Question

Explain the limitations of the Bernoulli theorem.

Answer 1

Hint: Bernoulli’s equation is written as:
 $ \dfrac{{{P_1}}}{\rho } + g{h_1} + \dfrac{1}{2}v_1^2 = \dfrac{{{P_2}}}{\rho } + g{h_2} + \dfrac{1}{2}v_2^2 $ .

Complete Answer:
We will consider an incompressible and non-viscous liquid flowing through a pipe with a cross-sectional area that varies.
These are the following limitations of the Bernoulli theorem:
-The above equation was derived by assuming that every element of the liquid has a uniform velocity across any cross-section of the pipe. It is, practically, not true. The elements of the liquid have the maximum velocity in the innermost layer. The liquid velocity decreases towards the pipe walls. We should therefore take the mean velocity of the liquid into account.
-The Bernoulli equation is derived from the assumption that when a liquid is in motion, there is no loss of energy. Some kinetic energy is converted into heat energy and, due to shear force, part of it is lost.
-The viscous drag of the liquid was not taken into account during the derivation of Bernoulli's equation. When a liquid is in motion, that viscous drag comes into play.
The energy due to the centrifugal force should also be taken into account if the liquid flows along a curved path.

Note:
The Bernoulli theorem is also known as Bernoulli's equation. In an optimal state, it can be applied to fluids. We already know that pressure and density are inversely proportional to each other, so a slow-speed fluid will exert more pressure than a slow-speed fluid that moves faster. In this case, fluid not only refers to liquids, but also to gases. The principle of Bernoulli forms the foundation of many applications in our daily lives.