  In this paper, we investigate the effects exerted by the interplay among
Laplacian diffusion, chemotaxis cross diffusion and the fluid dynamic mechanism
on global existence and boundedness of the solutions. The mathematical model
considered herein appears as \begin{align}\left\{ \begin{array}{l}
n_t+u\cdot\nabla n=\Delta n-\nabla\cdot( nS(n)\nabla c)-nm,\quad x\in \Omega,
t>0, \disp{ c_{ t}+u\cdot\nabla c=\Delta c-c+w},\quad x\in \Omega, t>0,
\disp{w_{t}+u\cdot\nabla w=\Delta w-nw},\quad x\in \Omega, t>0,\\ u_t+\kappa(u
\cdot \nabla)u+\nabla P=\Delta u+(n+m)\nabla \phi,\quad x\in \Omega, t>0,\\
\nabla\cdot u=0,\quad x\in \Omega, t>0,\\ \end{array}\right.\eqno(KSNF)
\end{align} in a bounded domain $\Omega\subset \mathbb{R}^2$ with a smooth
boundary, which describes the process of coral fertilization occurring in ocean
flow. Here $\kappa\in \mathbb{R}$ is a given constant, $\phi\in
W^{2,\infty}(\Omega)$and $S(n) $ is a scalar function satisfies $|S(n)|\leq
C_S(1+n)^{-\alpha}$ {for all} $n\geq 0$ with some $C_S>0$ and
$\alpha\in\mathbb{R}$. It is proved that if either $\alpha>-1,\kappa=0$ or
$\alpha\geq-\frac{1}{2},\kappa\in\mathbb{R}$ is satisfied,then for any
reasonably smooth initial data, the corresponding
Neumann-Neumann-Neumann-Dirichlet initial-boundary problem $(KSNF)$ possesses a
globally classical solution. In case of the stronger assumption
$\alpha>-1,\kappa = 0$ or $\alpha>-\frac{1}{2},\kappa \in\mathbb{R},$ we
moreover show that the corresponding initial-boundary problem admits a unique
global classical solution which is uniformly bounded on
$\Omega\times(0,\infty)$.
