Nonlinear behavior of electrohydrodynamic flow in viscoelastic fluids

This paper presents a numerical study of electrohydrodynamic flow for the problem of electroconvection in a viscoelastic dielectric liquid subjected to unipolar injection. The nonlinear evolution of flow for diverse characteristic Weissenberg numbers is examined in detail. The influences of elasticity on the flow pattern, oscillation amplitude, current transfer, and power-law spectral scaling accompanying oscillatory flow are also investigated. The results demonstrate that the effect of viscoelasticity leads to hydrodynamic behaviors that are absent in Newtonian fluids. The transition sequences to the chaos of electroconvection in viscoelastic fluids are quite different from those in Newtonian fluids, and the elasticity precipitates the onset of chaos. An asymmetric steady flow pattern resulting from elasticity is observed in a perfectly symmetric geometry. The fluctuation amplitude of fluid motion is found to be suppressed for weakly elastic fluids but amplified when the fluid elasticity exceeds a certain threshold. For viscoelastic fluids, electric current transfer is reduced in most cases, but for weakly elastic fluid, it may be enhanced.

Figure 1

Schematic illustration of the EHD system.

Figure 2

Oscillating state in a Newton fluid for (a) $T=230$ and (b) $T=277$. The left panels are the phase portraits of ($u$, $v$) at the point A, and the right panels are plots of the PSD with a logarithmic vertical scale.

Figure 3

Variation of the electrical Nusselt number $Ne$ with $M^{1/2}$.

Figure 4

Charge density isocontours (a) and streamlines (b) for $T=600$ and $Wi=0.05$.

Figure 5

Maximum velocity records (left panels), phase portraits of ($u$, $v$) at point A (middle panels), and power spectral densities (right panels) for $Wi=0.05$ showing the sequence of instabilities from steady to nonperiodic flow: (a) $T=600$, two-cell symmetric steady flow; (b) $T=750$, periodic state; (c) $T=800$, quasiperiodic state; (d) $T=900$, nonperiodic state.

Figure 6

Chaos judgment for $T=900$ and $Wi=0.05$: (a) fractal dimension computation; (b) lower bounds of the sums of positive Lyapunov exponents.

Figure 7

Phase portraits of ($u$, $v$) at point A (left panels) and power spectra density (right panels) for $Wi=0.05$: (a) $T=950$, quasiperiodic state; (b) $T=1100$, periodic state; (c) $T=1200$, quasiperiodic state; (d) $T=1300$, nonperiodic state.

Figure 8

Time evolution of the maximum velocity for $T=650$ and $Wi=0.2$. The insets show the charge density distribution, streamlines, and first principal normal stress difference in the final steady state.

Figure 9

Maximum velocity records (left panels) and power spectra density (right panels) for $Wi=0.2$: (a) $T=700$, periodic state; (b) $T=705$, quasiperiodic state; (c) $T=710$, nonperiodic state.

Figure 10

Maximum velocity records for $Wi=0.5$ at $T=200$, 250, 275, and 287.5.

Figure 11

Maximum velocity records for $Wi=0.5$ at $T=295$, $T=305$, and $T=330$.

Figure 12

Maximum velocity records (left panels) and power spectra density (right panels) for $Wi=0.5$: (a) $T=350$, quasiperiodic state; (b) $T=360$, nonperiodic state.

Figure 13

Influence of the viscoelastic effect on oscillation: (a) $T=600$; (b) $T=1000$.

Figure 14

Evolution of the electric Nusselt number with $T$ for different values of viscoelastic parameters.

Figure 15

Log-log plots of the power spectral density of electric Nusselt number fluctuations for $T=1500$: (a) Newtonian fluid; (b) $Wi=0.05$; (c) $Wi=0.2$; (d) $Wi=0.5$.