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Research Papers: Fundamental Issues and Canonical Flows

# Three-Dimensional Viscous Potential Electrohydrodynamic Kelvin–Helmholtz Instability Through Vertical Cylindrical Porous Inclusions With Permeable Boundaries

[+] Author and Article Information
Galal M. Moatimid

e-mail: gal_moa@hotmail.com

Mohamed A. Hassan

e-mail: m_a_hassan_gk@hotmail.com
Department of Mathematics,
Faculty of Education,
Ain Shams University,
Roxy 1575, Cairo, Egypt

Contributed by the Fluids Engineering Division of ASME for publication in the JOURNAL OF FLUIDS ENGINEERING. Manuscript received May 27, 2013; final manuscript received July 2, 2013; published online December 2, 2013. Assoc. Editor: Shizhi Qian.

J. Fluids Eng 136(2), 021203 (Dec 02, 2013) (10 pages) Paper No: FE-13-1341; doi: 10.1115/1.4025681 History: Received May 27, 2013; Revised July 02, 2013

## Abstract

In this paper, the electrohydrodynamic three-dimensional Kelvin–Helmholtz instability of a cylindrical interface with heat and mass transfer between liquid and vapor phases is studied. The liquid and the vapor are saturated, two coaxial cylindrical porous layers, and the suction/injection velocities for the fluids at the permeable boundaries are also taken into account. The dispersion relation is derived and the stability analysis is discussed for various parameters. It is found that the streaming velocity has a destabilizing effect, while the axial electric field has a stabilizing one. The suction for both the liquid and the steam has a destabilizing effect in contrast with the injection at both boundaries. The flow through porous structure is more stable than the pure flow. The case of the axisymmetric (for zero value of the azimuthal wave number $m$) and asymmetric (for nonzero value of the azimuthal wave number $m$) disturbances at large wavelength (at the wave number $k→0$) are always stable. Meanwhile, it is the same dispersion relation for the plane geometry at large wave number. Finally, our results are corroborated by comparing them with the previous published results.

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## Figures

Fig. 1

Physical model and coordinates

Fig. 2

Stability diagram for a system having the particulars: ρ1=1, ρ2=0.001, μ1=0.01, μ2=0.00001, r1=1, r2=2,ν1=0.5, ν2=0.5, α=10, ɛ1=78.64, ɛ2=1, R=1.5, m=1, E0=10

Fig. 3

Stability diagram for a system having the particulars: ρ1=1,ρ2=0.001,μ1   =0.01,μ2=0.00001,r1 =1,r2=2, ν1=0.1,ν2=0.3,α=20,ɛ1=78.64,ɛ2=1,R=1.5,m=1,U∧=1000

Fig. 4

Stability diagram for a system having the particulars: ρ1=1,ρ2=0.001,μ1=0.01,μ2=0.00001,r1=1,r2=2,V1=10,V2=-10,ν1=0.5, ν2=0.5, α,=10, ɛ1=78.64,ɛ2=1, R=1.5,E0=20

Fig. 5

Stability diagram for a system having the particulars: ρ1=0.689, ρ2=0.987, μ1=0.8, μ2=1.4, r1=1, r2=2, V1=0.5,V2=-0.5,ν1=0.5, ν2=0.5,α=2000, ɛ1=5.1,ɛ2=6.765,R=1.5,U∧=2000

Fig. 6

Stability diagram for a system having the particulars: ρ1=1, ρ2=0.001, μ1=0.01, μ2=0.00001, r1=1, r2=2, V1=0,V2=0,α=10,ɛ1=78.64,ɛ2=1,R=1.5,m=1,E0=20

Fig. 7

Stability diagram for a system having the particulars: ρ1=1, ρ2=0.001, μ1=0.01, μ2=0.00001, r1=1, r2=2, V1=0,V2=0,α=10,ɛ1=78.64,ɛ2=1,R=1.5,m=1,U∧=1000

Fig. 8

Stability diagram for a system having the particulars: ρ1=1, ρ2=0.001, μ1=0.01, μ2=0.00001, r1=1,r2=2,V1=0.5,V2=-0.5, ν1=0.1, ν2=0.1, ɛ1=78.64, ɛ2=1, R=1.5, m=1,E0=20

Fig. 9

Stability diagram for a system having the particulars: ρ1=1, ρ2=0.001, μ1=0.01, μ2=0.00001, r1=1, r2=2, V1=0,V2=0,ν1=0.3,ν2=0.3,ɛ1=78.64,ɛ2=1,R=1.5,m=1,U∧=2000

Fig. 10

Stability diagram for a system having the particulars: ρ1=1, ρ2=0.001, μ1=0.01, μ2=0.00001, r1=1, r2=2, V1=0,V2=0,ν1=0.5,ν2=0.5,α=0.5,k=0.5,R=1.5,m=3,U∧=10

Fig. 11

Stability diagram for the water, ammonia, and propane according to data of Table 1 at r1=1,r2=2,V1=0,V2=0,ν1=0,ν2=0,α=10,ɛ1=78.64,ɛ2=1,R=1.5,m=1,E0=10

Fig. 12

Stability diagram for the water, ammonia, and propane according to data of Table 1 at r1=1,r2=2,V1=0,V2=0,ν1=0,ν2=0,α=2,ɛ1=78.64,ɛ2=1,R=1.5,m=1,U∧=2000

Fig. 13

Stability diagram for the velocity potential φ against ωI for a system having the particulars: ρ1=1,α=2,A=1,k=0.5,z=0.5,m=0,θ=π/3,ωR=1,U1=5,r1=10,R=15

Fig. 14

Stability diagram for the electric potential ψ against ωI for a system having the particulars: ɛ1=78.64,ɛ2=1,E0=100,k=0.5, z=0.5, m=0, θ=π/3, ωR=1, r2=20, r1=10, R=15,A=1

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