Euler's equation:
Euler's equation is a fundamental equation in fluid dynamics that describes the motion of an inviscid fluid. It is derived from Newton's second law of motion and is applicable to ideal fluids. Euler's equation states that the rate of change of momentum of a fluid element is equal to the sum of the forces acting on it.

Reynolds equation:
Reynolds equation is a partial differential equation that describes the lubrication of fluid films between surfaces. It is derived from the Navier-Stokes equations by assuming that the fluid is incompressible and the pressure gradient is small compared to the viscous forces. Reynolds equation is used to analyze the flow of lubricants in various engineering applications.

Navier-Stokes equation:
The Navier-Stokes equation is a set of partial differential equations that describe the motion of viscous fluids. It is derived from the conservation laws of mass and momentum. The equation takes into account both the inertial and viscous effects on the fluid flow. The Navier-Stokes equation is widely used in fluid mechanics to study various flow phenomena, such as turbulence, laminar flow, and boundary layer behavior.

Hagen-Poiseuille equation:
The Hagen-Poiseuille equation is an empirical equation that describes the flow of a viscous fluid through a cylindrical pipe. It is derived from the Navier-Stokes equation under certain assumptions, such as steady laminar flow, incompressible fluid, and fully developed flow. The equation relates the flow rate to the pressure drop, viscosity, pipe length, and radius. The Hagen-Poiseuille equation is commonly used in fluid dynamics to analyze flow in pipes and tubes.

Explanation:
The correct answer to the given question is option 'c', Navier-Stokes equation. The Navier-Stokes equation is the most general and comprehensive equation that describes the motion of viscous fluids. It takes into account both the inertial and viscous effects on the fluid flow. The equation is derived from the conservation laws of mass and momentum and is widely used in fluid mechanics to analyze various flow phenomena.

Euler's equation is not applicable to viscous fluids as it assumes the fluid to be inviscid. Reynolds equation is specific to the lubrication of fluid films between surfaces and does not describe the motion of the fluid as a whole. The Hagen-Poiseuille equation is specific to the flow of a viscous fluid through a cylindrical pipe and does not consider the effects of inertia and other flow phenomena.

Therefore, the Navier-Stokes equation is the most appropriate equation for describing the motion of a viscous fluid as it takes into account all the relevant factors and provides a comprehensive description of the fluid flow.