ISSN: [Srivastava* et al., 5(11): November, 2016] Impact Factor: 4.116

Transcription

1 ISSN: [Srivastava* et al., 5(11): November, 16] Impact Factor: IJESRT INTERNATIONAL JOURNAL OF ENGINEERING SIENES & RESEARH TEHNOLOGY HEAT AND ASS TRANSFER IN STEADY POISEUILLE FLOW OF FLUID BETWEEN TWO PLATE PLAED IN AN INLINED AGNETI FIELD onika Srivastava*, Dr. Rajeev Khare * Dept. of athematics & Statistics, Sam Higginbottom Institute of Agriculture, Technology &Sciences Deemed University Allahabad-117 Dept. of athematics & Statistics, Sam Higginbottom Institute of Agriculture, Technology &Sciences Deemed University Allahabad-117 DOI: 1.581/zenodo ABSTRAT In this paper, the effect of heat and mass transfer of an electrically conducting incompressible fluid hich moves in presence of magnetic field beteen to parallel plates and under constant pressure gradient has been studied.after forming the governing equation of motion and choosing the suitable boundary conditions, the solution has been derived for the velocity of fluid and it has been studied analytically and velocity profile has been represented ith respect to magnetic field.. it has been found that the magnetic field is very effective in controlling the velocity of fluid and graphically justifying the result of the fluid motion. KEYWORDS: Poiseuille flo; Porous plate; onducting fluid; Pressure gradient; Heat and ass Transfer; inclined magnetic field; Hartmann number. INTRODUTION The flo of liquid metals, motel iron and ionize gasses etc. is very important in industrial field hich is studied in magneto-hydro dynamics and this field concerns the motion of an electrically conducting fluid in the presence of magnetic field.it is ell knon that an electric current is induced hen an electrically conducting fluid moves in a magnetic field. Thus HD generators, HD pumps, and electromagnetic flo meter are based on HD flo. any persons have orked in this field, but Kuri and Bhadur have concentrated on the effect of magnetic field on poiseuille flo beteen to parallel plates.in the present ork focus has been given on the steady HD to dimensions Poiseuille flo beteen to plates hich are placed in an inclined magnetic field. The concerned differential equations are formed and solved, after that the graphical representation beteen the magnetic field and velocity has been made and critical studied has been done to find the effect of magnetic on the velocity profile. FORULATION OF THE PROBLE Let us consider an electrically conducting, viscous, incompressible fluid move beteen to infinite parallel plates separated by a distance d and both plates are at rest. Such a flo indicates a plane Poiseuille flo ith constant pressure gradient. In the channel Heat and ass transfer. Suppose origin be at the centre of the channel ith x- axis parallel to the y-axis and perpendicular to the channel alls respectively. The plates are of perpetual figure of length, all the variables except the fluid pressure are function of y. The plates saturated at y= -1 to 1.fluid flo is horizontal along the x-axis. If u and v are components of velocity v v, for an incompressible fluid. The equation of continuity u,v =. Reduce the form, V V y =constant hich indicates, u=u(y). No, consider V be the characteristics velocity perpendicular to the fluid flo maintain a steady state flo at a constant pressure gradient, this V at the loer plate is one hich ill arrange a steady fluid flo against suction and injection. [78]

2 ISSN: [Srivastava* et al., 5(11): November, 16] Impact Factor: The governing differential equations are: u 1 u V u P g T T g (1) y y 1 p and () y Where, Bo: magnetic field Vo: characteristic velocity : kinematic viscosity σ: electrical conductivity ρ= fluid density. The equation () takes the fluid pressure P=P(x).Let us assume pressure gradient is constant i.e: dp/dx=constant=p d u V du 1 Sin u P g T T g dy dy d u V dy du dy 1 Sin u P g T T g Equation(3) represents the fluid under the influence of an inclined magnetic field. Differentiating (3) ith respect to x. The equation (3) reform: d u V du 1 Sin u g T T g (4) dy dy The to fields can be calculated in the nature at an angle θ for. ing folloing non-dimensional variables and parameters: d x y d x, y, P d d = * Sin or =Ha. G r L g T T ud p V, u, g, d, Sin, Where * = Ha d. /, G c L g = θ = (3) π. With boundary condition u= for y = (suction), λ (injection), T T T T / ;, T T T W T / Then equation (4) becomes: d u du u g G G r c (6) dy dy Normalize, boundary conditions are: u = at y = 1 and u = at y = -1. (7) SOLUTION OF THE PROBLE The equation (6) is ordinary differential ith constant coefficient. The corresponding boundary condition (7) then becomes: m 1y m y g Gr Gc uy 1e e (8) Where m1 and m are the roots of the equation obtained from the equation (6) and constants are computed ith boundary condition (7) and are obtained as:. (5) 1 & [79]

3 ISSN: [Srivastava* et al., 5(11): November, 16] Impact Factor: and 1 Sinh m Sinh m m 1 Sinh m1 Sinh m m1 (9) ing the constants values to obtained the solution of the equation (6). It ill be expressed as: u y Where 1 1 e Sinh( y1 y1 Sinh y 1 e Sinh y, Ha. Sin ; HaHartmann number. 1 (1) RESULT AND ONLUSION Fig.1 shos the variation of injection velocity of fluid ith respect to magnetic field expresed in term of Hartman number. On taking θ=15,λ=,graphs for y=.5,.1,. have been obtained. It is clear that as magnetic field increases, the velocity in each case decreases sharply up to a minimum value and then it increases rapidly but for certain value of magnetic field (=),dips in curves are obtained in each case. Also in the derived relation of velocity (U), the term of magnetic field present in denominator is also existing ith the functions of sine and cosine hich produce avy velocity profile. As the value of y increases, the graph shifts upards this means that the increasing value of y enhances the amplitude of the velocity because of increasing magnetic force. While fig. shos the variation of suction velocity of fluid ith respect to magnetic field expressed in terms of Hartman s number for same values of θ and y but in the case of suction λ= - and the result obtained shos that as magnetic field increases, the velocity decreases similarly as above. On comparing to velocity profiles, it is found that dips are shifting for higher values of magnetic field from suction velocity to injection velocity hich is obvious because of magnetic attraction. Also the velocity in each case of injection starts from a little higher value and also achieves higher values than in case of injection. Thus the magnetic field is quite capable of controlling the velocity of fluid in both cases hich finds application in industrial use Figure 1: θ= 15, λ = g =1. case of injection [8]

4 ISSN: [Srivastava* et al., 5(11): November, 16] Impact Factor: Figure : θ=, λ = - g=1. case of suction REFERENES [1] Das U. N. and Ahmed NFree onvective HD flo and heat transfer in a viscous incompressible fluid confined beteen a long vertical avy all and a parallel flat all, Indian Journal Pure Applied athematics, Vol. 3, pp ,(1999). [] Dutta N., Dalal D.. and ishra S.K.Unsteady heat transfer to palsatile flo of a dusty viscous incompressible fluid in a channel, International Journal of Heat and ass transfer, Vol. 36, No. 7, pp ,(1993) [3] Attia H. A.et.al, agnetic flo and heat transfer in rectangular channel ith variable viscosity. The Arabian Journal for Science and Engineering, Vol. (), No. A, pp. 1-1,(5). [4] Nahme,RBeitrage zur, hydro-dynamschen theory der lagerreibung, ing. Arch.,11,191,(194). [5] Kuiry.,Bahadur S., Effect of an inclined magnetic field on an steady poiseuille flo beteen to parallel plates, Vol.1,pp.9-96,(14). [6] J.S, Waterhouse, and J.G.Kingston, Plane magnetohydrodynamic flos ith constantly inclined magnetic and velocity fields, J. Appl.ath,Phys.Vol.4, ,(1973). [7] A.Shercliff, Entry of conducting and non-conducting fluids in pipes, J.ath.Proc.of the ambridge Philos.Soc.5, ,(195). [8] S. Ganesh and S.Krishnambal, Unsteady HDstokes flo of viscous fluid beteen to parallel porous plates, J. Appl.Sci.7, ,(7). [9] Bansal,J.L.and Jain.N.., Variable viscosity plane Poiseuille flo ith unequal all temprarure,)indian J. Pur Appl.ath,Vol.6(7),8.,(1975). [1] Alfven,H., Existenc of electromagnetic-hydrodynamics aves,nature,15,(385),45.(194). [11] Ghasemi E., Bayat. and Bayat. HD flo and heat transfer in a visco-elastic flo over a semiinfinite incompressible and non isothermal stretching sheet. International Journal of Physical Sciences, Vol. 6 (1), pp , (11). [1] Ghosh A. K., Khan A. R. and Debnath L. On a pulsatile flo of a to-phase viscous fluid is a tube of elliptic cross-section. International Journal of athematics and athematical Sciences, Vol. (13), No. 4, pp (199). [13] itra P. Unsteady flo of a dusty viscous incompressible gas through a hexagonal channel, Journal of Engineering and Applied Science, Vol., pp ,(1983). [14] itra P. and Bhattacharya, P. The flo of conducting gas beteen to plates in the influence of transverse magnetic field, Journal of Technology, India, Vol. 6, No.1, pp. 13-4,(1981). [81]

5 ISSN: [Srivastava* et al., 5(11): November, 16] Impact Factor: [15] Nag. Kumar S. and Dutta N. Flo of a dusty fluid through a rectangular channel. Indian Journal of athematics, Vol. 7 pp ,(1989). [16] Raptis A. et. al. Unsteady free convective flo through a porous medium, International Journal Engineering Science, Vol. 1, pp ,(1993). [17] Reddy Y. B.and et.al. Unsteady Laminar flo of a fluid ith uniform distribution of dust particles through a rectangular channel has been published in Defence Science, Journal, India, Vol. 3, pp ,(197). [18] S.Kumar,R.Khare, Heat and mass transfer in HDfree convective flo of a visco-elastic (alter s odel-b) dusty fluid through a porous medium, Vol.5,pp.44-58,(14). 19.S.Ahsan,R.Khare, HD flo of a Nn-Netonian fluid through an isosceles triangular channel, Vol.1,Is7,pp94-98,(15). [8]