Question

Using the cartesian coordinates, write down and discuss the meaning of each term and show all the differences you know between the Navier-Stoke's equations and the Euler's equations. Can we use Euler's equations to solve the flow in long pipes? Why? What is the relation between Euler's equations and Bernolli's equation? (20) Starting from the Navier-Stokes equations, drive the well known Bernoulli's equation. State all the assumptions made.

Answer 1

The Navier-Stokes equations include viscosity, while the Euler's equations do not. Euler's equations can be used to model flow in long pipes due to the assumption of inviscid flow. Bernoulli's equation is derived from the Navier-Stokes equations and describes the conservation of energy in fluid flow.

Step-by-step explanation:

The Navier-Stokes equations and the Euler's equations are mathematical equations that describe fluid flow. The Navier-Stokes equations take into account both the fluid's viscosity and the fluid's density, while the Euler's equations do not consider viscosity and assume that the fluid is inviscid. Therefore, the main difference between the two sets of equations is the inclusion of viscosity.

Euler's equations are often used to model flow in long pipes because the flow in pipes is usually smooth and laminar, which means that there is little to no turbulence. Therefore, the assumption of inviscid flow made in Euler's equations is reasonable for this situation.

Bernoulli's equation is derived from the equations of fluid flow, including the Navier-Stokes equations. It describes the conservation of energy for a fluid flowing along a streamline. Bernoulli's equation relates the pressure, velocity, and height of the fluid and is based on several assumptions, including the assumption of inviscid and incompressible flow.