Question

Which of the following represents the equation of continuity for a steady compressible fluid? \( \vec{V} \) represents the velocity vector; \( \rho \) represents density

• A. \( \nabla \cdot (\vec{V}) = 0 \)
• B. \( \nabla \times (\vec{V}) = 0 \)
• C. \( \nabla \times (\rho \vec{V}) = 0 \)
• D. \( \nabla \cdot (\rho \vec{V}) = 0 \)

Hint:

Remember that compressibility is a key factor. For incompressible fluids (\( \rho = \text{constant} \)), the density term can be taken out of the divergence operator, leading to \( \rho (\nabla \cdot \vec{V}) = 0 \), or \( \nabla \cdot \vec{V} = 0 \). For compressible fluids, density is a variable and must remain inside the divergence operator.

Answer 1

The Correct Option is D

Step 1: Understand the principle of conservation of mass.
The equation of continuity is a mathematical expression of the principle of conservation of mass in fluid dynamics. It states that mass can neither be created nor destroyed within a control volume. Any change in the mass within the control volume must be due to the net flow of mass into or out of the volume. Step 2: Recall the general form of the equation of continuity.
The general form of the equation of continuity for a fluid flow is given by:
$$\frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \vec{V}) = 0$$ where:
\( \rho \) is the density of the fluid
\( t \) is time
\( \vec{V} \) is the velocity vector of the fluid
\( \nabla \cdot \) is the divergence operator
Step 3: Apply the condition for steady flow.
A steady flow is defined as a flow in which the fluid properties at any point in the flow field do not change with time. Mathematically, this means that the partial derivative of any fluid property with respect to time is zero. In this case, for a steady flow, \( \frac{\partial \rho}{\partial t} = 0 \). Step 4: Simplify the equation of continuity for a steady flow.
Substituting \( \frac{\partial \rho}{\partial t} = 0 \) into the general equation of continuity, we get:
$$0 + \nabla \cdot (\rho \vec{V}) = 0$$
$$\nabla \cdot (\rho \vec{V}) = 0$$
This equation represents the conservation of mass for a steady compressible fluid. The term \( \rho \vec{V} \) represents the mass flux vector. The divergence of the mass flux vector being zero implies that there are no sources or sinks of mass within the flow field. Step 5: Analyze the other options.
Option (1), \( \nabla \cdot (\vec{V}) = 0 \), represents the equation of continuity for a steady incompressible fluid (where density \( \rho \) is constant).
Option (2), \( \nabla \times (\vec{V}) = 0 \), represents an irrotational flow (where the curl of the velocity vector is zero), not the equation of continuity.
Option (3), \( \nabla \times (\rho \vec{V}) = 0 \), does not generally represent the equation of continuity. It would imply that the mass flux vector is irrotational.
Therefore, the correct equation of continuity for a steady compressible fluid is \( \nabla \cdot (\rho \vec{V}) = 0 \).