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u_{t}&+&\\nabla P&=\\Delta u+n\\nabla\\phi,\\ &x\\in\\Omega,& t>0, &&\\nabla\\cdot u&=0,\\ &x\\in\\Omega,& t>0, \\end{array}\\right. \\end{align*} under homogeneous Neumann boundary conditions for $n$ and $c$ and homogeneous Dirichlet boundary conditions for $u$, where $\\Omega\\subset\\mathbb{R}^2$ is a bounded domain with smooth boundary and 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