The usual situation, if there is more than one part for the same instrument in a piece, is that the instruments do not play the same note. Such parts can be, and usually are, notated on the same staff in an orchestral score. Each such staff can be extracted into a single part or into multiple parts, as described by MattPutnam. Having a separate part for each player is the norm, as noted by Michael Seifert in a comment.
Since it's common for a single staff to carry two parts, it is not customary to write divisi on the staff when there is more than one line of notes. Instead, the composer or copyist must indicate what to do in the opposite case, when there's only a single line of notes.
If the single line of notes is supposed to be played by more than one part, the usual marking is not unison, but a due or, for a staff carrying three parts, a tre. This is frequently written as a 2 or a 3. An example may be found in the German Requiem of Johannes Brahms, in the bassoons, in the second movement, at measure 1 (I'm looking at the Dover reprint of the Breitkopf and Härtel complete works).
If there is a single note that only one part should play, the marking is primo or secondo, often written as 1º, Iº, 1., or I. and 2º, IIº, 2., or II., using Italian or German conventions for writing ordinal numbers. An example may be found in the Brahms, in the first movement, at measure 45, in the trombones. Another possibility is to write a rest under the note, if the note is for the first player, or above it, if the note is for the second player. An example of this is seen in the second movement at measures 33 and 34, in the oboes and bassoons.
If the parts combine for just a few notes, the convention is to write each note with both upward and downward pointing stems. This can be seen in the second movement of the Brahms, in the bassoons, from measure 13 through measure 22. For notes that have no stem, the convention is to write two notes side by side. This can be seen in the last measure of the same movement, on the third-trombone-and-tuba staff.
Finally, to underscore the fact that there is no need to mark divisi explicitly, look again at the beginning of the same movement, where the bassoons are a due. On the third beat of measure 12, they divide without comment, and on the downbeat of the next measure, where they are again in unison, the note has two stems. The parts remain divided until measure 42, where the staff is marked a 2.
I will close with an endorsement of Carl Witthoft's answer, which applies in the context of writing for wind and brass sections as is perhaps more common for bands. The assumption in orchestral writing is that each woodwind and brass part is assigned to a single player. There are certainly cases in which orchestral wind and brass parts have been doubled, such as at the premiere of Beethoven's ninth symphony. In such a case it would be for the conductor to decide whether any passage should be played by the entire section or as a solo. A composer could explicitly call for multiple-player sections in an orchestral piece, and then markings like solo, tutti, divisi, and unison would be useful, but that is not the convention.