Effects of Chemical Reaction and Radiation Absorption on MHD Flow of Dusty Viscoelastic Fluid

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1 Available at Appl. Appl. Math. ISSN: Vol. 9 Issue (June 4) pp Applications and Applied Mathematics: An International Journal (AAM) Effects of Chemical Reaction and Radiation Absorption on MHD Flow of Dust Viscoelastic Fluid J. Prakash A. G. Vijaa Kumar M. Madhavi 3 and S. V. K. Varma 4 Department of Mathematics Universit of Botswana Private Bag Gaborone Botswana prakashj@mopipi.ub.bw 3 Department of Mathematics R. V. P. Engineering College for Women Tadigotla Kadapa A.P India madhavi769@gmail.com Fluid Dnamics Division School of Advanced Sciences VIT Universit Tamil Nadu India agvijakumar79@gmail.com 4 Department of Mathematics S. V. Universit Tirupati A.P India svijaakumarvarma@ahoo.co.in Abstract This investigation is undertaken to stud the effects of heat source and radiation absorption on unstead hdro-magnetic heat and mass transfer flow of a dust viscous incompressible electricall conducting fluid between two vertical heated porous parallel plates in the presence of chemical reaction under the influence of a transverse applied magnetic field. Initiall the channel walls as well as the dust fluid are assumed to be at the same temperature and the mass is assumed to be present at low level concentration so that it is constant everwhere. It is also assumed that the dust particles are non-conducting solid spherical and equal in sizes these are uniforml and smmetricall distributed in the flow field. The governing equations are solved analticall using the perturbation technique. Non-dimensional velocit temperature concentration and skin-friction are discussed through graphs for various phsical parameters entering into the problem. It is found that velocit of the dust particles is less than that of the dust fluid and the skin-friction of the dust particles is greater than that of the dust fluid. It is observed that the temperature is minimal at the centre of the channel and decreases towards the plates whereas the concentration is minimal at the center of the channel but increases towards the plates. Kewords: MHD; heat and mass transfer; radiation absorption; heat source; chemical reaction MSC No.: 76T 76T5 4

2 4 J. Prakash et al. Nomenclature B - Magnetic field - Electric conductivit M - Magnetic field parameter C - Species concentration in the fluid P r - Prandtl number C - Concentration of the plate w S - Schmidt number c C - Initial uniform concentration at T T - Plate temperature w c - Specific heat at constant pressure p T - Initial temperature g - Acceleration due to gravit t - Time G r - Thermal Grashof number A - Deca parameter N - Number densit of the dust particles K - Resistance coefficient of dust particles D Coefficient of proportionalit for the absorption of radiation u v - Velocities of dust fluid and dust particles respectivel S - Dimensionless heat source or sink parameter Q - Dimensional heat source/sink parameter - Radiation absorption parameter G m - Modified Grashof number : - Co-ordinate axes normal to the plates k : - Dimensionless permeabilit parameter K : - Dimensional permeabilit of the porous medium K : - Dimensional chemical reaction parameter l K r :- Dimensionless chemical reaction parameter : - Thermal diffusivit : - Densit. Introduction In nature the availabilit of pure air or water is scarce. Air and water are contaminated with impurities like CO NH O etc. or salts in water. The influence of dust particles on free-forced convective flow of dust viscous fluids has attained some importance in man industrial applications such as wastewater treatment power plant piping combustion and petroleum transport. Flows with heat and mass transfer of electricall conducting fluids in channels under

3 AAM: Intern. J. Vol. 9 Issue (June 4) 43 the influence of a transverse magnetic field occur in MHD power generators the cooling of nuclear reactor boundar laer control in aerodnamics and crstal growth. Until recentl this stud has been largel concerned with the flow of heat and mass transfer characteristics in various phsical situations. On the other hand mass diffusion rates can be altered tremendousl with chemical reactions. The chemical reaction effects depend on whether the reaction is homogeneous or heterogeneous. This in-turn depends on whether the occur at an interface or as a single phase volume reaction. In well-mixed sstems the reaction is heterogeneous if it takes place at an interface and homogeneous if it takes place in the solution. In the majorit of cases a chemical reaction depends on the concentration of the species itself. A reaction is said to be of the first order if the rate of reaction is directl proportional to the concentration itself Cussler (988). A few representative areas of interest are those in which heat and mass transfer combined with chemical reaction pla an important role like in food processing and polmer production. Chambre and Young (958) have analzed a first order chemical reaction in the neighborhood of a horizontal plate. Alexander (98) studied the analtical solution for mass transfer with a chemical reaction of the first order. Das et al. (994) have studied the effects of the homogeneous first order chemical reaction on the flow past an impulsivel started vertical plate with uniform heat flux and mass transfer. The nature of dust viscous and viscoelastic fluids under the influence of different phsical conditions has been studied b several researchers notabl b Saffman (96) Micheal and Nore (968) Raptis and Perdikis (985). Singh () proposed the stud of free convection and mass diffusion of a dust viscoelastic (Walter s Liquid Model-B) fluid flowing between two heated porous plates in porous media in the presence of a magnetic field. Kumar and Srivastava (5) studied the effects of chemical reaction on MHD flow of dust viscoelastic (Walter s Liquid Model-B) liquid with heat source/sink. Mbeledogu and Ogulu (7) Patil and Kulkarni (8) proposed the stud of convective heat and mass transfer in an incompressible viscous Boussinesq fluid in the presence of a chemical reaction of the first order. Osalusi et al. (8) Afif (9) and Beg et al. (9) have studied the effects of thermo diffusion (Soret effects) on MHD mixed convection heat and mass transfer of an electricall conducting fluid. Nandeppanavar et al. () have proposed a stud on heat transfer in a Walter s Liquid B Fluid over an impermeable stretching sheet with heat source/sink. Makinde and Chinoka () analzed MHD transient flows and heat transfer of a dust fluid. In a channel with Navier slip condition. Sharma et al. () have discussed the unstead MHD heat and mass transfer free convective flow of a viscous fluid through a Darcian porous regime adjacent to a moving semi-infinite vertical plate. The governing equations were solved with Element free Galerkin method. Prakash et al. () have investigated the effects of thermal diffusion and chemical reaction on MHD flow of a dust viscous incompressible and electricall conducting fluid between two vertical heated porous plates with heat source/sink. Sivaraj and Kumar () analzed the problem of stead mixed convective and laminar flow of viscoelastic and viscous fluids in a vertical porous channel filled with viscoelastic fluid in one region and viscous fluid in another region. Mastanrao et al. (3) have studied chemical reaction and combined buoanc effects of thermal and mass diffusion on MHD convective flow

4 44 J. Prakash et al. along an infinite vertical porous plate in the presence of Hall current with variable suction and heat generation.. Mathematical Analsis In this problem we considered the effects of heat source (or sink) and radiation absorption on the unstead dust flow of a viscous incompressible slightl conducting viscoelastic fluid between two heated infinite vertical porous parallel plates (h distance apart) located at the = h and hplanes and extending from x to and from z to under the influence of a uniform magnetic field in the presence of chemical reaction of the first order. The x-axis is taken along the flow midwa of the plates ( = ) and -axis is taken normal to it. Let u and v be the velocities of dust fluid and dust particles respectivel in x-direction. Initiall the channel walls as well as the dust fluid are assumed to be at the same temperature T with concentration levels C (assumed to be present at low level and uniforml distributed everwhere). At time t the temperature of the walls is instantaneousl raised to T w and the species concentration is raised toc. It is also assumed that w p. The flow in the x-direction is driven b a constant pressure gradient with negligible bod x forces.. A uniform magnetic field with magnetic flux densit vector B is applied in the positive - direction. This is also assumed to be also the total magnetic field. 3. There is no externall applied electric field and the induced magnetic field is neglected b assuming a ver small magnetic Renolds number. 4. The dust particles are non-conducting solid spherical and equal in size. The are distributed uniforml and smmetricall everwhere in the flow field with number densit N in the entire flow. 5. There exists a chemical reaction in the mixture. 6. The plates are assumed to be infinite in the x and z directions so that the phsical quantities do not change in these directions. = h u = Bo h x x Flow geometr of the problem = h u =

5 AAM: Intern. J. Vol. 9 Issue (June 4) 45 Under these assumptions with Boussinesq s approximation the fluid motion temperature and concentration are governed b the following set of equations. KN g T T g C C K v u B u u u u v t t K v m K u v t T T Q D T T C C t cp cp cp () () (3) C t C D K C C l. (4) At t = the temperature and concentration level changes according to the following laws: w at w at T T T T e C C C C e with the following initial and boundar conditions t : u v T T d d at w at w at w at w. t : u v T T T T e C C C C e for d u v T T T T e C C C C e for d (5) On introducing the following non-dimensional quantities * * ud * vd * t u v t d d mn T * T T T T w * C C * C a C C w d a 3 dk k. We have the following governing equations which are dimensionless u u GT r GmC E vu M u t t w k (6)

6 46 J. Prakash et al. v w u v t (7) T T Pr ST C t (8) C Sc t C Sc Kr C. (9) Now initial and boundar conditions (5) according to new sstem reduce to t : u v T in t : u v T exp( at) C exp( at) for t : u v T exp( at) C exp( at) for () where B d M (Hartmann number) w G r G m m Kd g Tw T (Relaxation time parameter for particles) d g Cw C Sc (Schmidt number) D 3 (Grashof number) d 3 (Modified Grashof number) d Q S (Heat source/sink parameter) Kr Kld (Chemical reaction parameter) E K (Viscoelastic parameter) d c p Pr (Prandtl number)

7 AAM: Intern. J. Vol. 9 Issue (June 4) 47 c p d T C w w C T D (Radiation absorption parameter). 3. Method of Solution To solve the equations (6) to (9) subject to the boundar conditions () according to Pop (968) we assume at u t u u t e... at v t v v t e... at T t T T t e... at C t C C t e.... () Using () in equations (6) (7) (8) and (9) and comparing the coefficients of the harmonic and non-harmonic terms we obtain the following set of equations: u M u v u GT G C k w r m () ae um au vu GT r GmC k w u v u aw v TST C (3) (4) (5) (6) T SaPr T C (7) C ScKrC CSc Kra C (8) (9) where primes represent differentiation with respect to and further the boundar conditions are reduced to:

8 48 J. Prakash et al. u v u v T C T C at. and u v u v T C T C at. () B solving the equations () to (9) under the boundar conditions () we get the solutions for the velocit (dust fluid and dust particles) temperature and concentration fields as follows: u t A6 cosh A cosh A A5 cosh cosh S S A5 A6 cosh M cosh M cosh A cosh A cosh M A7cosh A3 A8cosh A A9cosh M e at 3 () v t A6 cosh A cosh A A5 cosh cosh S S A5 A6 cosh M cosh M cosh cosh cosh A7cosh A3 A8cosh A A9cosh M e at aw A3 A M () T t A cosh cosh S S A cosh A cosh A A4cosh A cosh A 3 A4 cosha3 e cosh A at (3) cosh A cosh A at C t e (4) cosh A cosh A where A A 3 S A S a Pr ScKr a A A S A4 M M A A k 3

9 AAM: Intern. J. Vol. 9 Issue (June 4) 49 a Gr Gm M M a Gr Gm ae aw ae ae GAG G A G A A A A r r m r SM A M A3 M G G A A A A A. m r A M 4. Skin-Friction Let f and p be the skin friction for dust fluid and dust particles respectivel. Then p f u A A tanha A S tanh S M A A tanhm at AAtanhM AAtanhA M Atanh M e v A A tanha A S tanh S M A A tanhm at AA 3 7 tanhm3 AA 3tanhAM A9 tanh Me. aw (5) (6) 5. Results and Discussion To get a phsical insight into the problem the numerical evaluation of the analtical results for velocit skin-friction for a dust fluid dust particles also temperature and concentration profiles for adust fluid have been calculated. The effects of various phsical parameters like magnetic parameter M heat source parameter S radiation absorption parameter chemical reaction parameter K r permeabilit parameter k the Schmidt number Sc and the Prandtl number P r on velocit temperature concentration and skin-friction are displaed through graphs. The concentration profiles for different values of the Schmidt number is considered.6 which corresponds to water-vapor. (Helium).78 (Carbon monoxide).96 (Methonal) and chemical reaction parameter K r (..3.5 &.7) with a =.9 S =. t = P r =.7(usuall for air) of the dust fluid are displaed through Figures and respectivel. From these Figures it is seen that increasing values of the Schmidt number S c and the chemical reaction parameter K r decreases the concentration. It is also observed that concentration is minimum at the centre of the channel ( = ) and increases towards the plates. The fluid temperature profiles for various

10 5 J. Prakash et al. values of Heat source/sink parameter S (..4.6 &.8) the Prandtl number P r (.5 &.7) and the radiation absorption parameter (.. 3. & 4.) with a =. and t =. are illustrated through the Figures 3 and 4. It is found that with the increase of the heat source/sink parameter and the Prandtl number the temperature decreases while it increases with the increase of the radiation absorption parameter. It is also observed that the temperature is minimum at the centre of channel ( = ) and decreases moving towards the plates. The behavior of the velocit of the liquid and the dust particles under the influence of the magnetic field parameter M (.. 3. & 4.) the radiation absorption parameter (.. 3. & 4.) and the heat source parameter S (... &.3) with S c =. K r =. a =.9 P r =.7 w =.5 E = d = k = G r = G m = 5 and t =. are shown in Figures 5-7 respectivel. From these Figures it is observed that b increasing the values of the magnetic field parameter we decrease the velocit of the dust fluid and dust particles absolutel while increasing the radiation absorption parameter and heat source/sink parameter. It is also noticed that the velocit is maximum at the centre of channel ( = ) and decreasing towards the plates. The velocit profiles for different values of the Schmidt parameter S c (..6 &.78) and the permeabilit parameter k (... &.3) with M = 4. K r =. a =.9 P r =.7 w =.5 E = = 5. d = G r = G m = 5 and t =. are displaed in Figures 8 and 9 respectivel. It is found that increasing the values of the Schmidt parameter and permeabilit parameter increases the velocit for both liquid and dust particles. It is also observed that the velocit is maximum at the centre of channel ( = ) and decreases towards the plates. In the above investigation it is ver interesting to note that the velocit of the dust particles is less than the velocit of the dust fluid. Skin-friction is presented in Figures and against the magnetic field parameter (M) and the heat source parameter (S) respectivel. From these Figures it is seen that the skin friction increases with increasing the magnetic field parameter M while it decreases with increasing heat source parameter S. It is also observed that the skin friction of the dust particles is greater than that of the dust fluid..9.8 Sc= Concentration Figure. Concentration profiles for different S c

11 AAM: Intern. J. Vol. 9 Issue (June 4) Concentration.6.5 Kr= Figure. Concentration profiles for different K r.3.. S= Pr=.7... Pr=.5 Temperature Figure 3. Temperature profiles for different S and P r Temperature pi= Figure 4. Temperature profiles for different

12 5 J. Prakash et al. M= (u v) Dust Fluid (u)... Dust Particles (v) Figure 5. Velocit profiles for different M 5 Dust Fluid (u)... Dust Partucles (v) -5 (u v) Pi= Figure 6. Velocit profiles for different Dust fluid (u)... Dust Particles (v) - - (u v) S= Figure 7. Velocit profiles for different S

13 AAM: Intern. J. Vol. 9 Issue (June 4) Velocit 6 4 Sc= Dust fluid (u)... Dust Particles (v) Figure 8. Velocit profiles for different S c (u v) 5 Kr=....3 Dust Fluid (u)... Dust Particles (v) Figure 9. Velocit profiles for different k Dust Fluid Dust Particles Skin-friction Skin friction for M (magnetic parameter) Figure. Skin-friction for different M (Magnetic parameter)

14 54 J. Prakash et al Skin-friction Dust Fluid Dust Particles Skin friction for S (Heat source parameter) Figure. Skin-friction for different S (Heat source parameter) 6. Conclusions It is found that with the increase of the heat source/sink parameter and the Prandtl number the temperature decreases while it increases with the increase of the radiation absorption parameter. It is also noticed that the temperature is minimum at the centre of the channel and decreases towards the plates whereas the concentration is minimum at the centre of the channel but increases towards the plates. It is observed that with increasing values of the magnetic parameter the velocit of dust fluid as well as velocit of dust particles decreases while it increases with the increase of the radiation absorption parameter and heat source parameter. It is also found that the velocit of the dust particles is less than that of the dust fluid and the kin-friction of the dust particles is greater than that of the dust fluid. Acknowledgements The authors are thankful to the reviewers for their suggestions which have significantl improved our paper. REFERENCES Afif A. A. (9). Similarit solution in MHD effects of thermal diffusion and diffusion thermo on free convective Heat and Mass transfer over a stretching surface considering suction or injection Comm in Nonlinear Science and Numerical Simulation Vol. 4 No. 5 pp. -4. Alexander Apelblat. (98). Mass transfer with a chemical reaction of the first order The Chemical Engineering Journal Vol. 9 pp. 9-37

15 AAM: Intern. J. Vol. 9 Issue (June 4) 55 Beg O. A. Baker A. Y. and Prasad V. R. (9). Numerical stud of free convection magneto hdrodnamic Heat and Mass transfer from a stretching surface to a saturated porous medium with Soret and Dufour effects Computational Materials Science Vol. 46 No. pp Chambre P. L. and Young J. D. (958). On the diffusion of chemicall reactive species in a laminar boundar laer flow The Phsics of Fluids Vol. 958 pp Cussler E. L. (988). Diffusion Mass Transfer in Fluid Sstems Cambridge Universit Press. Das U. N. Deka R. K. and Soundalgekar V. M. (994). Effects on mass transfer on flow past an impulsivel started infinite vertical plate with constant heat flux and chemical reaction Foushung im ingenieurwesen Vol. 6 pp Kumar D. and Srivastava R. K. (5). Effects of chemical reaction on MHD flow of Dust viscoelastic (Walter s Liquid Model-B) Liquid with heat source/sink Proc. of National Seminar on Mathematics and Computer Science Meerut pp.5-. Makinde O. D. and Chinoka T. (). MHD Transient flows and heat transfer of dust fluid in a channel with variable phsical properties and Navier slip condition Computers and Mathematics with Applications Vol. 6 pp Mastanrao S. Balamurugan K. S. Varma S. V. K. and Raju V. C. C. (3). Chemical reaction and Hall effects on MHD convective flow along an infinite vertical porous plate with variable suction and heat generation Applications and Applied Mathematics Vol. 8 Issue. pp Mbeledogu I. U. and Ogulu A. (7). Heat and Mass transfer of an unstead MHD natural convection flow of a rotating fluid past a vertical porous flat plate in the presence of radiative Heat transfer Int. J. of Heat and Mass Transfer Vol. 5 No. 9- pp Michael D. H. and Nore P. W. (968). The laminar flow of a dust gas between rotating clinders The Quarterl Journal of Mechanics and Applied Mathematics Vol. pp Nadeppanavar M. M Abel M. S. and Tawade J. (). Heat transfer in a Walter s Liquid B- Fluid over an incompressible stretching sheet with non-uniform Heat source/sink and elastic deformation Communications in Nonlinear Science and Numerical simulation Vol. 5 No. 7 pp Osalusi E Side J. and Harris R. (8). Thermal diffusion and Thermo effect on combined Heat and Mass transfer of a stud MHD convective and slip flow due to a rotating disk and viscous dissipation and ohmic heating Int. Comm. in Heat and Mass Transfer Vol. 35 No. 8 pp Patil P. M. and Kulkarni P. S. (8). Effects of chemical reaction on free convection flow of a polar fluid through a porous medium in the presence of thermal sciences Vol. 47 No. 8 pp Prakash O. Devendra Kumar Y.K. and Dwivedi Y. K. (). Effects of thermal diffusion and chemical reaction on MHD flow of dust viscoelastic (Walter s Liquid Model-B) fluid J. Electromagnetic Analsis and applications Vol. pp Pop I. (968). Rev. Roum. Phsics Vol. 3 pp. 4. Raptis A. and Perdikis C. P. (985). Oscillator flow through a porous medium b the presence of free convective flow Int. J. of Engineering Science Vol. 3 pp5-55. Saffman P. G. (96). On the stabilit of laminar flow of a dust gas Journal of Fluid Mechanics Vol. 3 No. pp.-9.

16 56 J. Prakash et al. Sharma R. Bhargava R. and Bhargava P. (). A numerical solution of stead MHD convection Heat and Mass transfer on a semi-infinite vertical porous moving plate using Element Free Galerkin Method Computational Materials Science Vol. 48 No. 3 pp Singh N. P Singh A. K Yadav M. K. and Singh A. K. (). Acta Ciencia Indica Vol. XXVIII M No. pp. 89. Sivaraj. R. and Rushi Kumar. B. (). MHD mixed convective flow of viscoelastic and viscous fluids in a vertical porous channel Applications and Applied Mathematics Vol.7 Issue. pp