{5}\left(0\right)=\ s_{2}\text{\ \ }\ \text{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\nonumber \\
\end{equation}
Using the unknown initial conditions \(s_{1}\) and
\(s_{2}\) in Eq. (17), Eq. (15) and Eq. (16) are integrated
numerically. The accuracy of the assumed missing initial conditions was
checked by comparing the calculated value of the dependent variable at
the terminal point with its given value there. If differences exist,
improved values of the missing initial conditions are obtained and the
process repeated. The accuracy and robustness for solving the boundary
value problems have been repeatedly confirmed previously
\(\mathbf{[23]}\). From the process of numerical computation, the fluid
velocity\(f^{\prime}(\eta)\) and temperature \(\theta(\eta)\) are
compared with the given boundary conditions.
Results and Discussion
The numerical computations are carried out for the various values of the
physical parameter with Runge-Kutta Fehlberg integration scheme. The
effects of the varying physical parameters -- magnetic field
parameter\((M)\), Casson parameter \((\beta)\),
velocity ratio parameter\((\lambda)\), Grashof number
\((Gr)\), Biot number \((Bi)\), Eckert number
\((Ec)\) and Prandtl number \((Pr)\) on velocity
an temperature profiles has been analyzed. The obtained computation
results are presented graphically in Figures (1) -- (8) and discussed.
The effects of various values of magnetic field parameter
\(M\) on the flow field velocity and temperature profiles
are displayed in Figures (1) and (2). As \(M\) increases,
the flow field velocity decreases and also increases with decreasing
values in \(M\). Due to the Lorentz force induced by the
dual actions of electric and magnetic fields, the velocity boundary
layer thickness decreases. Similarly, for\(\lambda=0.2\), the
temperature profiles increases with increasing values of
\(M\). The obtained result is in agreement
with\(\mathbf{[}\mathbf{22}\mathbf{,26]}\).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image2/image2}
\end{center}
\end{figure}
\textbf{Figure 1} Velocity profiles for different values of
\(M\).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image3/image3}
\end{center}
\end{figure}
\textbf{Figure 2} Temperature profiles for different values of
\(M\).
Figure (3) and (4) presents the effects of Casson parameter
\(\beta\) on the velocity and temperature fields. It is
observed from Figure (3) that the fluid velocity profiles decreases as
\(\beta\) increases. Thus, due to the increase of Casson
parameter \(\beta\), the yield stress\(P_{y}\) reduces
and consequently, the velocity boundary layer thickness reduces. Figure
(4) shows the influences of Casson parameter on the temperature
profiles. It shows that temperature decreases with increasing values in
\(\beta\). This implies that thermal boundary layer decreases
\(\mathbf{[}\mathbf{22}\mathbf{]}\). Figure (5) presents the effects of temperature
profiles for varying values of Biot number Bi. It
depicts that increasing values ofBi, decreases in
temperature profiles. However, as the flow field moves far from the
sheet within the thermal boundary layer, Biot number Bi
varnishes.\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image4/image4}
\end{center}
\end{figure}
\textbf{Figure 3} Velocity profiles for various values of Casson
parameter \(\beta\)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image5/image5}
\end{center}
\end{figure}
\textbf{Figure 4} Temperature profiles for different values of
\(\beta\)\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image6/image6}
\end{center}
\end{figure}
\textbf{Figure 5} Temperature profiles for different values of Biot
number Bi
The velocity profile for various values of velocity ratio
parameter\(\lambda\) is shown in Figure (6). As described
by\(\mathbf{[26,28]}\) for Newtonian fluid, the velocity of fluid inside
the boundary layer decreases from the surface towards the edge of the
layer for the first kind \((\lambda<1)\) and the fluid velocity
increases from the surface towards the edge for the second
kind\((\lambda>1)\). Similarly, it is important to note that the
stretching velocity and straining velocity are equal as such, there is
no boundary layer of Casson fluid flow near the sheet\(\mathbf{[9]}\).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image7/image7}
\end{center}
\end{figure}
\textbf{Figure 6} Velocity profiles for varying values of
\(\lambda\)
The velocity profiles for different values of Grashof
numberGr is displayed in Figure 7. It revealed that the
flow field velocity decreases with increasing values of Grashof
numberGr thereby reducing the thermal boundary layer
along the sheet.
The viscous dissipation effect on temperature profiles is shown in
Figure 8. It illustrates that temperature increases wit increase in
Eckert number (viscous dissipation parameter Ec). The
Eckert number produces heat due to drag between the fluid particles
causing an increase of the initial fluid temperature due to the extra
heat. However, Ec may not only cause thermal reversal
but also increases the thermal boundary layer \(\mathbf{[22]}\).
Figure 9 shows the effects of Prandtl number of temperature profiles. It
depicts that temperature initially increases with increasing values of
Prandtl number \(\Pr\) and later decreases with increased
values of\(\Pr\) towards the thermal boundary layer. The use
of Prandtl number in heat transfer problems reduces the relative
thickening of the momentum and the thermal boundary layer
\(\mathbf{[22]}\). Thus, the rate of heat transfer is enhanced with
\(\Pr\) causing the reduction of the thermal boundary layer
thickness \(\mathbf{[22]}\).\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image8/image8}
\end{center}
\end{figure}
\textbf{Figure 7} Effects of various values of Gr on
velocity profiles\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image9/image9}
\end{center}
\end{figure}
\textbf{Figure 8} Temperature profiles for various values
ofEc\selectlanguage{english}
\begin{figure}[H]
\begin{center}
\includegraphics[width=0.70\columnwidth]{figures/image10/image10}
\end{center}
\end{figure}
\textbf{Figure 9} Effects of different values of \(\Pr\) on
temperature profiles
\subsection*{conclusion}
{\label{conclusion}}
The magnetohydrodynamic stagnation-point flows of a Casson fluid towards
a convectively heated stretching/shrinking sheet are investigated taking
the buoyancy force into consideration. Using the similarity variables,
the governing differential equations are transformed to ordinary
differential equations and solved numerically by shooting method with
Runge-Kutta Fehlberg integration scheme. The effects of the various
governing parameters were analyzed and the following remarks can be
summarized below.
\begin{enumerate}
\tightlist
\item
The velocity boundary layer thickness reduces with magnetic field
parameter.
\item
The flow field velocity decreases with increase in Casson
parameter\(\beta\) as well as the thermal boundary layer
thickness.
\item
The Biot number decreases the thermal boundary layer thickness whereas
the Eckert number increases it.
\end{enumerate}
\textbf{ACKNOWLEDGEMENTS}
Authors wish to acknowledge all reviewers of the manuscript for their
time in evaluating the content towards a better output.
\textbf{REFERENCES}\selectlanguage{english}
\begin{longtable}[]{@{}ll@{}}
\toprule
{[}1{]} & B. C. Sakiadis, ``Boundary-layer behaviour of continuous solid
surfaces: I. boundary-layer equations for two-dimensional and
axisymmetric flow,'' \emph{AIChE Journal}, vol. 7, no. 1, pp. 26--28,
1961. https://doi.org/10.1002/aic.690070108\tabularnewline
\midrule
\endhead
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\bottomrule
\end{longtable}
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