To demonstrate the three scenarios involving the density field $\rho$, let's analyze them step by step, considering the respective definitions:

1. Uniform and steady density implies incompressibility:

   • Given:
     • Uniform density: $\nabla \rho = 0$, i.e., $\rho$ does not vary spatially.
     • Steady density: $\frac{\partial \rho}{\partial t} = 0$, i.e., $\rho$ does not vary with time.
   • Compressibility condition:
     • The rate of change of density is related to the divergence of velocity: $\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + (\mathbf{v} \cdot \nabla) \rho = 0.$
   • Substituting the given conditions:
     • $\frac{\partial \rho}{\partial t} = 0$ (steady state),
     • $\nabla \rho = 0$ (uniform density).
   • Therefore: $\frac{D\rho}{Dt} = 0 \implies \nabla \cdot \mathbf{v} = 0.$
   • The flow is incompressible.

2. Incompressible and steady density implies uniform density:

   • Given:
     • Incompressibility: $\nabla \cdot \mathbf{v} = 0$, meaning the flow conserves mass.
     • Steady density: $\frac{\partial \rho}{\partial t} = 0$, meaning $\rho$ does not vary with time.
   • Compressibility condition: $\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + (\mathbf{v} \cdot \nabla) \rho.$
   • Substituting $\frac{\partial \rho}{\partial t} = 0$: $0 = (\mathbf{v} \cdot \nabla) \rho.$
   • This implies $\nabla \rho = 0$, since $\mathbf{v} \neq 0$ in general.
   • Hence, $\rho$ is uniform.

3. Incompressible and uniform density implies steady density:

   • Given:
     • Incompressibility: $\nabla \cdot \mathbf{v} = 0$, meaning the flow conserves mass.
     • Uniform density: $\nabla \rho = 0$, meaning $\rho$ does not vary spatially.
   • Compressibility condition: $\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t} + (\mathbf{v} \cdot \nabla) \rho.$
   • Substituting $\nabla \rho = 0$: $\frac{D\rho}{Dt} = \frac{\partial \rho}{\partial t}.$
   • Since $\nabla \cdot \mathbf{v} = 0$, $\frac{\partial \rho}{\partial t} = 0$.
   • Thus, $\rho$ is steady.

Summary:

• (i) Uniform and steady density $\implies$ incompressible.
• (ii) Incompressible and steady $\implies$ uniform.
• (iii) Incompressible and uniform $\implies$ steady.

Let me know if you would like further clarification!

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Related Questions:

1. What is the mathematical definition of incompressible flow?
2. How does the material derivative relate to compressibility in fluid dynamics?
3. Can a flow be incompressible but unsteady? Provide examples.
4. How does uniform density affect the Navier-Stokes equations?
5. What are the physical implications of steady density in real-world scenarios?

Tip:

Always analyze the relationship between spatial, temporal, and velocity-related derivatives when exploring incompressibility, uniformity, or steadiness in density fields.