Alexandria Engineering Journal (2017) xxx, xxx–xxx
H O S T E D BY
Alexandria University
Alexandria Engineering Journal www.elsevier.com/locate/aej www.sciencedirect.com
ORIGINAL ARTICLE
Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate C. Sulochana *, G.P. Ashwinkumar, N. Sandeep Department of Mathematics, Gulbarga University, Gulbarga 585106, India Received 9 May 2017; revised 11 August 2017; accepted 15 August 2017
KEYWORDS Casson fluid; Thermal Radiation; Nanoparticles: MHD; Chemical reaction; Heat source/sink
Abstract The present study deals with the boundary layer analysis of a 2 D magnetohydrodynamic flow of chemically reacting Casson nanofluid flow over a semi-infinite inclined porous plate. The energy and diffusion equation are encompassed with frictional heating, heat generation/absorption, thermo diffusion and thermal radiation effects. For making the analysis more attractive we pondered two distinct type of nanofluids namely, TiO2-water and CuO-water. The transmuted governing PDEs are resolved analytically by employing regular perturbation method. The impact of pertinent flow parameters on momentum, thermal and mass transport behavior including the skin friction factor, thermal and mass transport rate are examined and published with the assistance of graphical and tabular forms. Results describe that thermal radiation and chemical reaction restrictions have a propensity to enhance thermal and mass transport rates, respectively. And also, water based TiO2 nanofluid possess higher velocity compared with water based CuO nanofluids. Ó 2017 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 studies of flow through a porous medium has achieved abundant prominence owing to enormous practical applications in diverse sections of science and engineering, few of them are polymer extrusion, heat exchangers, in geophysical and geothermal processes, etc. and also in ground water purification, petroleum reservoirs, in grain storage devices, etc. In recent days, the thermal and mass transport in MHD flow is a focusing area by the researchers, due to prominent applica* Corresponding author. E-mail address:
[email protected] (C. Sulochana). Peer review under responsibility of Faculty of Engineering, Alexandria University.
tions in engineering and industries. Particularly, in magnetic material processing, in geophysics and control of cooling rate, etc. Sandeep et al. [1] elaborated the impact of an inclined magnetic field on the natural convection flow on a vertical plate. A similar type of study by considering the radiation was deliberated by Ali et al. [2]. The literature focusing on this topic are revealed by few authors [3–6]. Very recently, Ahmed et al. [7] depicted the impact of frictional heating and chemical reaction flow along a vertical porous plate. The impact of cross diffusion on the 2D flow of power-law fluid past porous plate was examined by Pal and Chatterjee [8]. The fluid which disobeys Newton’s law of viscosity is abbreviated as non-Newtonian fluids. In these fluids, the ratio of shear stress is nonlinear to the strain rate. Past few years, the probes on thermal and mass transport behavior in non-
http://dx.doi.org/10.1016/j.aej.2017.08.006 1110-0168 Ó 2017 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/). Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
2 Newtonian fluids were improved, non-Newtonian fluids have inimitable properties because of this the usage of nonNewtonian fluids has convalesced than the Newtonian fluids. Casson fluid is the type of non-Newtonian fluid. Few regular life examples of Casson fluids involves blood, honey, fruit juices, etc. Casson fluid also possesses significant applications in science, engineering and medicine, such as cancer therapy, in fibrinogen, thermal insulation of buildings, oil recovery system, and cooling of electronic devices drying of solid systems. In view of these applications, few researchers [9–13]. They revealed that Casson fluid is more proficient in thermal transport phenomenon compared with the Newtonian fluid. Reddy et al. [14] explained the aligned magnetic field impact on Casson flow. Sulochana et al. [15] interrogated boundary layer behavior of the MHD radiative flow with joule heating effect. Recently, Rashid et al. [16] elaborated stagnation point-flow of Casson fluid past Stretching surface in the presence of thermal radiation effects. The erudition of chemical reaction is employed in enormous fields of engineering and technology, namely, in chemical industries, polymer processing, food processing and glass manufacturing industry, etc. Das et al. [17] illustrated the impact of chemical reaction on unsteady flow along a vertical plate by employing Laplace transform technique. Bhattacharyya and Layek [18] scrutinized the impact of chemical reaction and transpiration on the flow over a flat plate. Later on, copious investigators [19–22] studied the chemical reaction effects on dissimilar flow geometries. Recently, Hussain et al. [23] discussed the chemical reaction effects on flow past an accelerated moving plate in a rotating system. The phrase nanofluid was visualized by Choi [24], to improve the heat conduction process more successfully, he suspended the highly thermal conductive solid particles into the base liquids of lower conductivity. Sandeep and Sulochana [25] addressed the impact of induced magnetic field on a nanofluid along a stretchable cylinder. Sulochana et al. [26] extended this concept by the usage of transpiration effects. Reddy et al. [27] explored the heat transfer nature of the flow over a moving plate. They conveyed that, water based Silver nanoparticles are more effective in magnifying the local Nusselt number compared with titania nanoparticles. Recently, Sandeep et al. [28] deliberated the cross diffusion effects on the flow of a Casson and Maxwell fluid passed through stretched surface with thermophoretic and Brownian moment. Keeping in view of above-mentioned studies and applications we discussed the boundary layer analysis of a 2D magnetohydrodynamic flow of chemically reacting Casson nanofluid flow over a semi-infinite inclined porous plate. For making the analysis more attractive we pondered two distinct type of nanofluids namely, TiO2-water and CuO-water. The transmuted governing PDEs are resolved analytically by employing regular perturbation method. The impact of pertinent flow parameters on momentum, thermal and mass transport behavior including the skin friction factor, thermal and mass transport rate are examined and published with the assistance of graphical and tabular forms.
C. Sulochana et al. immersed in a porous medium under the consideration of thermal and concentration buoyancy effects and the plate is inclined vertically with an angle of inclination a as shown in Fig. 1. The plate is placed along the x-axis and it is perpendicular to the y-axis. Here Tw and Cw are the constant temperature and concentration applied at the wall which is higher than T1 and C1 . Where T1 and C1 are the ambient temperature and concentration respectively. It is also noticed that thermal diffusion and chemical reaction effects are taken into consideration. And also to make our study unique we introduced the concept of suspension of Silver and Titania nanoparticles into the base liquid. Under the presence of above physical assumptions, the flow governing equations are described as below: @v ¼ 0 ) v0 ¼ v0 @y v0
ð1Þ
@u0 1 1 @ 2 u0 lnf 0 ¼ l 1 þ u rB20 u0 nf @y0 qnf b @y02 K0
þðqbÞnf gðT 0 T1 Þ cos a þ ðqbÞnf gðC0 C1 Þ cos a ð2Þ
v0
(
2 @ 2 T0 1 @u0 @q þ l 1 þ r0 nf b @y0 @y02 @y þrB20 u02 Q0 ðT0 T1 Þ
@T0 1 ¼ v @y0 ðqcp Þnf 0
knf
2 0 @C0 @ C Dm K T @ 2 T 0 þ HðC0 C1 Þ ¼ D B @y0 @y02 Tm @y02
ð3Þ
ð4Þ
where u and v are the velocity towards the x y axis, respectively. And /, qnf , lnf , ðqcp Þnf knf are the volume fraction, effective density, effective dynamic viscosity, heat capacitance and thermal conductivity of the nanofluid which are stated as follows: qnf ¼ qf fð1 /Þ þ /qs =qf g;
ð5Þ
lnf ¼ lf ð1 /Þ5=2 ;
ð6Þ
ðqcp Þnf ¼ ðqcp Þf fð1 /Þ þ /ðqcp Þs =ðqcp Þf g;
ð7Þ
2. Mathematical formulation We presume the 2D boundary layer flow of electrically conducting Casson nanofluid over a permeable semi-infinite plate
Fig. 1
Flow geometry.
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
Effect of frictional heating on mixed convection
3
Ks þ 2Kf 2/ðKf Ks Þ knf ¼ Kf ; Ks þ 2Kf þ 2/ðKf Ks Þ
u ¼ 0; h ¼ 1; w ¼ 1 at y ¼ 0 ; u ! 0; h ! 1; w ! 1 as y ! 1
ð8Þ
The rheological equation of a Casson fluid is given by: ( pffiffiffiffiffiffi 2 lB þ py = 2p eij ; p > pc sij ¼ ð9Þ pffiffiffiffiffiffi 2 lB þ py = 2p eij ; p < pc
3. Solution methodology The set of PDEs (13)–(15) are solved analytically after reducing them into ODEs as given below:
The radiative heat flux is given by @qr ¼ 4ðT0 T1 ÞL ; ð10Þ @y0 R1 bk where L ¼ 0 Kkw @e dk, Kkw is the absorption coefficient at @T0 the wall and ebk is Plank’s function. The related boundary conditions are: 0 at y ¼ 0 u ¼ 0; T0 ¼ T1 ; C0 ¼ C1 ; ; ð11Þ u0 ! 0; T0 ! T1 ; C0 ! C1 ; as y ! 1 v0 y0 y¼ ; tf 0
ðC C1 Þ w¼ ; ðCw C1 Þ
ðT0 T1 Þ ; h¼ ðTw T1 Þ K¼
0
K V20 t2f
;
ð12Þ
u0 ¼ 0;
h0 ¼ 1;
w0 ¼ 1
u1 ! 0;
h1 ! 1;
at y ¼ 0;
w1 ! 1
and
hðyÞ ¼ h0 ðyÞ þ Ech1 ðyÞ þ OðEc2 Þ;
ð19Þ
wðyÞ ¼ w0 ðyÞ þ Ecw1 ðyÞ þ OðEc2 Þ;
ð20Þ
A2 u 0 þ A1 u0 A3 u0 ¼ Z1 h0 Z2 w0 ;
ð21Þ
A5 h 0 þ A6 h0 A7 h0 ¼ 0;
ð22Þ
w 0 þ Scw0 A10 w0 ¼ A11 h0 ;
ð23Þ
A2 u 1 þ A1 u1 A3 u1 ¼ Z1 h1 Z2 w1 ;
ð24Þ
2 2 A5 h 1 þ A6 h1 A7 h1 ¼ A8 ðu0 Þ A9 ðu0 Þ ;
ð25Þ
w 1 þ Scw1 A10 w1 ¼ A11 h1 ;
ð26Þ
where ‘‘dot” denotes the ordinary differentiation w.r.t. ‘y’ and Z1 ¼ Gr cos a, Z2 ¼ Gm cos a. The corresponding boundary conditions are:
ð13Þ
ð18Þ
Equating the coefficients of a first order of Eckert number, we get:
Using Eqs. (5)–(12) in basic Eqs. (2)–(4) reduces to the dimensionless form: ! ð1 þ 1=bÞ @ 2 u qs @u þ 1 / þ / qf @y ð1 /Þ5=2 @y2 ( ) ð1 /Þ5=2 A3 M2 þ K ¼ A44 Grh cos a A44 Gmw cos a;
uðyÞ ¼ u0 ðyÞ þ Ecu1 ðyÞ þ OðEc2 Þ;
Substituting Eqs. (18)–(20) in Eqs. (13)–(15) and equating the coefficients of zeroth order, we get
Now utilizing the suitable transmutations: u0 u¼ ; v0
u0 ¼ 0;
h0 ¼ 0;
u1 ! 0;
h1 ! 0;
( !) ðqcp Þs knf @ 2 h @h PrðR þ QH Þh þ Pr 1 / þ / @y kf @y2 ðqcp Þf ! 2 ð1 þ 1=bÞ @u ¼ PrM2 Ecu2 ; PrEc 5=2 @y ð1 /Þ
ð17Þ
w0 ¼ 0
w1 ! 0
as y ! 1;
ð27Þ
Using the boundary conditions (27), the solutions of the Eqs. (21)–(26) we obtain the followings: ð14Þ
@2w @w @2h þ Sc ScKrw ¼ ScSr 2 ; 2 @y @y @y
ð15Þ
u0 ðyÞ ¼ D4 eB3y D2 eB1y D3eB2y
ð28Þ
h0 ðyÞ ¼ eB1y
ð29Þ
w0 ðyÞ ¼ ð1 þ D1 ÞeB2y D1 eB1y
ð30Þ
where
M¼
rB20 t2f v0 l f
;
v2
Gr ¼
0 Ec ¼ cp ðTw T ; 1Þ
ðqbÞf gðTW T1 Þt2f
Kr ¼
v30 lf Htf v20
;
;
Gm ¼ 4t L0
R ¼ ðqcpfÞ v2 ; f 0
ðqbÞf gðCW C1 Þt2f v30 lf Q t
;
QH ¼ ðqcp0Þ fv2 ; f 0
Pr ¼
l f cp f kf t
;
Sc ¼ DfB ;
The corresponding transformed boundary conditions are given by:
9 pffiffiffiffiffiffiffi = b ¼ lB 2pc =py > ðTw T1 Þ > Sr ¼ DTmmktfTðC ;; w C1 Þ
ð16Þ
u1 ðyÞ ¼ D36 eB3y D28 eB1y D29 eB2y þ D30 e2B3y þ D31 e2B1y þ D32 e2B2y D33 eB4y þ D34 eB5y D35 eB6y
ð31Þ
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
4
C. Sulochana et al.
h1 ðyÞ ¼ D11 eB1y D5 e2B3y D6 e2B1y D7 e2B2y þ D8 eB4y D9 eB5y þ D10 eB6y
uðyÞ ¼ ðD4 eB3y D2 eB1y D3eB2y Þ ð32Þ
þ D31 e2B1y þ D32 e2B2y D33 eB4y
w1 ðyÞ ¼ D19 eB2y D12 eB1y þ D13 e2B3y þ D14 e2B1y þ D15 e2B2y D16 eB4y þ D17 eB5y D18 eB6y
þ EcðD36 eB3y D28 eB1y D29 eB2y þ D30 e2B3y
ð33Þ
Substituting the above solutions (28)–(23) in Eqs. (18)–(20), we get the final form of solutions for velocity, temperature, and concentration as given below:
þ D34 eB5y D35 eB6y Þ
ð34Þ
hðyÞ ¼ eB1y þ EcðD11 eB1y D5 e2B3y D6 e2B1y D7 e2B2y þ D8 eB4y D9 eB5y þ D10 eB6y Þ
Fig. 2
Velocity behavior with a.
Fig. 3
Velocity behavior with M.
ð35Þ
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
Effect of frictional heating on mixed convection
5
wðyÞ ¼ ðð1 þ D1 ÞeB2y D1 eB1y Þ þ EcðD19 eB2y B1y
D12 e
B4y
D16 e
2B3y
þ D13 e
B5y
þ D17 e
2B1y
þ D14 e
B6y
D18 e
2B2y
þ D15 e
Þ
ð36Þ
The physical quantities of practical interest are local skin friction factor; local Nusselt number and local Sherwood number are given as follows: Wall shear stress sw is given as: @u0 ¼ qf v20 u0 ð0Þ ð37Þ sw ¼ l 0 @y y0 ¼0
The local skin friction factor if given as sw Cfx ¼ ¼ u ð0Þ qf v20 ¼ ðB3 D4 þ B1 D2 þ B2 D3 Þ þ EcðB3 D36 þ B1 D28 þ B2 D29 2B3 D30 2B1 D31 2B2 D32 þ B4 D33 B5 D34 þ B6 D35 Þ The local surface heat flux is given as: @T0 qw ¼ k 0 @y y0 ¼0
Fig. 4
Velocity behavior with Gr.
Fig. 5
Velocity behavior with Gm.
ð38Þ
ð39Þ
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
6
C. Sulochana et al. The local Nusselt number is given as
Nux ¼
qw ; ðTw T1 Þ
ð40Þ
Shx @w ¼ ¼ w ð0Þ @y y¼0 Rex ¼ ðð1 þ D1 ÞB2 B1 D1 Þ þ EcðB2 D19 B1 D12 þ 2B3 D13 þ 2B1 D14 þ 2B2 D15 B4 D16 þ B5 D17
Nux @h ¼ ¼ h ð0Þ Rex @y y¼0
B6 D18 Þ where Rex ¼ vt0fx is the local Reynolds number.
¼ B1 þ EcðB1 D11 2B3 D5 2B1 D6 2B2 D7 þ B4 D8 B5 D9 þ B6 D10 Þ
ð42Þ
ð41Þ
4. Discussion of the results In this section we perpetrated impacts of sundry flow parameters namely, inclined angle a, Magnetic field M, Casson param-
The local Sherwood number is given as:
Fig. 6
Fig. 7
Velocity behavior with b.
Temperature behavior with b.
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
Effect of frictional heating on mixed convection
7
eter b, thermal Grashof number Gr, concentration Grashof number Gm, volume fraction /, Radiation R, heat source/sink parameter QH , Prandtl number Pr, Soret number Sr, chemical reaction parameter Kr on flow, thermal and concentration fields are interrogated and obtained with the assistance of graphical illustrations. Also, skin friction factor, heat and mass transport rates are displayed with tabular values. We deployed dual solutions for TiO2-water and CuO-water nanofluids. In the current analysis, we presumed the parametric values as constant in our complete analysis Pr ¼ 6; M ¼ 2:0; Sc ¼ 0:6; Sr ¼ 2:0; R ¼ 0:5; K ¼ 0:5; Gr ¼ 4:0; Gm ¼ 2:0; QH ¼ 2:0; Ec ¼ 0:01; a ¼ p=6; / ¼ 0:05; b ¼ 10. Except for the varied parameter values as shown in figures.
Fig. 8
Fig. 9
Figs. 2 and 3 depict the impact of escalating values of a and M on the nanofluid velocity distributions. As seen, increase in a lessens the velocity fields. This is due to the reason that a growth in the value of a improves the applied magnetic field. Generally, as a is equal to p=2 the applied aligned magnetic field acts as a transverse magnetic field. And rise in a from 0 to p=2 results in enlargement in the strength of magnetic field, due to this there develops an opposing force to the flow named as a Lorentz force. This force results in deceleration of velocity boundary layer. The similar kind of results has been perceived for magnetic field parameter M as shown in Fig. 3. Figs. 4 and 5 drawn to scrutinize the influence of Gr and Gm on velocity distributions. It is traced out that, the escala-
Velocity behavior with /.
Temperature behavior with /.
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
8
C. Sulochana et al.
tion in Gr and Gm, we witnessed inflation in velocity profiles of the flow. Physically, the ratio of buoyancy force to the viscous force is known as Grashof number, Disparity in the buoyancy forces regulates the momentum boundary layer. Due to this, we have noticed an enlargement in momentum boundary layer thickness. Figs. 6 and 7 reveal the impact of b on flow and thermal fields. It is obvious that raise in b diminish the velocity and thermal fields. Generally, elevation in b, the yield stress will be reduced. Hence, we depict deflation in momentum and thermal boundary layers. The nature of velocity and thermal distributions for ascending values of / are shown in Figs. 8 and 9. It manifest that,
Fig. 10
Fig. 11
hike in / decelerates the velocity fields and improves the thermal fields. Physically, raise in / enlarges the viscosity of the nanofluid as a result it slows down the velocity of the fluid. Hence, we have seen declination in velocity fields and enhancement in the temperature fields. Fig. 10 illustrates the impact of R on thermal fields. It is evident that, escalating in R inflates the thermal profiles. Generally, radiation is transmission of thermal energy. Rise in R contributes heat energy to the flow. The similar kind of outcomes are spotted for heat source/sink parameter QH as shown in Fig. 11. Fig. 12 is drawn to witness the impact of Pr on thermal profiles of the nanofluid. It is evident that the rise in Pr deflates the
Temperature behavior with R.
Temperature behavior with QH .
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
Effect of frictional heating on mixed convection
Fig. 12
Fig. 13
9
Temperature behavior with Pr.
Concentration behavior with Sr.
thermal fields. Physically, the ratio of momentum diffusivity to the thermal diffusivity delivers the Prandtl number, improvement in Pr possesses the lesser thermal conductivity. Hence, we have seen the declination in temperature profiles of the fluid. Fig. 13 is depicted to analyse the impact of Sr on concentration fields of the flow. It is clear that raise in Sr improves the concentration boundary layer. Physically, because of the thermal gradient, a mass flux is produced from the disparity in concentration species, this occurs in the case of Soret effect. Fig. 14 elaborates the influence of chemical reaction parameter Kr on concentration distributions of the flow. We reveal that,
increase in Kr declines the concentration fields. Generally, this may happen due to increase in the interfacial mass transfer. Tables 1 and 2 contrast the thermophysical properties of nanoparticles and base liquid, and numerical values for in skin friction factor, heat and mass transport rates for diverse values pertinent flow parameters for both TiO2-water and CuO-water nanofluids, respectively. From Table 2 notice that, M has a tendency to enhance thermal transfer rate and declines the mass transfer rate. The ascending Gr and Gm leads to boost the skin friction factor. A raise in R enhances the heat and mass transport rates. Chemical reaction parameter has ten-
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
10
C. Sulochana et al.
Fig. 14
Concentration behavior with Kr.
Table 1 Thermophysical properties of the base fluid (water), CuO and TiO2 nanoparticles. Physical properties
Water
CuO
TiO2
Cp ðJ=Kg KÞ q ðkg=m3 Þ j ðW=mKÞ b 105 ðK1 Þ
4179 997 0.613 21
531.8 6320 76.5 1.80
686.2 4250 8.9538 0.96
Present
Bvp4c
Shooting
Bvp5c
2 3 4 5
3.9932 1.4258 0.6498 0.1432
3.99322314 1.42584531 0.64986723 0.1432439
3.99322313 1.42584530 0.64986723 0.1432438
3.99322314 1.42584530 0.64986723 0.1432438
Variations in physical quantities of TiO2-water and CuO-water nanofluids.
Table 2 M
Validation of the results.
Table 3 M
Gr
Gm
R
QH
Kr
Sr
2 3 4 1 2 3 1 2 3 1 2 3 1 2 3 0.1 0.2 0.3 1 2 3
CuO-Water
TiO2-Water
Cfx
Nux =Rex
Shx =Rex
Cfx
Nux =Rex
Shx =Rex
3.9932 1.4258 0.6498 2.4417 2.9514 3.4680 2.2739 3.9932 5.7070 3.2751 3.4204 3.9932 3.4204 3.9932 5.9696 3.9932 4.0228 4.0332 2.6176 3.9932 5.3586
2.4258 2.4436 2.4500 2.4323 2.4304 2.4282 2.4487 2.4258 2.3895 2.0692 2.2881 2.4258 2.2881 2.4258 2.5921 2.4258 2.4823 2.4556 2.4427 2.4258 2.4030
1.3375 1.3600 1.3681 1.3455 1.3432 1.3405 1.3669 1.3375 1.2910 0.8786 1.1627 1.3375 1.1627 1.3375 1.5483 1.3375 1.2558 1.0898 0.1859 1.3375 2.4564
4.1429 1.2856 0.5505 2.4808 3.0251 3.5785 2.4159 4.1429 5.8576 3.1031 3.3204 4.1429 3.3204 4.1429 8.0168 4.1429 4.1940 4.1873 2.7442 4.1429 5.5249
2.4457 2.4633 2.4686 2.4529 2.4508 2.4484 2.4658 2.4457 2.4143 2.0826 2.3108 2.4457 2.3108 2.4457 2.5960 2.4457 2.5388 2.4731 2.4608 2.4457 2.4255
1.3629 1.3850 1.3918 1.3719 1.3692 1.3662 1.3886 1.3629 1.3228 0.8961 1.1919 1.3629 1.1919 1.3629 1.5536 1.3629 1.3284 1.1121 0.1974 1.3629 2.4996
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
Effect of frictional heating on mixed convection dency to regulate the Sherwood number. Table 3 depicts the validation of the present technique with numerical techniques.
11 Z4 ¼ Z2 ð1 þ D1 Þ; D2 ¼
5. Conclusions In the past few year’s investigations on the heat and mass transfer has been stimulating the researchers due to its perceptible significance in industries in addition to science and engineering applications. Owing to this prominence we are trying to study the boundary layer analysis of a 2D magnetohydrodynamic flow of chemically reacting Casson nanofluid flow over a semi-infinite inclined porous plate. The energy and diffusion equation are encompassed with frictional heating, heat generation/absorption, thermo diffusion and thermal radiation effects. For making the analysis more attractive we pondered two distinct type of nanofluids namely, TiO2-water and CuO-water. The conclusions are as follows: Higher the values of M inflate the thermal transport rate and deflate the mass transport rate. The improvement in Gr and Gm leads to regulating the skin friction factor. Raise R has tendency to upturn the thermal and mass transport rates. Higher values of Kr has a tendency to elevate the concentration fields and mass transfer rate. Dual solutions executed for CuO and TiO2 - water nanofluids.
D1 ¼
Z3 ; A2 B21 A1 B1 A3
D4 ¼ D2 þ D3 ;
D5 ¼
D24 ðA8 B23 þ A9 Þ ; A5 4B23 2A6 B3 A7
D6 ¼
D22 ðA8 B21 þ A9 Þ ; A5 4B21 2A6 B1 A7
D7 ¼
D23 ðA8 B22 þ A9 Þ ; A5 4B22 2A6 B2 A7
D9 ¼
2D2 D3 ðA8 B1 B2 þ A9 Þ ; A5 B25 A6 B5 A7
D10 ¼
D13 ¼
A11 B8 ; 4B23 2ScB3 A10
D15 ¼
A11 B10 ; 4B22 2ScB2 A10
Acknowledgement
D16 ¼
A11 B11 ; B24 ScB4 A10
The authors acknowledge the UGC – India for financial support under the UGC Dr. D. S. Kothari Postdoctoral Fellowship Scheme (No. F.4-2/2006 (BSR)/MA/13-14/0026).
D18 ¼
A11 B13 ; B26 ScB6 A10
D14 ¼
D17 ¼
A11 B9 ; 4B21 2ScB1 A10
A11 B12 ; B25 ScB5 A10
D19 ¼ D12 D13 D14 D15 þ D16 D17 þ D18 ;
Appendix A
D20 ¼ Z1 D11 Z2 D12 ; A1 ¼ 1 / þ /ðqs =qf Þ; A2 ¼ X1 =X2 ;
X1 ¼ 1 þ 1=b;
X2 ¼ ð1 /Þ5=2 ;
A3 ¼ M2 þ 1=KX2 ;
A44 ¼ 1 / þ /fðqbÞs =ðqbÞf g; A7 ¼ PrðR þ QH Þ; A10 ¼ ScKr;
A11 ¼ ScSr;
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A6 þ A26 þ 4A5 A7 2A5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi A1 þ A21 þ 4A2 A3 2A2
B5 ¼ B1 þ B2 ;
A5 ¼ knf =kf ;
A8 ¼ A2 Pr;
A9 ¼ PrM;
Sc þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Sc2 þ 4A10 ; 2
;
B2 ¼
;
B4 ¼ B1 þ B3 ;
B9 ¼ 4B21 D6 ;
A6 ¼ PrA6 ;
A12 ¼ A11 B21 ;
B6 ¼ B2 þ B3 ;
D22 ¼ Z1 D5 Z2 D13 ;
D21 ¼ Z2 D19 ; D23 ¼ Z1 D6 Z2 D14 ;
D24 ¼ Z1 D7 Z2 D15 ;
A4 ¼ 1 / þ /fðqcp Þs =ðqcp Þf g;
B8 ¼ 4B23 D5 ;
2D2 D4 ðA8 B1 B3 þ A9 Þ ; A5 B24 A6 B4 A7
D11 ¼ D5 þ D6 þ D7 D8 þ D9 D10 ; A11 B7 ; B21 ScB1 A10
B3 ¼
D8 ¼
2D3 D4 ðA8 B2 B3 þ A9 Þ ; A5 B26 A6 B6 A7
D12 ¼
B1 ¼
A12 ; ScB1 A10 Z4 D3 ¼ ; A2 B22 A1 B2 A3
B21
B7 ¼ B21 D11 ;
B10 ¼ 4B22 D7 ;
B11 ¼ B24 D8 ; B12 ¼ B25 D9 ; B13 ¼ B26 D10 ; Z1 ¼ Gr cos a; Z2 ¼ Gm cos a Z3 ¼ Z1 þ Z2 D1 ;
D25 ¼ Z1 D8 Z2 D16 ;
D26 ¼ Z1 D9 Z2 D17 ; D20 ; D27 ¼ Z1 D10 Z2 D18 ; D28 ¼ A2 B21 A1 B1 A3 D21 D22 ; D30 ¼ ; A2 B22 A1 B2 A3 4A2 B23 2A1 B3 A3 D23 ; D31 ¼ 2 4A2 B1 2A1 B1 A3 D29 ¼
D24 ; 4A2 B22 2A1 B2 A3 D26 ; D34 ¼ A2 B25 A1 B5 A3 D32 ¼
D35 ¼
D33 ¼
D25 ; A2 B24 A1 B4 A3
D27 ; A2 B26 A1 B6 A3
D36 ¼ D28 þ D29 D30 D31 D32 þ D33 D34 þ D35;
Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
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Please cite this article in press as: C. Sulochana et al., Effect of frictional heating on mixed convection flow of chemically reacting radiative Casson nanofluid over an inclined porous plate, Alexandria Eng. J. (2017), http://dx.doi.org/10.1016/j.aej.2017.08.006
