E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering              Research and Applic...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering                         Research...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering                    Research and ...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering           Research and Applicati...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering           Research and Applicati...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering           Research and Applicati...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering           Research and Applicati...
E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering           Research and Applicati...
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  1. 1. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892The Effects of Concentration and Hall Current on Unsteady Flow of a Viscoelastic Fluid in a Fixed Plate 1 E. Omokhuale, 1I. J. Uwanta,2A. A.Momoh, 2 A. Tahir. 1 Department of Mathematics, UsmanuDanfodiyo University, P. M. B. 2346, Sokoto, Nigeria.2 Department of Mathematics, Modibbo Adama University of Technology, P.M.B. 2076, Yola. Adamawa State. NigeriaABSTRACT The present Study deals with the effects numerically. Kai- Long (2010) studied Heat andof concentration and Hall current on mass transfer for a viscous flow with radiation effectunsteadyflow of a viscoelastic fluid in a fixed past a non-linear stretching sheet. Hall Effect onplate. The resultant equations have been solved MHD mixed convective flow of a viscousanalytically. The velocity, temperature and incompressible fluid past a vertical porous plateconcentration distributions are derived, and their immersed in a porous medium with Heatprofiles for various physical parameters are Source/Sink was studied by Sharma et al. (2007).shown through graphs. The coefficient of Skin Salehet al. (2010) examined Heat and mass transferfriction, Nusselt number and Sherwood number in MHD viscoclastic fluid flow through a porousat the plate are derived and their numerical medium over a stretching sheet with chemicalvalues for various physical parameters are reaction. Das and Jana (2010) examined Heat andpresented through tables. The influence of Mass transfer effects on unsteady MHD freevarious parameters such as the thermal Grashof convection flow near a moving vertical plate in anumber, mass Grashof number, Schmidt porous medium. The effects of chemical reaction,number, Prandtl number, Hall Parameter, Hall current and Ion – Slip currents on MHDHartmann number, Chemical Reaction micropolar fluid flow with thermal Diffusivity usingparameter and the frequency of oscillation on the Novel Numerical Technique was studied by Motsaflow field are discussed. It is seen that, the and Shateyi (2012). Aboeldahab and Elbarby (2001)velocity increases with the increase in Gc and examined Hall current effect onGr,and it decreases with increase in Sc, M, m, Kc, Magnetohydrodynamics free convection flow past aand Pr, temperature decreases with increase in Semi – infinite vertical plate with mass transfer. HallPr, Also, the concentration decreases with the current effect with simultaneous thermal and massincrease in Sc and Kc. diffusion on unsteady hydromagnetic flow near an accelerated vertical plate was investigated AcharyaetKEY WORDS: Hall Current, Viscoelastic, Porous al. (2001). Takhar (2006) studied Unsteady flow freemedium, Magnetohydrodynamics. convective flow over an infinite vertical porous platedue to the combined effects of thermal and1.0 INTRODUCTION mass diffusion, magnetic field and Hall current. There exist flows which are caused not only Exact Solution of MHD free convection flow andby temperature differences but also by concentration Mass Transfer near a moving vertical porous plate indifferences. There are several engineering situations the presence of thermal radiation was investigatedwherein combined heat and mass transport arise by Das (2010). Anjali Devi and Ganga examinedsuch as dehumidifiers, humidifiers, desert coolers, effects of viscous and Joules dissipation and MHDand chemical reactors etc. Magnetohydrodynamics is flow, heat and mass transfer past a Stretching porouscurrently undergoing a period of great enlargement surface embedded in a porous medium. Sonthet al.and differentiation of subject matter. It is important (2012) studied Heat and Mass transfer in ain the design of MHD generators and accelerators in viscoelastic fluid over an accelerated surface withgeophysics, underground water storage system, soil heat source/sink and viscous dissipation. Shateyietsciences, astrophysics, nuclear power reactor, solar al. (2010) investigated The effects of thermalstructures, and so on. Radiation, Hall currents, Soret and Dufour on MHD Moreso, the interest in these new problems flow by mixed convection over vertical surface ingenerates from their importance in liquid metals, porous medium. Effect of Hall currents andelectrolytes and ionized gases. On account of their Chemical reaction and Hydromagnetic flow of avaried importance, these flows have been studied by stretching vertical surface with internal heatseveral authors – notable amongst them are Singh generation / absorption was examined by Salem andand Singh (2012) they investigated MHD flow of El-Aziz (2008).Viscous Dissipation and Chemical Reaction over aStretching porous plate in a porous medium 2.0 MATHEMATICAL FORMULATION 885 | P a g e
  2. 2. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892 We consider the unsteady flow of a viscous volume expansion, k 0 is the kinematic incompressible and electrically conducting viscoelastic fluid over afixed porous plate with viscoelasticity,  is the density,  is the viscosity, oscillating temperature. The x- axis is assumed to be is the kinematic viscosity, KT is the thermal oriented vertically upwards along the plate and the conductivity, Cp is the specific heat in the fluid at y-axis is taken normal to the plane of the plate. It is constant pressure,  is the electrical conductivity of assumed that the plate is electrically non – the fluid,  e is the magnetic permeability, D is the conducting and a uniform magnetic field of straight molecular diffusivity, T0 is the temperature of the B0 is applied normal to the plate. The induced magnetic field is assumed constant. So that plane and Td is the temperature of the fluid far away  B   0, B0 ,0 ,. The plate is subjected to a constant from plane. C 0 is the concentration of the plane and suction velocity. Cd is the concentration of the fluid far away from1. The equation of conservation of charge J  0, gives constant. the plane. And v  v0 , the negative sign indicate that the       Pe  suction is towards the plane. 2. J  e e ( J  E )   V  B   (1) Introducing the following non-dimensionless  ene  parameters Equation (1) reduces to v0 y v 2t u * T  T C  C   B0   , t  0 , u  ,   ,  ,C   2  J x*  mu*   *    4 v0 v0 T  T C  C  (1  m )   g  (T  T ) g  c (C  C )  B0 2 Cp    B0 Gr  , Gc  ,M  , Pr   J y*  (1  m2 )  u  m  * *  (2) v0 2 v0 2  v0 2 KT     k1v0 2 k *v0 2 Where Sc  , K  2 , k  2  D 4    m  e e is the Hall parameter. (8) The governing equations for the momentum, energy Substituting the dimensionless variables in (8) into and concentration are as follows; (3) to (6), we get (dropping the bars) u u u  3u (u  m ) u 1 u u  2u K  3u M (u  m ) u  v0    k1 2   B02  g  (T  Td )  g  * (C  Cd )  *       Gr  GcC t y y y t  (1  m ) 2 k 4 t   2 4 2 t (1  m 2 ) k (3) (9)    2  3 (  mu ) u 1    2 K  3 M (  mu ) u  v0   2  k1 2   B0 2       t y y y t  (1  m 2 ) k * 4 t   2 4 2 t (1  m 2 ) k (4) (10) T T K T 2  v0  T 1   1  2 t y  Cp y 2   4 t  Pr  2 (5) (11) C C C 2 1 C C 1  C 2  v0  D 2  KrC   t y y 4 t  Sc  2 (6) (12) The boundary conditions of the problem are: The corresponding boundary conditions are u  0,   0, T  Td  (T0  Td )eit , C  Cd  (C0  Cd )eit aty  0 u(0, t )  0,  (0, t )  0, (0, t )  eit , C (0, t )eit aty  0 u  0,   0, T  0, C  0aty  1 u(1, t )   (1, t )   (1, t )  C (1, t )  0aty  1 (7) (13) Where u and v are the components of velocity in the x and y direction respectively, g is the acceleration Equations (9) and (10) can be combined into a single due to gravity,  and  are the coefficient of * equation by introducing the complex velocity. 886 | P a g e
  3. 3. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892 U  u , t   i , t  Solving (20) to (22) under the boundary conditions (23) and (25).And substituting the (14) obtained solutions into (17) to (19) respectively. Then Where the velocity distribution can be expressed as i  1 U (, t )   A5em5  A6em6  A8em3  A9em4  A10em1  A11em2   eint   Thus, (24)1 U U  2U K  3U M (1  im)U U And, the temperature field is given by       Gr  GcC4 t   2 4 2 t (1  m 2 ) k  ( , t )   A3em   A4e m    ei t 3 4 (15) (25) With boundary conditions: Similarly, the concentration distribution gives t  0 : U (0, t )  0, (0, t )  eit , C (0, t )eit at  0 * C ( , t )   A1em1  A2e m2   ei t U (1, t )   (1, t )  C (1, t )  0, at  1 (26) The Skin friction, Nusselt number and (16) Sherwood number is obtained by differentiating (24) where Gr is the thermal Grashof number, to (26) and at evaluated at   0 respectively. . Gc is the mass Grashof number, Sc is the Schmidt number, Pr is the Prandtl number, Rm is the Reynold U ( , t ) number, K is the viscoelastic Parameter and S is the    m5 A5  m6 A6  m3 A8  m4 A9  m1 A10  m2 A11   ei t pearmeability.   0 (27) 3.0 METHOD OF SOLUTIONS  ( , t ) To solve (11), (12) and (15) subject to the boundary    m3 A3  m4 A4   ei  t conditions (16), we assume solutions of the form   0 U (, t )  U1    eit (28) C ( , t ) (17)    m1 A1  m2 A2   ( , t )  1 ( ) e i t   0 (18) (29) C(, t )  C1    e it (19) 4.0 RESULTS AND DISCUSSION where U1   1   and C1   are to be The effects of concentration and Hall determined. current on unsteady flow of a viscoelastic fluid in a Substituting (17) to (19) into (11), (12) and (15), fixed plate has been formulated and solved Comparing harmonic and non harmonic terms, we analytically. In order to understand the flow of the obtain fluid, computations are performed for different parameters such as Gr, Gc, Sc, Pr,  , Kc, M, and U Gr GcC U    L3U   it  m. L1 4 L1e 4 L1eit 4.1 Velocity profiles (20) Figures 1 to 7connote the velocity profiles; 1 1  Pr 1  i1  0 figure 8 depicts the temperature profiles and figures9& 10display the concentration profiles with 4 varying parameters respectively. (21) The effect of velocity for different values of 1 (Pr = 0.71, 1, 3) is displayed in figure 1, the graph C1 ScC1  iC1  KcScC1  0 show that velocity decreases with increase in Pr. 4 (22) The effect of velocity for different values of (Sc = and boundary conditions give 0.18, 0.78, 2.01, 2.65) is given in figure 2, the graph   0 : U1 (0)   (0)  C (0)  1  show that velocity decreases with the increase in Sc. Figure 3 denotes the effect of velocity for different    1: U1 (1)  0, (1)  0, C (1)  0 values of (m = 2, 4, 6, 10), it is seen that velocity decreases with the increase in m. (23) The effect of velocity for different values of (M = 1, where the primes connotes differentiation with respect to  . 3, 6, 10) is shown in figure 4, it depict that velocity decreases with increase in M. 887 | P a g e
  4. 4. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892Figure 5 depicts the effect of velocity for (Gr = 1, 2, The effect of velocity for different values of (Kc = 1,3, 5), the graph connote that velocity increases with 2, 3, 4) is displayed in figure 7, it is clear thatthe increase in Gr. velocity decreases with the increase in Kc. The effect of velocity for different values of(Gc = 2, 3, 4, 8) is presented in figure 6, it is seenthat velocity increases with the increase in Gc. Figure 1. Velocity profiles for different values of Pr. Figure 2. Velocity profiles for different values of Sc. 888 | P a g e
  5. 5. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892Figure 3. Velocity profiles for different values of m. Figure 4. Velocity profiles for different values of M . Figure 5. Velocity profiles for different values of Gr. 889 | P a g e
  6. 6. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892Figure 6. Velocity profiles for different values of Gc.Figure 7. Velocity profiles for different values of Kc.4.2 Temperature profilesIn figure 8, the effect of temperature for differentvalues of (Pr = 0.71, 1, 3, 7) is given. The graphshow that temperature increases with increasing Pr. Figure 8. Temperature profiles for different values of Pr. 890 | P a g e
  7. 7. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-8924.3 Concentration profiles concentration for (Kc= 0, 0.5, 1, 3) is given in figureFigure 9 depicts the effect of concentration for (Sc = 10, it is seen that concentration decreases with the0.3, 0.78, 1, 2.01), it is seen that concentration decrease in Kc.increases with the increase in Sc. The effect ofFigure 9. Concentration profiles for different values of Sc.Figure 10. Concentration profiles for different values of Kc.Table 1 to 3 represent the Skin friction, Nusselt Table 2: Nusselt numbernumber and Sherwood number respectively.  Pr NuTable 1 depicts that the Skin friction increases with 1 0.71 0.8181increase in Gc and Grand decreases with increase in 4 0.71 0.7400Sc, m, M, Kc, and Pr. 1 3 0.7830Table 2 represents that the Nusselt number decreaseswith increase in Pr, and decreases with increase in  Table 3: Sherwood numberTable 3 shows that the Sherwood number decreases  Sc Shwith increase in Sc and  . 1 0.3 0.4015 5 0.3 0.3890Table 1: Skin friction  4 0.6 0.3977 Gc Sc Kc Gr M Pr M 1 1 0.3 0.1 1 1 0.71 0.5 6.1082 5.0 SUMMARY AND CONCLUSION3 1 0.3 0.1 1 1 0.71 0.5 5.5219 We have examined and solved the1 1 0.6 0.1 1 1 0.71 0.5 5.7548 governing equations for the effects of concentration1 4 0.3 0.1 1 1 0.71 0.5 6.5671 and Hall current on unsteady flow of a viscoelastic1 1 0.3 1 1 1 0.71 0.5 6.3654 fluid in a fixed plate analytically. In order to point1 1 0.3 0.1 4 1 0.71 0.5 6.8760 out the effect of physical parameters namely;1 1 0.3 0.1 1 3 3 0.5 5.9006 thermal Grashof number, mass Grashof number,1 1 0.3 0.1 1 1 0.71 2 5.0126 Schmidt number, Prandtl number, Chemical reaction1 1 0.3 0.1 1 1 0.71 0.5 6.8741 parameter, Hartmann number, Hall parameter and 891 | P a g e
  8. 8. E. Omokhuale, I. J. Uwanta, A. A. Momoh, A. Tahir. / International Journal of Engineering Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com Vol. 2, Issue5, September- October 2012, pp.885-892the frequency of oscillation on the flow field and the 8. Salem, A. M. and El-Aziz, M. Abd. (2008).following conclusions are drawn: Effect of Hall currents and Chemical*The velocity increases with the increase in Gc and reaction and HydromagneticGr. flow of a stretching vertical surface with*The velocity falls due to increasing in Sc,  , M, m, internal heat generation / absorption.Kc, and Pr. Applied Mathematical Modelling: 32, 1236*The temperature decreases with increase in Pr, – 1254.*The concentration reduces with the increase in Sc 9. Saleh, M.A.,Mohammed ,A.A.B. andand Kc. Mahmoud S.E. (2010). Heat and mass*The Skin friction increases with increase in Gc and transfer in MHD viscoclastic fluid flowGr and decreases with increase in Sc, M, m,Kc, and through a porous medium over a stretchingPr. sheet with chemical reaction. Journal of*The rate of heat transfer decreases with increase in Applied Mathematics, 1, 446-455.Pr and  ,*The Sherwood number decreases with the increase 10. Shateyi, S., Motsa, S. S. and Sibanda, P.in Sc and  . (2010). The effects of thermal Radiation, Hall currents, Soret andREFERENCES Dufour on MHD flow by mixed convection 1. Aboeldahab, E. M. and Elbarby, E. M. E. over vertical surface in porous medium. (2001). Hall current effect on Mathematical Problem in Engineering, Magnetohydrodynamics free convection 2010, Article ID 627475. flow past a Semi – infinite vertical plate 11. Sharma, B. K., Jha, A. K. and Chaudary, R. with mass transfer. Int. Journ. of C. (2007). Hall Effect on MHD mixed Engineering Sciences. 39. 1641 – 1652. convective flow of a viscous 2. Acharya, M., Dash, G. C., Singh, L. P. incompressible fluid past a vertical porous (2001). Hall current effect with plate immersed in a porous medium with simultaneous thermal and mass diffusion Heat Source/Sink. Rom. Journ. Phys. 52 (5- on unsteady hydromagnetic flow near an 7), 487 - 503. accelerated vertical plate. Indian Journal of 12. Singh, P. K. and Singh, J. (2012). MHD Physics, 75B(1), 168. flow of Viscous Dissipation and Chemical 3. Anjali Devi, S. P. and Ganga, B., effects of Reaction over a Stretching viscous and Joules dissipation and MHD porous plate in a porous medium. flow, heat and mass International Journal of Engineering transfer past a Stretching porous surface Research and Applications.2(2). 1556 – embedded in a porous medium. Non – 1564. Linear Analysis: Modelling and Control. 13. Sonth, R, M., Khan, S. K., Abel, M. S. and 14(3). 303 - 314. Prasad, K. V. (2012). Heat and Mass 4. Das, K. and Jana, S. (2010). Heat and Mass transfer in a viscoelastic fluid transfer effects on unsteady MHD free over an accelerated surface with heat convection flow near a source/sink and viscous dissipation. Journal moving vertical plate in a porous medium. of Heat and Mass Transfer, 38, 213 – 220. Bull. Soc. Math. 17, 15 - 32. 14. Takhar (2006). Unsteady flow free 5. Das, K. (2010). Exact Solution of MHD convective flow over an infinite vertical free convection flow and Mass Transfer porous platedue to the combined near a moving vertical porous effects of thermal and mass diffusion, plate in the presence of thermal radiation. magnetic field and Hall current. Journal of AJMP.8(1). 29 – 41. Heat and Mass Transfer. 39, 823 – 834. 6. Kai- Long, H. (2010). Heat and mass transfer for a viscous flow with radiation effect past a non-linear stretching sheet. World Academy of Science, Engineering and Technology, 62(3), 326-330. 7. Motsa, S. S. and Shateyi, S. (2012). The effects of chemical reaction, Hall current and Ion – Slip currents on MHD micropolar fluid flow with thermal Diffusivity using Novel Numerical Technique.Journal of Applied Mathematics. 10. 115 – 1165. 892 | P a g e

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