As you probably know from your research, diatoms can be classified by their shape as either 'centric' (round) or 'pennate' (mostly they look like ovals). As it turns out, the round centric diatoms appeared first in the fossil record, meaning that they are more primitive. Over geological time, the diatoms evolved into pennate forms. In general, you are correct and most centric diatoms cannot move. However, some centric diatoms can move and some pennates cannot!

How do the diatoms which can move do so? Scientists think that these diatoms secrete a sort of mucus and slide along a surface on that. It is possible that they also retract this mucus as they move. Pennate diatoms have developed a special slit in their silica shells called a 'raphe' through which the mucus is secreted. This structure is also important in classifying diatoms, or telling one pennate species from another.

How do the moving diatoms move?Do they have flagella or something to move them around? I didn't know that only round diatoms could move, but I don't know much about diatoms. Diatoms live free in the ocean, don't they? So do the diatoms that can't move still get pushed around by the ocean currents?