I was going over the derivation of the dimensionless Navier-Stokes equation, which is explained in this answer. By introducing dimensionless variables in the NS equation, one gets
$$ \frac{D\mathbf u}{Dt} = - \nabla P + \frac 1 {\text{Re}} \nabla^2 \mathbf u $$
where $\text{Re} \equiv \rho UL/\eta$ is the Reynolds number. For $\text{Re} \gg 1$ (high Reynolds number), the viscosity (the one with $\nabla^2$) term is negligible and thus
$$ \frac{D\mathbf u}{Dt} = - \nabla P \ \ \ (\text{Re} \gg 1) $$
For $\text{Re} \ll 1$, instead, the viscosity term dominates and we get
$$ \nabla P = \frac 1 {\text{Re}} \nabla^2 \mathbf u \ \ \ (\text{Re} \ll 1) $$
My question is: In all derivations I've seen, people keep the pressure gradient term $\nabla P$ for $\text{Re} \ll 1$. Why can't we also get rid of this term?