Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) HYDROAGNETIC DIVERGENT CHANNEL FLOW OF A VISCO- ELASTIC ELECTRICALLY CONDUCTING FLUID RITA CHOUDHURY Department o athematics, Gauhati University Guwahati-784, Assam, India Email: rchoudhury66@yahoo.in KAAL DEBNATH * Department o athematics, Gauhati University Guwahati-784, Assam, India Email: debnathtura@gmail.com Abstract A theoretical study or the two-dimensional boundary layer low through a divergent channel o a visco-elastic electrically conducting luid in presence o transverse magnetic ield has been considered. Similarity solutions are obtained by considering a special orm o magnetic ield. The analytical epressions or velocity and skin riction at the wall have been obtained and numerically worked out or dierent values o the low parameters involved in the solution. The velocity and the skin riction coeicient have been presented graphically to observe the visco-elastic eects or various values o the low parameters across the boundary layer. Key words and Phrases: Visco-elastic, divergent channel, electrically conducting luid, HD, similarity solution. AS athematics Subject Classiication: 76A5, 76 A. Introduction Jeery() has investigated the divergent low problem between two non-parallel planes by reducing the problem to an elliptic integral equation. Srivastava () has etended this problem to an electrically conducting luid in the presence o transverse magnetic ield. He has observed that with the application o magnetic ield it is possible to have purely divergent low without any secondary low or greater angle between the two planes. The solution o two-dimensional incompressible laminar low in a divergent channel with impermeable wall has been presented by Rosenhead (). Terril(4) has analyzed the slow laminar low in a converging or diverging channel with suction at one wall and blowing at the other wall. Hamel (5) has studied the preceding problem o calculating all three dimensional lows whose streamlines are identical with those o potential low. The numerical calculations o Jeery-Hamel lows between nonparallel plane walls were perormed by illsaps and Pohlhausen (6). Phukan(7) studied the hydromagnetic divergent channel low o a Newtonian electrically conducting luid. agnetohydrodynomic laminar low o a viscous luid in a converging or diverging channel with suction at one wall and equal blowing at the other wall has been studied by ahapatra et. al(8). The present work deals with the two-dimensional magnetohydrodynamic boundary layer low through a divergent channel o an electrically conducting visco-elastic luid characterized by Walters liquid (odel B') in presence o transverse magnetic ield. The eect o the visco-elastic luid across the boundary layer on the dimensional velocity component and skin riction coeicient have been presented graphically with the combination o other low parameters involved in the solution. ISSN : 975-546 Vol. No. October 7556
Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) The constitutive equation or Walters liquid (odel B') is σ pg + e k e where σ is the stress tensor, p is isotropic pressure, g is the metric tensor o a ied co-ordinate system i i, v is the velocity vector, the contravariant orm o e is given by ( τ ) e m k im i mk e + v e, m v, m e v, m e t It is the convected derivative o the deormation rate tensor e deined by i k e v, + v, Here is the limiting viscosity at the small rate o shear which is given by τ k N ( τ ) d and k τn ( τ ) i dτ N being the relaation spectrum as introduced by Walters (9, ). This idealized model is a valid approimation o Walters liquid (odel B') taking very short memories into account so that terms involving have been neglected. τ n N ( τ ) dτ, n athematical Formulation The basic equations or steady two-dimensional boundary layer low o Walters liquid (odel B') in the presence o a magnetic ield B() are given by u v + (.) y u u U u k u u u u v u B ( ) u v U u v ( U u) y y y y y y y + σ + + ν + (.) ρ ρ subject to the boundary conditions y : u, v y : u U ( ) (.) where -ais coincides with the wall o the divergent channel and y-ais perpendicular to it. U is the velocity component outsides the boundary layer, u and v are the low velocities in the direction o and y respectively, k the visco- ρ the luid density, ν the kinematic viscosity, σ the electrical conductivity o the luid and elastic parameter. In equation (.), the secondary eects o magnetic induction are ignored i.e. the induced magnetic ield is negligible as it is small in comparison to the applied magnetic ield. Furthermore, we assume that the eternal electric ield is zero and the eletric ield due to polarization o charges is also negligible. As in Sinha and Choudhury (), the potential low near the sources is taken to be u U ( ) With u > represents two dimensional divergent low and leads to similarity solution. We introduce the ollowing change o variables (Schlichting ()) and the stream unction ϕ U ( ) (, y) y ν y u (, y) U ( ) νf( ) νu F( ) Then, we obtain the velocity component as u ϕ U ( ) F ( ) y and v ϕ νu F ( ) y The equation o continuity (.) is identically satisied or the velocity component (.7) (.) (.) (.) (.4) (.5) (.4) (.5) (.6) (.7) ISSN : 975-546 Vol. No. October 7557
Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) Similarity solution eists i the magnetic ield B() has the special orm (Chiam ()) B B( ) Using the equation (.4) to (.8), the equation (.) takes the orm as ollows: F + F + k[ 4F F F ] + ( F ) (.9) k Here prime denotes the dierentiation with respect to. k and denote the modiied non-newtonian and hydrodynamic parameter respectively. The corresponding boundary conditions are F ( ), F (), F ( ) (.) ρ (.8) ethod o Solution We irst assume z, ( z) F( ) (.) which implies ( z ) F ( ), ( z ) F ( ), ( z ) F ( ) (.) Using the equation (.) in the equation (.9), we get the ollowing dierential equation ( z) + k[ 4 ( z) ( z) ( z) ] + ( ( z)) ε ( ( z)) (.) where ε The corresponding boundary conditions are ( ), (), ( ) (.4) The unknown unction (z) is epanded in terms o powers o the small parameterε as ollows: ( z) ( z) + ε ( z) + ε ( z) +... (.5) On substituting the equation (.5) in the equation (.) and equating the le powers o ε, we get + k[ 4 ] + ( ) (.6) k[ 4 ] (.7) The relevant boundary conditions are: ( ), ( ), ( ) ( ), ( ), ( ) (.8) Again, in order to solve equations (.6) and (.7), we consider very small values o k, so that and can be epressed as ( z) + k ( z) + ( k ) ( z) + k ( z) + ( k ) (.9) Now substituting (.9) into the equations (.6) and (.7), we get the ollowing sets o ordinary dierential equations oo (.) o 4 + (.) (.) 4[ ] (.) The appropriate boundary conditions are,, + + ( ) ( ) ( ) ISSN : 975-546 Vol. No. October 7558
Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) ( ), ( ), ( ) ( ), ( ), ( ) ( ), ( ), ( ) (.4) The solution o equations (.) to (.) satisying the respective boundary conditions (.4) are z + e (.5) ( 6 4 z ze + e + e 5 ) (.6) 6 ( 6ze + 4e + e 5) ( 9z e + e e ) 4 45 Substituting (.5) to (.8) into (.5) and ater dierentiation with respect to z, we obtain z z + ε z where ( z) ( e ) ( ) ( ) ( ) 4 + k e + e ( ) e z 4 4 z ( z) e e ( z) + e + k e + e + 6ze 8z e 5 5 F across the boundary layer. From (.) and (.9) we can easily ind the dimensionless velocity ( ) (.7) (.8) (.9) Results and Discussion The approimate skin riction coeicient is given by ( ) ( ) + ( ) τ ε + k 5 k. The corresponding results or Newtonian luid are obtained by setting k. (.) where ( ) k and ( ) The eects o visco-elastic parameter on the two-dimensional laminar HD boundary layer low in a divergent channel have been analyzed in this study The visco-elastic eect is ehibited through the nondimensionl parameter The igures, and demonstrate the variations o dimensionless velocity F ( ) against the variable across the boundary layer or dierent low parameters. The igures depict that the velocity decreases with the increasing values o the variable in both Newtonian and non-newtonian cases. Also, the igures show that an increase in the visco-elastic parameter in comparison with the Newtonian luid reduce the low velocity at all corresponding points in the low ield. Further, it is noticed rom the igures that F ( ) decreases when the magnetic parameter increases with the increase o visco-elastic parameter. Figure 4 depicts the variation o the shearing stress τ at the wall o the divergent channel against the magnetic parameter. It shows that the shearing stress decreases with the increasing values o the magnetic parameter in both Newtonian and non-newtonian cases. Further, the increase o visco-elastic parameter enhances the shearing stress at all corresponding points in the low ield. ISSN : 975-546 Vol. No. October 7559
Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) Graphs or Velocity Distribution and Skin riction Co-eicient k k. k.5 Figure : Velocity distribution against the variable or k k. k.5 ' () ' () Figure : Velocity distribution against the variable or k k. k.5 ' () Figure : Velocity distribution against the variable or 5 ISSN : 975-546 Vol. No. October 756
Rita Choudhury et al. / International Journal o Engineering Science and Technology (IJEST) k k. k.5 τ Figure 4: Skin riction co-eicient or various values o Reerences [] Jeery, G.B. (95): Two-dimensional steady motion o a viscous luid, Phil. ag., 9, pp. 455-465. [] Srivastava, A.C. (959): The eects o magnetic ield on the low between two non-parallel planes, Proc. Fith Cong. Theo. and Appl. ech., pp. 79-84,. [] Rosenhead (Ed.), L. (96): Laminar Boundary Layers, Oord. [4] Terril, R.. (965): Slow laminar low in a converging or diverging channel with suction at one wall and blowing at the other wall, Z. Angew. ath. Phys., 6, pp. 6-8. [5] Hamel, G. (96): Spiralrmige Bewegung Zher Flssigkeiten, Jahresber. d. Dt. athematar, 4. [6] Pohlhausen, K. (9): Zur nuherungsweisen integration der dierential gleichung der Grenzschicht, Z.A...,, pp. 5-59. [7] Phukan, D.K. (998): Hydromagnetic divergent channel low o a Newtonian electrically conducting luid, The Bulletin, GUA, 4, pp. 4-5. [8] ahapatra, T. R.; Dholey, S.; Gupta, A.S. (): agnetohydrodynomic laminar low o a viscous luid in a converging or diverging channel with suction at one wall and equal blowing at the other wall, The Bulletin, GUA,, pp. -. [9] Walters, K. (96): The motion o an elastico-viscous liquid contained between co-aial cylinders, Quart. J. ech. Appl. ath.,, pp. 444-46. [] Walters, K., (96): The solutions o low problems in the case o materials with memories, J. ecanique,, pp. 47-478. [] Sinha, K.D.; Choudhury, R.C. (969): Laminar incompressible boundary layer low in a divergent channel with homogenous suction at the wall, Proc. Nat. Inst. Sci., India, 5, pp. 66-7. [] Schlichting, H. (968): Boundary Layer Theory, (6 th edition) cgraw-hill, New York. [] Chiam, T.C. (995): Hydromagnetic low over a surace stretching with a power-law velocity, Int. J. Engg. Sci.,, pp. 49-45. ISSN : 975-546 Vol. No. October 756