Euler's
equation is the equivalent of Newton's second law for fluids, including
the effect of neighboring fluids and the force of gravity. "Euler"
is pronounced "oiler." In it, the vector v is the velocity field
a a point in space. v is not the velocity of a particular parcel of fluid,
except when it is at that point in space.
The left side of Euler's equation is the full derivative of the velocity, comprised of the "convective derivative" (the spatial component) and the partial derivative of the velocity field with respect to time. The right side includes the force on a unit volume due to the pressure gradient, grad P, and the force on a unit volume due to the gravitational field, g. The force of neighboring fluid points towards lower pressure; this is the origin of the minus sign on the pressure gradient term.
The continuity equation relates the partial derivative of the density with respect to time to the gradient of the momentum density. It amounts to conservation of mass, saying (in integrated form) that mass cannot enter or leave a volume in space without flowing across the boundaries of that volume.
The viscous force is a particular case: A plate of area A is pulled at velocity v (in the x direction) across the surface of a fluid. The large area, A, is in contact with the fluid. y is perpendicular to x. The Greek letter, eta, represents the viscosity of the fluid. In the case that the fluid is supported by a second, larger plate a distance Y below the surface, the derivative may be replaced with the ratio v/Y. As with friction for solid objects, the viscous force acts to oppose the motion in the x direction,
Poiseuille's law is an application of fluid dynamics to flow in a pipe of radius R. The volume flow rate may be measured in cubic meters per second. For simple flows through a pipe of length L, the derivative may be replaced by the ratio of the pressure difference from one end of the pipe to the other - the "head" - divided by the length L.