Heat and Mass Transfer Effects on MHD Oscillatory flow of a Couple Stress fluid in an Asymmetric Tapered channel

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1 IOP Conference Series: Materials Science and Engineering PAPER OPEN ACCESS Heat and Mass Transfer Effects on MHD Oscillator flow of a Couple Stress fluid in an Asmmetric Tapered channel To cite this article: J. Sasikumar et al 28 IOP Conf. Ser.: Mater. Sci. Eng View the article online for updates and enhancements. This content was downloaded from IP address on 3//29 at 5:44

2 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing Heat and Mass Transfer Effects on MHD Oscillator flow of a Couple Stress fluid in an Asmmetric Tapered channel J. Sasikumar,R.Gaathri 2,A.Govindarajan 3 23 Department of Mathematics, SRM IST, Kattankulathur, Tamil Nadu, Chennai sasikumar.j@ktr.srmuniv.ac.in, gaathri.ramachandran8@gmail.com 2, hod.maths@ktr.srmuniv.ac.in 3 23March 28 Abstract. In this paper it is proposed to discuss the effects of heat and mass transfer effects on oscillator flow of an incompressible electricall conducting viscous fluid under the influence of couple stress force in an asmmetric channel filled with porous medium. The fluid is assumed to be electricall conducting in the presence of externall applied uniform magnetic field. The governing equations of flow consisting of momentum, energ equation and concentration have been formulated under Boussinesq approximation and non-dimensionalized with suitable boundar conditions. The closed form solutions for velocit temperature and concentration are obtained. The effect of various parameters on the fluid velocit temperature and concentration are analsed using graphs and tables. Kewords: Oscillator flow, couple-stress, Asmmetric Channel, Heat, Mass Transfer.. Introduction Heat transfer in magnetohdrodnamic (MHD) flows emerge in various innovative applications. The MHD flow in the planar channels prompts a start-up procedure suggesting in this manner a viscous laer at the limit is all of a sudden set into movement and ends up imperative in the utilization of different branches of geophsics, astronom and liquid building. A field in which MHD will assume a basic part is atomic combination, where it is engaged with no less than two distinct issues: the control and elements of plasma, and the conduct of the fluid metal amalgams utilized in a portion of the right now considered outlines of tritium reproducing covers. A hpothetical investigation of the oscillator flow of a couple stress impacts, in a rotating, channel, has been directed b []. [2] discussed the consistent hdromagnetic flow of a couple stress liquid affected b a uniform magnetic field. A hpothetical investigation of the heat Content from this work ma be used under the terms of the Creative Commons Attribution 3. licence. An further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence b IOP Publishing Ltd

3 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing and mass transfer impacts on an unstead flow of a couple-stress liquid in a horizontal wav permeable space with travelling thermal waves emploed b [3]. [4] studied effect of chemical reaction of a couple stress fluid on an inclined assmetric channel. The oscillator flow in the opticall thin thermal radiation limit has been studied b [5]. The peristaltic motion of a couple stress fluid in a porous medium is studied b [6]. [7] studied the effects of chemical reaction in oscillator flow and mass transfer of in an asmmetric wav channel. Sunspots are caused b the solar magnetic fields, the sun power is also governed b MHD. The effect of MHD oscillator flow through porous medium with heat source has been investigated b[8]. [9] studied the effects heat and mass transfer in the oscillator flow of blood. The oscillator flow have extraordinar pertinence with applications in oil-penetrating, fabricating, preparing of nourishments, oil investigation, and polmer enterprises. The oscillator channel flow in viscous fluid is extended to a non-newtonian Jeffre fluid model discussed b []. [] analsed the oscillator flow in an asmmetric channel under the effects of chemical reaction and heat transfer. Couple pressure fluid theor, created b [2], is one among the polar fluid theor which considers couple stresses notwithstanding the traditional Cauch stress. The cardiovascular framework is delicate to changes in the earth, and flow qualities of blood are adjusted to fulfill changing requests of the life form. For numerous reasons, uses of MHD in phsiological stream issues are of developing interest. Oscillator flow of a fluid and heat transfer with porous under the magnetic field is discussed b [3]. The problem of an oscillator MHD convective flow in a vertical porous channel is discussed b [4]. Tapered channel ma fill in as a model for the intrauterine fluid movement in a sagittal cross-segment of the uterus under tumour treatment. [5] discussed the effects of magnetic field in the peristaltic flow in the tapered-asmmetric channel.[6-8] discussed effects of chemical reaction and thermal radiation on MHD oscillator flow in a porous medium. 2. Formulation of the problem Considering the viscous incompressible flow, of electricall conducting couple stress fluid in an asmmetric tapered wav channel. The geometr of the wall is given as [ 2π ] H = d m X a sin λ (X)+φ [ 2π ] H 2 = d + m + a 2 sin λ (X) () 2

4 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing Figure. Geometr of the asmmetric tapered channel in which a and a 2 are the amplitudes of the left and right of the walls and m ( ) is the non-uniform parameter of the channel, φ varies in the range φ π, when φ = corresponds to the smmetric channel. a,a 2,d and φ must satisf the condition of the channel a 2 + a a a 2 cos(φ) (2d) 2 The governing equations are given b Equation of momentum u t = p ρ x + ν 2 u η 4 u 2 ρ σb2 u + gβ 4 T (T T )+gβ C (C C )(2) ρ Energ equation ( ) T k 2 T = T ρc p + 4α 2 (T T 2 ) (3) ρc p Concentration equation C T = D 2 C (4) 2 The boundar conditions are = H : u =,T = T,C = C (5) = H 2 : u =,T = T 2,C = C 2 (6) The radiative heat flux is given as q =4α2 (T 2 T ) (7) where α 2 e bλ = K λw T dλ, K λw is the absorption coefficient,e bλ is a plank s function. We introduce the following non-dimensional quantities: x = X λ, = Y d,t = νt d 2,u = ud ν,h = H d,h 2 = H [ ] 2 σ d,m2 =, ρν 3

5 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing Gr = gβ T d 3 (T T ) ν 2,Gc= gβ Cd 3 (C C ) ν 2,Sc= D ν,θ = (T T ) (T T ) φ = (C C ) (C C ) (8) The channel wall equations in non-dimensional form becomes h =--mx -bsin(2πx + φ) h 2 = +mx +asin(2πx ) The governing equations in dimensionless form together with appropriate boundar conditions as follows: Re u t = 2 u 2 η 4 u 4 p x M 2 u + Grθ + Gcφ (9) Pe θ t = 2 θ 2 + N 2 θ () φ t = Sc φ 2 2 () The corresponding non-dimensional quantities are: Stress-free conditions = h : u =,θ =,φ= (2) = h 2 : u =,θ =,φ= (3) 2 u 2 =for = h,h 2 3. Solution of the problem The sstem of equations (9)-() can be reduced to dimensionaless form b assuming the following: p x = λeiωt (4) u(, t) =u ()e iωt θ(, t) =θ ()e iωt φ(, t) =φ ()e iωt (5) Now substituting equations (5) and (6) in equations (9)-() we obtain the following equations: 4

6 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing η d4 u d + d2 u 4 d +(Reiω + M 2 )u 2 = Grθ + Gcφ λ (6) d 2 θ d +(N 2 iωp e)θ 4 = (7) ( ) d 2 φ iω d φ 2 = (8) Sc subject to the boundar conditions = h : u =,φ =,θ = = h 2 : u =,φ =,θ = (9) Stress free conditions = h,h 2 : u = (2) substituting the above boundar conditions(2)-(2) in the equations (7)-(9) we obtain the following: Velocit distribution Temperature distribution [ ( u(, t) = A e α + B e α + C e β + D e β + ( Gcφ + η L 4 L 2 + Q 2 θ = Concentration distribution 5 Grθ η P 4 P 2 + Q 2 ) λq 2 ] e iωt (2) [ ] sin(p ( h2 )) e iωt (22) sin(p (h h 2 )) [ ] sinh(l( h2 )) φ = e iωt (23) sinh(l(h h 2 )) (24) The rate of heat transfer Nu = ( ) [ ] θ cosp ( h2 ) = e iωt =h,h 2 sinp (h h 2 ) )

7 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing The rate of mass transfer [ ] cosp (h h 2 ) Nu = e iωt (25) sinp (h h 2 ) [ ] Nu = e iωt (26) sinp (h h 2 ) [ ] [ ] φ Lcosh[L( h2 )] Sh = = e iωt =h,h 2 sinh[l(h h 2 )] [ ] cosh(l(h h 2 )) Sh = e iωt (27) sinh(l(h h 2 )) [ ] Sh = e iωt (28) sinh(l(h h 2 )) 4. Results and discussions The graphical distribution for the magnetic parameter M, Grashoff number Gr, radiation parameter N, frequenc of oscillation ω, Peclet number Pe, Renolds number Re, Schimdt number Sc on velocit u, temperature θ, and concentration φ. Fig.2 depicts the response of velocit to the increasing value of Gr. It shows that there is a backflow instigated in the left wall and significant flow in the right wall. Fig.3 demonstrates that increase in Sc instigates a perceivable decrease in the velocit over the walls near the left wall velocit stas positive further from the left wall as we approach the wall focus and from that point the right wall, there is impressive hindrance on the stream prompting to significant backflow. In fig.4 velocit profile for increase in Peclet number has been discussed. Increase in Peclet number accelerate the velocit in the left wall but decelerates the velocit in the right wall. In fig.5 we can see that there is a backflow of velocit induced in the left wall and significant flow in the right wall. Fig.6 shows the effect of radiation parameter N, on temperature profile. When there is an increase, there is an increase of the effect of N on temperature. Fig.7 depicts the Peclet number effect on temperature profile. When the Peclet number increases the slope decreases. Fig.8 discusses the frequenc of oscillation in temperature profile. It shows that increase in frequenc of oscillation decreases the temperature. Similarl increase in Sc increases the temperature profile. When there is an increase in frequenc of oscillation there is a decrease in concentration distribution which is shown in fig.9 and in fig. concentration increases for increasing Sc. In fig. and fig.2 for various values of Pe, there is a noticeable decrease of heat transfer on the wall =h and =h 2. Fig.3 and Fig.4 shows that increase in N increases the heat transfer at the walls =h and =h 2. In fig.5 and fig.6 increase in Sc increases the mass transfer at the walls =h and =h 2. 6

8 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing VELOCITY 3 2 a=.5;b=.2;m=.5;φ=π/4; x=.5; pe=.3; ω=;t=.;gc=;n=; Sc=.2;η=;Re=; M=;λ=.5; VELOCITY DISTRIBUTION Gr = 4 Gr = 5 Gr = 6 VELOCITY a=.5;b=.2;m=.5;φ=π/4; x=.5; pe=.3; ω=;t=;gc=;n=; Gr = ;η=;re=; M=;λ=.5; VELOCITY DISTRIBUTION Sc =.7 Sc =.9 Sc = (a) FIGURE:2 Velocit profile for various values of Gr (b) FIGURE:3 Velocit for various values of Sc 5 5 VELOCITY DISTRIBUTION Pe =.7 Pe =.8 Pe = a=.5;b=.2;gr=;φ=π/4; x=.5; pe=.3; ω=;t=;gc=;n=; Sc=.2;η=;Re=; M=;λ=.5; VELOCITY DISTRIBUTION VELOCITY a=.5;b=.2;m=.5;φ=π/4; x=.5; Gr=; ω=;t=;gc=;n=; Sc=.2;η=;Re=; M=;λ=.5; VELOCITY.5.5 m =. m =.2 m = (c) FIGURE:4 Velocit profile for various values of Pe (d) FIGURE:5 Velocit profile for various values of m 7

9 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing TEMPERATURE RADIATION N =.9 N = N =. N = TEMPERATURE DISTRIBUTION Pe =.4 Pe =.5 Pe =.6 Pe =.7 TEMPERATURE.2.8 TEMPERATURE a=.5;b=.2;m=.5;φ = π/4; x=.5;pe=.3;ω =;t=;.4.2 a=.5;b=.2;m=.5;φ = π/4; x=.5;n=.9;ω =;t=; (e) FIGURE:6 Temperature profile for various values of N (f) FIGURE:7 Temperature profile for various values of Pe.4 TEMPERATURE DISTRIBUTION.2 ω = ω = 2 ω = 3 ω = 4 TEMPERATURE a=.5;b=.2;m=.5;φ = π/4; x=.5;pe=.3;n=.9;t=;.5.5 (g) FIGURE:8 Temperature profile for various values of ω 8

10 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing.2.8 CONCENTRATION DISTRIBUTION ω = ω = 2 ω = 3 ω = CONCENTRATION DISTRIBUTION Sc =.2 Sc =.4 Sc =.6 Sc =.8 CONCENTRATION.6.4 a=.5;b=.2;m=.5;φ = π/4; x=.5;pe=.3;t=; Sc= CONCENTRATION.6.4 a=.5;b=.2;m=.5;φ = π/4; x=.5;t=;ω= (h) FIGURE:9 Concentration profile for various values of ω (i) FIGURE: Concentration profile for various values of Sc HEAT TRANSFER Pe =.5 Pe =.6 Pe =.7 Pe = HEAT TRANSFER Pe =.5 Pe =.6 Pe =.7 Pe =.8.84 =h a=.5;b=.2;m=.5;φ=π/4; x=.5;n=.;w=.5 =h 2 a=.5;b=.2;m=.5;φ=π/4; x=.5;n=.;w= Nu Nu t t (j) FIGURE: Heat transfer at the wall =h for various values of Pe (k) FIGURE:2 Heat transfer at the wall =h 2 for various values of Pe 9

11 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing HEAT TRANSFER =h N =. N =.2 N =.3 a=.5;b=.2;m=.5;φ=π/4; x=.5;pe=.3;w= HEAT TRANSFER = h 2 N =. N =.2 N =.3 a=.5;b=.2;m=.5;φ=π/4; x=.5;pe=.3;w= Nu Nu t t (l) FIGURE:3 Heat transfer at the wall =h for various values of N (m) FIGURE:4 Heat transfer at the wall =h 2 for various values of N.4.2 MASS TRANSFER Sc= Sc = 3 Sc = 5.8 a=.5; b=.2; x=.5; φ=π/4;m=.5;n=; Pe=.3;w=; MASS TRANSFER.2 = h a=.5; b=.2; x=.5; φ=π/4;m=.5;n=; Pe=.3;w=;.6.4 =h 2 Sh Sh Sc = Sc = 2 Sc = t t (n) FIGURE:5 Mass transfer at the wall =h for various values of Sc (o) FIGURE:6 Mass transfer at the wall =h 2 for various values of Sc 5. Conclusion A theoritical analsis on effects of heat and mass transfer in MHD oscillator flow of a couple stress fluid in an asmmetric tapered channel has been discussed in this paper. - Velocit profile increases for increasing value of non-uniform parameter m, Peclet number Pe, Schmidt number Sc, and Grashoff number Gr on the upper wall = h. - Velocit decreases for increasing value of m, Pe, Sc, and Gr on the lower wall = h 2. - Temperature decreases for increasing value of Pe, and increases for radiation N. - Concentration decreases for increasing value of and increases for increasing Sc. - At both the walls = h and = h 2 heat transfer increases while varing t and increasing value of N and decreases at both the walls for increasing Pe.

12 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing 6. Nomenclature a,a 2 - Amplitudes of right and left wall B - Uniform magnetic field g - Accelaration due to gravit Gr - Grashof number Gc - Modified Grashof number M - Hartmann number m,m - Non-uniform parameters q r - The radiative heat flux Re - Renolds number D - Mass diffusion coefficient N - Thermal radiation C,C - Concentration of the walls T,T - Temperature of the walls u - Dimensionless velocit component η - Couple stress parameter k - Porous permeabilit t,t - Dimensional and dimensionless time Pe - Peclet number 7. Greek smbol θ - Temperature κ - Permeable parameter λ - Wavelength φ - Phase difference β C - Coefficient of mass expansion ν - Kinematic viscosit ω - Frequenc of oscillation 8. Appendix α 2 = + 4η Q 2, β 2 = 4η Q 2, Q 2 =(Reiω + M 2 ), 2η 2η P 2 = N 2 iωp e, L 2 = iω Sc, Gr I = η P 4 P 2 + Q, J = Gc 2 η L 4 L 2 + Q, 2 A = eαh 2 β 2 JQ 2 Ie αh 2 β 2 JQ 2 + e αh 2 β 2 λ + e αh 2 L 2 JQ 2 Ie αh 2 P 2 Q 2 β 2 λe αh 2 P 2 Q 2 2(α 2 β 2 )sinh[β(h h 2 )]

13 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing [ λ B = 2sinh[α(h h 2 )] Q 2 (eαh 2 eα h )+e αh 2 (I + J) C e αh +βh 2 e αh 2+βh ] D e αh βh 2 e αh 2 βh λα 2 C = Q 2 (α 2 β 2 ) e βh 2 D (e 2βh 2 ) ( ) (α 2 + P 2 )I +(α 2 L 2 )J + λα2 e βh 2 λα2 e βh Q 2 Q 2 D = 2(α 2 β 2 )sinh[β(h h 2 )] References [] Sahin Ahmed, Anwar Beg O, Ghosh S K 24 A couple stress fluid modeling on free convection oscillator hdromagnetic flow in an inclined rotating channel, Ain Shams Engineering Journal. 5, [2] Ashmaw E A 26 Drag on a slip spherical particle moving in a couple stress fluid, Alexandria Engineering Journal. 55, [3] Samala Sarojini M, Veera Krishna M, and Uma Shankar C 2 MHD flow of a couple stress fluid through a porous medium in a parallel plate channel in presence of effect of inclined magnetic field, International Journal of Phsics and Mathematical Sciences ISSN: , 9 8 [4] Muthuraj R, Srinivas S, and Selvi R K 23 Heat and Mass Transfer Effects on MHD Flow of a Couple-Stress Fluid in a Horizontal Wav Channel with Viscous Dissipation and Porous Medium, Heat Transrfer Asian Research. 42, [5] Gnaneswara Redd M, Venugopal Redd K, Makinde O D 26 Hdromagnetic peristaltic motion of a reacting and radiating couple stress fluid in an inclined asmmetric channel filled with a porous medium, Alexandria Engineering Journal. 55, [6] Sankad G C, Pratima Nagathan S 26 Unstead MHD peristaltic flow of a couple stress fluid through porous medium with wall and slip effects, 55, [7] Sata Naraana P V, Venkateswarlu B, Devika B 26 Chemical reaction and heat source effects on MHD oscillator flow in an irregular channel, Ain Shams Engineering Journal. 7, [8] Sasikumar J, Govindarajan A 25 Free Convective MHD Oscillator flow Past Parallel Plates in a Porous Medium with Heat Source and Chemical Reaction, International Journal of Scientific & Engineering Research. 6, [9] Misra J C, Adhikar S D 26 MHD oscillator channel flow, heat and mass transfer in a phsiological fluid in presence of chemical reaction, Alexandria Engineering Journal. 55, [] Aamir Ali, Saleem Asghar 24 Analtic Solution for Oscillator Flow in a Channel for Jeffre Fluid,. Journal of Aerospace Engineering., [] Ogulu A 25 TOn the oscillating plate-temperature flow of a polar fluid past a vertical porous plate in the presence of couple stresses and radiation, International Communications in Heat and Mass Transfer. 32, [2] Stokes V K 966 Couple stress in fluid, Phs. Fluids. [3] Adhikar S D and Misra J C 2 Unstead two-dimensional hdromagnetic flow and heat transfer and heat transfer of a fluid, Int. J. of Appl. Math. and Mech. 7(4), 2 [4] Garg B P, Singh K D, Bansal A K 25 Oscillator MHD convective flow of second order fluid through porous medium in a vertical rotating channel in slip-flow regime with heat radiation, Int.J. of Applied Mechanics and Engineering. 2, [5] Kothandapani M, Prakash J 25 Effects of thermal radiation parameter and magnetic field on the peristaltic motion of Williamson nanofluids in a tapered asmmetric channel, International Journal of Heat and Mass Transfer. 8,

14 2nd International conference on Advances in Mechanical Engineering (ICAME 28) IOP Publishing [6] Sasikumar J, Govindarjan A 26 Effect of Heat and Mass transfer on MHD Oscillator flow with Chemical reaction and slip conditions in asmmetric wav channel, ARPN Journal of Engineering and Applied Sciences, 64 7 [7] Sasikumar J, Govindarajan A 28 Soret effect on Chemicall Radiating MHD Oscillator Flow with Heat Source through Porous Medium in Asmmetric Wav Channel, Journal of Phsics:Conf.series., 2 34 [8] Sasikumar J, Bhuvaneshwari, Govindarajan A 28 Diffusion of chemicall reactive species in MHD oscillator flow with thermal radiation in the presence of constant suction or injection, Journal of Phsics:Conf.series.,