Since the density remains constant throughout, the sampled mass also remains equivalent as long as the volumes also stay equivalent – the volumes used in the calculations above were differential/elemental i.e., dV, and hence equal. That being said, this affects the mass of the fluid considered for the Energy calculations. You may say that a liquid is much more incompressible than gas but not literally incompressible. Therefore, no fluid is perfectly incompressible. The change in density – even slightly if not so considerably – is quite commonplace in reality. How this affected our equation has been explained below. This property allowed us to make the density invariable in relation to the pressure at different points of the pipe. One quite significant assumption is the incompressibility of the fluid. In deriving this expression, we did make a few assumptions in order to crop out the dynamicity of the problem. This is in fact the Bernoulli’s Equation. The GPE is denoted by dU.ĭU = m 1gh 1 – m 2gh 2 = ρdVg (h 2 – h 1).īy equating the two sides of the equation (dW = dK + dU), we obtain the subsequent expression: Once again, we may simplify it further by expressing the masses alternatively i.e., Mass is Density times Volume. The associated Gravitational Potential Energy is rather straightforward. Since we would like to work with fewer terms, we may use the above to express masses of the fluid at points A and B alternatively. We shall use the standard expression for Kinetic Energy associated with a mass in motion i.e., dK = 0.5m 2v 2 2 – 0.5m 1v 1 2Ĭall to mind that we have been given the fluid density (ρ) and the differential volumes of fluid (dV). Meanwhile, the expression for the Kinetic Energy of the Fluid can be written using the masses and velocities of the fluid at A and B. Let’s say that dV is the differential volume. Note that A 1dx 1 = A 2dx 2 = differential volume (elemental/small volume), since any area multiplied by an adjacent length gives a volume. The expression for dW then can be modified.

Now, recall that Force is the Pressure experienced per unit Area. Since the fluid is displaced by dx 1 and dx 2, we rewrite the expression in terms of the forces experienced at points A and B.ĭW = F 1.dx 1 – F 2.dx 2 = P 1.A 1.dx 1 – P 2.A 2.dx 2į 1 and F 2 are the forces experienced by the fluid at points A and B, respectively.

The work done by the fluid when moving from Point A to Point B as dW = dW 1 – dW 2. We shall work out our derivation along with the following expression:ĭW = d(Kinetic Energy of Fluid) + d(Gravitational Potential of Fluid) If the fluid moves by dx 1 from point A and by dx 2 from point B, it shall do so by attaining a velocity of v 1 and v 2, respectively. Now, if some work is to be done, the fluid would have to move a certain distance, implying an associated velocity as well.

Also, observe that the cross-sectional area at Point A is A 1 and at Point B is A 2. It’s apparent that the gravitational potential energy at h 2 is greater. We assume that the fluid at points A and B is elevated at heights of h 1 and h 2, respectively. Role of Mahatma Gandhi in Freedom Struggle.

Definition of bernoulli principle