Final answer:
The field V = 2x i + y j is not an incompressible flow field because its divergence is 3, which does not satisfy the condition of zero divergence necessary for an incompressible flow.
Step-by-step explanation:
To determine if the field V = 2x i + y j is an incompressible two-dimensional flow field, we can employ the concept of divergence. An incompressible flow field should have a divergence of zero. Divergence in two dimensions is calculated as the partial derivative of the x-component of the field with respect to x plus the partial derivative of the y-component with respect to y:
∇ · V = ∂(2x)/∂x + ∂(y)/∂y = 2 + 1
Thus, the divergence is 3, which is not equal to zero. Therefore, the field V = 2x i + y j is not an incompressible flow field as the assumption of incompressibility requires the divergence to be zero, according to the continuity equation.
In an incompressible fluid, the density remains constant, therefore the continuity equation simplifies, and the density term cancels out. This principle is supported by the ideas presented in the continuity equation, where the fluid flow must be consistent at various points in a pipe, assuming the fluid is incompressible.