Free surface & isobars in rigid-body motion
\( \displaystyle \frac{\partial p}{\partial y} = -\,\rho\,a_y,\qquad
\frac{\partial p}{\partial z} = -\,\rho\,(g+a_z) \)
\( \displaystyle \text{Along an isobar }(dp=0):\quad
\frac{dz}{dy} = -\frac{a_y}{\,g+a_z\,}. \)
What this shows. A liquid moving with a container (no internal shear; “rigid-body motion”)
has a tilted free surface. \(y\) is horizontal, \(z\) vertical.
Changing \(a_y\) tilts the surface; \(a_z\) modifies the effective weight in the denominator.
Dashed lines are isobars inside the liquid. They are exactly parallel to the free surface and
spaced by equal perpendicular distance.
Fixed axes: \(a_y\in[-20,20]\), \(dz/dy\in[-4.5,4.5]\). The red dot marks the current \(a_y\).
g (m/s²)
9.81
Slope dz/dy
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Relation
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Schematic (\(y\) horizontal, \(z\) vertical). The blue region is the liquid below the free surface. Vertical scale reduced for readability; angles not to scale.