Axisymmetric instability in a thinning electrified jet

The axisymmetric stability of an electrified jet is analyzed under electrospinning conditions using the linear stability theory. The fluid is considered Newtonian with a finite electrical conductivity, modeled as a leaky dielectric medium. While the previous studies impose axisymmetric disturbances on a cylindrical jet of uniform radius, referred to as the base state, in the present study the actual thinning jet profile, obtained as the steady-state solution of the one-dimensional slender filament model, is treated as the base state. The analysis takes into account the role of variation in the jet variables like radius, velocity, electric field, and surface charge density along the thinning jet in the stability behavior. The eigenspectrum of the axisymmetric disturbance growth rate is constructed from the linearized disturbance equations discretized using the Chebyshev collocation method. The most unstable growth rate for the thinning jet is significantly different from that for the uniform radius jet. For the same electrospinning conditions, while the uniform radius jet is predicted to be highly unstable, the thinning jet profile is found to be unstable but with a relatively very low growth rate. The stabilizing role of the thinning jet is attributed to the variation in the surface charge density as well as the extensional deformation rate in the fluid ignored in the uniform radius jet analysis. The dominant mode for the thinning jet is an oscillatory conducting mode driven by the field-charge coupling. The disturbance energy balance finds the electric force to be the dominant force responsible for the disturbance growth, potentially leading to bead formation along the fiber. The role of various material and process parameters in the stability behavior is also investigated.

Figure 1

Schematic of the electrified jet in the electrospinning setup.

Figure 2

Steady-state profiles of the thinning jet: (a) radius; (b) velocity; (c) surface charge density; (d) axial electric field. Parameters: set I.

Figure 3

Variation of forces along the jet direction. As the jet propagates, the dynamics are governed mainly by the balance between the electric and the inertial forces. Parameters: set I.

Figure 4

Real part of the growth rate versus the disturbance wave number for a uniform jet superimposed on periodic perturbations. Parameters: set I.

Figure 5

Eigenspectrum of the disturbance growth rate: real part versus imaginary part of the growth rate for a uniform jet superimposed on nonperiodic perturbations. Parameters: set I.

Figure 6

Eigenspectrum of the disturbance growth rate: real part versus imaginary part of the growth rate for a thinning jet superimposed with nonperiodic perturbations. Parameters: set I.

Figure 7

Eigenspectrum of the disturbance growth rate: real part versus imaginary part of the growth rate for a uniform jet superimposed on nonperiodic perturbations. Parameters: set II.

Figure 8

Eigenspectrum of the disturbance growth rate: real part versus imaginary part of the growth rate for a thinning jet superimposed on nonperiodic perturbations. Parameters: set II.

Figure 9

Effect of the Weber number on the real part of the leading growth rate for both a uniform and a thinning jet: (a) set I; (b) set II.

Figure 10

Effect of the imposed electric field on the real part of the leading growth rate for both a uniform and a thinning jet: (a) set I; (b) set II.

Figure 11

Effect of the electrical conductivity of the fluid, nondimensionalized as the Péclet number, on the real part of the leading growth rate for both a uniform and a thinning jet: (a) set I; (b) set II.

Figure 12

Effect of the Reynolds number on the leading growth rate for both a uniform and a thinning jet: (a) set I; (b) set II.

Figure 13

Steady-state profiles of the interfacial electric stress, $E\sigma \left(z\right)$, and its variation with (a) the external electric field $E_{\infty }$ and (b) the Péclet number. Parameters: set II.

Figure 14

(a) Variation in the rate of change in the total kinetic energy for the most unstable disturbance, $d\left(KE\right)/dt$, with the external electric field strength. The leading growth rate, ${\omega }_r$, is also plotted to show that the change in sign for ${\omega }_r$ is coincidental with that for $d\left(KE\right)/dt$. (b) Various contributions to the disturbance energy as a function of the external field strength. The direction of the arrow signifies the corresponding $y$ axis.

Figure 15

(a) Variation in the rate of change in the total kinetic energy for the most unstable disturbance, $d\left(KE\right)/dt$, with the Reynolds number. The leading growth rate, ${\omega }_r$, is also plotted to show that the change in sign for ${\omega }_r$ is coincidental with that for $d\left(KE\right)/dt$. (b) Various contributions to the disturbance energy as a function of the Reynolds number. The direction of the arrow signifies the corresponding $y$ axis.