Relative equilibrium of liquid is a condition where the whole mass of liquid including the vessel in which the liquid is contained, is moving at uniform accelerated motion with respect to the earth, but every particle of liquid have no relative motion between each other. There are two cases of relative equilibrium that will be discussed in this section: linear translation and rotation. Note that if a mass of liquid is moving with constant speed, the conditions are the same as static liquid in the previous sections.
 

Rectilinear Translation - Moving Vessel

Horizontal Motion
If a mass of fluid moves horizontally along a straight line at constant acceleration a, the liquid surface assume an angle θ with the horizontal, see figure below.
 

007-fluid-mass-horizontal-motion.gif

 

007-fluid-mass-horizontal-motion-forces.gifFor any value of a, the angle θ can be found by considering a fluid particle of mass m on the surface. The forces acting on the particle are the weight W = mg, inertia force or reverse effective force REF = ma, and the normal force N which is the perpendicular reaction at the surface. These three forces are in equilibrium with their force polygon shown to the right.
 

From the force triangle
$\tan \theta = \dfrac{REF}{W}$

$\tan \theta = \dfrac{ma}{mg}$

$\tan \theta = \dfrac{a}{g}$

 

Inclined Motion
Consider a mass of fluid being accelerated up an incline α from horizontal. The horizontal and vertical components of inertia force REF would be respectively, x = mah and y = mav.
 

007-fluid-mass-inclined-motion.gif

 

From the force triangle above
$\tan \theta = \dfrac{x}{W + y}$

$\tan \theta = \dfrac{ma \cos \alpha}{mg + ma \sin \alpha}$

$\tan \theta = \dfrac{a \cos \alpha}{g + a \sin \alpha}$
 

but a cos α = ah and a sin α = av, hence
$\tan \theta = \dfrac{a_h}{g + a_v}$

$\tan \theta = \dfrac{a_h}{g \pm a_v}$

Use (+) sign for upward motion and (-) sign for downward motion.
 

Vertical Motion
The figure shown to the right is a mass of liquid moving vertically upward with a constant acceleration a. The forces acting to a liquid column of depth h from the surface are weight of the liquid W = γV, the inertia force REF = ma, and the pressure F = pA at the bottom of the column.
 

007-fluid-mass-vertical-motion.gif$\Sigma F_V = 0$

$F = W + REF$

$pA = \gamma V + ma$

$pA = \gamma V + \rho Va$

$pA = \gamma V + \dfrac{\gamma}{g} Va$

$pA = \gamma (Ah) + \dfrac{\gamma}{g} (Ah)a$

$p = \gamma h + \dfrac{\gamma}{g} ha$

$p = \gamma h \left(1 + \dfrac{a}{g}\right)$

$p = \gamma h \left(1 \pm \dfrac{a}{g}\right)$

Use (+) sign for upward motion and (-) sign for downward motion. Also note that a is positive for acceleration and negative for deceleration.
 

Rotation - Rotating Vessel

When at rest, the surface of mass of liquid is horizontal at PQ as shown in the figure. When this mass of liquid is rotated about a vertical axis at constant angular velocity ω radian per second, it will assume the surface ABC which is parabolic. Every particle is subjected to centripetal force or centrifugal force CF = mω2x which produces centripetal acceleration towards the center of rotation. Other forces that acts are gravity force W = mg and normal force N.
 

008-rotating-vessel.gif                     008-force-polygon.gif

 

$\tan \theta = \dfrac{CF}{W}$

$\tan \theta = \dfrac{m\omega^2x}{mg}$
 

$\tan \theta = \dfrac{\omega^2x}{g}$

Where tan θ is the slope at the surface of paraboloid at any distance x from the axis of rotation.
 

From Calculus, y’ = slope, thus
$\dfrac{dy}{dx} = \tan \theta$

$\dfrac{dy}{dx} = \dfrac{\omega^2x}{g}$

$dy = \dfrac{\omega^2}{g}x ~ dx$

$\displaystyle \int dy = \dfrac{\omega^2}{g} \int x ~ dx$

$y = \dfrac{\omega^2x^2}{2g}$

 

For cylindrical vessel of radius r revolved about its vertical axis, the height h of paraboloid is

$h = \dfrac{\omega^2r^2}{2g}$

 

Other Formulas
By squared-property of parabola, the relationship of y, x, h and r is defined by

$\dfrac{r^2}{h} = \dfrac{x^2}{y}$

 

Volume of paraboloid of revolution

$V = \frac{1}{2}\pi r^2h$

 

Important conversion factor

$1 \, \text{ rpm} = \frac{1}{30}\pi \, \text{ rad/sec}$

 

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