Question

Express the continuity equation for steady twodimensional flow with constant properties, and explain what each term represents.

Short answer

Answer: The simplified continuity equation for a steady, two-dimensional flow with constant properties is given by: ∇ ⋅ V = ∂u/∂x + ∂v/∂y = 0 It represents the conservation of mass in the fluid dynamics, stating that the mass entering and leaving the control volume is constant, under the assumptions of steady, incompressible, and two-dimensional flow with constant properties. The terms ∂u/∂x and ∂v/∂y represent the rate of change of the velocity components in their respective directions.

Step-by-step solution

1. Understanding the Continuity Equation

The continuity equation is a mathematical representation of the conservation of mass in fluid dynamics. It states that the mass flowing into a control volume is equal to the mass flowing out of it, assuming no accumulation or loss of mass within the volume. For a steady two-dimensional flow with constant properties, this continuity equation can be expressed in a simplified form.

2. Expressing the Continuity Equation

Considering incompressible flow and in terms of velocity components, we can express the continuity equation as: ∇ ⋅ V = ∂u/∂x + ∂v/∂y = 0 Where V represents the velocity vector (u, v), u and v are the velocity components in the x and y directions, respectively, and ∇⋅ represents the divergence operator.

3. Explaining Each Term

Now, let's break down each term in the continuity equation: - V = (u, v): This is the velocity vector of the fluid, which has two components, u in the x-direction and v in the y-direction. - ∇⋅: This is the divergence operator, which is used to calculate the rate of change of a fluid property (in this case, mass) in a given direction. It acts on the velocity vector V and results in a scalar value. - ∂u/∂x: This is the partial derivative of the x-component of the velocity with respect to the x-direction. It represents the rate of change of the x-component of velocity as you move in the x-direction. - ∂v/∂y: This is the partial derivative of the y-component of the velocity with respect to the y-direction. It represents the rate of change of the y-component of velocity as you move in the y-direction. - ∇ ⋅ V = ∂u/∂x + ∂v/∂y = 0: The sum of the rates of change of the velocity components in their respective directions is equal to zero. This implies that the mass entering and leaving the control volume is constant, which holds for a steady, incompressible, and two-dimensional flow with constant properties.