Global Journal of Pure and Applied Mathematics.
ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1575-1592
Β© Research India Publications
http://www.ripublication.com
Effect of Thermal Radiation on the Casson Thin
Liquid Film Flow over a Stretching Sheet
K. Kalyani, K. Sreelakshmi and G. Sarojamma*
Department of Applied Mathematics, Sri Padmavati Mahila Visvavidyalayam, Tirupati, 517502, A.P, India
*Corresponding Author
Abstract
The effect of thermal radiation and viscous dissipation on the characteristics of
flow in a chemically reactive Casson liquid thin film subject to a transverse
magnetic field is examined. By introducing appropriate similarity variables the
non-linear partial differential equations governing the flow are reduced into a
set of non-linear ordinary differential equations which are then solved using
the shooting technique along with the Runge-Kutta-Fehlberg method. The
velocity, temperature and species concentration, film thickness and free
surface velocity are evaluated numerically. It is observed seen that thinner
films are formed due to stronger magnetic field strengths. Increase in Casson
parameter reduced the film thickness. Free surface velocity is found to
enhance with unsteady parameter. Temperature distribution is found to be an
increasing function of Casson parameter while it reduces with increasing
Prandtl number. Species concentration is improved with Casson parameter
while a reversal trend is noticed for an increasing variation in Schmidtβs
number and chemical reaction parameter. Rate of heat transfer is favorably
enhanced due to thermal radiation and viscous dissipation. The Sherwood
number is increased significantly for increasing values of Schmidt number.
The present results are compared with the already published results and are
found to agree favorably with them.
Keywords: Unsteady flow, Casson thin liquid film, Thermal radiation,
Chemical reaction.
1576 K. Kalyani, K. Sreelakshmi and G. Sarojamma
INTRODUCTION
Several problems on flows in liquid thin films on stretching surfaces have been
extensively investigated due to their abundant applications in the last two decades. In
the melt spinning procedure when the extruded material is drawn through a die, the
flow caused due to stretching surface is very close to the extruded material is an
example. In all coating processes the objective is to obtain glossy smooth surface on
the end product with best finish in terms of low friction, transparency and good
strength. The properties of flow and heat transfer in a liquid thin film enable us to
achieve the expected finish of the coating and also in the design and development of
various heat exchangers as well as chemical processing equipments. Wang [1]
initiated the study of flow characteristics in a liquid thin film resting on an unsteady
stretching sheet. Dandapat et al. [2] extended this study to include heat transfer
analysis.
It is reported that in a heavily viscous fluid, considerable heat can be generated even
at low speeds of the fluid, for instance, in the case of extrusion of plastic sheets, and
thus rate of heat transfer may change appreciably due to viscous dissipation. Sarma
and Rao [3] obtained analytical solutions for the heat transfer in a steady laminar flow
of a viscoelastic fluid over a stretching surface in the presence of viscous dissipation
and internal heat generation. Sarojamma et al. [4] presented a mathematical model to
study the effect of viscous dissipation on the time dependent flow of a Casson fluid
due to a stretching sheet embedded in a rotating fluid subject to a uniform magnetic
field with thermal radiation and chemical reaction of nth order. Abel et al. [5]
examined the effect of viscous dissipation on the MHD flow and heat transfer in a
liquid film due to a stretching surface. Vajravelu et al. [6] carried out a mathematical
analysis of the effects of thermo physical properties on the thin film flow of an
Ostwald-de Waele liquid over a stretching surface in the presence of viscous
dissipation.
Majority of the fluids used for protective coatings are usually non-Newtonian. Hence,
the study of the non-Newtonian flow characteristics has significant relevance in
industry, for example in polymer and plastic fabrication and in coating equipment.
Chen [7] did a numerical investigation of heat transfer and flow characteristics in a
thin liquid film of a power law fluid due to an unsteady stretching sheet. Wang and
Pop [8] made a Homotopy analysis of the flow in a power-law fluid film on an
unsteady stretching surface. Mahmoud and Megahed [9] studied the effect of variable
viscosity and variable thermal conductivity on the flow and heat transfer of an
electrically conducting non-Newtonian power-law fluid within a thin liquid film over
an unsteady stretching sheet in the presence of a transverse magnetic field.
Effect of thermal radiation has significant applications in physics, space technology
and processes operated at very high temperature. For example, in polymer processing
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1577
industry when the whole system containing the polymer extrusion processes is placed
at high temperatures then radiation effect plays a vital role in controlling the heat
transfer process. As the quality of the end product greatly depends on the rate of heat
transfer, the knowledge of radiative heat transfer may be helpful to obtain the final
product with best quality. Hossain et al. [10] examined the influence of thermal
radiation on flow of viscous fluid over a heated vertical permeable plate with constant
surface temperature. Khader and Megahed [11] examined the flow and heat transfer in
a thin liquid film over an unsteady stretching sheet in a saturated porous medium with
thermal radiation. Prasad et al. [12] explored the effects of variable thermal
conductivity, thermal radiation and viscous heating on the MHD flow and heat
transfer of a non-Newtonian power-law liquid film at a horizontal porous sheet.
Khademinejad et al. [13] explored the effects of viscous dissipation, magnetic field
and thermal radiation to analyze the heat transfer characteristics of a thin liquid film
flow over an unsteady stretching sheet using HAM.
Studies on flows in liquid thin films are very limited. Casson fluid is a non-Newtonian
fluid initially proposed by Casson during this study on the flow curves of printing
inks. Subsequently this model was used to describe blood, varnishes, polymers etc.
Megahed [14] examined the impact of variable heat flux, viscous heating and velocity
slip flow on the heat transfer of a Casson fluid in thin film on a stretching sheet.
Vijaya et al. [15] investigated the heat transfer on the flow of a Casson fluid film in
the presence of viscous dissipation and temperature dependent heat source.
In this paper an analysis to study the effect of thermal radiation and first order
chemical reaction on the heat and mass transfer characteristics of flow in a Casson
thin liquid film is carried out.
MATHEMATICAL FORMULATION
We consider a chemically reactive non-Newtonian Casson liquid thin film with
thickness β(π‘) over a heated stretching sheet that emerges from a narrow slit at the
origin of the Cartesian coordinate system as shown schematically in figure 1. The
motion of the fluid within the film is due to the stretching of the sheet.
The continuous sheet is parallel to x-axis and moves in its own plane with a velocity
π(π₯, π‘) =ππ₯
1βπΌπ‘ (1)
where πΌ and π are positive constants with dimension per time. The stretching sheetβs
temperature and concentration ππ and πΆπ is assumed to vary with the distance x from
the slit as
ππ (π₯, π‘) = π0 β ππππ [ππ₯2
2π£] (1 β πΌπ‘)β3/2 (2)
1578 K. Kalyani, K. Sreelakshmi and G. Sarojamma
πΆπ (π₯, π‘) = πΆ0 β πΆπππ [ππ₯2
2π£] (1 β πΌπ‘)β3/2 (3)
where π0 and πΆ0 are the temperature and concentration at the slit, π£ is the kinematic
viscocity. A transverse magnetic field π΅ = π΅0(1 β πΌπ‘)β1/2 is applied to the thin
liquid film. Effect of thermal radiation is taken into account.
Figure 1.Physical model and coordinate system
The constitutive equation of the Casson fluid can be written as [16]
Οij = {2 (ΞΌB +
Py
β2Ο) eij , Ο > Οc
2 (ΞΌB +Py
β2Οc) eij , Ο < Οc
(4)
where Οij is the (i, j)th component of the stress tensor, ΞΌB is the plastic dynamic
viscosity of the non-Newtonian fluid, Py is the yield stress of the fluid, Ο is the
product of the component of deformation rate with itself, namely, Ο = eijeij, and eij
is the (i, j)th component of deformation rate, and Οc is the critical value of Ο depends
on non-Newtonian model.
Under these assumptions, equations of the flow in the liquid film are given by ππ’
ππ₯+
ππ£
ππ¦= 0 (5)
ππ’
ππ‘+ π’
ππ’
ππ₯+ π£
ππ’
ππ¦= π (1 +
1
π½)
π2π’
ππ¦2β
ππ΅2
ππ’ (6)
ππ
ππ‘+ π’
ππ
ππ₯+ π£
ππ
ππ¦=
π
ππΆπ
2π
π¦2 +16πβ π0
3
3ππππβ
π2π
ππ¦2+
π
πππ(1 +
1
π½) (
ππ’
ππ¦)
2
(7)
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1579
ππΆ
ππ‘+ π’
ππΆ
ππ₯+ π£
ππΆ
ππ¦= π·
π2πΆ
π¦2 β π1(πΆ β πΆβ) (8)
where π’ and π£ are the velocity components of fluid in x- and y- directions, T is
temperature, C is the fluid concentration, π is dynamic viscosity, π is electrical
conductivity, Ξ² = ΞΌBβ2Οc /Py is the Casson parameter, π is density, ππ is specific
heat at constant pressure, π is the thermal conductivity, πβ is the Stefen-Boltzman
constant, πβ is the absorption coefficient, D is the mass diffusivity and π1(π‘) =
π0/(1 β πΌπ‘) is the time dependent reaction rate.
The boundary conditions on the stretching sheet are no slip, no penetration and
imposed sheet temperature and concentration distributions and are represented
respectively as
π’ = π, π£ = 0, π = ππ , πΆ = πΆπ ππ‘ π¦ = 0, (9)
ππ’
ππ¦= 0,
ππ
ππ¦= 0,
ππΆ
ππ¦= 0, π£ =
πβ
ππ‘ ππ‘ π¦ = β(π‘) (10)
The following similarity transformations are introduced:
π = [π
π(1βπΌπ‘)]
1
2π¦ , π = π₯ [
ππ
1βπΌπ‘]
1
2π(π), (11)
π = π0 β ππππ [ππ₯2
2π(1βπΌπ‘)3/2] π(Ξ·), π(Ξ·) =Tβπ0
ππβπ0 (12)
πΆ = πΆ0 β πΆπππ [ππ₯2
2π(1βπΌπ‘)3/2] π(Ξ·), π(Ξ·) =CβπΆ0
πΆπβπΆ0 (13)
Also, π(π₯, π¦) is the stream function which automatically fulfils mass conservation
equation (5) and the velocity components are can be obtained as
π’ =ππ
ππ¦=
ππ₯
1βπΌπ‘πβ²(π) π£ = β
ππ
ππ₯= β (
ππ
1βπΌπ‘)
1/2
π(π) (14)
where prime denotes differentiation with respect to π.
METHOD OF SOLUTION
The mathematical problem defined through equations (5) β (8) are transformed to the
following non-linear boundary value problem on the finite range of 0 β πΎ:
(1 +1
π½) πβ²β²β² + [ππβ²β² β π(πβ² +
π
2πβ²β²) β πβ²2 β ππβ²] = 0 (15)
(1 +4
3ππ) πβ²β² + ππ (ππβ² β 2πβ²π β
π
2(ππβ² + 3π) + πΈπ (1 +
1
π½) πβ²β²2) = 0 (16)
πβ²β² + ππ (ππβ² β 2πβ²π βπ
2(ππβ² + 3π) β πΏπ) = 0 (17)
1580 K. Kalyani, K. Sreelakshmi and G. Sarojamma
subject to the boundary conditions
π(0) = 0, πβ²(0) = 1, π(0) = 1, π(0) = 1 (18)
π(πΎ) =1
2ππΎ, πβ²β²(πΎ) = 0, πβ²(πΎ) = 0, πβ²(πΎ) = 0 (19)
Where, π = πΌ πβ is the unsteadiness parameter, π = ππ΅02 ππβ is the magnetic field
parameter, ππ = ππππ πβ is the Prandtl number, ππ = 4πβπβ3 ππββ is the thermal
radiation parameter, πΈπ = π2 ππ(ππ β π0)β is the Eckert number, ππ = π π·β is the
Schmidt number and πΏ = π0 πβ is the chemical reaction parameter.
Further, πΎ denotes the value of the similarity variable π at the free surface so that the
first term of equation (11) gives
πΎ = (π
π(1βπΌπ‘))
1/2
β(π‘) (20)
Since πΎ is an unknown constant, which should be determined, as a whole, from the set
of the present boundary-value problem, the rate of change of the film thickness can be
obtained as follows:
πβ
ππ‘= β
πΌπΎ
2(
π
π(1βπΌπ‘))
1/2
(21)
Thus, the kinematic constraint at π¦ = β(π‘) given by equation (10) transforms to the
free surface condition (20).
The surface drag coefficient πΆππ₯ , Nusselt number ππ’π₯ and Sherwood number πβπ₯
which play a significant role in estimating the surface drag force, rate of heat and
mass transfer are defined respectively, as
πΆππ₯π ππ₯1/2 = β2 (1 +
1
π½) πβ²β²(0), ππ’π₯π ππ₯
β1/2 = πβ²(0), πβπ₯π ππ₯β1/2 = πβ²(0) (22)
where π ππ₯ = ππ₯/π is the local Reynolds number.
The coupled ordinary differential equations (15) β (17) are non-linear and
exact analytical solutions are not possible. Equations (15) β (17) with the appropriate
boundary conditions (18) and (19) are solved numerically by the efficient fourth order
Runge-Kutta-Fehlberg algorithm along with numerical shooting technique. These
equations are converted into a set of first order equations as follows:
ππ0
ππ= π1,
ππ1
ππ= π2,
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1581
(1 +1
π½)
ππ2
ππ= π (π1 +
π
2π2) + π1
2 β π0π2 + ππ1, (23)
ππ0
ππ= π1,
(1 +4
3ππ)
ππ1
ππ= ππ (
π
2(3π0 + ππ1) + 2π0π1 β π1π0 β πΈπ (1 +
1
π½) π2
2). (24)
ππ0
ππ= π1,
ππ1
ππ= ππ (
π
2(3π0 + ππ1) + 2π0π1 β π1π0 + πΏπ0) (25)
The associated boundary conditions take the form,
π0(0) = 0, π1(0) = 1, π0 = 1, π0 = 1 (26)
π0(πΎ) =1
2ππΎ, π2(πΎ) = 0, π1(πΎ) = 0, π1(πΎ) = 0. (27)
Here π0(π) = π(π) and π0(π) = π(π) and π0(π) = π(π) . This requires the initial
values π2(0), π1(0) and π1(0) and hence suitable guess values are chosen and later
integration is performed. A step size of βπ = 0.01 is chosen. The value of πΎ is
obtained in such a way that the boundary condition π0(πΎ) = ππΎ
2 is satisfied with an
error of tolerance of 10β6.
Table 1. Comparison of πΎ and πβ²β²(0) with published values when M = 0 and π½ β β
for various values of S
S Wang [1] Abel et al. [5] Megahed [14] Present study
πΎ πβ²β²(0)/πΎ πΎ πβ²β²(0) πΎ πβ²β²(0) πΎ πβ²β²(0)
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
5.122490
3.131250
2.151990
2.543620
1.127780
0821032
0.576173
0.356389
-1.307785
-1.195155
-1.245795
-1.277762
-1.279177
-1.233549
-1.491137
-0.867414
4.981455
3.131710
2.151990
1.543617
1.127780
0.821033
0.576176
0.356390
-1.134098
-1.195128
-1.245805
-1.277769
-1.279171
-1.233545
-1.114937
-0.867416
4.98145
3.131710
2.151994
1.543616
1.127781
0.821032
0.576173
0.356389
-1.134096
-1.195126
-1.245806
-1.277769
-1.279172
-1.233545
-1.114938
-0.867414
4.981455
3.131710
2.151990
1.543617
1.127780
0.821033
0.576176
0.356390
-1.134098
-1.195128
-1.245805
-1.277769
-1.279171
-1.233545
-1.114937
-0.867416
Accuracy of the present scheme is ensured by comparing the present results, viz., non
dimensional thickness of the film πΎ , surface drag coefficient πβ²β²(0) with the
corresponding values evaluated by Wang [1], Abel et al. [5] and Megahed [14] in the
1582 K. Kalyani, K. Sreelakshmi and G. Sarojamma
absence of magnetic field parameter (π = 0) for a Newtonian fluid (π½ β β) for
various values of unsteady parameter. Since Wang [1] used different similarity
variables, the values of πβ²β²(0) πΎβ evaluated by Wang [1], shall be same as πβ²β²(0) of
the present analysis. These values are presented in Table 1 and it is observed that they
are in excellent agreement.
RESULTS AND DISCUSSION
To obtain a flow in the thin film, numerical computations of flow velocity, temperature
and concentration for various sets of governing parameters have been obtained and
graphically illustrated.
Figures 2 - 4 depict the influence of non-Newtonian rheology of the fluid through the
Casson parameter (π½) on velocity, temperature and concentration. The velocity in the
vicinity of the boundary is seen to be a constant function of the Casson parameter.
However, a considerable reduction in the velocity of the fluid within the film away
from the boundary is observed for higher values of π½. Reduction in the velocity shall
be due to the non-Newtonian nature of the fluid as increase in Casson parameter
amounts to an increase in the plastic dynamic viscosity of the fluid. As a consequence
film thickness also decreases for higher values of π½. However, for the same variation
of π½, the temperature and concentration are found to increase as shown in Figures 3
and 4.
From figure 5 it is observed that in the absence of magnetic field (M), velocity steadily
decreases in the film. Presence of magnetic field leads to a rapid reduction of velocity
in the vicinity of the boundary due to the action of Lorentz force which opposes the
fluid motion. Figure 6 reveals that velocity distribution in the film decreases
monotonically for small values of unsteadiness parameter (S). As the unsteadiness
parameter assumes higher values fluid gets accelerated and hence higher velocities
occur. For increasing values of unsteadiness parameter the films become thinner.
When unsteadiness parameter S = 1.4, film thickness is found to be decreased by two
and half times than that of the film corresponding to S = 0.8.
Figure 7 illustrates the variation of thermal radiation parameter (Nr) on temperature.
Increasing values of Nr enhances the temperature prominently as the presence of
thermal radiation releases higher thermal energy. Figure 8 presents the temperature
profiles for a variation in Prandtl number (Pr). It is revealed that temperature falls from
its higher value on the wall to its minimum value on the surface. For higher values of
Pr, temperature decreases rapidly near the boundary. As higher values of ππ indicate
that the thermal conductivity of the fluid is smaller and hence lower temperatures are
resulted. Figure 9 illustrates the variation of Eckert number (Ec) on temperature.
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1583
Profiles of temperature reveal that increasing values of πΈπ heat up the fluid in the film
resulting in higher temperatures. This enhancement is due to internal heating in the
fluid layers. In particular near the boundary when πΈπ = 3.0 an over shoot of the
temperature occurs.
Figure 10 highlights the variation of Schmidtβs number (Sc) on species concentration.
It is noticed that concentration decreases considerably for increasing values of
Schmidtβs number which is in conformity with fact that higher values of Sc
corresponds to smaller mass diffusivity . From Figure 11 it is observed that the effect
of chemical reaction parameter (πΎ) on concentration is similar o that of Schmidtβs
number.
From Figure 12 it is clear that the free surface velocity remains almost steady for all
values of the magnetic field parameter (M). Increasing values of unsteadiness
parameter (S) is found to reduce the free surface velocity. As π various its value from
0.8 to 1.2 there is a twofold reduction in the free surface velocity is seen. Figure 13
presents the variation of film thickness versus magnetic field parameter for different
values of the unsteadiness parameter. It can be seen that film thickness reduces for
increasing values of π. Film thickness decreases rapidly for smaller values of magnetic
field and a further reduction is noticed for stronger magnetic field strength.
Figure 2.Velocity profiles for different values of π½
0 0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f ' (
)
M = 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 2.584269
= 2.238043
= 2.110047 = 2.043044
= 1.0
= 2.0
= 3.0
= 4.0
1584 K. Kalyani, K. Sreelakshmi and G. Sarojamma
Figure 3.Temperature profiles for different values of π½
Figure 4.Concentration profiles for different values of π½
0 0.5 1 1.5 2 2.5 30.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
M = 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 2.584269
= 2.238043 = 2.110047
= 2.043044
= 1.0
= 2.0
= 3.0
= 4.0
0 0.5 1 1.5 2 2.5 30.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
M = 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 2.584269
= 2.238043 = 2.110047
= 2.043044
= 1.0
= 2.0
= 3.0
= 4.0
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1585
Figure 5.Velocity profiles for different values of M
Figure 6.Velocity profiles for different values of S
0 0.5 1 1.5 2 2.5 3 3.5 40.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f ' (
)
= 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 0.5; S = 0.8; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 3.727362
= 2.800520 = 2.339794 = 2.051091
M = 0.0
M = 1.0
M = 2.0
M = 3.0
0 0.5 1 1.5 2 2.5 3 3.50.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
f ' (
)
M = 0.5; = 0.5; Pr = 1.0; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
= 3.165071
= 2.321451
= 1.727597
= 1.276756
S = 0.8
S = 1.0
S = 1.2
S = 1.4
1586 K. Kalyani, K. Sreelakshmi and G. Sarojamma
Figure 7.Temperature profiles for different values of Nr
Figure 8.Temperature profiles for different values of Pr
0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Ec = 0.1; Sc = 0.5; = 0.1
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Ec = 0.1; Sc = 0.5; = 0.1
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Ec = 0.1; Sc = 0.5; = 0.1
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Ec = 0.1; Sc = 0.5; = 0.1
Nr = 0.5
Nr = 1.0
Nr = 1.5
Nr = 2.0
0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
M = 0.5; = 0.5; S = 0.8; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
M = 0.5; = 0.5; S = 0.8; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
M = 0.5; = 0.5; S = 0.8; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
M = 0.5; = 0.5; S = 0.8; Nr = 0.5;
Ec = 0.1; Sc = 0.5; = 0.1
Pr = 0.7
Pr = 1.0
Pr = 2.0
Pr = 3.0
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1587
Figure 9.Temperature profiles for different values of Ec
Figure 10.Concentration profiles for different values of Sc
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
(
)
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Sc = 0.5; = 0.1
Ec = 0.0
Ec = 1.0
Ec = 2.0
Ec = 3.0
0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; = 0.1
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; = 0.1
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; = 0.1
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; = 0.1
Sc = 0.5
Sc = 1.0
Sc = 1.5
Sc = 2.0
1588 K. Kalyani, K. Sreelakshmi and G. Sarojamma
Figure 11.Concentration profiles for different values of πΏ
Figure 12.Variation of free surface velocity πβ²(πΎ) with M
0 0.5 1 1.5 2 2.5 3 3.50
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(
)
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; Sc = 0.5
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; Sc = 0.5
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; Sc = 0.5
M = 0.5; = 0.5; S = 0.8; Pr = 1.0;
Nr = 0.5; Ec = 0.1; Sc = 0.5
= 0.1
= 0.4
= 0.7
= 1.0
0 1 2 3 4 5 6
0.2
0.25
0.3
0.35
0.4
0.45
0.5
M
f ' ()
S = 0.8
S = 1.2
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1589
Figure 13.Variation of film thickness πΎ with M
Local skin friction coefficient, Nusselt number and Sherwood number on the stretching
surface for different variations of the governing parameters are presented in Table 2.
Surface drag coefficient is found to increase with elapse of time. Rate of heat transfer
is observed to reduce for an increase in the unsteadiness parameter which is in
conformity with the variation observed in temperature with unsteadiness parameter.
Sherwood number increases with higher values S. Larger values of Casson parameter
decrease skin friction coefficient due to smaller velocities. Rate of heat and mass
transfer is found to be smaller for a variation in Casson parameter. Surface drag
coefficient is significantly reduced due to stronger magnetic field strengths. Lorentz
force favors the rate of heat and mass transfer. Table 3 illustrates the Nusselt number
for different values of Pr, Ec and ππ. Increase in Prandtl number leads to an
enhancement in the temperature gradient. Eckert number and thermal radiation
parameter decrease the Nusselt number. Table 4 shows that Schmidtβs number
increases the mass concentration gradient significantly while the chemical reaction
parameter increases the mass concentration gradient moderately.
0 1 2 3 4 5 60.5
1
1.5
2
2.5
3
3.5
4
M
S = 0.8
S = 1.2
1590 K. Kalyani, K. Sreelakshmi and G. Sarojamma
Table 2. Variation of (1 +1
π½) πβ²β²(0) and βπβ²(0) for various values of S, π½ and M
S π½ M (1 +1
π½) πβ²β²(0) βπβ²(0) βπβ²(0)
0.8
1.0
1.2
1.4
0.5 0.5
-2.473527
-2.504972
-2.478698
-2.365328
1.355718
1.409314
1.450178
1.463985
1.212364
1.260450
1.295858
1.304118
0.8
1.0
2.0
3.0
4.0
0.5
-1.577477
-1.585218
-1.591939
-1.596655
1.289227
1.276834
1.269177
1.264258
1.159691
1.144804
1.135654
1.129799
0.8 0.5
0.0
1.0
2.0
3.0
-2.157798
-2.752908
-3.239791
-3.662303
1.365686
1.346290
1.327678
1.309009
1.229794
1.195827
1.163923
1.133122
Table 3. Variation of βπβ²(0) for various values of Pr, Ec and Nr
Pr Ec Nr βπβ²(0)
0.7
1.0
2.0
3.0
0.1 0.5
1.108487
1.355718
1.982332
2.465460
1.0
0.0
1.0
2.0
3.0
0.5
1.315220
0.910217
0.505218
0.100220
1.0 0.1
0.5
1.0
1.5
2.0
1.355718
1.121379
0.969474
0.860352
Effect of thermal radiation on the Casson thin liquid film flow over a stretching sheet 1591
Table 4. Variation of βπβ²(0) for various values of Sc and πΏ
Sc πΏ βπβ²(0)
0.5
1.0
1.5
2.0
0.1
1.212364
1.766484
2.191551
2.550359
0.5
0.1
0.4
0.7
1.0
1.212364
1.279212
1.341667
1.400595
CONCLUSIONS
Some of the significant conclusions of the study are:
Surface velocity is found to decrease with increasing values of magnetic field
parameter.
Higher values of unsteady parameter decreases film thickness. A qualitatively
similar trend is found for increasing values of magnetic and Casson parameters.
Temperature is an increasing function of Eckert number.
Species concentration is found to be a decreasing function of Schmidtβs number
and chemical reaction parameter.
Viscous heating and thermal radiation enhance the rate of heat transfer.
REFERENCES
[1]. Wang, C. Y., 1990, βLiquid film on an unsteady stretching surface,β Quarterly
of Applied Mathematics, 48(3), pp. 601β610.
[2]. Dandapat, B. S., Andersson, H. I., and Aarseth, J. B., 2000, βHeat transfer in a
liquid film on an unsteady stretching surface,β Int. J. Heat and Mass Transfer,
43(1), pp. 69β74.
[3]. Sarma, M. S., Rao, B. N., 1998, βHeat transfer in a viscoelastic fluid over a
stretching sheet,β J. Math. Anal. Appl., 1(1), pp. 268β275.
[4]. Sarojamma, G., Sreelakshmi, K., and Vasundhara, B., 2016, βMathematical
model of MHD unsteady flow induced by a stretching surface embedded in a
rotating Casson fluid with thermal radiation,β IEEE, 978-9-3805-4421-
2/16/$31.00_c 2016, pp. 1590β1595.
[5]. Abel, M. S., Tawade, J., and Nandeppanavar, M. M., 2009, βEffect of non-
uniform heat source on MHD heat transfer in a liquid film over an unsteady
stretching sheetβ, Int. J. Non-Linear Mechanics, 44(9), pp. 990β998.
1592 K. Kalyani, K. Sreelakshmi and G. Sarojamma
[6]. Vajravelu, K., Prasad K. V., and Raju, B. T., 2013, βEffects of variable fluid
properties on the thin film flow of Ostwald β de Waele fluid over a stretching
surface,β Journal of Hydrodynamics, 25(1), pp. 10β19.
[7]. Chen, C. H., 2003, βHeat transfer in a power-law film over an unsteady
stretching sheet,β Heat and Mass Transfer, 39(8), pp. 791β796.
[8]. Wang, C., and Pop, I., 2006, βAnalysis of the flow of a power-law fluid film on
an unsteady stretching surface by means of homotopy analysis method,β J. Non-
Newtonian Fluid Mechanics, 138(2), pp. 161β172.
[9]. Mahmoud, M. A. A., and Megahed, A. M., 2009, βMHD flow and heat transfer
in a non-Newtonian liquid film over an unsteady stretching sheet with variable
fluid properties,β Canadian Journal of Physics, 87(10), pp. 1065β1071.
[10]. Hossain, M. A., Khanafer, K., and Vafai, K., 2001, βThe effect of radiation on
free convection flow of fluid with variable viscosity from a porous vertical
plate,β Int. J. Therm. Sci., 40(2), pp. 115β124.
[11]. Khader, M. M. and Megahed, A. M., 2013, βNumerical simulation using the
finite difference method for the flow and heat transfer in a thin liquid film over
an unsteady stretching sheet in a saturated porous medium in the presence of
thermal radiation,β Journal of King Saud University β Engineering Sciences,
25(1), pp. 29β34.
[12]. Prasad, K. V., Vajravelu, K., Datti, P. S., and Raju, B. T., 2013, βMHD flow
and heat transfer in a Power-law liquid film at a porous surface in the presence
of Thermal radiation,β Journal of Applied Fluid Mechanics, 6(3), pp. 385β395.
[13]. Khademinejad, T., Khanarmuei, M. R., Talebizadeh, P., and Hamidi, A., 2015,
βOn the use of the homotopy analysis method for solving the problem of the
flow and heat transfer in a liquid film over an unsteady stretching sheet,β
Journal of Applied Mechanics and Technical Physics, 56(4), pp. 654β666.
[14]. Megahed, A. M., 2015, βEffect of slip velocity on Casson thin film flow and
heat transfer due to unsteady stretching sheet in presence of variable heat flux
and viscous dissipation,β Appl. Math. Mech. Engl. Ed., 36, pp. 1273β1284.
[15]. Vijaya, N., Sreelakshmi, K., and Sarojamma, G., 2016, βEffect of magnetic field
on the flow and heat transfer in a Casson thin film on an unsteady stretching
surface in the presence of viscous and internal heating,β Open Journal of Fluid
Dynamics, 6(4), pp. 303-320.
[16]. Eldabe, N. T. M., and Salwa, M. G. E., 1995, βHeat transfer of MHD non-
Newtonian Casson fluid flow between two rotating cylinders,β J. Phys. Soc.
Japan, 64, pp. 41.
Top Related