Dynamics of Fluid Flow: Energy Equation for Ideal Fluid Flow

For the complete study of fluid flow we have to also perform dynamic analysis along with kinematic analysis of the fluid flow. In the kinematic analysis we studied the position, velocity, and acceleration descriptions of the fluid particles in fluid flow. This we have done without taking into account the forces causing them. Now we will study these causal forces under the dynamics of fluid flow.

The motion of fluid particles, their position, velocity, and acceleration, is governed by the resultant of the forces acting on them. Typical forces acting on fluid particles in any flow are pressure forces, viscous forces, shear forces, fictional forces at boundaries, cohesive forces among fluid particles, and adhesive forces. Performing the dynamic analysis of fluid flow taking all these forces into account will become a bit complex, so we will start the analysis with a simplified approach.

To simplify the problem we will take such fluid for analysis which has negligible frictional resistance and compressibility effects. A fluid possessing such properties is called an Ideal or a Perfect fluid. The properties of water are in close agreement with those of an Ideal fluid. In civil engineering hydraulics we mainly deal with water, thus, the dynamic analysis for ideal fluids will provide a base for solving civil engineering hydraulics problems.

For the dynamic analysis of fluid flow in the simplified form we will first derive the energy equation for ideal flow by applying Newton’s second law of motion to an elemental streamtube. If we neglect frictional forces then the pressure and gravitational forces remain to act on the fluid governing its motion.

By applying Newton's second law of motion to the elemental streamtube along the streamline, in the form Force = Mass x Acceleration, we obtain a differential equation known as the Euler equation of motion for an ideal fluid flow, which on integration along the streamline gives an equation popularly known as Bernoulli's Equation.

z₁ + p₁/ρg + v₁²/2g = z₂ + p₂/ρg + v₂²/2g

All the terms on either sides of the equation have the dimensions of length. Both sides have similar terms. Summation on each side can be interpreted as the total energy of any fluid element of unit weight. That is why this equation is called the energy equation for ideal fluid flow. From this equation, we can say that for an ideal flow along a streamline, total energy remains constant.

This is the energy equation for an Ideal Fluid Flow. What do its different terms represent? And what are its physical interpretations? All this will be taken up in the next article followed by an article on the derivation of the energy equation for real fluid flow.