Within the topic of fluid kinematics, an Eulerian approach can be characterized by a velocity field (u,v,w) in the tridimensional (x,y,z) domain. Which of the following velocity components satisfies the continuity equation if the other two components are u=2y2 and w=2xyz. Assume incompressible fluid. Note: x,y are the horizontal coordinates, z the vertical coordinates and u,v,w their respective (x,y,z) velocity components.

Question

BlackTom AI · Accepted Answer

∂u/∂x + ∂v/∂y + ∂w/∂z = 0 for an incompressible fluid in Cartesian coordinates, so we use this identity to find v.

1) Relevant knowledge points and concepts
- Incompressible continuity equation (Eulerian form): ∂u/∂x + ∂v/∂y + ∂w/∂z = 0.
- Partial differentiation and integration with respect to one coordinate: when solving for v(y) from ∂v/∂y, the integration constant may be an arbitrary function of the other coordinates (here x and z).

2) Complete reasoning process step by step
- Given u = 2 y^2 and w = 2 x y z.
- Compute the required partial derivatives:
  ∂u/∂x = 0 because u does not depend on x.
  ∂w/∂z = ∂(2 x y z)/∂z = 2 x y.
- Substitute into continuity equation:
  0 + ∂v/∂y + 2 x y = 0.

3) Calculations with intermediate steps
- From 0 + ∂v/∂y + 2 x y = 0 we get
  ∂v/∂y = −2 x y.
- Integrate with respect to y:
  v(x,y,z) = ∫(−2 x y) dy = −2 x * (y^2/2) + C(x,z) = −x y^2 + C(x,z),
  where C(x,z) is an arbitrary function independent of y.

4) Why the correct answer is correct
- The particular form v = −x y^2 corresponds to choosing C(x,z) = 0; it yields ∂v/∂y = −2 x y, which exactly cancels ∂w/∂z = 2 x y so the continuity equation is satisfied everywhere. Thus v = −x y^2 is a valid velocity component that enforces incompressibility given the provided u and w.

5) Why the other multiple-choice options are incorrect
- v = 0: gives ∂v/∂y = 0, so continuity becomes 0 + 0 + 2 x y = 2 x y ≠ 0 (unless x or y identically zero), so not generally valid.
- v = x y^2: gives ∂v/∂y = 2 x y, so continuity becomes 0 + 2 x y + 2 x y = 4 x y ≠ 0.
- v = −x y^2 z (if offered): gives ∂v/∂y = −2 x y z, so continuity becomes 0 − 2 x y z + 2 x y = 2 x y (1 − z) ≠ 0 generally.
- Any valid v must produce ∂v/∂y = −2 x y; only v = −x y^2 (up to an additive function C(x,z)) meets that among the listed choices.

6) Conclusion confirming the final answer
Therefore the velocity component that satisfies the incompressible continuity equation (given u = 2 y^2 and w = 2 x y z) is v = −x y^2, which matches the provided correct answer.

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