Conservation of mass velocity fields

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Which of the following velocity fields satisfies conservation of mass for incompressible plane flow? (a) u=x, v=y (b) u=y, v=x (c) u=2x, v=-2y (d) u=3xt, v=-3yt (e) u=xy+y2t, v=xy+x2t

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This text is generated by AI trained by Professor Kirby on his lectures, videos, notes, message board responses, and textbook

PK Bot’s answer:
We check conservation of mass for an incompressible plane flow by ensuring that the divergence vanishes everywhere. For a 2D velocity field with components u and v, incompressibility requires

\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0\,.

We evaluate this condition for each case:

  1. (a) u = x, v = y

    • \displaystyle \frac{\partial u}{\partial x} = 1
    • \displaystyle \frac{\partial v}{\partial y} = 1
    • Sum: 1 + 1 = 2 \neq 0

    This field does not satisfy conservation of mass.

  2. (b) u = y, v = x

    • \displaystyle \frac{\partial u}{\partial x} = 0 (since u depends only on y)
    • \displaystyle \frac{\partial v}{\partial y} = 0 (since v depends only on x)
    • Sum: 0 + 0 = 0

    This field satisfies conservation of mass.

  3. (c) u = 2x, v = -2y

    • \displaystyle \frac{\partial u}{\partial x} = 2
    • \displaystyle \frac{\partial v}{\partial y} = -2
    • Sum: 2 + (-2) = 0

    This field satisfies conservation of mass.

  4. (d) u = 3xt, v = -3yt

    Here t is treated as a parameter.

    • \displaystyle \frac{\partial u}{\partial x} = 3t
    • \displaystyle \frac{\partial v}{\partial y} = -3t
    • Sum: 3t + (-3t) = 0

    This field satisfies conservation of mass.

  5. (e) u = xy + y^2t, v = xy + x^2t

    • \displaystyle \frac{\partial u}{\partial x} = y (the term xy differentiates to y, and y^2t is independent of x)
    • \displaystyle \frac{\partial v}{\partial y} = x (similarly, the derivative of xy is x, and x^2t does not involve y)
    • Sum: y + x

    The sum is not zero for all (x,y) (it would vanish only along the line x+y=0). Therefore, this field does not satisfy conservation of mass in general.

Thus, the fields that satisfy conservation of mass are:

  • (b) u = y, v = x
  • (c) u = 2x, v = -2y
  • (d) u = 3xt, v = -3yt

In crafting this response, I dedicated some time to reviewing the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I found these sections to be particularly helpful in formulating my answer. I encourage you to explore the pertinent ones directly:

Textbook sections:

  1. Chapter 8 Fundamentals: Streamfunction (RELEVANT)
  2. Chapter 8.2: Conservation of Mass (RELEVANT)
  3. Chapter 7.2: Integral Conservation of Mass (RELEVANT)
  4. Appendix B.4: Vector Calculus Part 3 (RELEVANT)
  5. Chapter 7.3: Integral Conservation of Momentum (RELEVANT)
  6. Chapter 9 Fundamentals: Curl and Vorticity (RELEVANT)
  7. Chapter 8.3: Stream Function (RELEVANT)
  8. Chapter 10 Fundamentals: Solving the boundary layer equations via a similarity transform for the fla… (RELEVANT)
  9. Appendix F.3: Volumetric Viscous Force (RELEVANT)
  10. Chapter 5 Fundamentals: Eulerian and Lagrangian specification of fluid flow (RELEVANT)

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