Heat and Mass Transfer 40 (2003) 47–57 DOI 10.1007/s00231-003-0428-x
Visco-elastic MHD flow, heat and mass transfer over a porous stretching sheet with dissipation of energy and stress work Sujit Kumar Khan, M. Subhas Abel, Ravi M. Sonth
47 Abstract The present paper deals with the study of momentum, heat and mass transfer characteristics in a viso-elastic fluid flow over a porous sheet, where the flow is generated due to linear stretching of the sheet and influenced by a uniform magnetic field applied vertically and a continuous injection of the fluid through porous boundary. In the flow region, heat balance is maintained with a temperature dependent heat source/sink, viscous dissipation, dissipation due to elastic deformation and stress work produced as the result of magnetic field on the non-Newtonian fluid. In mass transfer analysis we have taken into account the loss of mass of the chemically reactive diffusive species by means of first order chemical conversion rate. Using suitable similarity transformations on the highly non-linear partial differential equations we derive several closed form analytical solutions for nondimensional temperature, concentration, heat flux, mass flux profiles in the form of confluent hyper geometric (Kummer’s) functions and some other elementary functions as its special form, for two different cases of the boundary conditions, namely, (i) wall with prescribed second order power law temperature (PST) and prescribed second order power law concentration (ii) wall with prescribed second order power law heat flux (PHF) and prescribed second order power law mass flux. The effect of the non-dimensional magnetic parameter on momentum, heat and mass transfer characteristics for non-isothermal boundary condition and different physical situations of the fluid, having various degrees of visco-elasticity, Prandtl number, heat source/sink strength and Schmidt number, are discussed in detail. Some of the several
important findings reported in this paper are: (i) The combined effect of magnetic field, visco-elasticity and impermeability of the wall is to increase skin-friction largely at the wall; (ii) maximum enhancement of walltemperature profile due to the application of transverse magnetic field occurs when the boundary heating is maintained with prescribed heat flux, boundary wall is porous and Prandtl number of the fluid is low; (iii) the effect of transverse uniform magnetic field is to increase concentration in the flow region, however, enhancement of concentration is higher when the stretching wall is porous and subjected to injection and (iv) the reduction of concentration due to chemical conversion is of significant order near the wall in PHF case when the wall is maintained with prescribed power low mass flux, in comparison with the PST case when the wall is maintained with prescribed power law surface concentration.
1 Introduction Ever increasing industrial applications in the manufacture of plastic film and artificial fiber materials, in recent years, has led to a renewed interest in the study of visco-elastic boundary layer fluid flow, heat and mass transfer over a stretching sheet. Until recently, this study has been largely concerned with the mathematical analysis of the flow characteristics and heat transfer behaviours only in the visco-elastic fluid flow of the type Walters’ liquid B and second order fluid (Rajagopal et al. [1], Siddappa and Abel [2], Gupta and Sridhar [3], Siddappa and Abel [4], Rajagopal et al. [5], Bujurke et al. [6], Danapat and gupta [7], Rollins and Vajravelu [8], Lawrence and Rao [9], Received: 10 January 2002 Siddappa et al. [10], Abel and Veena [11], Prasad et al. Published online: 13 March 2003 [12], and Abel et al. [13]). Magnetic field might play an Springer-Verlag 2003 important role in controlling momentum, heat and mass transfers in the visco-elastic boundary layer flow over a S.K. Khan (&), M.S. Abel stretching sheet, having some specific industrial applicaDepartment of Mathematics, Gulbarga University, tions such as in polymer technology and metallurgy Gulbarga – 585 106, Karnataka, India E-mail:
[email protected] [14–16]. Keeping this view in mind, Andersson [14] studied a magneto-hydrodynamic (MHD) flow of Walters’ R.M. Sonth Department of Mathematics, K.C.T. Engineering College, liquid B past a stretching sheet without consideration of Gulbarga – 585 104, Karnataka, India suction/blowing. Char [15] presented an exact solution for the heat and mass transfer phenomena in a MHD flow Authors express their sincere thanks to the referee for giving valuable suggestions in the review process of the paper. One of of a visco-elastic fluid over a stretching surface. Subseqthe authors (RMS) is thankful to Sri. Qamar-ul-Islam, President, uently, Lawrence and Rao [16] presented an analysis on H.S.M.A Kallerawan Charitable Trust, Gulbarga and the non-uniqueness of the MHD flow of a visco-elastic Dr. M. Basavaraja, Principal, K.C.T. Engineering College, Gulbarga for inculcating invaluable inspirations in pursuing this fluid past a stretching surface and derived two closed form solutions. However, all these studies are restricted to the research work.
48
analysis of either flow characteristics or flow and heat transfer characteristics over an impervious stretching boundary. We know that the characteristic properties of the final product of the material depend to a great extent on the rate of cooling through the adjacent boundary. The rate of cooling associated with the heat transfer phenomenon may be controlled by suction/blowing thorough the porous boundary in presence of constant transverse magnetic field. To take into account this aspect, very recently, Abel et al. [17] presented a work on the momentum and heat transfer in a MHD visco-elastic fluid flow with suction/blowing. However, dissipation and stress work, which definitely have a significant role in the flow of nonNewtonian fluid like visco-elastic fluid, are excluded form their study. The very recent work of Sonth et al. [18] takes into consideration of the viscous dissipation term alongwith temperature dependent heat source/sink term on momentum, heat and mass transfers in a visco-elastic fluid flow over an acceleration surface. However, this work does not take into consideration of the dissipation energy due to elastic deformation of the fluid. Also this work deals with the situation when there is no application of magnetic field. When there may be a chemically reactive diffusive substance present in the boundary layer flow, it is also imperative to take into account the effect of chemical reaction, in addition to the effect of diffusion of the species in analyzing mass transfer phenomenon. Moreover, the presence of chemical reaction term would justify the presence of temperature dependent heat source/sink, depending upon the nature of the chemical reaction being exothermic or endothermic. Hence, in the present paper, we contemplate to study the momentum, heat and mass transfer phenomena in a visco-elastic boundary layer flow over a porous stretching surface, for two general cases of non-isothermal boundary heating, namely: (i) prescribed power law surface temperature (PST) with prescribed power law surface concentration and (ii) prescribed power law surface heat flux (PHF) with prescribed power law surface concentration flux, in the presence of an uniform transverse magnetic field, taking into consideration of the viscous dissipation, dissipation due to elastic deformation, stress work, heat source/sink and chemical reaction. Several closed form analytical solutions for the momentum, heat and mass transfer characteristics are obtained in the form of confluent hyper geometric functions (Kummer’s function). Solutions are also obtained in the form of some other elementary functions as the special cases of Kummer’s function. This is the generalization of the very recent work of Sonth et al. [18] to the case of visco-elastic fluid flow where the transverse uniform magnetic field is applied on it and the diffusive species is chemically reactive.
boundary sheet with the application of two equal and opposite forces along the x-axis, keeping the origin fixed. In addition to the consideration of temperature dependent heat source/sink and viscous dissipation, we assume that the flow is exposed under the influence of a transverse uniform magnetic field of strength B0. The magnetic Reynolds number is considered to be small and so the induced magnetic field is negligible. We also take the strength of electric field due to polarization of charges is negligibly small in order to deal the situation when no energy is added or extracted form the region by the electric field. In addition to these we also consider the presence of chemically reactive species in the flow field. Under these situations, the governing basic boundary layer equations for momentum, heat and mass transfers in Walters’ liquid B, in the presence of viscous dissipation, elastic deformation, internal heat generation and stress work, take the following form @u @v þ ¼0 @x @y
ð2:1Þ
@u @u þv @x @y @2u @3u @ 3 u @u @ 2 u @u @ 2 u rB20 u þ ¼ c 2 k0 u þ v @y @x@y2 @y3 @y @x@y @x @y2 q
u
2 Mathematical formulation and solution of the problem A steady state two-dimensional boundary layer flow of an electrically conducting incompressible visco-elastic fluid, in a region y > 0 over a stretching sheet, is considered for investigation (Fig. 1 in the work of Sonth et al. [18]). The Fig. 1. Variations of fg(g) for different values of Mn when flow is generated due to linear stretching of the adjacent (a) k1* = 0 and (b) k1* = 1.0
ð2:2Þ
@T @T k @ 2 T l @u 2 rB20 u2 Q u þv ¼ þ þ þ ðT T1 Þ @x @y qcp @y2 qcp @y qcp qcp k0 @u @ @u @u u þv ð2:3Þ @x @y cp @y @y u
@C @C @2C þv ¼ D 2 kc C n @x @y @y
ð2:4Þ
Where k is the thermal conductivity of the fluid medium, k0 the visco-elastic parameter, Q the heat source/sink parameter, r the electrical conductivity, D the molecular diffusivity of chemically reactive species, kc the rate of chemical conversion and T¥ is the free stream temperature. The other quantities have their usual meanings. The term Q represents the volumetric rate of heat generation when Q > 0 and heat absorption when Q < 0 to deal the situation of exothermic and endothermic chemical reaction respectively. The Eq. (2.2) represents the momentum balance of inertia, viscous, visco-elastic and magnetic forces acting in the flow field. The Eq. (2.2) is derived with the assumption that the normal stress is of the same order of magnitude as that of the shear stress, in addition to the usual boundary layer approximations. Boundary conditions on velocity: u ¼ bx; u ¼ 0;
v ¼ vw
at y ¼ 0
uy ¼ 0 as y ! 1;
ð2:5Þ
where subscript y represents differentiation w. r. to y; b is a constant, known as stretching rate; and vw is the suction velocity across the stretching sheet when vw < 0 and it is blowing velocity when vw > 0. Boundary conditions on temperature and concentration: The types of boundary conditions on temperature and concentration depend on the nature of the boundary heating processes and mass flow through the boundary. There are great deal of work considering constant surface temperature and constant surface concentration. However, in order to deal with the situation of variable temperature (non-isothermal) and variable concentration boundary conditions it is convenient to prescribe general power law temperature profiles of arbitrary degree on the boundary in the absence of dissipation terms (Soundalgokar and Ramana Murty [19], Rollins and Vajravelu [8], Vajravelu and Rollins [20], Vajravelu and Nayfeh [21], Vajravelu [22], Char MI [15], Chamkha and Issa [23], Acharya et al. [24]). When the dissipation term is taken into account in the linear stretching sheet problem it is essential to consider the power law of profile of degree one or two (solutions are different depending upon degree one or two) to deal with the non-isothermal boundary conditions (Vajravelu [22], Subhas and Veena [11], Abel et al. [13] and Sonth et al. [18]) in order to have closed form similar solution.In view of these we consider the following two general types of non-isothermal boundary heating processes, namely (i) prescribed second order power law surface temperature (PST) with prescribed second order power law concentration (ii) prescribed second order
power law heat flux (PHF) with prescribed second order power law concentration flux. In PST case, the appropriate boundary conditions on temperature and concentration are x2 x 2 T ¼ Tw ¼ T 1 þ A 0 ; C ¼ Cw ¼ C1 þ A1 at y ¼ 0 l l T ! T1 ; C ! C1 as y ! 1: ð2:6Þ Where Tw and T¥ are temperature at wall and temperature far away from the wall respectively. Cw and C¥ are species concentration at wall and species concentration far away from the wall respectively. A0 and A1 are constants whose values depend on the properties of the fluid. The constant l is chosen as characteristic length. In order to obtain the closed form analytical solutions of the differential Eqs. (2.3) and (2.4) we consider stretched boundary surface with prescribed power law temperature and prescribed power law concentration of second degree only. In PHF case, the corresponding boundary conditions on temperature and concentration are x 2 @T k ¼ qw ¼ E0 ; @y w l ð2:7Þ x 2 @C ¼ mw ¼ E1 ; at y ¼ 0 D @y w l T ! T1 ;
C ! C1
as y ! 1:
ð2:8Þ
Where E0 and E1 are constants whose values depend on the properties of the fluid.
3 Solution of the momentum equation Equations (2.1) and (2.2) admit self-similar solutions of the form pffiffiffiffiffiffiffi u ¼ bx fn ðgÞ; v ¼ ðbcÞ1=2 f ðgÞ; g ¼ b=cy; ð3:1Þ where f is the dimensionless stream function and g is the similarity variable. Substitution of Eq. (3.1) in the Eq. (2.2) results in a fourth order non-linear ordinary differential equation n o 2 fg2 ffgg ¼ fggg k1 2fg fggg ffgggg fgg M n fg ; ð3:2Þ where ¼ kc0 b is the dimensionless visco-elastic parameter rB2 and Mn ¼ bq0 is the dimensionless magnetic parameter. k1
The corresponding boundary conditions on f are of the form vw fg ¼ 1; f ¼ pffiffiffiffiffi at g ¼ 0 bc ð3:3Þ fn ¼ 0; fgg ¼ 0 as g ! 1: Making use of the boundary conditions of the Eq. (3.3), we derive the exact analytical solution of the Eq. (3.2) in the form
49
f ðgÞ ¼
1 eag vw pffiffiffiffiffi ; a bc
fg ðgÞ ¼ eag ;
ð3:4Þ
where a is a real positive root of the cubic algebraic equation: a3 þ
1 k1 k1
vw pffiffiffiffi bc
a2 þ
1 ð1 þ Mn Þ ¼0 a k k1 vw p1ffiffiffiffi
ð3:5Þ
bc
and making use of Eqs. (3.4) and (4.5) in the Eqs. (4.2) and (4.3), we obtain the governing non-dimensional equations of temperature and concentration, in the form Pr b nhnn þ ð1 a0 nÞhn þ 2 þ 2 h a n a2 E 2 a4 ¼ a þ Mn k 1 a 0 n ð4:6Þ Pr Pr
50
Sc c1 1/11 þ ð1 c0 1Þ/1 þ 2 2 / ¼ 0; a1
The limiting case of the expression (3.4) when vw = 0 yields the result of Andersson [14]. The quadratic equation for a, in such a case, may be deduced from the Eq. (3.5) in where, the limit vw fi 0. The dimensionless skin friction coefPr vw Pr Sc vw Sc Where; a0 ¼ 2 pffiffiffiffiffi ; c0 ¼ 2 pffiffiffiffiffi ficient Cf may be expressed as a a a bc bc a @u l @y 1 y¼0 ¼ fgg ð0Þ pffiffiffiffiffiffiffiffiffi ð3:6Þ Cf ¼ The corresponding boundary conditions are Re x qðbxÞ2 Pr 2 h 2 ¼ 1; hð0Þ ¼ 0 where Re x ¼ qbx a l is known as local Reynolds number.
4 Solutions of heat and mass transfer equations Case A: prescribed power law surface temperature (PST) In the PST case we define non-dimensional temperature and concentration variables as
Sc / 2 ¼ 1; a
/ð0Þ ¼ 0
ð4:7Þ
ð4:8Þ
ð4:9Þ
ð4:10Þ
The solution of the Eqs. (4.6) and (4.7), subject to the boundary conditions (4.9) and (4.10) respectively, are derived in the following form of confluent hypergeometric T T1 C C1 function of similarity variable g. hðgÞ ¼ ; /ðgÞ ¼ ð4:1Þ Tw T1 Cw C1 a0 þb0 c1 Pr2 hðgÞ ¼ 1 4 ðeag Þ 2 where the expressions for Tw – T¥ and Cw – C¥ are given in a Eq. (2.6). Now, we use the transformations given by the a0 þb0 2 Pr eag M 2; 1 þ b 2 0 2 a Pr Eqs. (3.1) and (4.1) in Eqs. (2.3) and (2.4). In order to þ c1 e2ag 2 obtain analytical solution we set n = 1 and C¥ = 0 for first a0 þb0 Pr a M 2 2; 1 þ b0 a2 order chemical reaction. This leads to the non-dimensional form of temperature and concentration equations as ð4:11Þ follows: c0 þd0 Sc ag c þd hgg þ Pr f hg Pr ð2fg bÞh ag 0 2 0 M 2 2; 1 þ d0 a2 e 0 ; ð4:12Þ /ðgÞ ¼ ðe Þ n o M c0 þd 2; 1 þ d0 aSc2 2 2 2 ð4:2Þ ¼ Pr E fgg k1 fgg ðfg fgg ffggg Þ þ Mn fg Where rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi /gg þ Sc f /g Scð2fg þ c1 Þ/ ¼ 0: ð4:3Þ a4 Ea2 Mn þ a2 k1 a0 Pr Prb ; b0 ¼ a20 4 2 and c1 ¼ lc Pr b a Where Pr ¼ kq is the Prandtl number, b ¼ qcQp b is the heat Pr 4 2a0 þ a2 b 2 l2 rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi source/sink parameter, E ¼ A0 cp is the Eckert number, c1 ¼ c kc Sc c is the first order chemical reaction rate and Sc ¼ is the b D d0 ¼ c20 þ 4 2 1 a Schmidt number. Using the dimensionless variables of the Eq. (4.1) in the ð4:13Þ Eq. (2.6) we get the corresponding dimensionless bounda0 and c0 are given by the Eq. (4.8). ary conditions as The Kummer’s function M [25] is defined by hð0Þ ¼ 1; hð1Þ ¼ 0; /ð0Þ ¼ 1; /ð1Þ ¼ 0: ð4:4Þ 1 X ða0 Þn zn Mða0 ; b0 ; zÞ ¼ 1 þ ðb0 Þn n! n¼1 Defining new variables ð4:14Þ ða Þ ¼ a ða þ 1Þða þ 2Þ ða þ n 1Þ 0 0 0 0 0 n Pr Sc n ¼ 2 eag ; 1 ¼ 2 eag ð4:5Þ ðb0 Þn ¼ b0 ðb0 þ 1Þðb0 þ 2Þ ðb0 þ n 1Þ a a
The expression for temperature in Eq. (4.11) is more reSc alistic than that of Abel et al. [17], where viscous dissi/g ð0Þ ¼ að3 þ d0 Þ ð4:17bÞ a pation, elastic deformation and stress work terms are omitted in the energy equation. Dimensionless wall temperature gradient hn(0) and wall Dimensional local heat flux qw and dimensional local concentration gradient /n(0) are obtained as mass flux mw are defined as sffiffiffi @T b Pr a0 þ b0 4Þ a0 þ b0 2 Pr ðTw T1 Þ½hg ð0Þ ¼k ð4:18aÞ qw ¼ k ; 2 þ b0 ; 2 M hn ð0Þ ¼ A @y w c 2a 1 þ b0 2 a aða0 þ b0 Þ a0 þ b0 4 Pr sffiffiffi M ; 1 þ b0 ; 2 2 2 a @C b 2 ðCw C1 Þ½/g ð0Þ ¼D ð4:18bÞ mw ¼ D Pr @y w c 2c1 a 2 ð4:15aÞ a Case B: prescribed power law heat flux (PHF) In the PHF aðc0 þ d0 Þ case we define dimensionless new variables /n ð0Þ ¼ 2 T T1 C C1 Sc ðc0 þ d0 4Þ M c0 þd20 2 ; 2 þ d0 ; aSc2 gðgÞ ¼ qffiffi ; hðgÞ ¼ qffiffi ð4:19Þ c0 þd0 4 þ : 2 2 c a 2ð1 þ d0 Þ M E0 xl kl b E1 xl D1 bc ; 1 þ d0 ; aSc2 2 ð4:15bÞ Where;
A¼ M
1 c1 Pr a4
a0 þb0 4 ; 2
2
and make use of the Eq. (3.1). This leads to the following non-dimensional form of the Eqs. (2.3) and (2.4) for temperature and concentration.
: 1 þ b0 ; Pr 2 a
We know that the Kummer’s function is related to other special forms of elementary functions. One of such special forms (Abramowitz and Stegun [25]) may be obtained by assigning special values to its argument. Hence, settinga0 þb20 4 ¼ 1 þ b0 and c0 þd20 4 ¼ 1 þ d0 , we get the expressions for non-dimensional temperature and concentration profiles in the special form: EPr a4 2 hðgÞ ¼ 1 þ 2 Mn þ a k1 a0 a ða0 5Þ Pr
Pr Exp ð3 a0 Þa g þ 2 ð1 eag Þ a EPr a4 2 Mn þ a k1 a0 Exp½2ag 2 a ða0 5Þ Pr ð4:16aÞ
Sc /ðgÞ ¼ Exp ð3 þ d0 Þag þ 2 ð1 eag Þ a
ð4:16bÞ
ggg þ Prf gg Prð2fg bÞg n o 2 k1 fgg k1 fgg ðfg fgg ffggg Þ þ Mn fg2 ¼ PrE fgg ð4:20Þ hgg þ Sc f hg Scð2fg þ c1 Þh ¼ 0:
ð4:21Þ
Corresponding boundary conditions in the PHF case are gg ð0Þ ¼ 1;
gð1Þ ¼ 0
ð4:22aÞ
hg ð0Þ ¼ 1;
hð1Þ ¼ 0:
ð4:22bÞ
Further, we introduce the transformations of the Eq. (4.5) and with this the transformed basic equations and boundary conditions of temperature and concentration, take form bPr ngnn þ ð1 a0 nÞgn þ 2 þ 2 g an 4 a2 E 2 a ¼ a þ Mn k1 a0 n ð4:23Þ Pr Pr
Expressions for non-dimensional wall temperature gradient hn(0) and wall concentration gradient /n(0) in the special forms are deduced as Sc c1 1 1h h¼0 þ ð1 c 1Þh þ 2 11 0 1 ( ) a2 1 4 2 EPr a Mn þ a2 k1 a0 hn ð0Þ ¼ 1 þ 2 a ða0 5Þ Pr Pr a gn 2 ¼ ; gð0Þ ¼ 0 Pr a Pr 3a a0 a þ a Sc a 2aEPr a4 2 h ¼ ; hð0Þ ¼ 0 Mn þ a k1 a0 ð4:17aÞ 1 þ 2 a2 Sc a ða0 5Þ Pr
ð4:24Þ
ð4:25aÞ
ð4:25bÞ
51
The analytical solutions of the Eqs. (4.23) and (4.24), subject to the corresponding boundary conditions of (4.25), are obtained in the following form of confluent hypergeometric functions of the similarity variable g. a þb a0 þ b0 Pr ag að 0 2 0 Þg gðgÞ ¼ c2 e 2; 1 þ b0 ; 2 e M a 2 2 Pr þ c1 e2ag 2 a
Expressions for non-dimensional wall temperature g(0) and wall concentration h(0) in the special form are deduced as
gð0Þ ¼
ð4:26Þ 52
n o EPr 2 a4 1 þ a2a M þ a k a 2 ða 5Þ n 0 1 0 Pr aða0 3Þ Pr a EPr a4 2 Mn þ a k1 a0 ; 2 a ða0 5Þ Pr
ð4:32Þ
Where 2 2c1 a Pr 1 a2 i c2 ¼ h a0 þb0 Pr þ a0 þb0 4 Pr M a0 þb0 1; 2 þ b ; Pr 0 2; 1 þ b ; M a a0 þb 2 2 0 0 2 2 a a 2 a 2ð1þb0 Þ
ð4:26aÞ
and expression for c1 is given by the Eq. (4.13). 0 c0 þd0 Exp ag c0 þd M 2 2; 1 þ d0 ; aSc2 eag 2 hðgÞ ¼ h c0 þd0 c0 þd0 4 Sc c0 þd0 i Sc Sc 0 a c0 þd 2; 1 þ d ; M 1; 2 þ d ; M 2 2 0 0 a a 2 2 2 2ð1þd0 Þ a Where the expressions for a0, c0 and b0, d0 are given by Eqs. (4.8) and (4.13) respectively. The expressions for dimensionless wall temperature and wall concentration are obtained as 2 a0 þ b0 Pr Pr gð0Þ ¼ c2 M 2; 1 þ b0 ; 2 þ c1 2 a a 2
hð0Þ ¼
1 að3 þ d0 Þ Sca
ð4:27Þ
ð4:33Þ
The expressions for wall temperature and wall concentration in dimensional form are
ð4:28Þ 2; 1 þ d0 ; aSc2 hð0Þ ¼ h c0 þd0 0 þd0 4 Sc c0 þd0 i Sc 0 a c0 þd M 1; 2 þ d ; M 2 2; 1 þ d0 ; aSc2 c2ð1þd 2 0 a a 2 Þ 2 0 M
c0 þd0 2
The limiting cases of our results, given by Eqs. (4.26)– (4.29), produce the corresponding results of Sonth et al. [18] when Mn = 0.0 and c1 = 0.0. The special forms of the results may be deduced from the Eqs. (4.26)–(4.27) by setting a0 þb20 4 ¼ 1 þ b0 and c0 þd0 4 ¼ 1 þ d0 and they are derived as 2 n o EPr 2 a4 1 þ a2a M þ a k a 2 ða 5Þ n 0 1 0 Pr gðgÞ ¼ aða0 3Þ Pr a
Pr ag Exp ð3 þ b0 Þag þ 2 ð1 e Þ a EPr a4 2 Mn þ a k 1 a 0 Exp½2ag 2 a ða0 5Þ Pr ð4:30Þ
Exp ð3 þ d0 Þag þ aSc2 ð1 eag Þ hðgÞ ¼ að3 þ d0 Þ Sca
ð4:31Þ
rffiffiffi E0 x2 c gð0Þ; b k l rffiffiffi E1 x2 c hð0Þ Cw ¼ C1 þ b D l
ð4:29Þ
Tw ¼ T 1 þ
ð4:34Þ
5 Discussion of the results A boundary layer problem for momentum, heat and mass transfer in a visco-elastic non-Newtonian fluid flow over a stretching sheet, in the presence of a transverse uniform magnetic field, is presented in this paper. Linear stretching of the porous boundary, dissipation of energy due to viscosity and elastic deformation, temperature dependent heat source/sink, stress work and chemical reaction of the species are taken into consideration in this study. The basic boundary layer partial differential equations, which are highly non-linear and in coupled form, have been converted into a set of non-linear ordinary differential
equations by applying suitable similarity transformations and their analytical solutions, are obtained in terms of the confluent hypergeometric function (Kummer’s function). Different analytical expressions are obtained for non-dimensional velocity, namely (i) prescribed second order power law surface temperature (PST) with prescribed second power law surface concentration and (ii) prescribed second order power law heat flux (PHF) with prescribed second order power law concentration flux. Analytical expressions are also obtained for dimensionless temperature gradient hg(0), wall concentration gradient /g(0) and local heat flux qw and local mass flux mw for general cases as well as special cases of different physical situations. Since, the present problem is the extension of our earlier problem (Sonth et al. [18] to the case of a flow region exposed by a transverse magnetic field, the species is chemically reactive and stress work occurs, we intend to restrict our analysis for the role of magnetic field and the rate of chemical reaction on various momentum, heat and mass transfer characteristics only. Numerical computations of the results are demonstrated in the Figs. 1–9 in order to have greater insight in the qualitative analysis of the results. In the process of computation we solve the cubic algebraic Eq. (3.5) for a using Graffe’s square root method. Through out the computation we assign b = 2 and c = 0.04. The effect of magnetic field parameter Mn on horizontal velocity profile fg(g) may be analyzed form the graphical representation of the result in the Figs. 1a and 1b. It is
found that the effect of magnetic parameter Mn is to reduce the horizontal velocity in both the cases of blowing and impermeability of the wall. Comparison study of Fig. 1a and Fig. 1b reveals that the magnetic parameter Mn decreases the horizontal velocity fg(g) significantly in the visco-elastic flow in comparison with viscous flow in the absence of injection of fluid through porous boundary. This is due to the fact that the increase of Mn signifies the increase of Lorentz force that opposes the horizontal flow in the reverse direction. Fig. 2 shows that the skin friction parameter increases on the wall with the application of magnetic field. This is because the magnetic force acts as a retarding force and causes the increase of shear stress. The combined effect of magnetic field, visco-elasticity and impermeability of the wall is to increase the skin-fraction at the wall largely. Here, the additional introduction of shear stress at the wall by the magnetic field, non-Newtonian nature of visco-elastic fluid and impermeability of the wall and thereby decrease of boundary layer thickness, leads to the increase of skin-friction. Figures 3a and 3b are plotted for temperature distribution in PST and PHF cases respectively, in the presence/ absence of magnetic field when Pr = 1 and E = 0.8. It is interesting to note that there would be a significant enhancement of temperature on the wall in the presence of magnetic field in PHF case when the wall is porous and subjected to injection of fluid. Whereas the temperature on
Fig. 3. Dimensionless temperature profiles for various values Fig. 2. Skin-friction fgg(0) Vs. visco-elastic parameter k1* for
different values of Mn
of Mn when E = 0.8, k1* = 0.4, b = –0.05 and Pr = 1.0 in (a) PST case and (b) PHF case
53
54
the wall would be minimum in the absence of magnetic field in PHF case when the wall is impermeable. This is quite consistent with the physical situation as the application of magnetic field introduces additional skin-frictional heating, due to stress work, which results in higher temperature on the wall with the increase of thermal boundary layer thickness. Figures 4a and 4b depict the graphs for the temperature profile in PST and PHF cases respectively for the same data values as used in Figs. 3a and 3b except for Pr = 3. Comparative observations of Figs. 3a and 3b with Figs. 4a and 4b demonstrate that there would be a higher temperature at a particular place in the flow region, due to an application of magnetic field, for lower values of Prandtl number Pr. Here, the increase of magnetic force causes significant increase of thermal boundary layer thickness in low Pradtl number fluid flow. Figures 5a and 5b depict the graphs for the temperature profile in PST and PHF cases respectively for the same data values as used in Figs. 4a and 4b except for E = 0.0. Comparison of the Figs. 4a and 4b with the Figs. 5a and 5b reveals that there would be a higher temperature for the situation when E = 0.8 in comparison with the case when E = 0.0. This is in conformity with the fact that energy is stored in the fluid region as a consequence of dissipation due to viscosity and elastic deformation.
Fig. 4. Dimensionless temperature profiles for various values
Concentration profiles across the flow field are plotted in the Figs. 6a and 6b pertaining to PST and PHF cases respectively for different values of Mn and vw. From these graphs we observe that the effect of introducing magnetic parameter Mn is to increase concentration in the flow field. However, the concentration would be higher in PHF case in comparison with the PST case. In conformity with a real situation, we note that the enhancement of concentration due to magnetic field is maximum when the wall is porous and it is subjected to injection of fluid. Figures 7a and 7b are plotted for the same data values as those are used in the Figs. 6a and 6b, except for the values of Sc. Comparison study of Fig. 6a and Fig. 7a shows that the concentration, away from the wall, would be significantly reduced with the increase of the values of Sc in PST case. The explanation for such behaviour of concentration field lies in the fact that the decay of concentration from the stretching sheet to the surrounding fluid takes place with lower rate due to lower values of molecular diffusivity, Sc being large. Consequently the decrease of concentration boundary layer occurs with the increase of Schmidt number Sc. Figures 8a and 8b are depicted for concentration distribution in PST and PHF cases respectively for different
Fig. 5. Dimensionless temperature profiles for various values of Mn when E = 0.8, k1* = 0.4, b = –0.05 and Pr = 3.0 in (a) PST of Mn when E = 0, k1* = 0.4, b = –0.05 and Pr = 3.0 in (a) PST case case and (b) PHF case and (b) PHF case
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Fig. 6. Dimensionless concentration profiles for values of Mn
when c1 = 0.0025, k1* = 0.001 and Sc = 0.2 in (a) PST case and (b) PHF case
Fig. 8. Dimensionless concentration profiles for various values
of k1* when c1 = 0.0025, Mn = 0.05 and Sc = 0.2 in (a) PST case and (b) PHF case
values of visco-elastic parameter k*1. As expected, concentration profile increases with the increase of visco-elastic parameter due to the increase of concentration boundary layer. Figures 9a and 9b show the effect of first order chemical conversion rate on the species concentration in PST and PHF cases respectively. It is noticed form both the graphs that the effect of increasing values of c1 is to decrease concentration in both the cases. This is due to the fact that the role of chemical conversion is to reduce the thickness of the concentration boundary layer. However, the reduction of concentration would be of significant order near the wall in PHF case when the wall is maintained with prescribed power law mass flux, in comparison with the PST case when the wall is maintained with prescribed power law surface concentration. This is owing to the reason that the prescribed wall concentration is maintained by external means in the first case, whereas it is not being done in the later case.
Fig. 7. Dimensionless concentration profiles for Mn = 0.05
when c1 = 0.0025, k1* = 0.001 and Sc = 1.0 in (a) PST case and (b) PHF case
6 Conclusion A mathematical model study on the influence of uniform magnetic field applied vertically in a visco-elastic fluid flow over an accelerating stretching sheet, where the flow is subjected to blowing through porous boundary, has been carried out. Linear stretching of the porous boundary, dissipation due to viscosity and elastic deformation, temperature dependent heat source/sink, stress work and chemical reaction of the species are taken into consideration in this study. Analytical solutions of the governing boundary layer partial differential equations, which are highly non-linear and in coupled form, have been obtained in terms of confluent hypergeometric function (Kummer’s function) and its special forms.
maximum enhancement of concentration due to the application of magnetic field occurs when the stretching wall is porous and subjected to injection of fluid. – The effect of increasing values of first order chemical conversion rate c1 is to decrease concentration in both the cases of PST and PHF. However, the reduction of concentration would be of significant order near the wall in PHF case when the wall is maintained with prescribed power law mass flux, in comparison with the PST case when the wall is maintained with prescribed power law surface concentration. – The limiting cases of the results of this paper are in excellent agreement with the results of Andersson [14] and Sonth et al. [18].
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References
Fig. 9. Dimensionless concentration profiles for various values
of c1 when k1* = 0.001, Mn = 0.05 and Sc = 0.2 in (a) PST case and (b) PHF case
Different analytical expressions are obtained for nondimensional velocity, temperature and concentration profiles for two general cases of boundary conditions, namely (i) prescribed second order power law surface temperature (PST) with prescribed second order power law surface concentration and (ii) prescribed second order power law heat flux (PHF) with prescribed second order power law concentration flux. Analytical expressions are also obtained for dimensionless temperature gradient hg(0), wall concentration gradient /g(0), local heat flux qw and local mass flux mw for general cases as well as special cases of different physical situations. The specific conclusions derived form this study can be listed as follows:
– The effect of magnetic parameter Mn is to decrease – –
– –
–
horizontal velocity significantly in a visco-elastic fluid flow in comparison with a viscous flow. The combined effect of magnetic field, visco-elasticity and impermeability of the wall is to increase skinfriction at the wall. There would be a significant enhancement of walltemperature profile due to the application of transverse magnetic field in PHF case when the wall is porous and subjected to injection of fluid. The combined effect of viscous dissipation energy and deformation energy due to the visco-elastic property of the fluid is to increase the wall temperature. There would be a higher temperature at a particular place in the flow region in the presence of a transverse magnetic field when the Prandtl number of the fluid is low. The effect of transverse uniform magnetic field is to increase concentration in flow field. However,
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