Effect of Thermal Radiation on the Casson Thin Liquid Film Flow over a Stretching … · 2017. 4....

Global Journal of Pure and Applied Mathematics.

ISSN 0973-1768 Volume 13, Number 6 (2017), pp. 1575-1592

© 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*

1Department 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 Schmidt’s

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 ℎ(𝑡) 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

𝑈(𝑥, 𝑡) =𝑏𝑥

1−𝛼𝑡 (1)

where 𝛼 and 𝑏 are positive constants with dimension per time. The stretching sheet’s

temperature and concentration 𝑇𝑠 and 𝐶𝑠 is assumed to vary with the distance x from

the slit as

𝑇𝑠 (𝑥, 𝑡) = 𝑇0 − 𝑇𝑟𝑒𝑓 [𝑏𝑥2

2𝑣] (1 − 𝛼𝑡)−3/2 (2)


𝐶𝑠 (𝑥, 𝑡) = 𝐶0 − 𝐶𝑟𝑒𝑓 [𝑏𝑥2

2𝑣] (1 − 𝛼𝑡)−3/2 (3)

where 𝑇0 and 𝐶0 are the temperature and concentration at the slit, 𝑣 is the kinematic

viscocity. A transverse magnetic field 𝐵 = 𝐵0(1 − 𝛼𝑡)−1/2 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]

τij = {2 (μB +

Py

√2π) eij , π > πc

2 (μB +Py

√2πc) eij , π < πc

(4)

where τij is the (i, j)th component of the stress tensor, μB is the plastic dynamic

viscosity of the non-Newtonian fluid, Py is the yield stress of the fluid, π is the

product of the component of deformation rate with itself, namely, π = eijeij, and eij

is the (i, j)th component of deformation rate, and πc is the critical value of π depends

on non-Newtonian model.

Under these assumptions, equations of the flow in the liquid film are given by 𝜕𝑢

𝜕𝑥+

𝜕𝑣

𝜕𝑦= 0 (5)

𝜕𝑢

𝜕𝑡+ 𝑢

𝜕𝑢

𝜕𝑥+ 𝑣

𝜕𝑢

𝜕𝑦= 𝜈 (1 +

1

𝛽)

𝜕2𝑢

𝜕𝑦2−

𝜎𝐵2

𝜌𝑢 (6)

𝜕𝑇

𝜕𝑡+ 𝑢

𝜕𝑇

𝜕𝑥+ 𝑣

𝜕𝑇

𝜕𝑦=

𝑘

𝜌𝐶𝑝

2𝑇

𝑦2 +16𝜎∗ 𝑇0

3

3𝜌𝑐𝑝𝑘∗

𝜕2𝑇

𝜕𝑦2+

𝜇

𝜌𝑐𝑝(1 +

1

𝛽) (

𝜕𝑢

𝜕𝑦)

2

(7)


𝜕𝐶

𝜕𝑡+ 𝑢

𝜕𝐶

𝜕𝑥+ 𝑣

𝜕𝐶

𝜕𝑦= 𝐷

𝜕2𝐶

𝑦2 − 𝑘1(𝐶 − 𝐶∞) (8)

where 𝑢 and 𝑣 are the velocity components of fluid in x- and y- directions, T is

temperature, C is the fluid concentration, 𝜇 is dynamic viscosity, 𝜎 is electrical

conductivity, β = μB√2πc /Py is the Casson parameter, 𝜌 is density, 𝑐𝑝 is specific

heat at constant pressure, 𝑘 is the thermal conductivity, 𝜎∗ is the Stefen-Boltzman

constant, 𝑘∗ is the absorption coefficient, D is the mass diffusivity and 𝑘1(𝑡) =

𝑘0/(1 − 𝛼𝑡) 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

𝑢 = 𝑈, 𝑣 = 0, 𝑇 = 𝑇𝑠, 𝐶 = 𝐶𝑠 𝑎𝑡 𝑦 = 0, (9)

𝜕𝑢

𝜕𝑦= 0,

𝜕𝑇

𝜕𝑦= 0,

𝜕𝐶

𝜕𝑦= 0, 𝑣 =

𝑑ℎ

𝑑𝑡 𝑎𝑡 𝑦 = ℎ(𝑡) (10)

The following similarity transformations are introduced:

𝜂 = [𝑏

𝜈(1−𝛼𝑡)]

1

2𝑦 , 𝜓 = 𝑥 [

𝜈𝑏

1−𝛼𝑡]

1

2𝑓(𝜂), (11)

𝑇 = 𝑇0 − 𝑇𝑟𝑒𝑓 [𝑏𝑥2

2𝜈(1−𝛼𝑡)3/2] 𝜃(η), 𝜃(η) =T−𝑇0

𝑇𝑆−𝑇0 (12)

𝐶 = 𝐶0 − 𝐶𝑟𝑒𝑓 [𝑏𝑥2

2𝜈(1−𝛼𝑡)3/2] 𝜙(η), 𝜙(η) =C−𝐶0

𝐶𝑆−𝐶0 (13)

Also, 𝜓(𝑥, 𝑦) is the stream function which automatically fulfils mass conservation

equation (5) and the velocity components are can be obtained as

𝑢 =𝜕𝜓

𝜕𝑦=

𝑏𝑥

1−𝛼𝑡𝑓′(𝜂) 𝑣 = −

𝜕𝜓

𝜕𝑥= − (

𝜈𝑏

1−𝛼𝑡)

1/2

𝑓(𝜂) (14)

where prime denotes differentiation with respect to 𝜂.

METHOD OF SOLUTION

The mathematical problem defined through equations (5) – (8) are transformed to the

following non-linear boundary value problem on the finite range of 0 – 𝛾:

(1 +1

𝛽) 𝑓′′′ + [𝑓𝑓′′ − 𝑆(𝑓′ +

𝜂

2𝑓′′) − 𝑓′2 − 𝑀𝑓′] = 0 (15)

(1 +4

3𝑁𝑟) 𝜃′′ + 𝑃𝑟 (𝑓𝜃′ − 2𝑓′𝜃 −

𝑆

2(𝜂𝜃′ + 3𝜃) + 𝐸𝑐 (1 +

1

𝛽) 𝑓′′2) = 0 (16)

𝜙′′ + 𝑆𝑐 (𝑓𝜙′ − 2𝑓′𝜙 −𝑆

2(𝜂𝜙′ + 3𝜙) − 𝛿𝜙) = 0 (17)


subject to the boundary conditions

𝑓(0) = 0, 𝑓′(0) = 1, 𝜃(0) = 1, 𝜙(0) = 1 (18)

𝑓(𝛾) =1

2𝑆𝛾, 𝑓′′(𝛾) = 0, 𝜃′(𝛾) = 0, 𝜙′(𝛾) = 0 (19)

Where, 𝑆 = 𝛼 𝑏⁄ is the unsteadiness parameter, 𝑀 = 𝜎𝐵02 𝜌𝑏⁄ is the magnetic field

parameter, 𝑃𝑟 = 𝜌𝑐𝑝𝜈 𝑘⁄ is the Prandtl number, 𝑁𝑟 = 4𝜎∗𝑇∞3 𝑘𝑘∗⁄ is the thermal

radiation parameter, 𝐸𝑐 = 𝑈2 𝑐𝑝(𝑇𝑠 − 𝑇0)⁄ is the Eckert number, 𝑆𝑐 = 𝜈 𝐷⁄ is the

Schmidt number and 𝛿 = 𝑘0 𝑏⁄ is the chemical reaction parameter.

Further, 𝛾 denotes the value of the similarity variable 𝜂 at the free surface so that the

first term of equation (11) gives

𝛾 = (𝑏

𝜈(1−𝛼𝑡))

1/2

ℎ(𝑡) (20)

Since 𝛾 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:

𝑑ℎ

𝑑𝑡= −

𝛼𝛾

2(

𝜈

𝑏(1−𝛼𝑡))

1/2

(21)

Thus, the kinematic constraint at 𝑦 = ℎ(𝑡) given by equation (10) transforms to the

free surface condition (20).

The surface drag coefficient 𝐶𝑓𝑥 , Nusselt number 𝑁𝑢𝑥 and Sherwood number 𝑆ℎ𝑥

which play a significant role in estimating the surface drag force, rate of heat and

mass transfer are defined respectively, as

𝐶𝑓𝑥𝑅𝑒𝑥1/2 = −2 (1 +

1

𝛽) 𝑓′′(0), 𝑁𝑢𝑥𝑅𝑒𝑥

−1/2 = 𝜃′(0), 𝑆ℎ𝑥𝑅𝑒𝑥−1/2 = 𝜙′(0) (22)

where 𝑅𝑒𝑥 = 𝑈𝑥/𝜈 is the local Reynolds number.

The coupled ordinary differential equations (15) – (17) are non-linear and

exact analytical solutions are not possible. Equations (15) – (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:

𝑑𝑓0

𝑑𝜂= 𝑓1,

𝑑𝑓1

𝑑𝜂= 𝑓2,


(1 +1

𝛽)

𝑑𝑓2

𝑑𝜂= 𝑆 (𝑓1 +

𝜂

2𝑓2) + 𝑓1

2 − 𝑓0𝑓2 + 𝑀𝑓1, (23)

𝑑𝜃0

𝑑𝜂= 𝜃1,

(1 +4

3𝑁𝑟)

𝑑𝜃1

𝑑𝜂= 𝑃𝑟 (

𝑆

2(3𝜃0 + 𝜂𝜃1) + 2𝜃0𝑓1 − 𝜃1𝑓0 − 𝐸𝑐 (1 +

1

𝛽) 𝑓2

2). (24)

𝑑𝜙0

𝑑𝜂= 𝜙1,

𝑑𝜙1

𝑑𝜂= 𝑆𝑐 (

𝑆

2(3𝜙0 + 𝜂𝜙1) + 2𝜙0𝑓1 − 𝜙1𝑓0 + 𝛿𝜙0) (25)

The associated boundary conditions take the form,

𝑓0(0) = 0, 𝑓1(0) = 1, 𝜃0 = 1, 𝜙0 = 1 (26)

𝑓0(𝛾) =1

2𝑆𝛾, 𝑓2(𝛾) = 0, 𝜃1(𝛾) = 0, 𝜙1(𝛾) = 0. (27)

Here 𝑓0(𝜂) = 𝑓(𝜂) and 𝜃0(𝜂) = 𝜃(𝜂) and 𝜙0(𝜂) = 𝜙(𝜂) . This requires the initial

values 𝑓2(0), 𝜃1(0) and 𝜙1(0) and hence suitable guess values are chosen and later

integration is performed. A step size of ∆𝜂 = 0.01 is chosen. The value of 𝛾 is

obtained in such a way that the boundary condition 𝑓0(𝛾) = 𝑆𝛾

2 is satisfied with an

error of tolerance of 10−6.

Table 1. Comparison of 𝛾 and 𝑓′′(0) with published values when M = 0 and 𝛽 → ∞

for various values of S

S Wang [1] Abel et al. [5] Megahed [14] Present study

𝛾 𝑓′′(0)/𝛾 𝛾 𝑓′′(0) 𝛾 𝑓′′(0) 𝛾 𝑓′′(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 𝛾 , surface drag coefficient 𝑓′′(0) with the

corresponding values evaluated by Wang [1], Abel et al. [5] and Megahed [14] in the


absence of magnetic field parameter (𝑀 = 0) for a Newtonian fluid (𝛽 → ∞) for

various values of unsteady parameter. Since Wang [1] used different similarity

variables, the values of 𝑓′′(0) 𝛾⁄ evaluated by Wang [1], shall be same as 𝑓′′(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 (𝛽) 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 𝛽. 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 𝛽. However, for the same variation

of 𝛽, 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 𝑃𝑟 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.


Profiles of temperature reveal that increasing values of 𝐸𝑐 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 𝐸𝑐 = 3.0 an over shoot of the

temperature occurs.

Figure 10 highlights the variation of Schmidt’s number (Sc) on species concentration.

It is noticed that concentration decreases considerably for increasing values of

Schmidt’s 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 (𝛾) on concentration is similar o that of Schmidt’s

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 𝑆 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 𝑆. 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 𝛽

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


Figure 3.Temperature profiles for different values of 𝛽

Figure 4.Concentration profiles for different values of 𝛽

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


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


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


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


Figure 11.Concentration profiles for different values of 𝛿

Figure 12.Variation of free surface velocity 𝑓′(𝛾) 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


Figure 13.Variation of film thickness 𝛾 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 𝑁𝑟. 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 Schmidt’s 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


Table 2. Variation of (1 +1

𝛽) 𝑓′′(0) and −𝜃′(0) for various values of S, 𝛽 and M

S 𝛽 M (1 +1

𝛽) 𝑓′′(0) −𝜃′(0) −𝜙′(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 −𝜃′(0) for various values of Pr, Ec and Nr

Pr Ec Nr −𝜃′(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


Table 4. Variation of −𝜙′(0) for various values of Sc and 𝛿

Sc 𝛿 −𝜙′(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 Schmidt’s number

and chemical reaction parameter.

Viscous heating and thermal radiation enhance the rate of heat transfer.

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