Euler’s degree of vortex comes under the dynamics of fluid. In this we study the velocity and acceleration at a point in a fluid flow, taking into consideration the forces causing the flow.
According to the Newton’s second law of motion, the net force Fx acting on a fluid element in the direction of x is equal to the mass of the fluid element multiplied by the acceleration ax in the x-direction.
Fx =m × ax.
Dynamics of fluid flow is the study of fluid motion with the forces causing the flow. The following forces are present in the fluid in motion.
- F g – Gravity Forces.
- F p – Pressure Forces.
- F v – Forces due to Viscosity.
- F t – Forces due to Turbulence.
- F c –Forces due to Compressibility.
The net forces acting on the fluid due to motion are:
F x = (F g) + (F p) + (F v) + (F t) + (F c).
1. From the above equation, if the (F c) is forces due to compressibility and is negligible the resulting net force is:
F x = (F g) + (F p) + (F v) + (F t).
And the equation is called the Reynolds’s Equation of Motion.
2. For the flow where (F t) forces due to turbulence is negligible the resulting equation of the motion are known as the Navier- Strokes Equation.
F x = (F g) + (F p) + (F v).
3. If the flow is assumed to be ideal viscous force (F v) is zero, the equation of motion is known as Euler’s equation of Motion.
F x = (F g) + (F p).
In the Euler’s equation of motion the forces due to the gravity and pressure are taken into consideration. This derived by considering the motion of a fluid element along a stream line flow.
Pressure force =(P + (∂p/∂s × ds)) × dA.
Pressure force in flow direction = P × dA.
Let Θ be the angle of flow .
The resultant flow on the fluid element in the direction of s must be equated to the mass of the fluid x acceleration in the direction of s.
CosΘ = dz/ds.
(∂p / p∂s)+(g × cosΘ) + (V× dv /∂s) =0
thus the Euler Equation of Motion :
(∂p/p) + (g× dz) + (V × dv ) = 0