Vertical pressure gradient

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 ) 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... 

Consider an open bucket of water resting on a turntable that is rotating at an angular velocity on (see Fig. 4-5). The (inward) radial acceleration due to the rotation is o r, 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... 

Because the vanor in the reservoir is assumed to be at rest, i.e., to be stagnant, the pressure distribution in the vapor reservoir will be given by the hydrostatic pressure equation. The vertical pressure gradient in the vapor is therefore given by ... 

This simplification is commonly called the hydrostatic approximation. The approximation states that, in the vertical direction, the most important forces acting on a parcel of air are the vertical pressure gradient and gravity. This approximation is common in dynamical models, although it neglects some phenomena such as sound waves, and is not appropriate for high resolution models. The ideal gas law can be used to substitute for p in the above equation to obtain... 

On the basis of a new method correlating the biomass concentration with the vertical pressure gradient of the fluidized bed, a comprehensive optimization and modeling procedure has been elaborated for fluidized bed bioreactors. 

The neglect of vertical acceleration and the Coriolis effect relative to the differences between the vertical pressure gradient force and gravity (the hydrostatic assumption). 

As previously mentioned, the vertical pressure gradient in a spouting bed increases from zero at the base to a maximum near the top of the bed. A total spouting pressure drop APgg at the maximum spoutable bed depth can be estimated by the following relation ... 

Development of numerical models for time-domain wave transformation has been tried by many researchers, but random wave-breaking process has not been well reproduced in these models. One of the exceptions is the Boussinesq-type model developed by Hirayama et and Hirayama and Hiraishi, who employed the breaking criterion of the vertical pressure gradient by Nadaoka et alJ They raised the threshold gradient from 0 to 0.5 to compensate the insufficiency in numerical accuracy due to the features of weak nonlinearity inherent to the Boussinesq equation. They have succeeded in reproducing the pc /variation across the surf zone. 

A more thermally complex system involves an atmosphere that is vertically variable in temperature and in hydrostatic equilibrium under the force of gravity. The vertical pressure gradient in such an atmosphere is given by... 

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