Relative Equilibrium - Pressure

Relative Equilibrium of Fluids:

If a fluid is contained in a vessel which is at rest, or moving with constant linear velocity, it is not affected by the motion of the vessel; but if the container is given a continuous acceleration, this will be transmitted to the fluid and affect the pressure distribution in it. Since the fluid remains at rest relative to the container, there is no relative motion of the particles of the fluid and, therefore, no shear stresses, fluid pressure being everywhere normal to the surface on which it acts. Under these conditions the fluid is said to be in relative equilibrium.

2.1.0    Pressure:

For a static fluid, the only stress is the normal stress since by definition a fluid subjected to a shear stress must deform and undergo motion. Normal stresses are referred to as pressure P. For the general case, the stress on a fluid element or at a point is a tensor.

                   F = normal force acting over A.

As already noted, p is a scalar, which can be easily demonstrated by considering the equilibrium of forces on a wedge-shaped fluid element.

2.1.1    Pressure Transmission:

Pascal's law: in a closed system, a pressure change produced at one point in the system is transmitted throughout the entire system.

v Absolute Pressure, Gage Pressure, and Vacuum

For P_A > P_a ,                P_g = P_A  -  P_a = gage pressure.

For P_A < P_a ,                P_vac = -P_g = P_a  -  P_A = vacuum pressure.

2.1.2    Pressure Variation with Elevation:

Basic Differential Equation

           For a static fluid, pressure varies only with elevation within the fluid. This can be shown by consideration of equilibrium of forces on a fluid element.

Newton's law (momentum principle) applied to a static fluid:

Basic equation for pressure variation with elevation.

For a static fluid, the pressure only varies with elevation z and is constant in horizontal xy planes. The basic equation for pressure variation with elevation can be integrated depending on whether  = constant or = (z), i.e., whether the fluid is incompressible (liquid or low-speed gas) or compressible (high-speed gas) since g  constant.

2.1.3 Pressure with Depth:

           Suppose we had an object submerged in water with the top part touching the atmosphere. If we were to draw an FBD for this object we would have three forces:

1.      The weight of the object.

2.      The force of the atmosphere pressing down.

3.   The force of the water pressing up.

But recall, pressure is force per unit area. So if we solve for force we can insert our new equation in.

Where,

P = Absolute Pressure.

P. = Initial Pressure – may or may NOT be atmospheric pressure.

2.1.4 Pressure Measurements:

            Pressure is an important variable in fluid mechanics and many instruments have been devised for its measurement. Many devices are based on hydrostatics such as barometers and manometers, i.e., determine pressure through measurement of a column (or columns) of a liquid using the pressure variation with elevation equation for an incompressible fluid.

More modern devices include Bourdon-Tube Gage (mechanical device based on deflection of a spring) and pressure transducers (based on deflection of a flexible diaphragm/membrane). The deflection can be monitored by a strain gage such that voltage output is p across diaphragm, which enables electronic data acquisition with computers.

2.1.5 Manometer:

A change in elevation (Z₂ – Z₁) of a liquid is equivalent to a change in pressure (P₂ – P₁) / γ. Thus a static column of one or more liquids can be used to measure differences between two points. Such a device is called a manometer. If multiple fluids are used, we must change the specific weight in the equation as move from one fluid to another. Fig. 2.8 illustrates the use of the equation with a column of multiple fluids. The pressure change through each fluid is calculated separately. If we wish to know the total change (P₅ – P₁), we add successive changes (P₂ – P₁), (P₃ – P₂), (P₄ – P₃), and (P₅ – P₄). The intermediate values of P cancel, and we have, for the example of Fig 

        When calculating hydrostatic pressure changes, engineers work instinctively by simply having the pressure increase downward and decrease upward.

Thus, without worrying too much about which point is Z₁ and which is Z₂, the equation simply increases or decreases the pressure according to whether one is moving down or up. For example, Equ above could be written in the following “multiple increase” mode:

That is, keep adding on pressure increments as you move down through the layered fluid. A different application is a manometer, which involves both “up” and “down” calculations.

Fig 19 

  

The Fig.19 shows a simple manometer for measuring P_A in a closed chamber relative to atmospheric pressure P₀, in other words, measuring gage (relative) pressure. The chamber fluid  is combined with a second fluid , perhaps for two reasons:

 1) To protect the environment from a corrosive chamber fluid or,

2) Because a heavier fluid  will keep Z₂ small and the open tube can be shorter.

One can apply the basic hydrostatic 2nd Equ. Or, more simply, one can begin at A, apply 1st Equ.“down” to Z₁, jump across fluid 2 to the same pressure P₁, and then use 2nd Equ “up” to level Z₂:

The physical reason that we can “jump across” at section 1 in that a continuous length of the same fluid connects these two elevations. The hydrostatic relation requires this equality as a form of Pascal’s law: Any two points at the same elevation in a continuous mass of the same static fluid will be at the same pressure. This idea of jumping across to equal pressures facilitates multiple-fluid problems.

2.1.6    Manometer:

              Atmospheric pressure is measured by a device called a Barometer. Thus, the atmospheric pressure is often referred to as the barometric pressure. A force balance in vertical direction gives: