Exact solution for thermal boundary layer in Casson fluid flow over permeable shrinking sheet with variable wall temperature and thermal radiation

ORIGINAL ARTICLE

Exact solution for thermal boundary layer

in Casson fluid flow over permeable shrinking

sheet with variable wall temperature and thermal

radiation

Krishnendu Bhattacharyya

a,b,*

, M.S. Uddin

c

, G.C. Layek

b

a

Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, Uttar Pradesh, India

b

Department of Mathematics, The University of Burdwan, Burdwan 713104, West Bengal, India1

c

Mathematics Discipline, Khulna University, Khulna 9208, Bangladesh Received 19 June 2013; revised 30 January 2016; accepted 13 March 2016 Available online 21 April 2016

KEYWORDS

Thermal boundary layer; Casson fluid;

Shrinking sheet; Exact dual solutions; Variable wall temperature; Thermal radiation

Abstract An analysis of thermal boundary layer in the flow of Casson fluid over a permeable shrinking sheet with variable wall temperature and thermal radiation is made. Using similarity transformations, self-similar nonlinear ODEs are obtained from the governing equations. Dual exact solutions of transformed velocity and energy equations are obtained. From the plotted results it can be observed that the temperature inside the boundary layer decreases with Casson parameter and wall mass transfer parameter in first solution and it increases in second solution. Whereas, temperature decreases for larger values of Prandtl number, radiation parameter and power-law exponent for inverse variation along the sheet in both solutions and it enhances with power-law exponent for direct variation along the surface. Also, thermal boundary layer thickness reduces with stronger thermal radiation and inverse variation of wall temperature along the surface and it becomes thicker with direct variation of wall temperature. The rate of heat transfer is less with increasing values of power-law exponent for direct variation along the sheet and for inverse varia-tion it is higher. In graphical representavaria-tion of temperature field, temperature overshoot is observed in certain cases. So, in some situations heat absorption at surface occurs instead of heat transfer from surface.

Ó2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

* Corresponding author at: Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi 221005, Uttar Pradesh, India. Tel.: +91 9474634200.

E-mail addresses:[email protected],[email protected](K. Bhattacharyya). 1 _{Previous affiliation of K. Bhattacharyya.}

Peer review under responsibility of Faculty of Engineering, Alexandria University.

H O S T E D BY

Alexandria University

Alexandria Engineering Journal

www.elsevier.com/locate/aej

www.sciencedirect.com

http://dx.doi.org/10.1016/j.aej.2016.03.010

1110-0168Ó2016 Faculty of Engineering, Alexandria University. Production and hosting by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

The boundary layer flow due to a shrinking sheet is emerged as an interesting problem in fluid dynamics. The shrinking sheet flows occur in some practical situations, such as, for rising shrinking balloon and it is very useful in packaging of bulk products. The heat transfer plays a vital role in the flow due to shrinking. Miklavcˇicˇ and Wang[1]demonstrated that the steady flow of Newtonian fluid over a shrinking sheet is possi-ble only when the adequate amount of mass suction through the porous sheet is applied. Actually, the mass suction sup-presses the generated vorticity due to shrinking inside the boundary layer and maintains boundary layer flow. Later, Fang and Zhang[2]explained the influence of external mag-netic field on the shrinking sheet flow and found that the strong magnetic field guarantees the steady boundary layer flow. Thereafter, the Newtonian fluid flow past a shrinking sheet is investigated by many researchers[3–15]under various physical aspects.

On the other hand, Hayat et al.[16–18]discussed the non-Newtonian fluid flows over a shrinking sheet. In reality, the majority of the fluids appeared in the technological processes shows non-Newtonian fluid properties[19–25]. So, the studies

of non-Newtonian flows are very crucial in technological point of view. Some significant investigations on the shrinking sheet flow of non-Newtonian fluids were reported by Ishak et al. [26,27], Yacob et al. [28,29] and Rosali et al. [30]. Later, Nadeem et al.[31]investigated MHD flow of a Casson fluid over an exponentially shrinking sheet. Recently, Bhat-tacharyya et al.[32,33]discussed the flow dynamics of Casson fluid over stretching/shrinking sheet with and without effect of magnetic field.

The thermal radiation effect on the boundary layer flow is relevant in many engineering problems because of its applica-tions especially in high temperature engineering processes. Heat radiation effect is important in controlling the quality of the final product as it affects the rate cooling. Due to the above fact, important works on thermal radiation effect had been done by some researchers, viz., Hossain and Takhar [34], Elbashbeshy [35], Mukhopadhyay [36], Mukhopadhyay et al.[37]Akbar et al.[38]and Bhattacharyya[39]. The radia-tion effects on the boundary layer flow of Newtonian fluid and heat transfer over shrinking sheet were investigated by Bhat-tacharyya and his co-workers[40–42]and Ali et al.[43]. The radiation effect becomes more interesting when the wall tem-perature distribution is variable. Pal et al. [44] showed the effect of thermal radiation on convection–dissipation heat First solution Second solution = 0.5 = 0.7 = 1 = 2 = 5 8 = = 3.5 S β β β β β β (η) f η / −1 −0.8 −0.6 −0.4 −0.2 0 0 2 4 6 8 10 12 14

Figure 1 Effect ofbon the velocity profilesf0(g).

η η

,

1

=

n

=

1

R

S

R

=

1

,

n

=

1

S

R

=

1

,

n

=

1

S

1.7, 1.9, 1.95, 2,

= 0.5, 0.7, 1, 1.4,

2.05

β

θ (η)

,

1

=

n

=

1

R

S

θ (η)

β

= 0.5, 0.7, 1, 2, 5

Second dolution

First solution

,

1

=

=

1

= 1.5,

= 3.5,

Pr

R

n

S

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4

Figure 2 Effect ofbon the temperature profilesh(g).

= 2.5 = 2.7 = 3 = 3.5 = 4 f (η) η / = 2 β S S S S S First solution Second solution −1 −0.8 −0.6 −0.4 −0.2 0 0 2 4 6 8 10 12 14 16

transfer over stretching/shrinking sheet with variable surface temperature.

But very little is known about the simultaneous effects of radiation and variable wall temperature on the boundary layer flow of non-Newtonian fluid and heat transfer. So, the effects of thermal radiation and the variable surface temperature on the thermal boundary layer of Casson fluid flow over a shrink-ing sheet are still unknown. The significance of heat radiation effects on the heat flow when wall temperature is variable motivates us to study this paper. Hence, in the present paper the heat transfer in boundary layer flow of non-Newtonian Casson fluid over a shrinking sheet with variable wall temperature and thermal radiation is investigated. Similarity transformations are used and the exact solutions for velocity and temperature distributions are obtained.

2. Flow analysis

Consider the steady two-dimensional incompressible flow of Casson fluid bounded by a shrinking sheet aty= 0, the flow being confined iny> 0. We assume also that the rheological equation of state for an isotropic and incompressible flow of a Casson fluid can be written as follows (see Nakamura and Sawada[45]): sij¼ lBþpy= ffiffiffiffiffiffi 2p p 2eij; p>pc lBþpy= ffiffiffiffiffiffiffi 2pc p 2eij; p<pc; ( ð1Þ where lB is plastic dynamic viscosity of the non-Newtonian fluid,pyis the yield stress of fluid,pis the product of the com-ponent of deformation rate with itself, namely,p=eijeij,eijis the (i,j)-th component of the deformation rate andpcis critical value ofpbased on non-Newtonian model.

Under these conditions the boundary layer equations for the steady flow of Casson fluid may be written as follows[33]: @u @xþ @v @y¼0; ð2Þ u@u @xþv @u @y¼t 1þ 1 b @2 u @y2; ð3Þ

whereuandvare the velocity components inxandydirections respectively, t is the kinematic fluid viscosity, q is the fluid density, b¼l_B ffiffiffiffiffiffiffi2pc

p

=p_y is the non-Newtonian (Casson) parameter.

The boundary conditions for the velocity components are u¼ Uw; v¼ vw at y¼0;u!0 as y! 1; ð4Þ whereUw=cx is shrinking velocity of the sheet withc(>0) being the shrinking constant. Herevwis the wall mass transfer velocity with vw>0 for mass suction and vw<0 for mass injection.

The following similarity transformations are introduced: w¼ ffiffiffiffiffiffiffiffiffiffiffitxUw p fðgÞandg¼y ffiffiffiffiffiffiffi Uw tx r ; ð5Þ wherewis the stream function defined in the usual notation as u¼@_@w_y and v¼ @_@w_xandgis the similarity variable.

In view of the relations in(5), the Eq.(2)satisfies automat-ically and the Eq. (3) reduces to the following self-similar equation:

1þ1 b

f000þff00f02¼0; ð6Þ where primes denote differentiation with respect tog.

The boundary conditions become

fðgÞ ¼S;f0ðgÞ ¼ 1 at g¼0;f0ðgÞ !0 as g! 1; ð7Þ where S = vw/(ct)1/2 is wall mass transfer parameter with

S> 0 (i.e. vw>0) for mass suction andS< 0 (i.e. vw<0) for mass injection.

3. Heat transfer analysis

To know the behaviour of temperature distribution in the abovementioned flow field, the following energy equation using boundary layer approximation (neglecting viscous dissipation) needs to be solved:

u@T @xþv @T @y¼ j qcp @2_T @y2 1 qcp @q_r @y; ð8Þ η η

3.4, 3.5, 3.55

S

= 2.5, 2.7, 3, 3.3,

θ (η)

S

= 2.5, 2.7, 3, 3.5 ,4

θ (η)

= 1.5,

Pr

= 2,

β

R

= 1,

n

= 1

First solution

Second solution

0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 6

whereTis the temperature,jis the thermal conductivity,cpis the specific heat andqris the radiative heat flux.

The appropriate boundary conditions are

T¼Tw¼T1þT0xn at y¼0;T!T1 as y! 1; ð9Þ

whereTwis the variable temperature along the sheet,T1is the free stream temperature assumed to be constant, T0 is a

constant depended on the thermal properties of the fluid and nis a power-law exponent.

Using Rosseland approximation for radiation [46], qr=(4r/3k1)oT

4

/oy is obtained, where r is the Stefan– Boltzmann constant andk1 is the absorption coefficient. We

presume that the temperature variation within the flow is such thatT4may be expanded in a Taylor’s series. ExpandingT4 about T₁ and neglecting higher order terms we get, T4= 4T3₁T3T4₁.

Now Eq.(8)reduces to

u@T @xþv @T @y¼ j qcp @2 T @y2 þ 16rT3 1 3k1qcp @2 T @y2: ð10Þ

Next, the dimensionless temperaturehis introduced as hðgÞ ¼ TT1

TwT1: ð11Þ

Using the relations in(5) and (11), the Eq.(10)reduces to

ð3Rþ4Þh00þ3RPr½fh0nf0h ¼0; ð12Þ where primes denote differentiation with respect to g, Pr = cpl/j is the Prandtl number andR=j*k1/4rT13 is the

thermal radiation parameter.

The boundary conditions for h are obtained from (9) as follows: hðgÞ ¼1 at g¼0;hðgÞ !0 as g! 1: ð13Þ

1

=

,

5

.

3

=

3

.

5

,

=

1

=

3

.

5

,

=

1

=

3

.

5

,

=

1

=

3

.

5

,

=

1

=

3

.

5

,

=

1

=

3

.

5

,

=

1

=

= 1.49, 1.5, 1.52,

1.55, 1.6, 1.7, 2,

2.5

Pr

θ (η)

= 0.98, 0.99, 1, 1.04,

R

1.1, 1.2, 1.5, 2.5, 5

θ (η)

= 0.5, 0.7, 1, 1.5,

2.5, 5

R

θ (η)

= 3.5,

S

= 2,

β

Second solution

First solution

= 1

n

= 1.5,

Pr

η η

1

=

,

5

.

3

=

3

.

5

,

=

1

=

2.5, 4

= 0.5, 0.7, 1, 1.5,

Pr

Second solution

First solution

S

= 2,

β

=

3

.

5

,

R

= 1,

n

=

1

θ (η) η (b) (a) 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 5 10 15 20 η 0 5 10 15 20 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 0 2 4 6 8 10

η η η η

= 0, −3, −8, −15

n

θ (η)

= 0, −3, −8, −15

First solution

θ (η)

n

Second solution

= 3.5,

S

= 2,

β

Pr

= 1.5,

R

= 1

= 0, 0.5, 0.8, 0.9,

0.95, 0.98, 0.99,

1, 1.01

n

θ (η)

= 0, 1, 3, 6, 10,

12, 14

n

θ (η)

Pr

= 1.5,

R

= 1

= 3.5,

S

= 2,

β

Second solution

First solution

(b) (a) 0 0.2 0.4 0.6 0.8 1 0 1 2 3 4 5 0 0.2 0.4 0.6 0.8 1 0 0.5 1 1.5 2 2.5 3 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 0 4 8 12 16 0 0.2 0.4 0.6 0.8 1 1.2 0 1 2 3 4

Figure 6 Effect ofn(a.npositive and b.nnegative) on the temperature profilesh(g).

(0) Second solution First solution Solid line 8 S f // = 0.5, 0.7, 1, 2, 5, β Broken line 0 0.5 1 1.5 2 2.5 3 3.5 4 2 2.5 3 3.5 4

Figure 7 Values off00(0) againstSfor various values ofb.

= 0.5, 0.7, 1, 2, 5, 8 β = 1 n = 1, = 1.5, −θ (0) / S Pr R First solution Second solution Broken line Solid line −2 −1 0 1 2 3 4 2 2.5 3 3.5 4 4.5 5 5.5 6

4. Solution procedure

Exact closed form solutions of Eqs. (6) and (12) with the boundary conditions(7) and (13)will be obtained.

Let us assume that the solution of Eqs.(6)with(7)has the form[2]fðgÞ ¼aþbekg, wherea,bandk(>0) are constants. Substituting in Eqs.(6) and (7), the following is obtained:

b¼1 k; a¼S 1 k and k¼ S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2_4ð1þ1=bÞ q 2ð1þ1=bÞ : ð14Þ Now the closed form solution is obtained as

fðgÞ ¼S1 kþ 1 kekg i:e:; fðgÞ ¼S 2ð1þ1=bÞ S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2_4ð1þ1=bÞ q þ 2ð1þ1=bÞ S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S24ð1þ1=bÞ q e SpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS24ð1þ1=bÞ 2ð1þ1=bÞ g: ð₁₅Þ So,f0ðgÞ ¼ e SpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiS24ð1þ1=bÞ 2ð1þ1=bÞ g _andf00ð₀Þ ¼k¼S ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi S2_4ð1þ1=bÞ p 2ð1þ1=bÞ .

For physically feasible solution of the steady Casson fluid flowkbe positive, the steady flow is possible only if wall mass suction parameterS(>0) satisfies the following:

S2P4ð1þ1=bÞ: ð16Þ

The similarity solution is unique ifS2_{¼4ð1þ1=bÞ, it is of}

dual nature ifS2_{>4ð1þ1=bÞand no similarity solution exists}

forS2_{<4ð1þ1=bÞ.}

Substituting the solution of the velocity field, the Eq.(12) becomes h00_þ 3RPr ð3Rþ4Þ S 1 kþ 1 ke kg h0_þ 3nRPr ð3Rþ4Þe kg_h¼ 0: ð17Þ = 2 = 1 = 1 Broken line First solution Second solution Solid line Pr R n β −θ (0) / = 1, 1.5, 2, 3, 4 −2 0 2 4 6 8 S 2.5 3 3.5 4 4.5 5 5.5 6

Figure 9 Values ofh0(0) againstSfor various values ofPr.

−θ = 0.5, 1, 2, 5 R Second solution First solution Solid line Broken line Pr= 1.5 = 2 β = 1 n (0) / S −2 0 2 4 6 8 2.5 3 3.5 4 4.5 5 5.5 6

Figure 10 Values ofh0(0) againstSfor various values ofR.

= 0, 0.2, 0.5, 0.7, 0.8, 0.9 First solution Second solution = 2 = 1.5 β Pr R = 1 −θ S (0) / n Solid line Broken line = 1, 1.5, 2, 2.5, 3 n = 2 = 1.5 = 1 β Pr R −θ Second solution Broken line

Solid line First solution

(0) /

(b)

(a)

0.5 1 1.5 2 2.5 3 3.5 2.5 3 3.5 4 4.5 5 5.5 6 −2 0 2 4 6 8 S 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Figure 11 Values ofh0(0) against S for various values of n

(>0). = −1, −3, −7, −13, −20 n = 1 R Pr β −θ S (0) /

Solid line First solution Broken line Second solution = 2 = 1.5 1 2 3 4 5 6 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7

Figure 12 Values ofh0(0) against S for various values of n

Now, a new variablee¼ 3RPr

ð3Rþ4Þk2e

kg _{is introduced and so}

the Eq.(17)reduces to e@ 2_h @e2þ ðheÞ @h @egh¼0; ð18Þ whereh¼1 3RPr kð3Rþ4ÞðS1kÞandg¼ n.

The boundary conditions are h 3RPr

ð3Rþ4Þk2

¼1 and hð0Þ ¼0: ð19Þ

Therefore, the solution for Eq.(18)is given by

hðeÞ ¼A1Mðg;h;eÞ þA2e1hMðgþ1h;2h;eÞ ð20Þ

whereMis the confluent hypergeometric function of the first kind or Kummer function with A1 and A2 arbitrary being

constants.

Using(19)and submittinghandgwe obtain from(20)

hðeÞ ¼ ð3Rþ4Þk2_e 3RPr 3RPr ð3Rþ4Þkð ÞS1k M 3RPr ð3Rþ4ÞkðS1kÞ n;1þð33RRPrþ4Þk S1k ;e M 3RPr ð3Rþ4Þk S1k n;1þ 3RPr ð3Rþ4Þk S1k ; 3RPr ð3Rþ4Þk2 :

Then the temperature solution becomes

hðgÞ ¼e 3RPr ð3Rþ4Þð ÞS1kg_M 3RPr ð3Rþ4Þk S1k n;1þ 3RPr ð3Rþ4Þk S1k ; 3RPr ð3Rþ4Þk2e kg M 3RPr ð3Rþ4Þk S1k n;1þ 3RPr ð3Rþ4Þk S1k ; 3RPr ð3Rþ4Þk2 : ð21Þ

The wall temperature gradient is given as

5. Results and discussion

To visualize the effects of various physical parameters on the velocity and temperature fields the exact solutions are plotted in some figures.

The effects of Casson parameterbon the velocity and tem-perature profiles are presented in Figs. 1and 2, respectively. The velocityf0(g) at a point increases withbfor first solution

and the effect ofb is opposite for second solution. Also, the velocity boundary layer thickness decreases with increasing values of b for first solution and it increases for second solution. Similarly, the thermal boundary layer thickness decreases with b for first solution and increases for second solution. Moreover, it is important to note that for large values ofbtemperature overshoot is observed in second solution. For second solution, it occurs because of the reverse heat flow near the wall, i.e., heat transfers from ambient fluid to the sheet.

The dimensionless velocity and temperature profiles for various values of wall mass transfer parameter are depicted inFigs. 3and4, respectively. Similar to that of Casson param-eter, it is observed that for increasing mass suction (S> 0) the velocity and thermal boundary layer thicknesses reduce for first solution and those become thicker for second solution. For stronger mass suction temperature overshoot is obtained in second solution only.

InFig. 5, the temperature profiles for various values of the Prandtl numberPrand the radiation parameterRare plotted. Due to increase in the Prandtl number the temperature at a point decreases and the thermal boundary layer thickness reduces. Similar effects are noticed for the variation inR. In both cases, it is worth noting that for smaller values of Pr andRthermal overshoots are found in second solution.

The variations in dimensionless temperature profiles for dif-ferent values of power-law index n are exhibited in Fig. 6. When the wall temperature directly varies with the distance along the sheet, i.e., whenn> 0 the temperature in the flow field increases withnfor both solutions. On the other hand,

for inverse variation of wall temperature (n< 0) with the dis-tance along the sheet, the temperature decreases with the increasing magnitude ofn. More importantly, thermal over-shoot is viewed in some situations for direct variation (n> 0) in first solution (for largen) and in addition, for sec-ond solution the overshoot is always found (even for small n). The thermal overshoot implies that there heat transfers from the ambient fluid to the surface, i.e., heat absorption.

Table 1 Values off00(0) for various values ofbandS.

b S 3 3.5 4 4.5 5 5.5 6 0.5 1st – 0.66667 1.00000 1.22871 1.43426 1.62867 1.81650 2nd – 0.50000 0.33333 0.27129 0.23241 0.20467 0.18350 1 1st 1.00000 1.39039 1.70711 2.00000 2.28078 2.55425 2.82288 2nd 0.50000 0.35961 0.29289 0.25000 0.21922 0.19575 0.17712 2 1st 1.57735 2.00000 2.38743 2.75831 3.11963 3.47481 3.82574 2nd 0.42265 0.33333 0.27924 0.24169 0.21370 0.19186 0.17426 5 1st 2.10391 2.59561 3.06110 3.51277 3.95602 4.39367 4.82737 2nd 0.39609 0.32105 0.27223 0.23723 0.21065 0.18967 0.17263 1 1st 2.61803 3.18614 3.73205 4.26556 4.79129 5.31174 5.82843 2nd 0.38197 0.31386 0.26795 0.23444 0.20871 0.18826 0.17157 h0_{ð0Þ ¼} 3RPr ð3Rþ4Þ S 1 k þ_ð 3RPr 3Rþ4Þk 3RPr ð3Rþ4Þk S1k n 1þ 3RPr ð3Rþ4Þk S 1 k ! M 1þ 3RPr ð3Rþ4Þk S1k n;2þ 3RPr ð3Rþ4Þk S1k ; 3RPr ð3Rþ4Þk2 M 3RPr ð3Rþ4Þk S 1 k n;1þ 3RPr ð3Rþ4Þk S 1 k ; 3RPr ð3Rþ4Þk2 :

The quantities f00(0) and h0(0) related to the local skin friction coefficient and wall temperature gradient (rate of heat transfer) respectively are plotted inFigs. 7–12and also pre-sented inTables 1 and 2for several values involved parame-ters. The value of f00(0) increases with Casson parameter b and wall mass transfer parameterS for first solution and it decreases for both parameter in second solution (Fig. 7). Con-sequently, the local skin friction coefficient increases (decreases) for first (second) solution. The quantity h0(0) related to the wall temperature gradient increases with Casson parameter, wall mass transfer parameter, Prandtl number and radiation parameter, which implies that the rate of heat trans-fer increases. For some values of wall mass transtrans-fer parameter

the values ofh0(0) are negative, which means heat absorption occurs in some situations. The heat absorption is found for higher valuesband smaller values ofPrandR. Whereas, when b, Pr and R (2, 1.5 and 1 respectively) are fixed, the heat absorption is observed when nP1 (approximately). Also, the values of h0(0) decrease with increasing values of n (>0, i.e., for direct variation) and the reduction is very promi-nent for second solution while minor effect is found for first solution. For inverse variation, i.e., for n< 0 the value of

h0_{(0) increases with the magnitude of n. In addition to the}

abovementioned figures, two tables (Tables 1 and 2) are presented to show the effects and to demonstrate the current results.

Table 2 Values ofh0(0) for various values ofb,Pr,R,n, andS.

b Pr R n S 3 3.5 4 4.5 5 5.5 6 0.5 1.5 1 1 1st – 1.15415 1.78223 2.23840 2.64831 3.03443 3.40597 2nd – 0.96380 1.31949 1.82664 2.29774 2.72924 3.13457 1 1st 1.27345 1.77480 2.18333 2.56113 2.92315 3.27560 3.62169 2nd 0.73385 0.95647 1.47881 2.00401 2.45817 2.87018 3.25974 2 1st 1.52040 1.92819 2.30189 2.65958 3.00804 3.35056 3.68898 2nd 0.32261 0.70403 2.40233 2.18650 2.54154 2.93656 3.31900 5 1st 1.61279 1.99439 2.35484 2.70420 3.04682 3.38496 3.71995 2nd 3.22798 1.50380 1.73848 2.15325 2.57610 2.97424 3.35358 1 1st 1.66269 2.03187 2.38537 2.73016 3.06950 3.40514 3.73817 2nd 1.38753 1.37235 1.73036 2.17636 2.60288 2.99914 3.37629 2 1 1st 0.99025 1.26781 1.52010 1.76066 1.99449 2.22400 2.45053 2nd 0.80381 1.33692 1.06088 1.20424 1.47168 1.77028 2.05714 2 1st 2.01310 2.55825 3.05796 3.53609 4.00169 4.45919 4.91110 2nd 1.06965 1.76075 2.44503 3.02919 3.56267 4.06967 4.56004 3 1st 2.84541 3.68883 4.45800 5.19025 5.90053 6.59641 7.28221 2nd 2.21600 3.18540 4.04281 4.83436 5.58781 6.31684 7.02906 4 1st 3.47592 4.63497 5.69264 6.69557 7.66446 8.61041 9.54002 2nd 3.28267 4.50579 5.59699 6.61968 7.60148 8.55648 9.49280 1.5 0.5 1st 0.94112 1.20701 1.44838 1.67838 1.90185 2.12113 2.33754 2nd 0.87748 1.28930 1.00366 1.12463 1.37276 1.66002 1.93920 2 1st 2.10581 2.67945 3.20506 3.70779 4.19721 4.67803 5.15289 2nd 1.19420 1.92145 2.61176 3.21212 3.76638 4.29513 4.80745 5 1st 2.66733 3.43777 4.14111 4.81156 5.46258 6.10093 6.73044 2nd 1.95648 2.86601 3.67156 4.40980 5.10981 5.78567 6.44502 1 0 1st 1.54501 1.94247 2.31174 2.66703 3.01401 3.35553 3.69323 2nd 1.17786 1.60393 2.02243 2.41719 2.79388 3.15840 3.51452 0.5 1st 1.53090 1.93439 2.30624 2.66293 3.01076 3.35285 3.69096 2nd 0.91515 1.35857 1.83537 2.27052 2.67138 3.05198 3.41989 0.8 1st 1.52413 1.93044 2.30349 2.66083 3.00907 3.35143 3.68974 2nd 0.66089 1.06784 1.65305 2.16096 2.58978 2.98369 3.36019 1.5 1st 1.51406 1.92405 2.29874 2.65702 3.00586 3.34866 3.68728 2nd 1.21697 1.03657 1.36012 1.86543 2.36534 2.80620 3.21008 2 1st 1.51255 1.92215 2.29688 2.65527 3.00424 3.34716 3.68589 2nd 0.39617 0.25604 0.01527 5.69389 2.68637 2.70790 3.09406 2.5 1st 1.51663 1.92270 2.29638 2.65438 3.00321 3.34608 3.68481 2nd 0.33990 8.39716 1.86128 1.58297 1.94817 2.47395 2.95479 1 1st 1.58224 1.96368 2.32588 2.67735 3.02200 3.36198 3.69862 2nd 1.53217 1.92956 2.30223 2.65980 3.00822 3.35071 3.68911 3 1st 1.68316 2.02275 2.36521 2.70566 3.04353 3.37903 3.71255 2nd 2.02040 2.38722 2.72287 3.04648 3.36491 3.68109 3.99638 7 1st 1.94372 2.18707 2.47789 2.78750 3.10559 3.42774 3.75187 2nd 2.69807 3.03843 3.34503 3.63955 3.92964 4.21866 4.50821 13 1st 2.38533 2.49594 2.70212 2.95568 3.23538 3.53043 3.83488 2nd 3.43725 3.76094 4.04923 4.32443 4.59458 4.86339 5.13272 20 1st 2.90145 2.89304 3.00878 3.19541 3.42562 3.68375 3.96031 2nd 4.11493 4.42893 4.70643 4.96992 5.22765 5.48348 5.73944

6. Conclusions

The heat transfer in boundary layer flow of Casson fluid past a shrinking sheet with variable wall temperature and thermal radiation effect is investigated. The transformed self-similar nonlinear ordinary differential equations are obtained and then those are solved analytically. It is found that the velocity increases (decreases) with Casson parameter and wall mass transfer parameter for first (second) solution and the effect is opposite for the temperature. Consequently, the thermal boundary layer thickness decreases with Casson parameter and wall mass transfer parameter for first solution and it increases for second solution. On the other hand, the thermal boundary layer thickness reduces with the Prandtl number and the radiation parameter for both solutions. The local skin fric-tion coefficient increases with Casson parameter and wall mass transfer parameter for first solution and the effect is reverse for second solution. The rate of heat transfer decreases with increasing values of power-law exponent for direct variation along the sheet and for inverse variation it increases. Importantly, heat absorption is found for second solution and in certain cases for first solution.

Acknowledgements

The authors express their sincere thanks to the esteemed referees for their constructive suggestions. One of the authors, K. Bhattacharyya gratefully acknowledges the financial support of National Board for Higher Mathematics (NBHM), Department of Atomic Energy, Government of India, for pursuing this work. G.C. Layek would like to acknowledge the assistance and support received from UGC, DSA-I in the Department of Mathematics, The University of Burdwan. References

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