Symbols:
t := time
r := density (from rho)
v := velocity vector
a := dv/dt = acceleration
p := pressure
f := force (vector field)
 := viscosity
-2/3 = Lame's first parameter (to avoid another symbol)

x, y, z := coordinate directions (linear independent)
d := delta, the infinitesimal operator (d/dt is the partial (not the total) derivative with respect to the given variable)
D := del operator = (d/dx, d/dy, d/dz) (a vector)
L := D*D = laplace perator = d/dx + d/dy + d/dz

vx, fx etc. means the component of the vector in x direction. Same for y and z.
X means the cross product

Navier-Stokes-Equasion
r(dv/dt + (v*D)v) = -Dp + *Lv + (-2/3 + )*D(D*v) + f
<=>
r(dv/dt + (v*D)v) = -Dp + *Lv + 1/3*D(D*v) + f

Simplifications (well-grounded):
Assuming D*v = 0 for water is (quite) incompressible
r(dv/dt + (v*D)v) = -Dp + *Lv + f
Assuming r is constant and 1 for water
dv/dt + (v*D)v = -Dp + *Lv + f
Assuming  is constant and 1 for water (even if it only is constant and not 1 it's ok because we only need proportionality, not equality)
dv/dt + (v*D)v = -Dp + Lv + f
Switching to components (not entirely sure):
dvx/dt + (vx*d/dx + vy*d/dy + vz*d/dz)vx = -dp/dx + (d/dx + d/dy + d/dz)vx + fx
dvy/dt + (vx*d/dx + vy*d/dy + vz*d/dz)vy = -dp/dy + (d/dx + d/dy + d/dz)vy + fy
dvz/dt + (vx*d/dx + vy*d/dy + vz*d/dz)vz = -dp/dz + (d/dx + d/dy + d/dz)vz + fz
Assuming fx = fy = 0 for gravity only directs in z direction
(Assuming fz = cons. for gravity lowers with the distance to the center of mass but the distance to the center of mass of earth >> z difference of the surface of earth)
dvx/dt + (vx*d/dx + vy*d/dy + vz*d/dz)vx = -dp/dx + (d/dx + d/dy + d/dz)vx
dvy/dt + (vx*d/dx + vy*d/dy + vz*d/dz)vy = -dp/dy + (d/dx + d/dy + d/dz)vy
dvz/dt + (vx*d/dx + vy*d/dy + vz*d/dz)vz = -dp/dz + (d/dx + d/dy + d/dz)vz + fz

Simplifications (experimantal):
Assuming (v*D)v = 1/2*D(v) for vX(DXv) = 0 (mainly) if wirls only exist in the x-y plane

Assuming Lv = 0 (Euler Equasion???)


---- Approach: Conservation of energy ----
m/2*v + p + rgz = const.
Assuming g = 1, r = 1
v/2 + p + z = const.

Assuming p = const.
D(v) proportional to -Dz

Assuming dz const.
v/2 + p = const.
Intergation through space
Dv proportional to -Dp

Assuming both (but calculating each tile seperately so z does matter in the end) leads to
const.*Dv + const.*Dp = 0[break]


---- Energetic approach ----
F = m*g
a = F/m = g (acaled with dh)