The Navier-Stokes equations were derived by Navier, Poisson, Saint-Venant, and Stokes between 1827 and 1845. These equations are always solved together with the continuity equation: The Navier-Stokes equations represent the conservation of momentum, while the continuity equation represents the conservation of mass.

Navier-Stokes equation represents the conservation of mass.

The Navier-Stokes equation is a fundamental equation in fluid mechanics that describes the motion of fluid substances, such as gases and liquids. It is based on the principle of conservation of mass and momentum.

Conservation of Mass:
The conservation of mass principle states that mass cannot be created or destroyed, only transferred or transformed. In the context of fluid mechanics, this means that the total mass of a fluid within a given volume remains constant over time, regardless of any changes in velocity or pressure.

The Navier-Stokes equation incorporates this principle by including a term that represents the conservation of mass. This term is known as the continuity equation, which states that the rate of change of mass within a control volume is equal to the net mass flow rate into or out of the control volume.

Continuity Equation:
The continuity equation is a mathematical expression of the conservation of mass principle. It can be written as:

∂ρ/∂t + ∇ · (ρv) = 0

Where:
- ∂ρ/∂t represents the rate of change of mass density with respect to time.
- ∇ · (ρv) represents the divergence of the mass flux vector ρv, which is the product of mass density ρ and velocity vector v.

This equation states that the change in mass density with time is equal to the negative divergence of the mass flux vector. In simpler terms, it means that any increase or decrease in mass density within a control volume must be balanced by a corresponding flow of mass into or out of the control volume.

Conclusion:
The Navier-Stokes equation represents the conservation of mass through the inclusion of the continuity equation, which ensures that the total mass of a fluid remains constant over time within a given control volume. This equation is fundamental in fluid mechanics and is used to study and analyze various fluid flow phenomena.

According to me option d is correct.