Equation fluid statics

Fluid statics, discussed in Sec. 10 of the Handbook in reference to pressure measurement, is the branch of fluid mechanics in which the fluid velocity is either zero or is uniform and constant relative to an inertial reference frame. With velocity gradients equal to zero, the momentum equation reduces to a simple expression for the pressure field, Vp = pg. Letting z be directed vertically upward, so that g, = —g where g is the gravitational acceleration (9.806 mVs), the pressure field is given by... 

Having obtained two simultaneous equations for the singlet and doublet correlation functions, X and, these have to be solved. Furthermore, Kapral has pointed out that these correlations do not contain any spatial dependence at equilibrium because the direct and indirect correlations of position in an equilibrium fluid (static structures) have not been included into the psuedo-Liouville collision operators, T, [285]. Ignoring this point, Kapral then transformed the equation for the singlet density, by means of a Laplace transformation, which removes the time derivative from the equation. Using z as the Laplace transform parameter to avoid confusion with S as the solvent index, gives... 

As described in Problem CLM.3, the fundamental equation of fluid statics indicates that the rate of change of the pressure P is directly proportional to the rate of change of the depth Z, or... 

We have said that the Young Laplace equation cannot be satisfied unless V n = constant. The proof is more or less trivial. Let us suppose that u = 0. Then, according to the equation of fluid statics, (2 61),... 

The above results may be summarized as follows. To obtain microscopic particle equations of static equilibrium like Eqs. (3.10) and (3.11), one merely replaces U x) with W S x) in the corresponding macroscopic particle equations, in this case Eqs. p.4) and (3.6). (Of course this simple correspondence holds only if lT[5 x] obeys the conditions required of U[x], as stated after Eq. (3.1). This occurs, for example, for a spherical particle moving in a homogeneous fluid.)... 

This equation and its applications are almost the whole of fluid statics. 

This is the basic equation of fluid statics, also called the barometric equation. It is correct only if there are no shear stresses on the vertical faces of the cube in Fig. 2.1. If there are such shear stresses, then they may have a component in the vertical direction, which must be added to the sum of forces in Eq. 2.1. For simple newtonian fluids, shear stresses in the vertical direction can exist only if the fluid has a different vertical velocity on one side of the cube from that on the other side (see Eq. 1.5). Thus this equation is correct if the fluid is not moving at all, which is the case in fluid statics, or if it is moving but only in the X and y directions, or if it has a uniform velocity in the z direction. In this chapter, we apply it only when a fluid has no motion relative to its container or to some set of fixed coordinates. In later chapters, we apply it to flows in which there is no motion in the z direction or a motion with a uniform z component. 

For simple fluids at rest, the pressure-depth relationship is given by the basic equation of fluid statics dPIdz -pg. This equation is found by considering the weight of a small element of fluid and the pressure change with depth necessary to support that weight. 

The basic equation of fluid statics is a limited form of Eq. 5.7. If we apply that equation between any two points in a fluid at rest, there is no external work or friction so... 

As the extent of S is arbitrary, the integral of Equation 1.37 must itself vanish, which leads to the differential equation of fluid statics ... 

The subjects in this chapter will include fluid statics, fluid flow phenomena, categories of fluid flow behavior, the equations of change relating the momentum transport, and the macroscopic approach to fluid flow. 

An important equation relating to fluid statics is the barometric equation... 

The method selected to determine the effective density was to utilize the equation of fluid statics relating pressure, density, and vertical distance in a fluid. Since it would haye been difficult to determine the surface level accurately, the difference in pressure between stations in the tank was measured while the liquid level dropped, due to boil-off, from above to below the upper station. 

In Example 14.1, the calculated pressure at the bottom of the well is 2.155 MPa. What pressure would we calculate for that depth using the basic equation of fluid statics (barometric equation) and assuming that the average molecular weight of the gas in the well was the same as at that the top Assume isothermal, ideal gas behavior. 

Equation (5-1) is used as a basis for derivation of the unsteady-state three-dimensional energy equation for solids or static fluids ... 

This equation applies to any incompressible or compressible static fluid. For an incompressible hquid, pressure varies linearly with depth. For compressible gases, p is obtained by integration accounting for the variation of p with z. 

For the gravity discharge case, the height of the fluid at maximum vacuum, which is the point at which air would begin to backflow into the tank, is determined by Eq. (26-54). Equation (26-55) calculates the corresponding vacuum in the tank s headspace at this hquid height. Since the drain nozzle is open to the atmosphere, this solution is a static force balance that is satisfied when the sum of the internal pressure and the remaining fluid head is equal to the atmospheric pressure. 

The static properties of an isolated chain constitute a good starting point to study polymer dynamics many of the features of the chain in a quiescent fluid could be extrapolated to the kinetics theories of molecular coil deformation. As a matter of fact, it has been pointed out that the equations of chain statistics and chain dynamics are intimately related through the simplest notions of graph theory [16]. 

Equations 6.11 and 6.12 can be used for the calculation of the fluid velocity and the impact pressure in terms of the static pressure a short distance upstream. The two sections are chosen so that they are sufficiently close together for frictional losses to be negligible. Thus Pi will be approximately equal to the static pressure at both sections and the equations give the relation between the static and impact pressure—and the velocity — at any point in the fluid. 

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