Euler's equation for motion
Key FactsGyroscopic Couple: The rate of change of angular momentum () = (In the limit).
- = Moment of Inertia.
- = Angular velocity
- = Angular velocity of precession.
Blaise Pascal (1623-1662) was a French mathematician, physicist, inventor, writer and Catholic philosopher.
Leonhard Euler (1707-1783) was a pioneering Swiss mathematician and physicist.
OverviewThe Euler's equation for steady flow of an ideal fluid along a streamline is a relation between the velocity, pressure and density of a moving fluid. It is based on the Newton's Second Law of Motion. The integration of the equation gives Bernoulli's equation in the form of energy per unit weight of the following fluid. It is based on the following assumptions:
- The fluid is non-viscous (i,e., the frictional losses are zero).
- The fluid is homogeneous and incompressible (i.e., mass density of the fluid is constant).
- The flow is continuous, steady and along the streamline.
- The velocity of the flow is uniform over the section.
- No energy or force (except gravity and pressure forces) is involved in the flow.
Derivation Of EquationLet us consider a steady flow of an ideal fluid along a streamline and small element AB of the flowing fluid as shown in figure. Let,
- dA = Cross-sectional area of the fluid element
- ds = Length of the fluid element
- dW = Weight of the fluid element
- P = Pressure on the element at A
- P+dP = Pressure on the element at B
- v = velocity of the fluid element
From the geometry of the figure, we find that the component of the weight of the fluid element in the direction of flow,
Dividing both sides by
or in other words,
which proves the Bernoulli's equation.