This paper investigates a chemotaxis-generalized Navier–Stokes system with rotational flux 0.1 \(\begin{aligned} \left\{ \begin{array}{ll} n_{t}+\textbf{u}\cdot \nabla n=\Delta n-\nabla \cdot (nS(x,n,c)\cdot \nabla c),~& (x,t)\in \mathbb {T}^2\times (0,\infty ),\\ c_{t}+\textbf{u}\cdot \nabla c=\Delta c-nc,~& (x,t)\in \mathbb {T}^2\times (0,\infty ),\\ \textbf{u}_{t}+(\textbf{u}\cdot \nabla )\textbf{u}=-(-\Delta )^{\alpha }\textbf{u}+\nabla P+n\nabla \phi ,~& (x,t)\in \mathbb {T}^2\times (0,\infty ),\\ \nabla \cdot \textbf{u}=0,~& (x,t)\in \mathbb {T}^2\times (0,\infty ),\\ n(x,0)=n_{0}(x),c(x,0)=c_{0}(x),\textbf{u}(x,0)=\textbf{u}_{0}(x),~& x\in \mathbb {T}^2.\\ \end{array} \right. \end{aligned}\) Here, \(\alpha \in (0,1)\) and the matrix-valued sensitivity function S(x, n, c) represents the rotational effect and satisfies \(|S(x,n,c)|\le C_{S}\) , where \(C_{S}\) is a positive constant. Suppose that the initial data \((n_{0},c_{0},\textbf{u}_{0})\in C^{0}(\mathbb {T}^2)\times W^{1,q}(\mathbb {T}^2)\times C^{0}(\mathbb {T}^2)\) satisfy a smallness condition on \(\Vert c_{0}\Vert _{L^{\infty }(\mathbb {T}^2)}\) , we establish the existence and uniqueness of global classical solution to (0.1). Furthermore, we show that the solution \((n,c,\textbf{u})\) converge to \((\bar{n},0,0)\) as \(t\rightarrow \infty \) , where \(\bar{n}=\frac{1}{|\mathbb {T}^2|}\int _{\mathbb {T}^2}n_{0}(x)\) . Our results cover and optimize the conclusions regarding classical Laplacian diffusion problem.
