Mechanism of Wave-Shape Pattern Formation

We analyze the mechanism of wave-shape pattern formation based on the proposed model and verify it by comparing the numerical computation and the experimental data. The model is represented by a rather simple nonlinear first-order partial differential equation of equation (2.7). It describes that the effect of the electric field to the gel is determined solely by the geometry of the equipo-tential surface and the gel surface. We rewrite the equation as follows in order to represent the geometry of the experimental setup. 

The constant electric field is generated by the pair of parallel electrodes of infinite length. The equipotential surfaces of the electric field run parallel to the electrodes. The current density vector, a gradient of the electric potential, is vertical to the equipotential surface. Since the electrodes were placed parallel to the x-axis, the current density vector is written by  

We can obtain the specific expression of equation (2.7) by substituting equations (7.3) and (7.4) into it, 

Note that the curvature 1/i is a spatial differentiation along the beam-shaped 

By substituting equation (7.6) into equation (7.5) to eliminate a, we have 

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