The continuity equation, ρ₁A₁V₁ = ρ₂A₂V₂ is based on ______assumption.(where ρ1, ρ2 are densities, A₁, A₂ areas and V₁, V₂ are velocities)

Explanation:

Continuity Equation:

The continuity equation is based on the principle of conservation of mass. It states as follows: "If no fluid is added or removed from the pipe in any length then the mass passing across different sections shall be same."

From the law of conservation of mass;

ρ₁A₁V₁ = ρ2A2V2

Where ρ₁ and ρ₂ are densities at sections 1 and 2 respectively, A1, and A2 cross-section areas conduit/pipe at sections 1 and 2 respectively, and V1, and V2 are flow velocities at sections 1 and 2 respectively.

The above equation is valid for both compressible flow as well as incompressible flow of fluids.

Steady flow:

The type of flow in which the fluid characteristics like velocity, pressure, density, etc. at a point do not change with time is called steady flow.

\(({∂ u \over ∂ t})_{x_o,y_o,z_o}=0\,, ({∂ v \over ∂ t})_{x_o,y_o,z_o}=0\,, ({∂ w \over ∂ t})_{x_o,y_o,z_o}=0\)

\(({∂ p \over ∂ t})_{x_o,y_o,z_o}=0\,, ({∂ ρ \over ∂ t})_{x_o,y_o,z_o}=0\,\)

Where (x₀, y₀, z₀) is a fixed point in a fluid field where these variables are being measured w.r.t time.

Uniform flow:

The type of flow, in which the velocity at a given time does not change with respect to space is called uniform flow.

\(({∂ V \over ∂ t})_{t=constant}=0\,\)

∂V = change in velocity, and ∂s = displacement in any direction.

Incompressible flow:

It is that type of flow in which density is constant for the fluid flow.

ρ = constant

Hence option (1) is the correct answer.