Horizontally Accelerating Free Surface

Consider a pool of water in the bed of your pickup truck. If you accelerate from rest, the water will slosh toward the rear, and you want to know how fast you can accelerate (ax) without spilling the water out of the back of the truck (see Fig. 4-4). That is, you must determine the slope (tan 6) of the water surface as a function of the rate of acceleration (ax). Now at any point within the liquid there is a vertical pressure gradient due to gravity [Eq. (4-5)] and a horizontal pressure gradient due to the acceleration ax [Eq. (4-23)]. Thus at any location within the liquid the total differential pressure

Figure 4-4 Horizontally accelerating tank.

dP between two points separated by dx in the horizontal direction and dz in the vertical direction is given by dP = — dx H--dz

Since the surface of the water is open to the atmosphere, where P = constant (1 atm),

which is the slope of the surface and is seen to be independent of fluid properties. A knowledge of the initial position of the surface plus the surface slope determines the elevation at the rear of the truck bed and hence whether or not the water will spill out.

C. Rotating Fluid

Consider an open bucket of water resting on a turntable that is rotating at an angular velocity ! (see Fig. 4-5). The (inward) radial acceleration due to the rotation is !2r, which results in a corresponding radial pressure gradient at all points in the fluid, in addition to the vertical pressure gradient due to gravity. Thus the pressure differential between any two points within the fluid separated by dr and dz is dP =

Just like the accelerating tank, the shape of the free surface can be determined from the fact that the pressure is constant at the surface, i.e.,

Figure 4-5 Rotating fluid.

This can be integrated to give an equation for the shape of the surface:

!2r2

2g which shows that the shape of the rotating surface is parabolic.

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