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CHAPTER 5

Radiation effects on Unsteady Free Convection Heat and Mass Transfer in a

Walters-B Viscoelastic flow past an Impulsively started Vertical plate with

Uniform Heat and Mass Flux.

Published in Mathematical Sciences International Research Journal

Vol. 2, Issue 1 with ISSN 2278-8697 Jan/Feb-2013

& Paper presented in the International conference INTERNATIONAL CONFERENCE ON

MATHEMATICS, STATISTICS & COMPUTER ENGINEERING, Sep 13-14, Vijayawada, A.P, India.

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5.1. INTRODUCTION The flow of a viscous incompressible fluid past an impulsively started an infinite

horizontal plate, moving in its own plane was first studied by Stokes [1]. It is often called

Rayleigh’s problem in the literature. Such a flow past an impulsively started semi-infinite

horizontal plate was first presented by Stewartson [2]. Hall [3] considered the flow past

an impulsively started semi-infinite horizontal by finite-difference method of a mixed

explicit-implicit type. Following Stoke’s analysis, Soundalgekar [4], presented an exact

solution to the flow of a viscous fluid past an impulsively started infinite isothermal

vertical plate. The solution was derived by the Laplace transform technique and the

effects of heating or cooling of the plate on the flow field were discussed through

Grashof number. Soundalgekar and Patil [5] studied Stoke’s problem for an infinite

vertical plate with constant heat flux.

Free convection flow occurs frequently in the nature. It occurs not only due to the

temperature differences, but also due to the concentration differences or combination of

these two. For example, in atmospheric flows, there exists differences in 퐻 표

concentration and hence the flow is affected by such concentration differences. Flows in

bodies of water are driven through the comparable effects upon density, of temperature,

concentration of dissolved materials and suspended particulate matter. Many transport

process exist in nature and in industrial applications in which the simultaneous heat and

mass transfer occur as a result of combined buoyancy effects of discussion of chemical

species.

A few representative fields of interest in which combined heat and mass transfer

plays an important role are design of chemical processing equipment, formation and

dispersion of fog, distribution of temperature and moisture over agricultural fields and

groves of fruit trees, damage of crops due to freezing, food processing and cooling

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towers. Cooling towers are the cheapest way to cool large quantities of water. They are

very common in industries, especially in power plants. The tower is packed with the

internal material, commonly with wooden slats. Hot water sprayed into the top of the

trickles down through wood, evaporating as it goes. Air enters the bottom of the tower

and rises up through the packing. In smaller towers, the air can be pumped with the fan;

in larger ones, it is often allowed to rise by natural convection. In this context,

Soundalgekar [6] extended his own problem of [4] to mass transfer effects.

Muthucumaraswamy and Ganesan [7] studied first order chemical reaction on the flow

past an impulsively started vertical plate with uniform heat and mass flux.

Soundalgekar [8] first studied mathematical investigations for such a fluid

considering the oscillatory two dimensional viscoelastic flow along an infinite

porous wall, showing that an increase in the Walters viscoelasticity parameter and the

frequency parameter reduces the phase of the skin-friction. Raptis and Takhar [9] studied

flat thermal convection boundary layers flow of Walters-B fluid using numerical

shooting quadrature. The study of the Walters-B (1962) viscoelastic model [10] was

developed to simulate the viscous fluids are possessing short memory elastic effects and

can simulate accurately in many complex polymeric fluids, biotechnological and tri-

biological fluids. The Walters-B viscoelastic model has been studied extensively in many

flow problems. Elbashbeshy [11] studied the heat and mass transfer along a vertical plate

under the combined buoyancy effects of thermal and species diffusion, in the presence of

magnetic field. Helmy [12] presented an unsteady two-dimensional laminar free

convection flow of an incompressible electrically conducting (Newtonian or polar) fluid

through a porous medium bounded by an infinite vertical plane surface of constant

temperature.

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In the context of space technology and in the process involving high

temperatures. The effect of radiation are of vital importance in recent developments in

hypersonic flights, missile re-entry, rocket combustion chambers, power plants for inter

planetary flight and gas cooled nuclear reactors, have focused attention on thermal

radiation as a mode of energy transfer, and emphasize the need for improved

understanding of radiative transfer in these processes. The interaction of radiation with

laminar free convection heat transfer from a vertical plate was investigated by Cess [13]

for an absorbing, emitting fluid in the optically thick region, using the singular

perturbation technique.

Arpaci [14] considered a similar problem in both the optically thin and optically

thick regions and used the approximate integral technique and first order profiles to solve

the energy equation. Cheng and Ozisik [15] considered a related problem for an

absorbing, emitting and isotropically scattering fluid, and treated the radiation part of the

problem exactly with the normal mode expansion technique. Raptis [16] analyzed the

thermal radiation and free convection flow through a porous medium by using

perturbation technique. Hossain and Takhar [17] studied the radiation effects on mixed

convection along a vertical plate with uniform surface temperature using the Keller Box

finite difference method. In all these papers, the flow is considered to be steady. The

unsteady flow past a moving plate in the presence of free convection and radiation were

studied by Mansour [18].

Raptis and Perdikis [19] studied the effect of thermal radiation and free

convection flow past moving plate. Das et al [20] analyzed the radiation effects on the

flow past an impulsively started infinite isothermal vertical plate. Chamkha et al [21]

considered the effect of radiation on free convection flow past a semi-infinite vertical

plate with mass transfer. Rajagopal et al [22] obtained exact solutions for the combined

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various hydro magnetic flow, heat and mass transfer phenomena in a conducting

viscoelastic Walters-B fluid percolating a porous regime adjacent to a stretching sheet

with heat generation, viscous dissipation and wall mass flux effects, by using confluent

hyper geometric functions for different thermal boundary conditions at the wall. Rath

and Bastia [23] used a perturbation method to analyze the steady flow and heat transfer

of Walters-B model viscoelastic liquid between two parallel uniformly porous disks

rotating about a common axis, showing that drag is enhanced with suction but reduced

with injection and that heat transfer rate is accentuated with a rise in wall suction or

injection.

More recently Beg at al [24] studied the effects of thermal radiation on

impulsively started vertical plate using network simulation method [NSM]. V. R. Prasad

et al [25] studied the radiation and mass transfer effects on two-dimensional flow past an

impulsively started isothermal vertical plate. It is observed that an increase in the value of

radiation parameter the velocity and temperature decreases accompanied by simultaneous

reductions in both momentum and thermal boundary layers and also the time taken to

reach the steady state increases when the radiation parameter increases. V. Ramachandra

Prasad et al [26] studied Transient radiative hydro-magnetic free convection flow past an

impulsively started vertical plate with uniform heat and mass flux.

The above studies did not consider combined viscoelastic momentum, heat and

mass transfer from a impulsively started vertical plate, either for the steady case or

unsteady case. Owing to the significance of this problem in chemical and medical

biotechnological processing (e.g. medical cream manufacture) we study the transient case

of such a flow in the present paper using the robust Walters-B viscoelastic rheological

material model. A Crank-Nicolson finite difference scheme is utilized to solve the

unsteady dimensionless, transformed velocity, thermal and concentration boundary layer

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equations in the vicinity of the impulsively started vertical plate. Another motivation of the

study is to further investigate the observed high heat transfer performance commonly

attributed to extensional stresses in viscoelastic boundary layers.

5.2. MATHEMATICAL MODEL

An unsteady two-dimensional laminar natural convection radiative flow of a

viscoelastic fluid past an impulsively started semi-infinite vertical plate is considered.

The x-axis is taken along the plate in the upward direction and y-axis is taken normal to

it. The physical model is shown in fig.1.

Initially assumed that the plate and the fluid are at the same temperature 푇 and

concentration level퐶 everywhere in the fluid. At time 푡 > 0, the plate starts moving

impulsively in the vertical direction with constant velocity 푢 against the gravitational

field. Temperature of the plate and concentration level near the plate are raised to 푇 and

퐶 respectively and maintained constantly thereafter. It is assumed that the concentration

퐶 of the diffusing species in the mixture is very less in comparison to the other

chemical species, which are present and hence the Soret and Dufour effects are

negligible. It is also assumed that there is no chemical reaction between the diffusing

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species and the fluid. Under the above assumptions, the governing boundary layer

equations with Boussinesq’s approximation are

휕푢휕푥 +

휕푢휕푦 = 0(5.2.1)

휕푢휕푡 + 푢

휕푢휕푥 + 푣

휕푢휕푦 = 푔훽(푇 − 푇 ) + 푔훽∗(퐶 − 퐶 ) + 휈

휕 푢휕푦 − 푘

휕 푢휕푦 휕푡 (5.2.2)

휕푇휕푡 + 푢

휕푇휕푥 + 푣

휕푇휕푦 = 훼

휕 푇휕푦 −

1휌푐

휕푞휕푦 (5.2.3)

휕퐶휕푡 + 푢

휕퐶휕푥 + 푣

휕퐶휕푦 = 퐷

휕 퐶휕푦 (5.2.4)

The initial and boundary conditions are:

푡 ≤ 0:푢 = 0,푣 = 0,푇 = 푇 ,퐶 = 퐶

푡 > 0:푢 = 푢 ,푣 = 0,휕푇휕푦 = −

푞 (푥)푘 ,

휕퐶 휕푦 = −

푞∗ (푥)퐷 푎푡푦 = 0

푢 = 0, 푇 = 푇 , 퐶 = 퐶 ,푎푡푥 = 0(5.2.5)

푢 → 0, 푇 → 푇 ,퐶 → 퐶 푎푠푦 → ∞

By using the Rosseland approximation the radiative heat flux 푞 is given by

푞 = −4휎3푘

휕푇휕푦 (5.2.6)

Where휎 is the Stefan-Boltzmann constant and 푘 is the mean absorption coefficient.

It should be noted that by using the Rosseland approximation, the present analysis is

limited to optically thick fluids. If temperature differences within the flow are

significantly small, then equation [5.2.6] can be linearised by expanding푇 into the

Taylor series about 푇 , which after, neglect higher order terms, takes the form:

푇 ≅ 4푇 푇 − 3푇 (5.2.7)

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In view of equations [6] and [7], equation [3] reduces to:

휕푇휕푡 + 푢

휕푇휕푥 + 푣

휕푇휕푦 = 훼

휕 푇휕푦 +

16휎 푇3푘 휌푐

휕 푇휕푦 (5.2.8)

On introducing the following non-dimensional quantities:

푋 =푥푢휈 ,푌 =

푦푢휈 ,푡 =

푡 푢휈 ,푈 =

푢푢 , 푉 =

푣푢

Γ =k Gr

L ,퐹 =푘 푘

4휎 푇 , 퐺푟 =

푔훽푞 휈푘푢

; 퐺푟 =푔훽∗푞∗ 휈퐷푢

(5.2.9)

푇 =(푇 − 푇 )푞 휈푘푢

,퐶 =(퐶 − 퐶 )푞∗ 휈퐷푢

,푃푟 =휈훼 , 푆푐 =

휈퐷 , 휈 =

휇휌 ,

1푃푟 =

푘휇푐 .

Equations (5.2.1) – (5.2.4) with (5.2.9) are reduced to the following non-dimensional

form:

휕푈휕푋 +

휕푈휕푌 = 0(5.2.10)

휕푈휕푡 + 푈

휕푈휕푋 + 푉

휕푈휕푌 =

휕 푈휕푌 + 퐺푟. 푇 + 퐺푚.퐶 − Γ

∂ U∂Y ∂t (5.2.11)

휕푇휕푡

+ 푈휕푇휕푋

+ 푉휕푇휕푌

=1푃푟 1 +

43퐹

휕 푇휕푌

(5.2.12)

휕퐶휕푡 + 푈

휕퐶휕푋 + 푉

휕퐶휕푌 =

1푆푐

휕 퐶휕푦 (5.2.13)

The corresponding initial and boundary conditions are:

푡 ≤ 0 ∶ 푈 = 0,푉 = 0, 푇 = 0,퐶 = 0

푡 > 0:푈 = 1, 푉 = 0,휕푇휕푦 = −1,

휕퐶휕푦 = −1푎푡푌 = 0(5.2.14)

푈 = 0, 푇 = 0,퐶 = 0,푎푡푋 = 0

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푈 → 0,푇 → 0,퐶 → 0푎푠푌 → ∞

L – is the length of the plate, 퐺푟 –the Grashof number , 푃푟–the Prandtl

number, 푆푐 - Schmidt number, 퐹- is the radiation parameter, 푁- is the buoyancy ratio

parameter, Γ- is the viscoelasticity parameter. Knowing the velocity, temperature, and

concentration fields, it is interesting to find the local skin-friction, Nusselt number and

Sherwood numbers are in the non-dimensional equation model, computed with the

following mathematical expressions.

휏 = 퐺푟 휕푈휕푌

(5.2.15)

푁푢 = −푋퐺푟 휕푇휕푌

푇(5.2.16)

푆ℎ = −푋퐺푟 휕퐶휕푌

퐶(5.2.17)

We note that the dimensionless model defined by equations (5.2.10) –(5.2.13) under

conditions (5.2.14) reduces to Newtonian flow in the case of vanishing viscoelasticity,

i.e. when Γ → ∞.

5.3. NUMERICAL TECHNIQUE:

In order to solve these unsteady, non-linear coupled equations (5.2.10) –(5.2.13)

under conditions (5.2.14) an implicit finite difference scheme of The Crank- Nicolson

type has been employed. The finite difference equations corresponding to equations

(5.2.10) –(5.2.13) are as follows.

푈 , −푈 , + 푈 , −푈 , + 푈 , −푈 , + 푈 , − 푈 ,

4∆푋

+ 푉 , − 푉 , + 푉 , −푉 ,

2∆푌= 0(5.3.1)

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푈 , − 푈 ,

∆푡+ 푈 ,

푈 , − 푈 , + 푈 , − 푈 ,

2∆푋+푉 ,

푈 , −푈 , +푈 , − 푈 , 4∆푌

=

푈 , − 2푈 , + 푈 , + 푈 , − 2푈 , + 푈 ,

2(∆푌)

−Γ푈 , − 2푈 , + 푈 , +푈 , − 2푈 , + 푈 ,

2(∆푌) ∆푡+ 퐺푟.

푇 , + 푇 ,

2

+퐺푚.퐶 , + 퐶 ,

2 (5.3.2)

푇 , − 푇 , Δ푡 + 푈 ,

푇 , − 푇 , + 푇 , − 푇 ,

2Δ푋 +푉 ,푇 . − 푇 , + 푇 , 푇 ,

4∆푌

=1푃푟

1 +4

3퐹푇 , − 2푇 , + 푇 , + 푇 , − 2푇 , + 푇 ,

2(∆푌) (5.3.3)

퐶 , − 퐶 ,

∆푡+푈 ,

퐶 , − 퐶 , + 퐶 , − 퐶 ,

2∆푋+푉 ,

퐶 , − 퐶 , + 퐶 , − 퐶 ,

4∆푌=

퐶 , − 2퐶 , + 퐶 , + 퐶 , − 2퐶 , + 퐶 ,

2푆푐(∆푌) (5.3.4)

The boundary condition at Y= 0 for the temperature in the finite difference form is

푇 , −푇 , + 푇 , −푇 ,

4.∆푌= −1(5.3.5)

At Y= 0 (i.e., j = 0), equation (5.3.3) becomes

푇 , − 푇 , Δ푡 + 푈 ,

푇 , − 푇 , + 푇 , − 푇 ,

2Δ푋

+푉 ,푇 , − 푇 , + 푇 , − 푇 ,

4∆푌

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=1푃푟 1 +

43퐹

푇 , − 2푇 , + 푇 , + 푇 , − 2푇 , + 푇 ,

2(∆푌) (5.3.6)

After eliminating 푇 + 푇 , by using the equation (5.3.5), equation (5.3.6) reduces

to the following form

푇 , − 푇 , Δ푡 + 푈 ,

푇 , − 푇 , + 푇 , − 푇 ,

2Δ푋 − 푉 ,

=1푃푟 1 +

43퐹

푇 , − 푇 . + 푇 , − 푇 , + 2∆푌(∆푌) (5.3.7)

The boundary conditions at 푌 = 0 for the concentration in the finite difference form is

퐶 , − 퐶 , + 퐶 , − 퐶 ,

4∆푌 = −1(5.3.8)

At 푌 = 0 (i.e., j= 0), equation (5.3.4) becomes

퐶 , − 퐶 ,

∆푡+ 푈 ,

퐶 , − 퐶 , + 퐶 , − 퐶 ,

2∆푋+푉 ,

퐶 , − 퐶 , + 퐶 , − 퐶 ,

4∆푌=

퐶 , − 2퐶 , + 퐶 , + 퐶 , − 2퐶 , + 퐶 ,

푆푐(∆푌) (5.3.9)

After eliminating 퐶 , + 퐶 , by using the equations (5.3.8), equation (5.3.9)

reduces to the following +form

퐶 , − 퐶 ,

∆푡 + 푈 ,퐶 , − 퐶 , + 퐶 , − 퐶 ,

2∆푋 −푉 ,

= 1푆푐

퐶 , − 퐶 , + 퐶 , − 퐶 , + 2∆푌(∆푌) (5.3.10)

The region of integration is considered as a rectangle with sides 푋 (1)and

푌 (= 14) where 푌 corresponds to 푌 = ∞, which lies very well outside the

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momentum, energy and concentration boundary layers. The grid system is shown in the

following figure.

After exercising some preliminary numerical experiment the mesh sizes have been

fixed as ∆푋 = 0.05, ∆푌 = 0.25 with the time step ∆푡 = 0.01. The computations are

executed initially, reducing the spatial mesh sizes by 50% in one direction, and later in

both directions by 50%. The results are compared. It is observed that in all cases, the

results differ in only fifth decimal places. Hence, the choice of mesh sizes is verified as

extremely efficient. During any one time step, the coefficients 푈 , and 푉 , appearing

in the difference equations are treated as constants. Here 푖designates the grid point

along the X- direction, j- along the Y-direction. The values of 퐶,푇,푈 and 푉 are known

at all grid points when 푡 = 0 from the initial conditions, the computations for 퐶,푇,푈

and 푉 at a time level (푛 + 1), using the values at previous time level (푛) are carried out

as follows.

The finite difference equations at every internal nodal point on a particular

푖 level constitutes a tridiagonal system of equations and is solved by using Thomas

algorithm. Thus the values of C are known at every nodal point at a particular

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푖- level at (푛 + 1)th time level. In a similar way computations are carried out by moving

along 푖- direction. After computing values corresponding to each 푖 at a time level, the

values at the next time level are determined in a similar manner. Computations are

repeated until the steady state is reached. Here the steady state solution is assumed to

have reached when the absolute difference between the values of the velocity 푈 ,

temperature 푇, as well as the concentration 퐶 at two consecutive time steps are less

than 10 at all grid points. The scheme is unconditionally stable. Here the local

truncation error given by 푂(∆푡 + ∆푋 + ∆푌 ) and it tends to zero as ∆푡,

∆푋푎푛푑∆푌 tend to zero. It follows that the CNM scheme is compatible and stable

ensure the convergence.

5.4. RESULTS AND DISCUSSION

In order to check the accuracy of our numerical results, the present study is

compared with the available theoretical solution of Soundalgekar and Patil [4], and they

are found to be in good agreement. A representative set of numerical results is shown

graphically in figs.5.2(a)–5.7(c), to illustrate the influence of physical parameters, viz.

viscoelasticity parameter () = 0.005, thermal Grashof parameter (퐺푟) = 1.0, species

Grashof parameter (퐺푚) = 1.0, Schmidt number(푆푐) = 0.25 (oxygen diffusing in the

viscoelastic fluid), radiation parameter(퐹) = 0.5 and Prandtl number (푃푟) = 0.71 (water-

based solvents). All graphs therefore correspond to these values unless specifically

otherwise indicated. For the special case of 푆푐 = 1, the species diffuses at the same rate

as momentum in the viscoelastic fluid. Both concentration and boundary layer

thicknesses are the same for this case. For 푆푐 < 1, species diffusivity exceeds

momentum diffusivity, so that mass is diffused faster than momentum. The opposite case

applies when 푆푐 > 1. An increase in 푆푐will suppress concentration in the boundary

layer regime since higher 푆푐values will physically manifest in a decrease of molecular

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diffusivity (퐷) of the viscoelastic fluid, i.e. a reduction in the rate of mass diffusion.

Lower 푆푐 values will exert the reverse influence since they correspond to higher

molecular diffusivities. Concentration boundary layer thickness is generally greater for

푆푐 < 1 than for 푆푐 > 1.

Figs. 5.2a to 5.2c we have presented the variation of velocity (푈), temperature

function (푇) and concentration (퐶) versus (푌) with on viscoelasticity () at 푋 = 1.0. An

increase in from 0 to 0.001, 0.003, 0.005 and the maximum value of 0.007, as depicted

in figure 5.2a, clearly enhances the velocity,푈 which ascends sharply and peaks in close

vicinity to the plate surface (푌 = 0). With increasing distance from the plate wall however

the velocity 푈 is adversely affected by increasing viscoelasticity, i.e. the flow is

decelerated. Therefore close to the plate surface the flow velocity is maximized for the

case of a Newtonian fluid (vanishing viscoelastic effect, i.e. =0) but this trend is reversed

as we progress further into the boundary layer regime. The switchover in behaviour

corresponds to approximately 푌 = 1. With increasing (푌) velocity profiles decay smoothly

to zero in the free stream at the edge of the boundary layer. The opposite effect is caused

by an increase in time. A rise in ‘푡’ from 4.310 through 4.370, 4.470, 4.530 to 4.940 causes

a decrease in flow velocity, 푈 nearer the wall in this case the maximum velocity arises for

the least time progressed. With more passage of time (푡 = 4.940) the flow is decelerated.

Again there is a reverse in the response at 푌~1, and thereafter velocity is maximized with

the greatest value of time. In figure 5.2(b), increasing viscoelasticity is seen to decrease

temperature throughout the boundary layer. All profiles decay from the maximum at the

wall to zero in the free stream. The graphs show therefore that increasing viscoelasticity

cools the flow. With progression of time, however the temperature, 푇 is consistently

enhanced, i.e. the fluid is heated as time progresses. A similar response is observed for the

concentration field, 퐶, in fig. 5.2(c). Increasing viscoelasticity again reduces concentration,

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showing that species diffuses more effectively in Newtonian fluids (=0) than in strongly

viscoelastic fluids. Once again with greater elapse in time the concentration values are

reduced throughout the boundary layer regime (0 < 푌 < 14).

Figs. 5.3(a) to 5.3(c) depict the distributions of velocity (푈) and concentration (퐶)

versus coordinate (푌) for various Schmidt numbers (푆푐) and time (푡), close to the leading

edge at 푋 = 1.0, are shown. These figures again correspond to a viscoelasticity parameter

() = 0.005, i.e. weak elasticity and strongly viscous effects. An increase in푆푐 from 0.1

(low weight diffusing gas species) through 0.5 (oxygen diffusing) to 1.0 (denser

hydrocarbon derivatives as the diffusing species), 3.0 and 5.0, clearly strongly decelerates

the flow. In fig. 5.3(a), the maximum velocity arises at 푆푐 = 0.1 very close to the wall.

All profiles then descend smoothly to zero in the free stream. With higher Sc values the

gradient of velocity profiles is lesser prior to the peak velocity but greater after the peak.

An increase in time, t, is found to accelerate the flow. However with progression of time in

fig. 5.3(b), the temperature is found to be decreased in the boundary layer regime. Fig.

5.3(c), shows that a rise in 푆푐 strongly suppresses concentration levels in the boundary

layer regime.푆푐 defines the ratio of momentum diffusivity () to molecular diffusivity (퐷).

An increase in Schmidt number effectively depresses concentration values in the boundary

layer regime. Since higher 푆푐 values will physically manifest in a decrease of molecular

diffusivity (퐷) of the viscoelastic fluid, i.e. a reduction in the rate of mass diffusion. Lower

푆푐 values will exert the reverse influence since they correspond to higher molecular

diffusivities. Concentration boundary layer thickness is therefore considerably greater for

푆푐 = 0.1 than for 푆푐 = 5.0.

In Figs. 5.4(a) to 5.4(c) it is of great interest to show how the radiation-conduction

interaction effects on velocity, temperature and concentration. It is observed that, initially

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for lower values of the radiation parameter 퐹, the heat transfer is dominated by conduction,

as the values of 퐹 increases the radiation absorption in boundary layer increases, i.e. an

increase in the radiation parameter 퐹 from 0.1 through 0.5, 1.0, 3.0, 5.0 and 10.0 results

in a decrease in the temperature and increase in concentration shown in figs.5.4(b) and

5.4(c).Here the velocity profiles increases initially and then slowly decreases to zero due to

viscoelasticity shown in fig. 5.4(a), i.e. when the radiation parameter 퐹 is increasing,

observed that the time progress initially increases and slowly decreases.

The effects of thermal Grashof number 퐺푟 are shown in Figs. 5.5(a) to 5.5(c) for

the velocity 푈, temperature 푇 , and species concentration 퐶 , distributions. figure 5.5(a)

indicates that an increasing 퐺푟from 0.1 through 1, 2, 4 to 6 strongly boosts velocity in

particular over the zone 0 < Y < 2. There is a rapid rise in the velocity near the wall

especially for the cases 퐺푟= 4 and 퐺푟 = 6. Peak value of 퐺푟 = 6 is about 푌~0.75 the

profiles generally descend smoothly towards zero although the rate of descend is greater

corresponding to higher thermal Grashof numbers. 퐺푟 defines the ratio of thermal

buoyancy force to the viscous hydrodynamic force and as expected does accelerate the

flow. Temperature distribution 푇 versus 푌 is plotted in figure 5.5b and is seen to decrease

with a rise in thermal Grashof number, results which agree with fundamental studies on

free convection (Schlichting boundary layer theory). Increasing 퐺푟 values figure 5.5c is

seen to considerably reduce concentration function values (퐶) throughout the boundary

layer.

The effects of species Grashof number 퐺푚 on velocity, temperature function and

concentration function are presented in figs. 5.6(a) to 5.6(c). Velocity, 푈, is observed to

increase considerably with rise in 퐺푚 from 0.1 to 6. Hence species Grashof number

boosts velocity of the fluid indicating that buoyancy as an accelerating effect on the flow

field. Temperature 푇, undergoes marked decrease in value however with a rise in species

151

Grashof number as illustrated in fig 7b. All temperatures descend from unity at the wall to

zero in the free stream. The depression in temperature is maximized by larger species

Grashof number in the vicinity 푌~2.1 , indicating a similar trend to the influence of

thermal Grashof number (figure 5.6b). Dimensionless concentration, 퐶as depicted in

figure 5.6c, is also seen to be reduced by increasing the species Grashof number. All

profiles decay smoothly from unity at the vertical surface to zero as 푦 ⟶ ∞ .

In figs. 5.7(a) to 5.7(c) the variation of dimensionless local skin friction (surface

shear stress), x, Nusselt number (surface heat transfer gradient), Nux and the Sherwood

number (surface concentration gradient), Shx, versus axial coordinate (푋 ) for various

viscoelasticity parameters () and time (푡) are illustrated. Shear stress is clearly enhanced

with increasing viscoelasticity, i.e. the flow is accelerated, a trend consistent with our

earlier computations in figure 5.6(a). The ascent in shear stress is very rapid from the

leading edge (푋 = 0) but more gradual as we progress along the plate surface away from

the plane. With an increase in time, t, shear stress, x, is however increased. Increasing

viscoelasticity () is observed in figure 5.7(b) to enhance local Nusselt number, Nux values

whereas they are again decreased with greater time. Similarly in figure 5.7(c), the local

Sherwood number Shx values are elevated with an increase in elastic effects, i.e. a rise in

from 0 (Newtonian flow) through 0.001, 0.003 to 0.005 but depressed finally in slightly

with time.

Table: It is observed that an increase in the radiation parameter 퐹 from 0.1 through

1.0, 10.0 and 100 results decrease in the temperature and increase in concentration, i.e. the

amount of heat transfer decreases (Nux), skin-friction increases gives shear stress in the

flow causes reduction in mass transfer rate observed.

152

5.5. CONCLUSION

A two-dimensional, unsteady laminar incompressible boundary layer model has

been presented for the external flow, heat and mass transfer in a Walters-B viscoelastic

buoyancy-driven radiative flow from an impulsively started vertical plate with uniform

heat and mass flux. The Walters-B viscoelastic model has been employed which is valid

for short memory polymeric fluids. The dimensionless conservation equations have been

solved with the well-tested, robust, highly efficient and implicit Crank-Nicolson finite

difference numerical method. The present computations have shown that increasing

viscoelasticity accelerates the velocity and enhances shear stress (local skin friction), local

Nusselt number and local Sherwood number, but reduces temperature and concentration in

the boundary layer. Here an increase of radiation, effects the amount of heat transfer

decreases (Nux), skin-friction increases gives shear stress in the flow causes reduction in

mass transfer rate observed. The temperature decreases with increase in 푃푟 or radiation

parameter 퐹 and also time required to reach steady state increases, when the radiation

parameter 퐹increases. This phenomenon is of interest in very high temperature (e.g. glass)

flows in the mechanical and chemical process industries and is currently under

investigation.

153

5.6. GRAPHS:

154

155

156

157

158

159

160

161

162

TABLE: 횪 = ퟎ.ퟎퟎퟓ, 퐒퐜 = ퟎ.ퟔ, 퐆퐫 = 퐆퐦 = ퟎ.ퟐ

Pr

F

X = 0.1 X = 0.5 X = 1.0

휏 푁푢 푆ℎ 휏 푁푢 푆ℎ 휏 푁푢 0.59805

0.71

0.1 3.34991 0.10941 0.18077 0.51558 0.55349 0.41042 0.01284 1.10697 0.59805

1.0 3.35116 0.07542 0.18075 0.51561 0.55348 0.41041 0.01284 1.10697 0.59804

10 3.35279 0.00432 0.18072 0.51560 0.55344 0.41041 0.01286 1.10696 0.59804

100 3.35315 -0.0132 0.18071 0.51568 0.55340 0.41040 0.01286 1.10696 0.59805

2.0

0.1 3.35033 0.10164 0.18076 0.51559 0.55348 0.41041 0.01284 1.10697 0.59804

1.0 3.35388 0.05071 0.18070 0.51571 0.55318 0.41040 0.01287 1.10696 0.59803

10 3.35724 0.00866 0.18065 0.51602 0.53870 0.41038 0.01291 1.10687 0.59803

100 3.35784 -0.0370 0.18064 0.51614 0.52943 0.41037 0.01292 1.10674 0.59804

7.0

0.1 3.35211 0.13625 0.18073 0.51564 0.55348 0.41041 0.01286 1.10697 0.59801

1.0 3.36016 0.00117 0.18060 0.51691 0.43854 0.41034 0.01301 1.10237 0.59796

100 3.36379 -0.4230 0.18054 0.52041 0.34453 0.41025 0.01389 0.93232 0.59805

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