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Identify the velocity field \((u, v)\) which satisfies conservation of mass for incompressible plane flow :

  • A.\(u = 3x^{2} + y^{2},\ v = x^{3} - 4xy^{2}\)
  • B.\(u = x^{2},\ v = -y^{2}\)
  • C.\(u = 2xy - x^{2},\ v = 2xy - y^{2}\)
  • D.\(u = x^{2}y,\ v = xy^{2}\)

Answer: C

The field satisfying conservation of mass is \(u = 2xy - x^{2}\), \(v = 2xy - y^{2}\).

For incompressible plane flow the continuity equation is \(u_{x} + v_{y} = 0\). Checking each option:

  • A: \(6x + (-8xy) \neq 0\).
  • B: \(2x - 2y \neq 0\) in general.
  • C: \((2y - 2x) + (2x - 2y) = 0\) everywhere.
  • D: \(2xy + 2xy = 4xy \neq 0\).

Only option C has zero divergence at every point, so only it describes a possible incompressible flow.

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