Non metallic materials can consist of molecules that are polar in nature forming dipoles, or that can have dipoles induced by an external field. This is why insulators called dielectrics, which are polar in nature, are used in capacitors. When an electric field is applied between the capacitor plates, the dipoles line up with the field. In a capacitor, the aligning of the dipoles increases the capacitance of the capacitor. In your example the two rings, one negatively charged and one positively charged act as the plates of a capacitor and the oil/sawdust in between the rings serves as a dielectric.
In your case it is possible that the sawdust consists of polar molecules, though I am not familiar with the molecular structure of wood. But some oils, such as alcohols, esters, and triglycerides are polar in nature. If that is the case here, the application of the electric field between the inner and outer rings could cause the oil dipoles to align with the field. Since the sawdust sits on the oil molecules, the dust may also align with the field on with the oil. Its kind of like seeing iron filings line up with an applied magnetic field.
Whether it is the sawdust or the oil that has polar molecules, they will align with the field. The lower left diagram shows the dipoles randomly arranged in the absence of an applied field. The lower right shows alignment of the dipoles with the application of the field, except inside the inner ring since there would be no field there. For clarity, the dipoles are not shown superimposed on the sawdust, but the sawdust would be aligned with the dipoles, except inside the inner ring, because there is no field there.
Hope this helps.