Question

Bernoulli’s theorem is applicable in the case of:
A) Compressible liquid in streamlined flow
B) Compressible liquid in turbulent flow
C) Incompressible liquid in streamlined flow
D) Incompressible liquid in turbulent flow

Answer 1

Before we understand the principle of Bernoulli’s theorem to solve this question, it is necessary to understand the concepts of compressibility of fluid flow and the differences between the streamline and non-streamline flow and the criteria to define them.

Complete answer:
Firstly, let us understand the principle of compressibility in flow.
When a fluid is in flow, if the density of the fluid changes from one point to the other, the flow is said to be compressible. If the density remains constant, the flow is referred to as incompressible flow.
Secondly, let us understand the principle of streamline and turbulent of flow.
Any fluid flow is characterized by a number known as Reynold’s number. If the Reynold’s number of a fluid flow is less than 2000, the flow is said to be laminar or streamlined and beyond the number 2000, the flow is said to be turbulent.
The Reynold’s number is calculated by the formula –
$R = \dfrac{{\rho vL}}{\mu }$
where, $\rho$= density of fluid
$v$ = velocity of the fluid
L = linear dimension associated with the flow
$\mu$= dynamic viscosity
Therefore, Reynold's number is dependent on density.
Now, in a compressible flow, where the density changes with the flow, it is very difficult to associate a Reynold’s number for it.
Hence, the streamline or turbulent flow is applicable only to the incompressible fluid.
Bernoulli's principle is based on the principle of the conservation of energy. It states that the total sum of the pressure energy, kinetic energy and potential energy of the fluid flow is constant.
$P + \dfrac{1}{2}\rho {v^2} + \rho gh = C$
Now, this is only applicable to streamline flow because only in a streamline flow, we can be able to predict, calculate the fluid flow properties such as pressure, velocity at any point. This relation can only hold true for streamline fluids because their fluid flow can be predictable.
Given that Bernoulli's principle is only applicable to the streamline flow, we have inferred from the facts above that any streamline flow is necessarily incompressible.
Therefore, Bernoulli’s principle is only applicable to incompressible and streamline flows.

Hence, the correct option is Option C.

Note: The Reynold’s number is obtained from the fact that it is the ratio of the actual inertial force acting on the fluid flow to the viscous forces acting in the liquid. Thus, Reynold's number is centred on the concept of viscosity. So, it is very difficult to achieve high Reynold’s number if the fluid’s viscosity is of very high value.