Physics · Mechanical Properties Of Fluids · NEET
When fluid moves from a wide low part of a pipe to a narrow high part, the pressure behind it pushes it forward (does work), gravity changes its potential energy, and its speed changes its kinetic energy. The work-energy theorem says net work equals change in kinetic energy. Writing this out for a small fluid mass and rearranging gives Bernoulli's equation. So it is not a new law, it is energy conservation applied to a flowing fluid.
Every term is an energy per unit volume (its unit is pascal, same as pressure). P is the pressure energy per unit volume, coming from the push of the fluid behind. (1/2) rho v^2 is the kinetic energy per unit volume, from the fluid's motion. rho g h is the potential energy per unit volume, from its height. Their sum is constant along one streamline.
The common form P + (1/2) rho v^2 + rho g h = constant is per unit VOLUME, because each term has units of pascal (J/m^3). If you divide every term by rho, you get P/rho + (1/2) v^2 + g h = constant, which is per unit MASS (units J/kg). Both are correct; NEET usually uses the per-unit-volume form.
You start by writing the work done by pressure and the changes in KE and PE for the same small mass m of fluid. Because m appears in every term (using m = rho V and dividing through), it cancels. This is why the final equation only contains density rho, not mass, and applies to any amount of fluid on that streamline.
Four main assumptions: (1) the fluid is incompressible, so density rho is constant; (2) the flow is steady and streamline (non-turbulent); (3) the fluid is non-viscous, so no energy is lost to internal friction; and (4) it applies along a single streamline. Real fluids lose some energy to friction, so Bernoulli is an ideal-fluid result.
The continuity equation A v = constant tells you the speed at each point (narrow pipe means higher speed). Bernoulli then tells you the pressure at that point. You almost always use them together: continuity gives v, Bernoulli gives P. In NEET pipe problems, first apply A1 v1 = A2 v2, then plug the speeds into Bernoulli.
The venturi-meter works on:
Water flows in streamline motion through a horizontal pipe. The pressure difference of water between P and Q is 15 N/m^2. The areas of cross-section at P and Q are 40 cm^2 and 20 cm^2. The rate of flow of water (in cm^3/s) is: [density of water = 1000 kg/m^3]
Try the real previous-year questions from this chapter — each with the answer and a full solution.
For a steady, streamline flow of an ideal incompressible fluid, P + (1/2) rho v^2 + rho g h = constant along a streamline, where P is pressure, rho is density, v is speed and h is height.
No. The derivation assumes a non-viscous fluid so that no energy is lost to internal friction. For real viscous fluids there is some energy loss, so Bernoulli's equation is only approximate.
The standard form assumes an incompressible fluid with constant density. It works well for liquids and for gases only at low speeds where density change is small.
When height is the same at both points, rho g h cancels, so P + (1/2) rho v^2 = constant. This shows that where the fluid moves faster the pressure is lower.
It means the constant on the right side is the same only for points lying on one continuous flow line. Two different streamlines may have different constant values.
