Question

Consider a velocity field \(\mathbf{u}(x, y, z, t)\) From conservation of mass and a spaceindependent density show that it follows that \(\operatorname{div} \mathbf{u}=0\).

Short answer

For a space-independent constant density, the divergence of the velocity field is zero: \(\nabla \cdot \mathbf{u} = 0\).

Step-by-step solution

Understand the continuity equation

The continuity equation for fluid motion expresses the conservation of mass. For a fluid with density \(\rho\), the equation is:\[\frac{\partial \rho}{\partial t} + abla \cdot (\rho \mathbf{u}) = 0\]Here, \(abla \cdot (\rho \mathbf{u})\) is the divergence which considers both density and velocity effects.

Consider constant density

We are given that the density \(\rho\) is space-independent and constant. Hence, \(\frac{\partial \rho}{\partial t} = 0\). The continuity equation simplifies to:\[abla \cdot (\rho \mathbf{u}) = 0\]

Apply constant density into the continuity equation

Substitute the constant \(\rho\) into the simplified continuity equation:\[abla \cdot (\rho \mathbf{u}) = \rho abla \cdot \mathbf{u} = 0\]

Conclude the result of divergence

Since \(\rho\) is constant and non-zero, we can divide the equation by \(\rho\):\[abla \cdot \mathbf{u} = 0\]This shows that for space-independent and constant density, the divergence of the velocity field \(\mathbf{u}\) is zero, thus proving the result.

Key concepts

Continuity Equation

The continuity equation is a fundamental principle in fluid dynamics that ensures the conservation of mass in a fluid flow. It can be expressed as:\[ \frac{\partial \rho}{\partial t} + abla \cdot (\rho \mathbf{u}) = 0 \]Here:

• \(\rho\) represents the density of the fluid.
• \(\mathbf{u}\) is the velocity vector field of the fluid.
• The term \(abla \cdot (\rho \mathbf{u})\) is the divergence operator applied to the product of density and velocity.

This equation implies that any change in fluid density within a specific volume over time should balance with the flow of mass into or out of the volume. When the fluid density is constant, this equation aids in determining the behavior of the velocity field under those conditions.

Constant Density

In many fluid flow scenarios, assuming constant density simplifies the analysis and calculations significantly. When density is constant, \[ \frac{\partial \rho}{\partial t} = 0 \]This means that the density does not change with time. Substituting this fact into the continuity equation, the equation simplifies to:\[ abla \cdot (\rho \mathbf{u}) = 0 \]With constant density \(\rho\) being non-zero, you can factor it out of the divergence term:\[ \rho abla \cdot \mathbf{u} = 0 \]This is a critical step because it allows solving for the divergence of the velocity field. If the density \(\rho\) is constant and non-zero, dividing through by \(\rho\) simplifies the result further. This leads to the conclusion that the divergence of the velocity field must itself be zero for mass conservation to hold in a constant density flow.

Divergence of Velocity Field

The divergence of a velocity field \(\mathbf{u}\) is an operation that measures the rate at which 'stuff' expands or contracts in the vicinity of a point. Mathematically, it is expressed as:\[ abla \cdot \mathbf{u} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} + \frac{\partial w}{\partial z} \]where \(u, v, \text{and } w\) are the components of the velocity field \(\mathbf{u}\) in the respective \(x, y, \text{and } z\) directions.

• If the divergence is zero, the fluid is said to be incompressible. This means the volume of a fluid element does not change as it moves through the field, adhering to the conservation of mass for constant density.
• A non-zero divergence indicates either a local expansion (positive divergence) or contraction (negative divergence) of the fluid.

Thus, the condition \(abla \cdot \mathbf{u} = 0\), achieved when density is constant and non-zero, is a crucial result as it enforces the concept of incompressibility in the context of fluid dynamics.