Possible Fluid Flow: u = x, v = -y

Explanation:

Introduction:
In fluid dynamics, a velocity field describes the motion of a fluid at each point in space. It consists of two components, the x-component (u) and the y-component (v), representing the velocity in the x and y directions, respectively. To determine if a velocity field represents a possible fluid flow, we need to examine its properties and see if they satisfy the conditions of a valid flow.

Properties of a Valid Fluid Flow:

1. Continuity Equation:
The continuity equation states that the rate of mass flow into a region must equal the rate of mass flow out of that region. Mathematically, it can be written as:

div(u) = ∂u/∂x + ∂v/∂y = 0

2. Irrotationality:
An irrotational flow implies that the fluid particles do not rotate as they move. This condition is expressed by the vorticity equation:

∂v/∂x - ∂u/∂y = 0

3. Incompressibility:
An incompressible flow means that the fluid density remains constant. In terms of the velocity field, this condition can be written as:

div(v) = ∂u/∂x + ∂v/∂y = 0

Analysis of the Given Velocity Field:

The given velocity field is u = x and v = -y. Let's analyze this field to see if it satisfies the properties of a valid fluid flow.

1. Continuity Equation:
Calculating the divergence of the velocity field:

div(u) = ∂u/∂x + ∂v/∂y = ∂(x)/∂x + ∂(-y)/∂y = 1 - 1 = 0

The divergence is zero, satisfying the continuity equation.

2. Irrotationality:
Calculating the vorticity of the velocity field:

∂v/∂x - ∂u/∂y = ∂(-y)/∂x - ∂(x)/∂y = 0 - 0 = 0

The vorticity is zero, indicating an irrotational flow.

3. Incompressibility:
Calculating the divergence of the v-component of the velocity field:

div(v) = ∂u/∂x + ∂v/∂y = ∂(x)/∂x + ∂(-y)/∂y = 1 - 1 = 0

The divergence is zero, satisfying the incompressibility condition.

Conclusion:
The given velocity field u = x, v = -y satisfies all the properties of a valid fluid flow. It satisfies the continuity equation, is irrotational, and represents an incompressible flow. Therefore, this velocity field represents a possible fluid flow.