**The Bernoulli Principle** was formulated by Bernoulli, who obtained a relation between the pressure and velocity at different parts of a moving incompressible fluid. If the viscosity is negligibly small, there are no frictional forces to overcome. Hence the work done by the pressure difference per unit volume of a fluid flowing along a pipe steadily is equal to the gain in kinetic energy per unit volume plus the gain in potential energy per unit volume according to The Bernoulli Principle. Engineering in Kenya has more information.

# Mathematical Definition of The Bernoulli Principle

The work done by pressure in moving a fluid through a distance = force × distance moved = pressure × area × distance moved = pressure × volume. At the beginning of the pipe where the pressure is P_{1}, the work done per unit volume on the fluid according to The Bernoulli Principle is P_{1}. At the other end, if the pressure is P_{2}, then the work done per unit volume according to The Bernoulli Principle is P_{2}.

Hence the net work done per unit volume is the difference of P_{2} and P_{1}. The kinetic energy per unit volume = ½mass per unit × velocity^{2} = ½ρ × velocity^{2} where ρ is the density of the fluid. Thus if v1 and v2 are the final and initial velocities at the end and the beginning of the pipe respectively, then the kinetic energy gained per unit volume = ½ρ(v_{2}^{2} – v_{1}^{2}).

Further, if h_{2} and h_{1} are the respective heights measured from a fixed level at the end and the beginning of the pipe respectively, the potential energy gained per unit volume according to The Bernoulli Principle is = mass per unit volume × g × (h_{2} – h_{1}) = ρg(h_{2} – h_{1}). Thus from the conservation of energy.

P_{1} – P_{2} = ½ρ (v_{2}^{2} – v_{1}^{2}) + ρg (h_{2} – h_{1}).

P_{1} + ½ρv_{1}^{2} + h_{1}ρg = P_{2 }+ ½ρv_{2}^{2} + h_{2}ρg.

Therefore;

P + ½ρv^{2} + hρg = constant where P is the pressure at any part and v is the velocity at that part. Hence for streamline motion of an incompressible non-viscous fluid, The Bernoulli Principle is proven and it states that; the sum of the pressure at any part plus the kinetic energy per unit volume plus the potential energy per unit volume is always a constant. The Bernoulli Principle shows that at points in a moving fluid where the potential energy change hρg is very small, or zero as in flows through a horizontal pipe, the pressure is low where the velocity is high; conversely, the pressure is high where the velocity is low according to The Bernoulli’s Principle.