Conservation of mass calcs

Homework Help
1

Which of the following velocity fields satisfies conservation of mass for incompressible plane flow: u=x, v=y;u=y, v=x; u=2x; v=-2y; u=3xt, v=-3yt; u=xy+y^2t, v=xy+x^t; u=3x^2y^2, v=-2xy^3

2

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

PK Bot’s answer:
Problems like this are a bit cavalier with units – when they write u=x or u=3xt, it has to be understood that the speed, position in the coordinate plate, and time are all actually nondimensional values, otherwise these relations are all dimensionally inconsistent. So this answer implicitly treats all of these parameters as unitless. t doesn’t have to mean time in this context, because the divergence-free condition is an instantaneous spatial condition, so I just treat it as a parameter. Whoever wrote the question probably meant time.

Below is the computation for each velocity field:

  1. For u=x, v=y:

    \frac{\partial u}{\partial x} = 1,\quad \frac{\partial v}{\partial y} = 1,\quad \nabla\cdot\boldsymbol{u}=1+1=2\,.

    This does not satisfy \nabla\cdot\boldsymbol{u}=0.

  2. For u=y, v=x:

    \frac{\partial u}{\partial x} = 0,\quad \frac{\partial v}{\partial y} = 0,\quad \nabla\cdot\boldsymbol{u}=0+0=0\,.

    This field satisfies conservation of mass.

  3. For u=2x, v=-2y:

    \frac{\partial u}{\partial x} = 2,\quad \frac{\partial v}{\partial y} = -2,\quad \nabla\cdot\boldsymbol{u}=2-2=0\,.

    This field satisfies conservation of mass.

  4. For u=3xt, v=-3yt (with t as a parameter):

    \frac{\partial u}{\partial x} = 3t,\quad \frac{\partial v}{\partial y} = -3t,\quad \nabla\cdot\boldsymbol{u}=3t-3t=0\,.

    This field satisfies conservation of mass.

  5. For u=xy+y^2t, v=xy+x^t (interpreting x^t as x raised to the power t, with t a parameter):

    \frac{\partial u}{\partial x} = y,\quad \frac{\partial v}{\partial y} = x,\quad \nabla\cdot\boldsymbol{u}=y+x\,.

    In general x+y\neq0, so this field does not satisfy conservation of mass.

  6. For u=3x^2y^2, v=-2xy^3:

    \frac{\partial u}{\partial x} = 6xy^2,\quad \frac{\partial v}{\partial y} = -6xy^2,\quad \nabla\cdot\boldsymbol{u}=6xy^2-6xy^2=0\,.

    This field satisfies conservation of mass.

Thus, the velocity fields that satisfy conservation of mass for incompressible plane flow are:

  • u=y, v=x
  • u=2x, v=-2y
  • u=3xt, v=-3yt
  • u=3x^2y^2, v=-2xy^3

To formulate this response, I reviewed the 2025 edition of Professor Kirby’s textbook (Vol 1 | Vol 2), and I have ranked their significance below. If they are marked as relevant, I encourage you to take a look at them:

Textbook sections:

  1. Appendix F.3: Volumetric Viscous Force Part 1 (HIGHLY RELEVANT)
  2. Chapter 8.4: Differential Conservation of Mass Equation Part 2 (HIGHLY RELEVANT)
  3. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 5 (HIGHLY RELEVANT)
  4. Chapter 8.2: Conservation of Mass (HIGHLY RELEVANT)
  5. Chapter 9.3: Why We Use Irrotational Flow Analysis Part 3 Part 1 (HIGHLY RELEVANT)

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