Study of Surface Instability of Kelvin-Helmholtz Type in a Fluid Layer Bounded above by a Porous Layer and below by a Rigid Surface

Author(s)
Nanjundappa Rudraiah,
Krishna Basappa Chavaraddi,
Inapura Siddagangaiah Shivakumara,
Bangalore Mahadevaiah Shankar

ABSTRACT

The surface instability of Kelvin-Helmholtz type bounded above by a porous layer and below by a rigid surface is investigated using linear stability analysis. Here we adopt the theory based on electrohydrodynamic as well as Stokes and lubrication approximations. We replace the effect of boundary layer with Beavers and Joseph slip condition. Here we have studied the combined effect of electric and magnetic fields on Kelvin-Helmholtz instability (KHI) in a fluid layer bounded above by a porous layer and below by a rigid surface. The dispersion relation is obtained using suitable boundary and surface conditions and results are depicted graphically. Also the ratio Gm is numerically computed for different values of We and M given in the Table 1. From this it is clear that the combined effect of electric and magnetic fields with porous layer are more effective than the effect of compressibility in reducing the growth rate of RTI. Also, these results shows that with a proper choice of magnetic field it is possible to control the growth rate of Electrohydrody-namic KHI (EKHI) and hence can be restored the symmetry of IFE target.

The surface instability of Kelvin-Helmholtz type bounded above by a porous layer and below by a rigid surface is investigated using linear stability analysis. Here we adopt the theory based on electrohydrodynamic as well as Stokes and lubrication approximations. We replace the effect of boundary layer with Beavers and Joseph slip condition. Here we have studied the combined effect of electric and magnetic fields on Kelvin-Helmholtz instability (KHI) in a fluid layer bounded above by a porous layer and below by a rigid surface. The dispersion relation is obtained using suitable boundary and surface conditions and results are depicted graphically. Also the ratio Gm is numerically computed for different values of We and M given in the Table 1. From this it is clear that the combined effect of electric and magnetic fields with porous layer are more effective than the effect of compressibility in reducing the growth rate of RTI. Also, these results shows that with a proper choice of magnetic field it is possible to control the growth rate of Electrohydrody-namic KHI (EKHI) and hence can be restored the symmetry of IFE target.

Cite this paper

nullN. Rudraiah, K. Chavaraddi, I. Shivakumara and B. Shankar, "Study of Surface Instability of Kelvin-Helmholtz Type in a Fluid Layer Bounded above by a Porous Layer and below by a Rigid Surface,"*World Journal of Mechanics*, Vol. 1 No. 6, 2011, pp. 267-274. doi: 10.4236/wjm.2011.16033.

nullN. Rudraiah, K. Chavaraddi, I. Shivakumara and B. Shankar, "Study of Surface Instability of Kelvin-Helmholtz Type in a Fluid Layer Bounded above by a Porous Layer and below by a Rigid Surface,"

References

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[2] S. N. Shore, “An Introduction to Astrophysical Hydro- dynamics,” Academic Press, New York, 1992.

[3] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Sta- bility,” Dover Publications, New York, 1961.

[4] L. Kelvin, “Hydrokinetic Solutions and Observations, On the Motion of Free Solids through a Liquid,” Mathemati- cal and Physical Papers IV, Hydrodynamics and General Dynamics, Cambridge, 1910, pp. 69-75.

[5] L. Kelvin, “Influence of Wind and Capillary on Waves in Water Superposed Frictionless,” Mathematical and Phy- sical Papers IV, Hydrodynamics and General Dynamics, Cambridge, 1910, pp. 76-85.

[6] J. R. D. Francis, “Wave Motions and the Aerodynamic Drag on a Free Oil Surface,” Philosophical Magazine Se- rials 7, Vol. 45, No. 366, 1954, pp. 695-702.

[7] R. C. Sharma and K. M. Srivastava, “Effect of Horizontal and Vertical Magenetic Fields on Rayleigh-Taylor Insta- bility,” Australian Journal of Physics, Vol. 21, No. 6, 1968, pp. 923-930. doi:10.1071/PH680923

[8] J. R. Melcher, “Field Coupled Surface Waves,” MIT Press, Cambridge, 1963.

[9] A. R. F. Elhefnawy, “Nonlinear Electrohydrodynamics Kelvin-Helmholtz Instability under the Influence of an Oblique Electric Field,” Physica A, Vol. 182, No. 3, 1992, pp. 419-435. doi:10.1016/0378-4371(92)90352-Q

[10] V. Mehta and P. K. Bhatia, “Kelvin-Helmholtz Instability of two Viscous Superposed Rotating and Conducting Flu- ids,” Astrophysics and Space Science, Vol. 141, No. 1, 1998, pp. 151-158. doi:10.1007/BF00641921

[11] R. P. Singh and H. C. Khare, “Stability of Rotating Gravi- tating Superposed Streams in a Uniform Magnetic Field,” Proceedings of National Academy of Sciences, New Delhi, Vol. 43, 1991, pp. 49-55.

[12] P. K. Bhatia and A. B. Hazarika, “Gravitational Instabil- ity of Partially Ionized Plasma in an Oblique Magnetic Field,” Physica Scripta, Vol. 51, No. 6, 1995, pp. 775- 779.

[13] T. B. Benjamin and T. J. Bridges, “Reappraisal of the Kel- vin-Helmholtz Problem. Part-2: Interaction of the Kelvin- Helmholtz Superharmonic and Benjamin-Feir Instabili- ties,” Journal of Fluid Mechanics, Vol. 333, 1997, pp. 327-373. doi:10.1017/S0022112096004284

[14] R. C. Sharma and P. Kumar, “On the Stability of Two Su- perposed Walters B' Viscoelastic Liquid,” Czechoslovak Journal of Physics, Vol. 47, No. 2, 1996, pp. 197-204.

[15] M. H. O. Allah, “The Effects of Magnetic Field and Mass and Heat Transfer on Kelvin-Helmholtz Stability,” Pro- ceedings of National Academy of Sciences, Vol. 68, No. 2, 1998, pp. 163-173.

[16] J. A. M. McDonnel, “Cosmic Dust,” John Wiley and Sons, Toronto, 1978.

[17] N. Rudraiah and P. K. Srimani, “Thermal Convection of a Rotating Fluid through a Porous Media,” Vignana Bharti, Vol. 2, No. 2, 1976, pp. 11-17.

[18] Sunil and T. Chand, “Effects of Rotation on the Rayleigh- Taylor Instability of Two Superposed Magnetized Con- ducting Plasma,” Indian Journal of Physics, Vol. 71, 1997, pp. 95-105.

[19] A. J. Babchin, A. L. Frenkel, B. G. Levich and G. I. Si- vashinsky, “Nonlinear Saturation of Rayleigh-Taylor Instability in Thin Films,” Physics of Fluids, Vol. 26, No. 11, 1983, pp. 3159-3161. doi:10.1063/1.864083

[20] N. Rudraiah, R. D. Mathad and H. Betigeri, “The RTI of Viscous Fluid Layer with Viscosity Stratification,” Cur- rent Science, Vol. 76, No. 6, 1997, pp. 391-398.

[21] G. S. Beavers and D. D. Joseph, “Boundary Conditions at a Naturally Permeable Wall,” Journal of Fluid Mechanics, Vol. 30, No. 1, 1967, pp. 197-207. doi:10.1017/S0022112067001375

[22] M. F. El-Sayed, “Effect of Variable Magnetic Field on the Stability of a Stratified Rotating Fluid Layer in Po- rous Medium,” Czechoslovak Journal of Physics, Vol. 50, No. 5, 2002, pp. 607-614. doi:10.1023/A:1022854217365

[23] Y. O. El-Dib and R. T. Matoog, “Electrorheological Kel- vin-Helmholtz Instability of a Fluid Sheet,” Journal of Colloid and Interface Science, Vol. 289, No. 1, 2005, pp. 223-241. doi:10.1016/j.jcis.2005.03.054

[24] A. E. Khalil Elcoot, “New Analytical Approximation Forms for Non-Linear Instability of Electric Porous Media,” In- ternational Journal of Non-Linear Mechanics, Vol. 45, No. 1, 2010, pp. 1-11. doi:10.1016/j.ijnonlinmec.2009.08.011

[1] H. Helmholtz, “On Discontinuous Movements of Fluids,” Philosophical Magazine Series 4, Vol. 36, 1868, pp. 337- 346.

[2] S. N. Shore, “An Introduction to Astrophysical Hydro- dynamics,” Academic Press, New York, 1992.

[3] S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Sta- bility,” Dover Publications, New York, 1961.

[4] L. Kelvin, “Hydrokinetic Solutions and Observations, On the Motion of Free Solids through a Liquid,” Mathemati- cal and Physical Papers IV, Hydrodynamics and General Dynamics, Cambridge, 1910, pp. 69-75.

[5] L. Kelvin, “Influence of Wind and Capillary on Waves in Water Superposed Frictionless,” Mathematical and Phy- sical Papers IV, Hydrodynamics and General Dynamics, Cambridge, 1910, pp. 76-85.

[6] J. R. D. Francis, “Wave Motions and the Aerodynamic Drag on a Free Oil Surface,” Philosophical Magazine Se- rials 7, Vol. 45, No. 366, 1954, pp. 695-702.

[7] R. C. Sharma and K. M. Srivastava, “Effect of Horizontal and Vertical Magenetic Fields on Rayleigh-Taylor Insta- bility,” Australian Journal of Physics, Vol. 21, No. 6, 1968, pp. 923-930. doi:10.1071/PH680923

[8] J. R. Melcher, “Field Coupled Surface Waves,” MIT Press, Cambridge, 1963.

[9] A. R. F. Elhefnawy, “Nonlinear Electrohydrodynamics Kelvin-Helmholtz Instability under the Influence of an Oblique Electric Field,” Physica A, Vol. 182, No. 3, 1992, pp. 419-435. doi:10.1016/0378-4371(92)90352-Q

[10] V. Mehta and P. K. Bhatia, “Kelvin-Helmholtz Instability of two Viscous Superposed Rotating and Conducting Flu- ids,” Astrophysics and Space Science, Vol. 141, No. 1, 1998, pp. 151-158. doi:10.1007/BF00641921

[11] R. P. Singh and H. C. Khare, “Stability of Rotating Gravi- tating Superposed Streams in a Uniform Magnetic Field,” Proceedings of National Academy of Sciences, New Delhi, Vol. 43, 1991, pp. 49-55.

[12] P. K. Bhatia and A. B. Hazarika, “Gravitational Instabil- ity of Partially Ionized Plasma in an Oblique Magnetic Field,” Physica Scripta, Vol. 51, No. 6, 1995, pp. 775- 779.

[13] T. B. Benjamin and T. J. Bridges, “Reappraisal of the Kel- vin-Helmholtz Problem. Part-2: Interaction of the Kelvin- Helmholtz Superharmonic and Benjamin-Feir Instabili- ties,” Journal of Fluid Mechanics, Vol. 333, 1997, pp. 327-373. doi:10.1017/S0022112096004284

[14] R. C. Sharma and P. Kumar, “On the Stability of Two Su- perposed Walters B' Viscoelastic Liquid,” Czechoslovak Journal of Physics, Vol. 47, No. 2, 1996, pp. 197-204.

[15] M. H. O. Allah, “The Effects of Magnetic Field and Mass and Heat Transfer on Kelvin-Helmholtz Stability,” Pro- ceedings of National Academy of Sciences, Vol. 68, No. 2, 1998, pp. 163-173.

[16] J. A. M. McDonnel, “Cosmic Dust,” John Wiley and Sons, Toronto, 1978.

[17] N. Rudraiah and P. K. Srimani, “Thermal Convection of a Rotating Fluid through a Porous Media,” Vignana Bharti, Vol. 2, No. 2, 1976, pp. 11-17.

[18] Sunil and T. Chand, “Effects of Rotation on the Rayleigh- Taylor Instability of Two Superposed Magnetized Con- ducting Plasma,” Indian Journal of Physics, Vol. 71, 1997, pp. 95-105.

[19] A. J. Babchin, A. L. Frenkel, B. G. Levich and G. I. Si- vashinsky, “Nonlinear Saturation of Rayleigh-Taylor Instability in Thin Films,” Physics of Fluids, Vol. 26, No. 11, 1983, pp. 3159-3161. doi:10.1063/1.864083

[20] N. Rudraiah, R. D. Mathad and H. Betigeri, “The RTI of Viscous Fluid Layer with Viscosity Stratification,” Cur- rent Science, Vol. 76, No. 6, 1997, pp. 391-398.

[21] G. S. Beavers and D. D. Joseph, “Boundary Conditions at a Naturally Permeable Wall,” Journal of Fluid Mechanics, Vol. 30, No. 1, 1967, pp. 197-207. doi:10.1017/S0022112067001375

[22] M. F. El-Sayed, “Effect of Variable Magnetic Field on the Stability of a Stratified Rotating Fluid Layer in Po- rous Medium,” Czechoslovak Journal of Physics, Vol. 50, No. 5, 2002, pp. 607-614. doi:10.1023/A:1022854217365

[23] Y. O. El-Dib and R. T. Matoog, “Electrorheological Kel- vin-Helmholtz Instability of a Fluid Sheet,” Journal of Colloid and Interface Science, Vol. 289, No. 1, 2005, pp. 223-241. doi:10.1016/j.jcis.2005.03.054

[24] A. E. Khalil Elcoot, “New Analytical Approximation Forms for Non-Linear Instability of Electric Porous Media,” In- ternational Journal of Non-Linear Mechanics, Vol. 45, No. 1, 2010, pp. 1-11. doi:10.1016/j.ijnonlinmec.2009.08.011