#### General solution for MHD-free convection flow over a vertical plate with ramped wall temperature and chemical reaction

Arab. J. Math.
General solution for MHD-free convection flow over a vertical plate with ramped wall temperature and chemical reaction
Nehad Ali Shah 0 1
Azhar Ali Zafar 0 1
Shehraz Akhtar 0 1
Mathematics Subject Classification 0 1
0 A. A. Zafar Department of Mathematics, GC University , Lahore 54600 , Pakistan
1 N. A. Shah (
The aim of the article is to study the unsteady magnetohydrodynamic-free convection flow of an electrically conducting incompressible viscous fluid over an infinite vertical plate with ramped temperature and constant concentration. The motion of the plate is a rectilinear translation with an arbitrary time-dependent velocity. Closed-form solutions for the temperature, concentration and velocity fields of the fluid are obtained. The influence of transverse magnetic field that is fixed relative either to fluid or plate is studied. Furthermore, the effects of system parameters on the fluid velocity are analyzed through numerical simulations and graphical illustrations.
1 Introduction
Magnetohydrodynamics (MHD) boundary layer flows of an electrically conducting fluid with heat and mass
transfer have applications in engineering, geophysical environments, chemical engineering, etc. Several flow
problems involving the heat and mass transfer with various mechanical, thermal, and species concentration
boundary conditions have been studied. For example, Singh and Kumar [15] studied the fluctuating heat and
mass transfer on unsteady MHD-free convection flow of a radiating and reacting fluid with slip on a porous
plate by considering periodic functions for the suction velocity, temperature, and concentration at the wall.
The unsteady-free convection flow of a Newtonian fluid past an impulsively started infinite vertical plate in
a porous medium with ramped wall temperature, ramped wall concentration, and ramped plate velocity has
been recently studied by Ahmed and Dutta [1].
Ghara et al. [5] have studied the MHD-free convection flow past an impulsively moving vertical plate with
ramped wall temperature. Seth et al. [14] studied Hall and Soret effects on unsteady MHD-free convection flow
of radiating and chemically reactive fluid past a moving vertical plate with ramped temperature in a rotating
system. The unsteady MHD-free convection flow past a porous plate under oscillatory suction velocity was
analyzed by Reddy [10]. MHD-free convection flow of a second grade fluid was studied by Samiulhaq et. al.
in [11]. Narahari and Debnath [8] have studied the unsteady MHD-free convection flow past an accelerated
vertical plate with constant heat flux and heat source. They considered two cases, namely, the magnetic lines of
force held fixed relative to the fluid or held fixed relative to the plate. For more interesting and related results,
see [6, 12, 13, 16] and the references therein.
In the present paper, the unsteady MHD-free convection flow near a vertical plate is considered. The plate
has a translational motion in its plane with an arbitrary time-dependent velocity. The wall temperature varies
as a ramped law and the wall concentration is constant. The heat generation or absorption and the chemical
reaction are also considered. The fluid is electrically conducting and regarding the applied magnetic field two
cases are considered, namely, when the magnetic lines of force are held fixed to the fluid or to the plate [8].
The difference between fluid velocities in the two cases is studied and some properties are highlighted. The
governing linear partial differential equations are written into a non-dimensional form and solved by means
of the Laplace transform method. The influence of the system parameters (e.g., magnetic parameter, Grashof
numbers, chemical reaction, and heat absorption parameters) on the fluid velocity are also analyzed through
graphical illustrations.
2 Mathematical formulation of the problem
Consider the unsteady-free convection flow of an incompressible viscous, electrically conducting fluid. The
fluid is near an infinite vertical plate with ramped temperature and constant concentration. The motion of the
plate is a rectilinear translation with an arbitrary time-dependent velocity. We introduce a coordinate system
with x -axis along the plate in vertical upward direction and y-axis normal to the plate. A uniform transverse
magnetic field of strength Bo is applied. Initially, at time t = 0, the plate and the fluid are at rest with the same
temperature T∞ and species concentration C∞.
After time t = 0+, the plate moves with the velocity Uo f (t ) in its own plane along the x -axis. Here, U0
is a constant velocity and f (·) is a dimensionless piecewise continuous function, whose value f (0) = 0. Heat
is supplied to the plate as a time-ramped function in the presence of heat source and chemical reaction. The
species concentration at the plate is constant Cw. We further assume that:
1o. The magnetic Reynolds number is small, so that the induced magnetic field is negligible in comparison
with the applied magnetic field.
2o. Viscous dissipation, radiative effects, and Joule heating terms are neglected in the energy equation (usually
in free convection flows the velocity has small values). However, according to Magyari and Pantokratoras
[7], the radiative effects can be easily included by a simple rescaling of the Prandal number.
3o. No external electric field is applied and the effect of polarization of ionized fluid is negligible; therefore,
electric field is assume to be zero.
4o. There exits a first-order chemical reaction between the fluid and species concentration. The level of species
concentration is very low, so that the heat generated by chemical reaction can be neglected.
Since the plate is infinite extended in x and z directions, therefore, all physical quantities are functions of
the spatial coordinate y and time t only.
Under usual Boussinesq’s approximation, the flow is governed by the following system of equations [8, 15]:
∂ u(y, t )
∂ t
= ν
∂ 2u(y, t )
∂ y2
+ gβT (T (y, t ) − T∞) + gβC (C (y, t ) − C∞) −
ρ
σ B2
o u(y, t ),
ρC p
∂C (y, t )
∂t
= Dm
∂2C (y, t )
Into above equations, u(y, t ), T (y, t ), and C (y, t ) are velocity, temperature, and species concentration of the
fluid, respectively, ν is the kinematic viscosity, g is the acceleration due to gravity, βT is the thermal expansion
coefficient, ρ is fluid density, βC is the volumetric coefficient of expansion with species concentration, σ is
electrical conductivity, C p is the specific heat at constant pressure, k is the thermal conductivity, Q is the
heat generation or absorption coefficient, Dm is the chemical molecular diffusivity, and R is chemical reaction
parameter.
Equation (
1
) is valid when the magnetic lines of force are fixed relative to the fluid. If the magnetic field
is fixed relative to the plate, the momentum Eq. (
1
) is replaced by [8,17]
∂u(y, t )
∂t
= ν
∂2u(y, t )
Equations (
1
) and (
4
) can be combined as
∂u(y, t )
∂t
= ν
∂2u(y, t )
where
ε =
0, if Bo is fixed relative to the fluid;
1, if Bo is fixed relative to the plate.
The appropriate initial and boundary conditions are
Introducing the dimensionless variables
u(y, 0) = 0, T (y, 0) = T∞, CT∞(y,+0)(T=w C−∞T,∞)ytto ≥, 00, < t ≤ to; , C (0, t ) = Cw,
u(0, t ) = Uo f (t ), T (0, t ) = Tw, t > to.
u(y, t ) < ∞, T (y, t ) → T∞, C (y, t ) → C∞
as y → ∞.
y∗ = √yνto , t ∗ = tto , u∗ = Uuo , T ∗ = TTw−−TT∞∞ , C ∗ = CCw−−CC∞∞ ,
Gr =
Pr =
gβT to(Tw − T )
∞ ,
Uo
Gm =
gβC to(Tw − T )
∞ ,
Uo
μC p
k
,
Qto ν
Q∗ = ρC p , Sc = Dm ,
M = √νto Bo
σ
μ
and dropping out the star notation, we get
∂u(y, t )
∂t
=
∂2u(y, t )
+ Gr T (y, t ) + Gm C (y, t ) − M 2(u(y, t ) − ε f (t )),
− QT (y, t ),
− RC (y, t ),
∂t ∂ y2
u(y, 0) = 0, T (y, 0) = 0, C (y, 0) = 0,
u(0, t ) = f (t ), T (0, t ) =
t, 0 ≤ t ≤ 1; = t H (t ) − (t − 1)H (t − 1), C (0, t ) = 1,
1, t > 1.
u(y, t ) < ∞, T (y, t ) → 0, C (y, t ) → 0 as y → ∞.
(
3
)
(
4
)
(
5
)
(
6
)
(
7
)
(
8
)
(
9
)
(
10
)
(
11
)
(
12
)
(13)
(14)
(15)
Into above relations, Gr , Gm , M , Pr , Q, R, and Sc denote thermal Grashof number, Grashof number,
Hartman number, Prandtl number, the dimensionless heat generation or absorption coefficient, the dimensionless
chemical reaction parameter, and the Schmidt number, respectively, whereas
H (t ) =
0, t ≤ 0;
1, t > 0
is the Heaviside unit step function.
3 Solution of the problem
In the following, the solution of partial differential equations (
10
)–(
12
) with initial and boundary conditions
(13)–(15) will be determined by Laplace transform. The momentum equation depends upon energy and
concentration equation; therefore, we shall first establish the exact solutions for temperature and concentration.
3.1 Temperature field
In a recent paper, Ghara et al. [5] studied a similar MHD-free convection flow with thermal radiation and
ramped wall temperature but without heat source. Due to their assumption regarding the relative heat flux,
the dimensionless form of the energy equation is identical to our Eq. (
11
) with the radiation parameter Ra
instead of Q. As the corresponding initial and boundary conditions are also identical to our conditions (14)2
and (15)2, we directly present the temperature field:
T (y, t ) = H (t ) (y, t ; Pr, Q) − H (t − 1) (y, t − 1; Pr, Q),
and its Laplace transform [2]
to be used later for velocity. The function
(y, t ; a, b), defined by
T (y, s) =
s2
1 − e−s e−y√Pr(s+Q),
(y, t ; a, b) =
2t √b − y√a
4√b
+
2t √b4√+by a ey√aber f c
√
e−y√aber f c
y√a
2 t −
√
y√a
2 t +
√
√
bt
√
bt ,
is identical to that of Ghara et al. [5, Eq. (21)] for Q = Ra.
3.2 Species concentration Applying the Laplace transform to Eq. (12) and using the corresponding initial and boundary conditions, we find that
∂2C (y, s)
∂ y2
− Sc(s + R)C (y, s) = 0;
1
C (0, s) = s , C (y, s) → 0 as y → 0,
where C (y, s) is the Laplace transform of C (y, t ). The solution of the differential equation (19)1 with the
adjoining boundary conditions is
C (y, s) = 1s e−y√Sc√s+R .
By now applying the inverse Laplace transform to Eq. (20), we can easily obtain the concentration
distribution and is known in the literature.
(16)
(17)
(18)
(19)
(20)
3.3 Calculation of fluid velocity
Applying the Laplace transform to Eq. (
10
), and using the initial and boundary conditions (13)1, (14)1 and
(15)1 as well as Eqs. (17), (20), we find that
∂2u(y, s)
∂ y2
− (s + M2)u(y, s) = −Gr
s2
1 − e−s e−y√Pr√s+Q
− Gm 1 e−y√Sc√s+R − ε M2 F(s),
s
u(0, s) = F(s), u(y, s) < ∞
as y → ∞,
where u(y, s) and F(s) are the Laplace transforms of u(y, t) and f (t), respectively. The solution of the problem
(21), (22), is
√
u(y, s) = F(s)e−y s+M2 + ε M2 F(s)
√
1 − e−y s+M2
s + M2
− m
,
e−y√Pr√s+Q
s2
(y, t; a, b, c) = L−1 e−y√c√s+b
s − a
eat
e−y√c√a+ber f c
+ ey√c√a+ber f c
y√c
2√t −
y√c
2√t +
(a + b)t
(a + b)t
.
(21)
(22)
(23)
(24)
(25)
(26)
(27)
(28)
(29)
where
where
Applying the inverse Laplace transforms to Eq. (23), we find that
m =
Q Pr − M2
Pr − 1
, n =
R Sc − M2
Sc − 1
for Pr = 1 and Sc = 1.
u(y, t) = um (y, t) + uT (y, t) + uC (y, t),
y t f (t − τ ) e − 4yτ2 −M2τ t
um (y, t) = 2√π 0 τ √τ dτ + ε M2 0
Gr (Pr − 1)
uT (y, t) = (Q Pr − M2) {H (t)W (y, t) − H (t − 1)W (y, t − 1)},
Gm
uC (y, t) = − R Sc − M2 { (y, t; 0, R, Sc) −
+ (y, t; −n, R, 1)}.
denotes that the mechanical, thermal, and concentration components of velocity.
Into above relations
+ (y, t; −m, Q, Pr ) −
(y, t; −m, M2, 1),
W (y, t) = m (y, t; Pr, Q) − m (y, t; 1, M2) −
(y, t; 0, Q, Pr ) +
(y, t; 0, M2, 1)
where as the functions (y, t; a, b, c) as defined in Appendix (A.3) of [4]
f (t − τ )e−M2τ er f
y
√
2 τ
dτ,
(y, t; 0, R, 1) −
(y, t; −n, R, Sc)
(30)
Moreover, to compare the fluid velocities corresponding to the two ways of implementing magnetic field, we
note that the two velocities differ through the term:
Now, it is important to highlight the following aspects:
(
1
) If f (·) is a piecewise continuous and positive function, V (y, t ) has positive values. Consequently, for the
magnetic field is fixed relative to the plate, the fluid velocity is larger than the fluid velocity corresponding
to the case when the magnetic field is fixed to the fluid. An opposite trend appears if the function f (·) is
negative.
(
2
) The term V (y, t ) is an increasing function of the spatial variable y. Therefore, the fluid velocity is
significantly changed in the neighborhood of the plate if the magnetic field is fixed relative to the plate.
V (y, t ) = M 2
Equation (25) seems to not satisfy the boundary condition (14)1. To eliminate this drawback, we rewrite (25)
in the equivalent form:
lim u(y, t ) = L −1
y→∞
= L −1
ε M 2
F (s)
s + M 2
lim u(y, s)
y→∞
t
0
= ε M 2
f (t − τ )e−M2τ dτ = h(t ) =
0,
M 2 t
0 f (t − τ )e−M2τ dτ,
ε = 0
ε = 1
Consequently, unlike the case when the magnetic field is fixed to the fluid, the fluid at infinity does not
remain at rest if the magnetic field is fixed relative to the plate.
4 Special cases
As thermal and concentration components of velocity do not depend on the plate motion. However, heat and
mass transfer can influence the fluid motion and we have to know if their influence is significant or it can be
neglected in some motions with possible engineering applications.
4.1 Motion of the plate with constant velocity ( f (t ) = H (t ))
We take f (t ) = H (t ) (the Heaviside unit step function) into Eq. (25). After lengthy but straightforward
computation, the mechanical component take the form:
um (y, t ) = (1 − ε) (y, t ; 0, M 2, 1) + ε 1 − e−M2t er f
As it was to be expected the corresponding results are identical to those obtained by Narahari and Debnath [8,
Eq. (11a) with ao = 0] and Tokis [17, Eq. (
12
)].
where
4.2 Motion of the plate with exponential acceleration ( f (t ) = H (t )ebt )
Introducing f (t ) = H (t )ebt into Eq. (25), we find that
um (y, t ) =
ε M 2
1 − b + M 2
ε M 2
(y, t ; b, M 2, 1) + b + M 2
ebt − e−M2t er f
,
The results are identical to those obtained by Pattnaik et al. [9, Eq. (13)] with a = b, λ = M 2, K1p = 0 and
γ = 0 in the absence of thermal and concentration effects and for the case when magnetic field is held fixed
relative to the fluid.
Indeed, assigning to f (·) suitable forms, we can determine exact solutions for any motion with technical
relevance of this type.
5 Numerical results and discussion
The unsteady-free convection flow of an incompressible viscous, electrically conducting fluid is studied.
The fluid is near an infinite vertical plate with ramped temperature and constant concentration. The motion
(33)
(34)
of the plate is a rectilinear translation with an arbitrary time-dependent velocity. Closed-form solutions for
the dimensionless temperature, concentration, and velocity fields of the fluid are obtained. The influence of
transverse magnetic field that is fixed relative to fluid or plate is studied. It is important to note that the influence
of the flow parameters is significantly different on fluid flow, depending on how the magnetic lines of force
are fixed relative to the fluid or the plate. Finally, two particular cases for the translational motion of the plate,
namely, the translation with constant velocity, respectively, exponential accelerated motion are considered.
Numerical results for velocity have been computed for several values of magnetic parameter M , Grashof
number Gr , mass Grashof number Gm , chemical reaction parameter R, and heat source parameter Q.
The velocity profiles versus the spatial variable y at time t = 15 and for constant plate velocity ( f (t ) =
H (t )) are shown in Figs. 1, 2, 3, 4, and 5. The velocity proflies u(y, t ), are plotted when the magnetic field is
being fixed to the fluid (ε = 0) and to the moving plate (ε = 1).
The influence of magnetic field M on the velocity profiles is presented in Fig. 1. It is known that, under the
influence of a magnetic field on an electrically conducting fluid, a resistive force arises (so called the Lorentz
force). This force has tendency to slow down the fluid motion in the boundary layer. It can be seen from Fig. 1
that the fluid velocity decreases as the magnetic field M increases. In addition, it is noted that if the magnetic
field is fixed to the fluid (ε = 0), the values of the fluid velocity are lower than in the case when the magnetic
field is fixed to the plate (ε = 1).
The effects of the thermal Grashof number Gr and the mass Grashof number Gm on the fluid velocity
are shown in Figs. 2 and 3. For both parameters, velocity has a maximum value in the vicinity of the plate
and tends towards the value h(t ) for large values of the spatial coordinate y. It is further observed that the
values of the fluid velocity increase with increasing values of Gr and Gm . The influence of the dimensionless
chemical reaction parameter R on the fluid velocity is presented in Fig. 4. and it is observed that the fluid
velocity decreases with increasing values of the parameter R.
In Fig. 5, we have plotted the velocity profiles versus y for two values of the heat absorption parameter Q.
It can be seen from these curves that the velocity decreases with increasing of the values of parameter Q.
In Fig. 6, we have plotted velocities um (y, t ), um (y, t ) + uC (y, t ) and um (y, t ) + uC (y, t ) + uT (y, t )
versus y to investigate the contributions of mechanical, thermal, and concentration components of velocity on
the fluid motion. It is observed that contributions of mechanical, thermal, and concentration components of
velocity on the fluid motion are significant and they cannot be neglected.
6 Conclusions
The unsteady-free convection flow of an incompressible viscous and electrically conducting fluid is studied.
The fluid is near an infinite vertical plate with ramped temperature and constant concentration. Closed-form
solutions for dimensionless temperature, concentration, and velocity field of the fluid are obtained. The
influence of transverse magnetic field that is held fixed relative to fluid or to the plate is studied.
The motion of the plate is a rectilinear translation with an arbitrary time-dependent velocity f (·); assigning
to this f (·) suitable forms, we can determine exact solutions for any motion with technical relevance of this
type. The plate temperature changes as a time-ramped function and the concentration on the plate is constant.
The resulting coupled partial differential equations are written in the non-dimensional form and are solved by
means of the Laplace transforms.
Through graphical illustrations for the case when plate is moving with uniform velocity, the influence of
magnetic field and of the system parameters on the fluid velocity is brought to light. Some useful conclusions
of the study are as under:
• If the magnetic field is fixed relative to the moving plate, the fluid velocity differs significantly from the
case when the magnetic field is fixed relative to the fluid.
• At large distances from the plate, the fluid will be at rest when the magnetic field is fixed relative to the
fluid, but it attains a finite non zero value if the magnetic field is fixed relative to the plate.
• For increasing values of the Hartmann number, the fluid velocity decreases; therefore, stronger magnetic
field leads to slower flows.
• The fluid velocity increases with increasing values of the thermal Grashof and mass Grashof number and
decreases with respect to the increasing values of chemical reaction and heat absorption parameters R and
Q.
• Contributions of mechanical, thermal, and concentration components of velocity on the fluid motion are
significant and they cannot be neglected.
Acknowledgements The authors are thankful to Abdus Salam School of Mathematical Sciences, GC University Lahore and
Higher Education Commission of Pakistan for facilitating the research work.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://
creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided
you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate
if changes were made.
13. Seth, G.S.; Ansari, M.S.; Nandkeolyar, R.: MHD natural convection flow with radiative heat transfer past an inpulsively
moving plate with ramped wall temperature. Heat Mass Transf. 47, 551–561 (2011)
14. Seth, G.S.; Kumbhakar, B.; Sarkar, S.: Soret and Hall effects on unsteady MHD free convection flow of radiating and
chemically reactive fluid past a moving vertical plate with ramped temperature in rotating system. Int. J. Eng. Sci. Technol.
7(
2
), 94–108 (2015)
15. Singh, K.D.; Kumar, R.: Fluctuating heat and mass transfer on unsteady MHD free convection flow of radiating and reacting
fluid past a vertical porous considered plate in slip-flow regime. J. Appl. Fluid Mech. 4(
4
), 101–106 (2011)
16. Srinivasacharya, D.; Surender, O.: Non-Darcy natural convection from a vertical plate with a uniform wall temperature and
concentration in a doubly stratified porous medium. J. Appl. Mech. Tech. Phys. 56(
4
), 590–600 (2015)
17. Tokis, J.N.: A class of exect solutions of the unsteady magnitohydrodynamic free convection flows. Astrophys. Space Sci.
112, 413–422 (1985)
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
1. Ahmed , N.; Dutta , M.: Transient mass transfer flow past an impulsively started infinite vertical plate with ramped plate velocity and ramped temperature . Int. J. Phys. Sci . 8 ( 7 ), 254 - 263 ( 2013 )
2. Fetecau , C. ; Rana , M. ; Fetecau , C. : Radiative and porous effects on free convection flow near a vertical plate that applies shear stress to the fluid . Z. Naturforsch. A 68 ( 1-2 ), 130 - 138 ( 2013 )
3. Fetecau , C. ; Fetecau , C. ; Nigar , N. : Magnetohydrodynamic natural convection flow with Newtonian heating and mass diffusion over an infinte plate that applies shear stress to a viscous fluid . Z. Naturfororsch. A 69 ( 12 ), 714 - 724 ( 2014 )
4. Fetecau , C. ; Vieru , D. ; Fetecau , C. ; Pop , I. : Slip effects on the unsteady radiation MHD free convection flow over a moving plate with mass diffusion and heat source . Eur. Phys. J. Plus . 130 , 6 ( 2015 ). doi: 10 .1140/epjp/i2015-15006-8
5. Ghara , N. ; Das , S. ; Maji , S.L. ; Jana , R.N. : Effect of radiation on MHD free convection flow past an impulsively moving vertical plate with ramped wall temperature . Am. J. Sci. Ind. Res . 3 ( 6 ), 376 - 386 ( 2012 )
6. Khalid , A. ; Khan , I. ; Khan , A. ; Shafie , S. : Unsteady MHD free convection flow of Casson fluid past over an oscillating vertical plate embedded in a porous medium . Eng. Sci. Technol . Int. J. 18 , 309 - 317 ( 2015 )
7. Magyari , E.; Pantokratoras , A. : Note on the effect of thermal radiation in the linearized Rosseland approximation on the heat transfer characteristics of various boundary layer flow . Int. Commun. Heat Mass Transf . 38 , 554 - 556 ( 2011 )
8. Narahari , M. ; Debnath , L. : Unsteady magnetohydrodynamic free convection flow past an accelerated vertical plate with costant heat flux and heat generation or absorption . ZAMM Z. Angew. Math. Mech . 93 ( 1 ), 38 - 49 ( 2013 )
9. Pattnaik , J.R. ; Dash, G.C. ; Singh , S. : Radiation and mass transfer effects on MHD flow through porous medium past an exponentially accelerated inclined plane with variable temperature . Ain Shams Eng. J . 8 , 67 - 75 ( 2017 )
10. Reddy , B.P.: Effects of thermal diffusion and viscous dissipation on unsteady MHD free convection flow past a vertical porous plate under oscillatory suction velocity with heat sink . Int. J. Appl. Mech. Eng . 19 ( 2 ), 303 - 320 ( 2014 )
11. Samiulhaq; Vieru, D. ; Khan , I. ; Shafie, S. : Unsteady magnetohydrodynamic free convection flow of a second grade fluid in a porous medium with ramped wall temperature . PLoS One 9 ( 5 ), e88766 ( 2014 ). doi: 10 .1371/journal.pone. 0088766
12. Sarkar , B.C. ; Das , S. ; Jana, R.N. : Hall effects on unsteady MHD free convective flow past an accelerated moving vertical plate with viscous and Joule dissipations . Int. J. Comput. Appl . 70 ( 24 ), 19 - 28 ( 2013 )