Ideal Fluids in Motion — Question 1

Question 1

The ideal-flow assumptions are separate claims

Three observations are made in a laboratory: (A) velocity at each fixed probe is constant in time, although probes at different positions report different velocities; (B) the volume of each tracked fluid parcel is constant; (C) a slowly moving plate requires a nonzero tangential force proportional to its speed. Assume these are resolved effects rather than measurement noise.

1. Which observations support steady flow, incompressibility, or nonzero viscosity? Can all three occur in one flow?

2. If an ideal model neglects viscosity, does observation C automatically make every prediction of that model useless? State what comparison is needed.

Question 1 - Solution

The ideal-flow assumptions are separate claims

(a) Derivation and calculation

The three observations concern different quantities:$$A:\quad\partial\mathbf v/\partial t=0\quad\text{(steady)};$$$$B:\quad DV_{\mathrm{parcel}}/Dt=0\quad\text{(incompressible)};$$$$C:\quad F_t\propto U,\ F_t\ne0\quad\text{(viscous response)}.$$They can coexist in one steady incompressible viscous flow.

(b) Result and interpretation

An inviscid approximation neglects viscous forces. Its usefulness requires, for the chosen observable,$$\frac{\text{neglected viscous contribution}}{\text{relevant retained contribution}}\ll1$$$$\text{and a resulting error below the required tolerance}.$$Observation C rules out exact zero viscosity in the test, but does not determine the size of every correction elsewhere.

Physical check

Steady does not mean spatially uniform, and incompressible does not mean inviscid.