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\AtBeginDocument{{\noindent\small
International Conference on Applications of Mathematics to Nonlinear Sciences,\newline
\emph{Electronic Journal of Differential Equations},
Conference 24 (2017), pp. 85--101.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{8mm}}
\begin{document} \setcounter{page}{85}
\title[\hfilneg EJDE-2017/conf/24\hfil
Free surface dynamics]
{Free surface dynamics of thin MHD second-grade fluid over a
heated nonlinear stretching sheet}
\author[K. K. Patra, S. Panda, M. Sellier \hfil EJDE-2017/conf/24\hfilneg]
{Kiran Kumar Patra, Satyananda Panda, Mathieu Sellier}
\address{Kiran Kumar Patra \newline
Department of Mathematics,
National Institute of Technology Calicut,
NIT(P.O)-673601, Kerala, India}
\email{kirankumarpatra1984@gmail.com}
\address{Satyananda Panda (corresponding author)\newline
Department of Mathematics,
National Institute of Technology Calicut,
NIT(P.O)-673601, Kerala, India}
\email{satyanand@nitc.ac.in }
\address{Mathieu Sellier \newline
Department of Mechanical Engineering,
University of Canterbury, Private Bag 4800,
Christchurch 8140, New Zealand}
\email{mathieu.sellier@canterbury.ac.nz}
\thanks{Published November 15, 2017.}
\subjclass[2010]{76A05, 76A10, 76A20, 76M12, 80A20}
\keywords{Thin liquid film; heat transfer; second-grade fluid;
free surface flow;
\hfill\break\indent magnetic field; long-wave theory}
\begin{abstract}
This article presents a long-wave theory for the free surface dynamics
of magnetohydrodynamics (MHD) second-grade fluid over a non-uniform heated
flat elastic sheet. An evolution equation for the film thickness is derived
from the instationary Navier-Stokes equations using regular asymptotic
expansion with respect to the small aspect ratio of the flow domain.
The derived thin film equation is solved numerically using finite volume
method on a uniform grid system with implicit flux discretization.
The finding reveals the dependency of the thinning behavior of the fluid
film on the stretching speed and the non-Newtonian second-grade parameter.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\allowdisplaybreaks
\section{Introduction}\label{sec:sect1}
Many theoretical studies on the flow of fluid and heat transfer due to stretching
sheet in the context of polymer extrusion, continuous casting, drawing of
the plastic sheets, cable coating, etc., can be found in the literature, about, e.g.
Newtonian fluid
\cite{Andersson2000, Aziz2010, Liu2008,dandapat,vajravelu, Khan2011,salleh, Wang},
non-Newtonian fluid
\cite{andersson1996flow, hayat2007analytic,sajid2009unsteady, sajid2008influence} and
MHD effect \cite{hayat2008influence,ali,Noor2010} as well as derivation
of boundary layer equations. In general, such model reductions are based
on the uniform film thickness assumption
which enables the similarity transformation. Thereby the set of partial
differential equations are reduced to a more tractable one of ordinary
differential equations.
Recognizing the restrictions of the plane interface assumptions,
Dandapat and co-workers were the first to
extend the formulation to account for local deformation of the free surface
in \cite{dandapatZAMP2006, dandapatPoF2006}. The authors exploit the slenderness
of the flow domain to derive a long-wave approximation of
the Navier-Stokes equations and solve the resulting governing
equation using the matched asymptotic method. Lately, this
work was extended to include the heat transfer problem
\cite{ dandapat2011thin,santra2009}.
In this work, the nonuniform temperature distribution at
the stretching sheet induces an inhomogeneous temperature
field in the film. Consequently, a surface temperature gradient
develops at the film free surface. As a result of the surface
tension gradients, the film thickness varies along the flow, and
these deformations are advected in the stretching direction. Also,
a free surface model based on a long-wave theory for the thin film
dynamics of Casson fluid over a nonlinear
stretching sheet including magnetic effect has been
recently deduced in \cite{singh}.
This work focuses on the systematic derivation, in the spirit of
\cite{dandapat2011thin}, of the thin film equation for
a second-grade non-Newtonian MHD fluid over a heated steady stretching sheet without
the restriction of the plane interface assumption. One motivation for this study
is the flow of mucus in biological tissues which undergo expansion or contraction.
A particular example is pulmonary alveoli which are
covered with a lining of non-Newtonian fluid \cite{levy} and which undergo periodic
expansion and contraction.
This article is organized as follows. The mathematical model for the flow of
second-grade fluid is described in Sec.~\eqref{sec:sect2}.
The long-wave theory by using the standard expansion technique with
respect to a small aspect ratio of the
flow domain for model reduction is presented in Sec.~\eqref{sec:sect3}. The
thin film model equation is given in Sec.~\eqref{sec:sect4}.
The numerical procedure for the numerical solution of the thin
film equation is explained in Sec.~\eqref{sec:sect5}.
In Sec.~\eqref{sec:sect6} we
discuss the numerical investigation of the thin film
equation, and give some conclusions from this work.
\section{Physical model and problem formulation}\label{sec:sect2}
We consider here an unsteady planar fluid which lies over a heated
stretching sheet in the presence
of transverse magnetic field $\mathbf{B}_0$ as shown in
Figure \eqref{fig:flow_geometry}.
The elastic sheet lies at $z=0$, and
the liquid-gas interface lies at $z=h(x,t)$, where the $x$-axis is directed
along the stretching layer, and the
$z$-axis is normal to the sheet in the outward direction toward
the fluid. Gravity acts along the negative z-direction.
Further, the
surface at $z$ = 0 starts stretching from rest and within a very
short time attains the stretching velocity $u = U(x)$.
The evaporation and buoyancy effects are neglected considering
the liquid is non-volatile and thin.
The elastic surface is heated with non-uniform temperature $T_{s}$ a function of
$x$ alone, and the ambient gas phase is at constant temperature $T_{a} $.
A constant uniform magnetic field of strength $B_0$ is applied transversely
in the parallel direction to the $z$-axis. Since the sheet is nonconducting,
an uniform electric field is applied along
the z-direction to generate Lorentz force.
The fluid is assumed to be incompressible and non-Newtonian second-grade fluid with
constant viscosity $\mu$, density $\rho$, specific heat $c_{p}$, thermal
conductivity $k$ and the electric conductivity $\lambda$. The surface tension
of the liquid-gas interface decreases linearly with temperature $T$ according to
\begin{equation}\label{eq:st}
\sigma = \sigma_{a}\big(1-\gamma(T - T_{a})\big),
\end{equation}
where $\sigma_{a}$ is the surface tension at $T=T_{a}$, and
$\gamma=-\frac{1}{\sigma_a}\big(\frac{d\sigma}{dT}\big)_{T=T_a}$
a positive constant specific to the fluid.
\begin{figure}[hbt]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1} % pd.eps Prob_desc}
\end{center}
\caption{Sketch of flow geometry}
\label{fig:flow_geometry}
\end{figure}
Neglecting viscous dissipation and radiation effects,
the motion of the second-grade fluid due to the heated stretching sheet is
governed by the continuity equation of mass flow, the momentum equation
and the temperature equation.
\begin{itemize}
\item Continuity equation:
\begin{equation}\label{eq:contd}
\nabla \cdot \mathbf{V} = \mathbf{0};
\end{equation}
\item Momentum equation:
\begin{equation}\label{eq:momd}
\rho \frac{D\mathbf{V}}{Dt} = \nabla \cdot {\boldsymbol{\tau}}
+ \rho \mathbf{g} + \mathbf{F};
\end{equation}
\item Temperature equation:
\begin{equation}\label{eq:tempd}
\rho c_p \frac{DT}{Dt}=k\nabla^2T;
\end{equation}
\item Free surface boundary conditions i.e. at $z=h(x,t)$:
\begin{gather}\label{eq:bc2d}
p_{a} + \hat{\mathbf{n}}\cdot \boldsymbol{\tau}\cdot \hat{\mathbf{n}}
= -\sigma(\boldsymbol{\nabla}\cdot \hat{\mathbf{n}}), \\
\label{eq:bc3d}
\hat{\mathbf{n}}\cdot \boldsymbol{\tau} \cdot \hat{\mathbf{t}}
= \nabla \sigma \cdot \hat{\mathbf{t}}, \\
\label{eq:bc4d}
h_{t} + u h_{x} = w, \\
q_i = -k \nabla T \cdot \hat{\textbf{n}} = \alpha\left(T-T_{a}\right)
\end{gather}
\item Boundary conditions on the stretching sheet i.e. at $z=0$:
\begin{equation}\label{eq:bc1d}
u(x, 0, t) = U(x), \quad w(x, 0, t) = 0, \quad T(x, 0, t) = T_{s}(x);
\end{equation}
\item Initial condition:
\begin{equation}\label{eq:ic}
u(x,z,0) = w(x, z,0) = 0, \quad h(x,0) = h_0 +\delta(x)
\end{equation}
\end{itemize}
The symbols $\mathbf{V}(x,z,t)=(u(x,z,t), w(x,z,t))$,
${\boldsymbol{\tau}}$, $\mathbf{g}$ and $\mathbf{F}$
denote the fluid velocity at position $(x,z)$ and
time $t$, the Cauchy stress tensor, the acceleration due to gravity and the
Lorentz force.
The free surface boundary conditions are due to balance of stresses,
the kinematic condition and the convective heat flux at the interface.
Here, $\hat{\mathbf{n}}$ and $\hat{\mathbf{t}}$ are the unit normal and
tangential vectors on the surface, respectively. The $p_{a}$ stands for the
atmospheric pressure at the free surface, $\sigma$ is the surface tension of the
fluid, $q_i$ is the heat flux, $\alpha$ is the rate of heat transfer from
the liquid to the ambient gas phase and the symbol $\nabla$ stands for the
gradient operator. The subscripts $x$ and $t$ stand for the partial
differentiation with respect to $x$ and $t$ respectively. The symbol
$h_0$ is the characteristic height of the free surface and $\delta$
is the initial small disturbance from $h_0$.
The \emph{Cauchy-stress tensor}, given by Rivlin and Ericksen \cite{rivlin}
for a second-grade fluid can be written as
\begin{equation}\label{eq:cst}
\mathbf{\boldsymbol{\tau}} = - p\mathbf{I} + \mu \mathbf{A}_1
+ \alpha_1 \mathbf{A}_2
+ \alpha_{2} \mathbf{A}_1^{2},
\end{equation}
where $p$ is the pressure, $\mathbf{I}$ is the identity tensor.
The material constants $\alpha_1$ and $\alpha_{2}$ are the first and
second normal stress coefficients.
The quantities $\mathbf{A}_1$ and $\mathbf{A}_2$ are the first two
Rivlin-Ericksen tensors and they are defined as
\begin{equation} %\label{eq:r-e-tensor}
\mathbf{A}_1 = \left(\nabla \mathbf{V}\right)
+ \left(\nabla \mathbf{V}\right)^{*}, \quad
\mathbf{A}_2 = \frac{D}{Dt} \mathbf{A}_1
+ \mathbf{A}_1 \cdot \left(\nabla \mathbf{V}\right) +
\left({\nabla \mathbf{V}}\right)^{*} \cdot \mathbf{A}_1\,,
\label{eq:an}
\end{equation}
where $D/Dt$ is the material time derivative, and the superscript $(*)$
is used for the transpose.
The constitutive model \eqref{eq:cst} is derived by considering second order
approximation of retardation parameter. Dunn and Fosdick~\cite{dunn}
have shown that this model equation is invariant under transformation
and therefore the material constants must meet the following
restriction
\begin{equation}\label{eq:cstres}
\mu \ge 0, \quad \alpha_1 \ge 0, \quad \alpha_1 + \alpha_{2} = 0.
\end{equation}
Fluids characterized by these restrictions \eqref{eq:cstres} are
called second-grade fluid. The fluid model represented by \eqref{eq:cst}
with the relationship \eqref{eq:cstres} is compatible with the hydrodynamics.
The third relations of \eqref{eq:cstres} is the
consequence of satisfying the Clausis-Duhem inequality by fluid motion and
a second relation arises due to
the assumptions that specific Helmholtz free energy of the fluid takes its
minimum value in equilibrium.
The fluid satisfying model \eqref{eq:cst} with $\alpha_i < 0 $;
($\alpha_i = 1, 2$) is termed as
second-order fluid and with $\alpha_i > 0$ is termed as second-grade fluid.
Although second-order fluid is obeying model
\eqref{eq:cst} with $\alpha_1< \alpha_{2}, \alpha_1 <0$,
exhibits some undesirable instability characteristic
(Fosdick and Rajagopal~\cite{fosdicketal}).
The second order approximation is valid at low shear rate
(Dunn and Rajagopal~\cite{dunnetal}).
The fluid is flowing under the environment of uniform transverse magnetic field,
therefore momentum of the fluid is influenced by the Lorentz force
$\mathbf{F}=\lambda(\textbf{E}+\mathbf{V}\times\mathbf{B}_0)\times\mathbf{B}_0$,
where $\textbf{E}+\mathbf{V}\times\mathbf{B}_0$ represents the total current
density with magnetic Reynolds number
$R_m<<1$. The electric field $\textbf{E}=\mathbf{0}$, as there is no
electric current present during the fluid flow.
The term $\mathbf{V}\times\mathbf{B}_0$ is the potential difference across
the fluid.
Using basic vector calculus the expression for Lorentz force can be
calculated as follows
\begin{equation}\label{eq:efd}
\mathbf{F}=\lambda(\mathbf{V}\times\mathbf{B}_0)\times\mathbf{B}_0
= -\lambda[\mathbf{V}(\mathbf{B}_0\cdot\mathbf{B}_0)
-\mathbf{B}_0(\mathbf{V}\cdot\mathbf{B}_0)]=-\lambda\left(uB_0^2,0,0\right)
\end{equation}
Scaling the film thickness with the characteristic height of the flow
($h = h_0\tilde{h}$, $\delta = h_0 \tilde{\delta}$), the coordinates
by the characteristic length of the domain
$(x,z)=L(\tilde{x}, \epsilon \tilde{z})$ and the
velocity $(u, w) = (\nu/h_0\tilde{u}$, $\epsilon \nu/h_0 \tilde{w})$,
$U= (\nu/h_0)\tilde{U}$, the time $t=(h_0^2/\epsilon \nu)\tilde{t}$, the pressure
with $p = p_{a} + (\rho \nu^2/\epsilon h_0^2)\tilde{p}$ and the temperature
$T = T_{a} + \left(T_{s_0} - T_{a}\right)\tilde{T}$,
where $\epsilon = h_0/L$ is the aspect ratio, $\nu = \mu/\rho$ is the
kinematic viscosity of the fluid, $T_{s_0}$ is the temperature of the sheet at
the origin, and using the constitutive
relation \eqref{eq:cst} with Eqs.~\eqref{eq:r-e-tensor}, \eqref{eq:cstres}, and
the external force \eqref{eq:efd},
the non-dimensional form of the governing
\eqref{eq:contd}-\eqref{eq:ic}, after dropping the tilde symbol in
explicit form are:
\begin{gather}\label{eq:contnd}
u_x+w_z=0 \\
\label{eq:momndx}
\begin{aligned}
&\epsilon \left(u_t+uu_x+wu_z\right) \\
&= -p_x+\epsilon^2 u_{xx}+u_{zz}+
\mathrm{K}\bigl(\epsilon^3 u_{xxt}+ \epsilon u_{zzt} +
\epsilon^3uu_{xxx} - \epsilon u w_{zzz} \\
&\quad + \epsilon^3 u_x u_{xx} -\epsilon u_xu_{zz} +\epsilon^3wu_{xxz}
+\epsilon wu_{zzz} - 4\epsilon^3w_xw_{zz}-2\epsilon^5w_xw_{xx} \\
&\quad + \epsilon^3 u_z w_{xx} -\epsilon u_zw_{zz} +2\epsilon w_zu_{zz}\bigr)
-\epsilon M^2 u\,,
\end{aligned} \\
\label{eq:momndz}
\begin{aligned}
&\epsilon^3 \left(w_t+uw_x+ww_z\right) \\
&= -p_z+\epsilon^4w_{xx}+\epsilon^2w_{zz}+
\mathrm{K}\epsilon\bigl(\epsilon^4w_{xxt}+\epsilon^2w_{zzt}
+ \epsilon^2 \,u \,w_{xzz} \\
&\quad + \epsilon^4\, u\, w_{xxx} + 2\epsilon^4\,u_x\,w_{xx}
+\epsilon^2\,w_x\,u_{zz}- \epsilon^4 \,w_x\, u_{xx} + \epsilon^2\, w\, w_{zzz} \\
&\quad - \epsilon^4wu_{xxx}+\epsilon^2w_zw_{zz}-\epsilon^4w_zw_{xx}
-4\epsilon^2u_zu_{xx} -2u_zu_{zz}\bigr)-\epsilon \mathrm{Fr},
\end{aligned} \\
\label{eq:tempnd}
\epsilon Pr \left(T_t+uT_x+wT_z\right) =\epsilon^2T_{xx}+T_{zz}, \\
\label{eq:bc2nd}
\begin{aligned}
&\epsilon S h_{xx} \left(1- M_w Ca T\right) (\epsilon^2 h_x^2 + 1)^{-1/2}\\
& = - (\epsilon^2 h_x^2 + 1)p + 2\epsilon^2 \bigl(\epsilon^2 h_x^2 u_x
- \epsilon^2 h_x w_x - h_xu_z+w_z\bigr) \\
&\quad + K\epsilon^3\bigl(2\epsilon^2u_{tx}h_{x}^2 -2\epsilon^2h_xw_{tx}-
2h_xu_{tz}+2w_{tz}\bigr)\\
&\quad + K\epsilon\Bigl(\epsilon^2 h_x^2\bigl(2\epsilon^2uu_{xx}
+ 2\epsilon^2\,wu_{xz} + u_z^2-\epsilon^4\,w_x^2\bigr)
- 2\epsilon^2\,h_x\bigl(\epsilon^2\,uw_{xx} +uu_{xz} \\
&\quad + \epsilon^2 w w_{xz}+ w u_{zz} + \epsilon^2\,u_xw_x
- \epsilon^2w_xw_z + u_zw_z-u_xu_z\bigr) \\
&\quad + 2\epsilon^2uw_{xz}+2\epsilon^2ww_{zz}+\epsilon^4 w_x^2- u_z^2\Bigr),
\end{aligned} \\
\label{eq:bc3nd}
\begin{aligned}
&-\epsilon M_w\left(T_x+h_xT_z\right)\left(1+\epsilon^2h_x\right)^{1/2} \\
&=(\epsilon^2w_x+u_z)(1-\epsilon^2h_x^2)
+ 2\epsilon^2 h_x(w_z-u_x) + K\bigl((1-\epsilon^2h_x^2)(\epsilon^3w_{tx}
+\epsilon u_{tz}) \\
&\quad +2\epsilon^3h_x(w_{tz}-u_{tx})\bigr)
+K\Bigl((1-\epsilon^2h_x^2) \bigl(\epsilon^3uw_{xx}+ \epsilon uu_{xz}
+ \epsilon^3ww_{xz} \\
&\quad + \epsilon wu_{zz}+ \epsilon^3u_xw_x
-\epsilon^3w_xw_z +\epsilon u_zw_z-\epsilon u_xu_z\bigr)
+2\epsilon h_x(\epsilon^2uw_{xz} \\
&\quad +\epsilon^2ww_{zz} - \epsilon^2uu_{xx}-\epsilon^2wu_{xz}
+ \epsilon^4w_x^2-u_z^2)\Bigr),
\end{aligned}\\
\label{eq:bc4nd}
h_t = w - u h_x, \\
\label{eq:bc5nd}
T_z - \epsilon^2h_xT_x=-B\left(1+\epsilon^2h_x^2\right)^{1/2} T, \\
\label{eq:bcnd1}
\text{and at }z=0: \quad u=U(x), \quad w=0, \; T=\theta(x),\quad
\text{where } \theta(x)=\frac{T_{s}-T_{a}}{T_{s_0}-T_{a}}
\end{gather}
The non-dimensional parameters are the second grade parameter
$K=\alpha_1 / \rho h_0^2$, the Hartmann number
$M=\sqrt{\lambda B_0^2h_0L/\rho\nu}$,
the Froude number $Fr=gh_0^3/\nu^2$, and the
Prandtl number $Pr=\rho c_p\nu/k$.
The symbol $S$ stands for the non-dimensional surface tension parameter
defined as $S=\epsilon^2 \sigma_a h_0 / \rho \nu^2$.
Here $Ca=\epsilon^2/S$, $B=\alpha h_0/k$ and
$M_w=\sigma_a h_0\gamma\left(T_{S_0}-T_a\right)/\rho\nu^2$
are the Capillary number, Biot number, and the effective Marangoni number,
respectively. Equations \eqref{eq:bc2nd} and \eqref{eq:bc3nd} are obtained
after using the expression for unit normal vector
$\hat{\textbf{n}}=\big(-h_{x}/\sqrt{1+h_{x}^2}, \;1/\sqrt{1+h_{x}^2}\big)$ and
the unit tangent vector
$\hat{\textbf{t}}=\big(1/\sqrt{1+h_{x}^2}, \;h_{x}/\sqrt{1+h_{x}^2}\big)$.
The initial conditions read:
\begin{equation}\label{eq:icnd}
u = 0, \quad w =0, \quad h(x, 0) = 1 + \delta(x).
\end{equation}
\section{Long-wave approximation}\label{sec:sect3}
The derivation of the one-dimensional thin film equation is based on long wave theory.
The stated asymptotic analysis uses the
techniques of \cite{dandapat2011thin} but extends it to incorporate the
complex rheological effects.
For the underlying velocity and pressure variables, regular power series
expansion in $\epsilon$ are set up.
To obtain the equation of the thin film from the above problem, we expand
the variables as follows:
\begin{equation}\label{eq:as1}
(u,w,p)=(u_0,w_0,p_0)+\epsilon(u_1,w_1,p_1)+\epsilon^2(u_2,w_2,p_2)+\dots
\end{equation}
Then the temperature variable $T$ is expanded as
\begin{equation}\label{eq:as2}
T=T_0+\epsilon^{1-n}T_1+\dots
\end{equation}
where $0