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In accelerated fluids, fluid in a container can orient itself in a direction due to acceleration. In that case, pressure at different heights (at the surface) is same (atm). Then at the same height, pressure is different. How does this pressure increase along the horizontal?enter image description here

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  • $\begingroup$ hint: use a reference frame moving with the container, now instead of $\vec{g}$ you have $\vec{g}-\vec{a}$, the surface will be perpendicular to this vector $\endgroup$ – user83548 Dec 15 '15 at 16:18
  • $\begingroup$ Yea I got that, and so surface inclination will be a/g. But still, what do we know about the pressure increment due to the pseudo force? $\endgroup$ – Shodai Dec 15 '15 at 16:24
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    $\begingroup$ Because it is a contant across space you can use it the same way as g, so from the surface "down" the pressure will raise as $p=p_0+\rho |\vec{g}-\vec{a}| H $ where H is measured perpendicular to the surface $\endgroup$ – user83548 Dec 15 '15 at 16:42
  • $\begingroup$ so surface inclination will be a/g. Correction: $\tan\alpha=a/g$, with $\alpha$ the inclination of the surface with respect to the horizontal. $\endgroup$ – Gert Dec 15 '15 at 16:55
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Pseudo-forces and/or accelerated frames simplify this mathematically, but here’s a way to understand it conceptually:

Consider a little bit of fluid in the middle. The pressure it exerts to the right has to be enough to accelerate all the fluid to the right. (“Has to” in the sense that fluid will flow around & seek levels so that this is true)

Now consider the bit of fluid just to the left of that. It must be exerting more pressure: it has to accelerate the bit to its right (the bit in the middle) plus all the stuff to the right of that.

So as you go from right to left across the middle, the pressure goes up.

From here you can create an exact model with $\Delta x$ etc, but this is the basic idea.

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Considering the frame of reference in which the vessel is at rest,

Every molecule in the liquid will experience a pseudo force in the direction opposite to that of the acceleration of the vessel, but since the liquid as a whole is at rest, it means that there must be an equal and opposite force acting on the molecules, this equal and opposite force comes from the pressure difference between heights of the fluid and thus the surface inclines to an angle correctly mentioned in the comments, or, the pressure along the horizontal increases (to make liquid move as a whole).

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