THE BASIC EQUATION OF FLUID STATICS

Consider a cylindrical region of arbitrary size and shape within a fluid, as shown in Fig. 4-1. We will apply a momentum balance to a slice of the 

Because this is a vector equation, we apply it to the z vector components. 

Fz is the sum of all of the forces acting on the system (the slice ) in the z direction, m is the mass of the system, and az is the acceleration in the z direction. Because the fluid is not moving, az = 0, and the momentum balance reduces to a force balance. The z forces acting on the system include the (—) pressure on the bottom (at z) times the (—) z area, the (—) pressure on the top (at z + Az) times the (+) z area, and the z component of gravity, i.e., the weight of the fluid (—pgA Az). The first force is positive, and the latter two are negative because they act in the —z direction. The momentum (force) balance thus becomes 

If we divide through by AzAz, then take the limit as the slice shrinks to zero (Az — 0), the result is 

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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... 

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. 

The principles of the statics of fluids, treated in Section 2.2, are almost an exact science. On, the other hand, the principles of the motions of fluids are quite complex. The basic relations describing the motions of a fluid are the equations for the overall balances of mass, energy, and momentum, which will be covered in the following sections. 

The Froude number described above is frequently used for the description of radial and axial flotvs in liquid media when the pressure difference along a mixing device is important. When cavitation problems are present, the dimensionless group (Pj — p,) /pw - called the Euler number - is commonly used. Here p is the liquid vapour saturation pressure and p is a reference pressure. This number is named after the Swiss mathematician Leonhard Euler (1707-1783) who performed the pioneering work showing the relationship between pressure and flow (basic static fluid equations and ideal fluid flow equations, which are recognized as Euler equations). 

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