Question

Consider a three-dimensional incompressible steady flow. Two of the three velocity components for an incompressible flow are:

u = x²+2x z , v = y² +2y z
What is the general form of the velocity component w(x,y,z) that satisfies the continuity equation?

Answer 1

To find the general form of the third velocity component w(x,y,z) for an incompressible steady flow with given u and v components, we apply the continuity equation, which requires zero divergence, and then solve for w.

Step-by-step explanation:

The question asks for the general form of the third velocity component, w(x,y,z), for a three-dimensional incompressible steady flow where the other two velocity components are given by u = x²+2x z and v = y² +2y z. The continuity equation for an incompressible flow states that the divergence of the velocity field must be zero. In other words, ∇ · ο{V} = 0, where ο{V} is the velocity vector field. Applying the divergence operator to the given components, we obtain the equation: ∂u/∂x + ∂v/∂y + ∂w/∂z = 0. Substituting the given expressions for u and v, taking the derivatives with respect to x and y, and rearranging the terms, we can find the partial derivative of w with respect to z, leading us to the general form of w(x, y, z) that satisfies the continuity equation.