Let $ G$ be a classical group with natural module $ V$ over an algebraically closed field of good characteristic. For every unipotent element $ u$ of $ G$, we describe the Jordan block sizes of $ u$ on the irreducible $ G$-modules which occur as composition factors of $ V \otimes V^*$, $ \wedge ^2(V)$, and $ S^2(V)$. Our description is given in terms of the Jordan block sizes of the tensor square, exterior square, and the symmetric square of $ u$, for which recursive formulae are known.