Journal of Multidisciplinar Engineering Science and Technolog (JMEST) Magnetohdrodnamic Fluid Flow Between Two Parallel Infinite Plates Subjected To An Inclined Magnetic Field Under Pressure Gradient Agnes Mburu Department of Mathematics and Computer Science, Catholic Universit of Eastern Africa Nairobi,Kena. gnsnjeri@ahoo.com Jackson Kwanza 2 Department of Pure and Applied Mathematics, Jomo Kenatta Universit of Agriculture and Technolog.,Nairobi,Kena kwanzakioko@ahoo.com ThomasOnango 3 Department of Industrial and Engineering Mathematics, Technical Universit of Kena, Nairobi. onangottm@gmail.com Abstract Magnetohdrodnamic (MHD) flow between two parallel infinite plates with inclined magnetic field under applied pressure gradient has been investigated in this paper. The differential equations governing the flow are nondimensionalised and solved analticall with appropriate boundar conditions. The effects of Hartmann number, the angle of magnetic field inclination, pressure gradient, and Renolds number on the flow field have been presented graphicall. It was found that decrease in angle of inclination of magnetic field increases the velocit profiles while increase in the Hartmann number decreases the velocit. Increase in pressure gradient leads to increased velocit. Kewords Magnetohdrodnamic fluid flow, Hartmann number, Renolds number, Pressure Gradient. Nomenclature B Magnetic flux E Electric field F b Bod force H o Applied magnetic field µ Dnamic viscosit υ Kinematic viscosit J Current densit L Characteristic length M Hartmann number P Pressure Re Renolds number t Time V (u, v. w) Velocit in the x,,z direction ρ Densit σ Electrical conductivit of the fluid I. INTRODUCTION When a conducting fluid flows past a magnetic field, an electric field and consequentl an electric current is induced. In turn the current interacts with the magnetic field to produce a bod force on the fluid.such phenomenon occurs both in nature and in manmade devices and is termed as Magnetohdrodnamics(MHD).It is an important branch of fluid dnamics. In natural phenomena MHD occurs in the sun, the Earth s interior, the ion sphere the stars and their atmosphere to mention a few examples. In the laborator or in man-made phenomena man new devices have been made which utilizes the MHD interaction directl such as propulsion units and power generators. In the recent past, several investigations have been carried out on MHD. Chaturani and Saxena [] investigated a twolaered MHD model for parallel plate haemodialsis, under the influence of uniform transverse magnetic field. Agwal, Ram and Yadav [2] carried an investigation of MHD unstead viscous flow through a porous straight channel. The stud was carried out with two parallel porous flat walls in the presence of transverse magnetic field under a time varing pressure gradient. Ganesh and Krishnambal [3] studied the unstead stoke s flow of an electricall conducting viscous incompressible fluid between two parallel porous plates of a channel in the presence of a transverse magnetic field. Singh [4] investigated on hdromagnetic stead flow of viscous incompressible fluid between two parallel infinite plates under the influence of inclined magnetic field. Singh and Okwoo [5] carried out a stud of couette flow between two parallel infinite plates in the presence of a transverse magnetic field. Krishna [6]carried out an investigation on unstead incompressible free convective flow of an electricall conducting fluid between two heated parallel horizontal plates under the action of magnetic field applied transversel to the flow. Saed, et al [7] did a stud of time dependent pressure gradient effect on unstead MHD couette JMESTN4235864 59
flow and heat transfer of a casson fluid, while Kumar et al [8] investigated on an unstead periodic flow of a viscous incompressible fluid through a porous planer channel in the presence of transverse magnetic field. Despite the investigations done on MHD flows, flow past parallel plates subjected to inclined magnetic field in presence of pressure gradient has received little attention. Hence the main objective of the present investigation is to obtain an exact solution of MHD fluid flow between two parallel infinite plates subjected to inclined magnetic field in presence of pressure gradient. Journal of Multidisciplinar Engineering Science and Technolog (JMEST) = ρ = ρ μ 2 u F x ρ 2 ρ F ρ (4a) (4b) We know that the bod forces or the ponder motive forces F = F(F x, F, F z ) F = J B therefore (4a) becomes = ρ μ 2 u J B ρ 2 ρ since F = (4b) becomes = P (5a) (5b) Since we are considering the flow in x-direction then the flow will be affected b the magnetic flux which is perpendicular to the flow. Since we want to stud the effect of different angles of inclination of the magnetic field then the velocit and magnetic flux profiles will be; V = V (u,,) and Fig. Geometric configuration II.MATHEMATICAL ANALYSIS We consider the stead viscous flow along the x- axis of an electricall conducting fluid between two horizontal parallel infinite plates situated at = L and =L. The plate at = L is stationer while the plate at =L is moving with uniform velocit U. An inclined magnetic field at an angle α is applied to the flow. There is no external applied electric field and we assume stead flow conditions have been attained. It is required to analse the velocit profiles taking into consideration the variable pressure gradient. The equations governing the flow field are: ρ( u 2 u z 2) F x ρ ( u v w z ) = () u u u u v w ) = u z µ( 2 2 u 2 2 (2a) ρ( v v v v u v w ) = μ v z ( 2 2 v 2 2 2 v z 2) F (2b) ρ( w 2 w z 2 ) F z u w w w v w ) = μ w z z ( 2 2 w 2 2 (2c) The flow is along the x-direction so the velocit profile along the and z-axis v=w= and the velocit u along the x-axis depends on onl. So () becomes; u = (3) Since the flow is stead (does not depend on time), and v=, w= and using (3) then (2) above reduces to LB σ μ B = B (, Bsinα, ) Respectivel. Now F x = J B and since J = σe, and E = V B Then we have V B = σubsinαk Again i j k V B = σ u Bsinα = i () j () k (σubsinα) i j k J B = σubsinα Bsinα = i ( σub 2 sin 2 α) j () k () F x = J B = σub 2 sin 2 αi (6) Using (6), then (5a) becomes = ρ υ 2 u σub2 sin 2 α 2 ρ Using the following non-dimensional quantities x = x L, = L, p = P ρu 2, (7) becomes u = u U, UL υ (7) = Re,M= = Re 2 u 2 M2 sin 2 αu (8) Suppose we let pressure gradient to be a variable sa ( P(x)) so that we can write = P and (8) becomes JMESTN4235864 59
Journal of Multidisciplinar Engineering Science and Technolog (JMEST) = 2 u 2 M2 sin 2 αu or d 2 u d 2 M2 sin 2 αu = (9) Equation (9) is the ordinar differential equation governing the flow problem which will be solved subject to the boundar conditions. When = L u= When = L u=u The non-dimensional form of these boundar conditions is; =,u= =, u= () III.METHOD OF SOLUTION The ordinar differential equation (9) is inhomogeneous therefore we will solve the homogenous part and the inhomogeneous part separatel and then add them together to get the general solution. The auxiliar equation for the homogenous part can be written as D 2 u M 2 sin 2 u = () Or (D 2 ) = D= ± D= ±Msinα (2) The complimentar solution is then given b u c = Ae Msinα Be Msinα (3) Where A and B are constants to be determined using the boundar conditions. The particular integral of equation (9) is given b u p = ( D 2 Q ) where Q= M2 sin 2 α = Q u p = (4) The general solution is got b adding (3) and (4) as follows u= Ae Msinα Be Msinα (5) The above equation will be solved subject to the boundar conditions () Let M sinα be m and becomes be R so that (5) u= Ae m Be m R (6) The boundar conditions are Solving (6) subject to the boundar conditions we have, = Ae m Be m R (7a) = Ae m Be m R (7b) Multipling (7a) b e m and (7b) b e m and subtracting we get e m = Ae 2m Ae 2m Re m Re m e m = A(e 2m e 2m ) R(e m e m ) A= em R(e m e m ) Similarl we multipl (7a) b e m and (7b) b e m and subtract to get e m = Be 2m Be 2m Re m Re m e m = B(e 2m e 2m ) R(e m e m ) B= e m R(e m e m ) (e 2m e 2m ) Substituting the values of A and B we get u= em R(e m e m ) e m e m R(e m e m ) e m R (e 2m e 2m ) (8) u= (emm ) Rsinhme m Rsinhme m U(e m m ) R u= (emm e (mm) ) Rsinhm(e m e m ) R u= (em() e m() ) Rsinhm(e m e m ) R u= sinhm() Rsinhm (coshm) (9) Putting back the values of m and R we have u = sinh(msinα()) IV.RESULTS sinh(msinα)cosh(msinα) sinh(2msinα) (2) Using (2) we analse the effects of different parameters on velocit. The analsis is done using graphs as shown below: Throughout the four figures we have taken the standard values for the Hartmann number to be 2, the angle of inclination to be 6, the Pressure gradient to be 2 and the Renolds number to be 2. JMESTN4235864 592
Journal of Multidisciplinar Engineering Science and Technolog (JMEST).2.8 M=.5 M2=. M3=.5 M3=2..8.6 Re= Re= Re=.6 Re=.8 Velocit (u).6 Velocit (u) - - - -.8 -.6 - -.6.8 Fig. 2 Velocit distributions for different values of Hartmann Number (M) Velocit (u).2.8.6 3 45 6 9 - - -.8 -.6 - -.6.8 Fig.3 velocit profiles for different inclination of magnetic field Velocit (u).2.8.6 P=2 P= P=- P=-2 - - -.8 -.6 - -.6.8 Fig.4 Velocit profiles for different values of pressure gradient (P) - - -.8 -.6 - -.6.8 Figure 5.velocit distribution for different values of Renolds number (Re) V. DISCUSSIONS From Figure 2 it is observed that the increase in Hartmann number leads to a decrease in velocit. This is because the Hartmann number is the ratio of magnetic force to viscous force and so the larger the Hartmann number the stronger the magnetic force and the smaller the viscous force. So it is clear from the figure that increasing the Hartmann number reduces the velocit due to the action of Lorentz force. Figure 3 shows the effect of the angle of inclination of magnetic field on the flow velocit. The smallest angle of inclination is 3 and from the figure we observe that when the angle of inclination is small then the velocit is high and as the angle of inclination increases the velocit decreases. At 9 it indicates a special case where the effect of fluid flow is observed at transverse magnetic field which is the normal MHD flow. Figure 4 illustrates the effect of pressure gradient on the flow velocit. The figure shows that due to the force the pressure will have on the fluid an increase from negative to positive pressure gradient leads to an increase in flow velocit. Figure 5 shows the effects of the Renolds number on the velocit of the flow. We observe that an increase in Renolds number leads to a decrease in the velocit of the flow while a decrease of the Renolds number leads to an increase in the flow velocit. This is due to the viscous force of the fluid since the Renolds number is the ratio of viscous force to inertia force. VI.VALIDATION OF RESULTS Our results were compared with those of Singh [4] who considered hdromagnetic flow of viscous incompressible fluid flow between two parallel infinite plates under the influence of inclined magnetic field. He used analtical method to obtain the velocit profiles. The results coincides with our findings that the velocit profiles decreased as the strength of JMESTN4235864 593
magnetic field was increased and also with the increase However he did not consider the effect of pressure gradient and the Renolds number on the flow. We found that increase in pressure gradient leads to increase in velocit while increase in Renolds number leads to a decrease in velocit. VII.CONCLUSION From this investigation we conclude that increasing the magnetic force (M) reduces the velocit of the flow. This is due to the action of the Lorentz force on the fluid. Reducing the angle of inclination of the magnetic field and the strength of the magnetic field leads to an increase in velocit this is mainl because the magnetic field slows down the motion of the fluid so the stronger the magnetic field the slower the motion of the fluid. Increasing the pressure gradient on the flow leads to an increase of the flow velocit. This is because the pressure gradient will tend to push the fluid thus increasing its velocit. REFERENCES [] P. Chaturani, & S. Saxena, Two laered MHD flow through parallel plates with applications. Indian Journal of Pure applied Mathematics, vol.32(),pp.55-68,2. [2] H.Agwal, P. Ram. and N. Yadav, Hdromagnetic unstead viscous flow through a porous channel. The journal of scientific research- 225,25. Journal of Multidisciplinar Engineering Science and Technolog (JMEST) [3] S.Gannesh and Krishnambal, Magnetohdrodnamic stoke s flow of viscous fluid between two parallel porous plates.journal of applied science,vol. l7, pp.374-379, 27. [4] C. Singh, Hdromagnetic stead flow of viscous incompressible fluid between two parallel infinite plates under the influence of inclined magnetic field. Kena Journal of Sciences Series A, vol.2, No,27. [5] C.Singh and J. Okwoo Couette flow between two parallel infinite plates in the presence of transverse magnetic field. Journal of Kena Meteorol.Soc.,2(2) 9-94, 28. [6] G.Krishna, Analtical solution to the problem of MHD free convective flow of an electricall conducting fluid between two heated parallel plates. International journal of Applied Mathematics and Computation, vol. (4),pp. 83-93, 29. [7] M.Saed, A.Hazem and M. Karem, Time depedent pressure gradient effect on unstead MHD couette flow and heat transfer of Casson Fluid.Journal of Engineering, vol 3,pp 38-49,2. [8] A. Kumar, C.Varshne and L.Sajjan. Perturbation technique to unstead MHD periodic flow of viscous fluid through a planer channel. Journal of Engineering and Technolog Research, Vol. 2(4), pp. 73-8, 2. JMESTN4235864 594