We consider the following chemotaxis-Stokes system with rotation {n(t )=( )Delta n - del . (nS(x,n,c) . del c) - u . del n, c(t )= Delta c - f(x,n,c) - u . del c, u(t = Delta u + del P + n del phi,) del . u = 0 in Omega x (0, T), subject to the non-flux boundary conditions for n andc, as well as the Dirichlet boundary condition for u, where the bounded smooth domain Omega subset of R-3, the matrix-valued function S is an element of C-2 ((Omega) over bar x [0, infinity)(2); R-3x3) fulfills vertical bar S(x, n, c)vertical bar