I suppose this is a basic fluid mechanics problem but I have one thing I do not understand.
I am about to solve a problem with steady, incompressible, parallel, laminar flow of viscous fluid falling between two infinite walls, inclined at an angle $\theta$. There is no applied pressure driving the flow, it falls by gravity alone. And I want to calculate the velocity field.
In order to solve it I use Navier-Stokes equation for incompressible fluids:
$$\rho\left[\frac{\partial v}{\partial t}+\left(v\nabla \right) v\right]=-\nabla p + \eta\Delta v$$
and with the "keywords" describing the problem it can be simplified to
$$\nabla p=\eta \Delta v$$
The next step is to divide the equation in the $\hat{x}$ and $\hat{y}$ direction . When I do it I just got
$$\frac{\partial p}{\partial x}=\eta \frac{\partial^2v }{\partial y^2} $$
in $\hat{x}$ direction (parallel to the flow).
But there should be a term with the gravitational force component, which reads $\rho g \sin \theta$ here as well I think. Where does it come from? Someone who can help me?