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In this note, we investigate partial regularity of weak solutions of thethree dimensional chemotaxis-Navier-Stokes equations, and obtain the$\frac53$-dimensional Hausdorff measure of the possible singular set isvanishing at the first blow-up time. The new ingredients are to establishcertain type of local energy inequality and deal with the non-scaling invariantquantity of $n\ln n$, which seems to be the first description for the singularset of weak solutions of the chemotaxis-fluid model, which is motivated byCaffarelli-Kohn-Nirenberg's partial regularity theory \cite{CKN}.

Partial regularity of solutions to the 3D chemotaxis-Navier-Stokes equations at the first blow-up time