Incompressibility of liquids

M. Amon and C. D. Denson [33-34] attempted a theoretical and experimental examination of molding a thin plate from foamed thermoplastic. In the first part of the series [33] the authors examined bubble growth, and in the second [34] — used the obtained data to describe how the thin plate could be molded with reference to the complex situation characterized in our third note. Here, we are primarily interested in the model of bubble growth per se, and, of course, the appropriate simplification proposals [33]. Besides the conditions usual for such situations ideal gets, adherence to Henry s law, negligible mass of gas as compared to mass of liquid, absence of inertia, small Reynolds numbers, incompressibility of liquid, the authors postulated [33] several things that require discussion ... 

Demonstrate the compressibility of gases and the incompressibility of liquids by using a syringe to show that the volume of a gas (e.g. air) decreases with pressure whereas that of a liquid (e.g. water) does not. 

An integrated pTAS system for the detection of bacteria including lysis, DNA purification, PCR and fluorescence readout has also been published recently [113]. A microfluidic plastic chip with integrated porous pol5mier monoliths and silica particles for lysis and nucleic acid isolation was used for detection (Fig. 8). A custom-made base device provided liquid actuation and off-chip valving by stopping liquid flow from the exits of the chip, utilizing the incompressibility of liquids. Detection of 1.25 x 10 cells of B. subtilis was demonstrated with all assay steps performed on-chip. 

The PV work in this example is much less than that calculated in Example 2.7. This is due to the relative incompressibility of liquids. Because the volume of the liquid does not )deld much under pressure, the displacement of the force exerted by the outside pressure is very small. 

Liquids can be expanded or compressed just as gases can. For technical reasons, the design of expansion and compression units for liquids is different from that of gases, but the thermodynamics analysis is the same. A turbine for liquids is usually called an expander, and a compressor for liquids, a pump. The analysis is based on the same equations as for gases, but we treat the subject separately because the relative incompressibility of liquids allows us to use short-cut approximations for the enthalpy and entropy changes of the fluid. The starting point is eqs. (5.20) and (5.30). 

The incompressibility of liquids is utilised in many ways in industry particularly in hydraulic power transmission. Two common applications are as power sources for machines and vehicles and in hydraulic cylinders. Normally the hydraulic medium is an oil and hazards can arise where leaks occur either as oil pools on a floor or from high-pressure lines as a fine mist which is highly flammable. 

FIGURE 11.6 Incompressibility of liquids Liquids are not compressible because there is so little space between the liquid particles. 

Many transient flows of liquids may be analyzed by using the full time-dependent equations of motion for incompressible flow. However, there are some phenomena that are controlled by the small compressibility of liquids. These phenomena are generally called hydraulic transients. 

The molecules of liquids are separated by relatively small distances so the attractive forces between molecules tend to hold firm within a definite volume at fixed temperature. Molecular forces also result in tlie phenomenon of interfacial tension. The repulsive forces between molecules exert a sufficiently powerful influence that volume changes caused by pressure changes can be neglected i.e. liquids are incompressible. 

The original spherical surface increases as a result of an increase in its volume at a constant velocity u. The area of the surface of the sphere at time t is given by the relationship A = (4jv)1/3(3ut)2/3 = AQt2/3, where A0 is the area at time t— 1. From the incompressibility of the liquid it follows for volume AoJtf2/3 that... 

A storage tank is shown in Figure 4-5. A hole develops at a height hL below the fluid level. The flow of liquid through this hole is represented by the mechanical energy balance (Equation 4-1) and the incompressible assumption, as shown in Equation 4-2. 

All fluids are compressible to some extent but the compressibility of liquids is so low that they can be treated as being incompressible. Gases... 

The part of this process that is described by a force balance is the liquid flowing through the pipe. It will have a mass equal to the volume of the pipe (j4j,L) times the density of the liquid p. This mass of liquid will have a velocity v (ft/s) equal to the volumetric flow divided by the cross-sectional area of the pipe. Remember we have assumed plug-flow conditions and incompressible liquid, and therefore all the liquid is moving at the same velocity, more or less like a solid rod. If the flow is turbulent, this is not a bad assumption. 

Example 2.11. As an example of a force balance for a microscopic system, let us look at the classic problem of the laminar flow of an incompressible, newtonian liquid in a cylindrical pipe. By newtonian we mean that its shear force (resistance that adjacent layers of fluid exhibit to flowing past each other) is proportional to the shear rate or the velocity gradient. 

Solve the second-order ODE describing the steadystate flow of an incompressible, newtonian liquid through a pipe ... 

The remaining approximations of liquid phase incompressibility and a discontinuous mass distribution can be removed through the use of the Yvon-Born-Green (YBG) equation (3), which is simply a... 

Six viscosity coefficients required for a description of the dynamics of an incompressible, nematic liquid crystal. 

Note 3 The Finger strain tensor for a homogeneous orthogonal deformation or flow of incompressible, viscoelastic liquid or solid is... 

Equation relating stress and deformation in an incompressible viscoelastic liquid or solid. Note 1 A possible general form of constitutive equation when there is no dependence of stress on amount of strain is... 

This natural circulation occurs by a direct transfer of momentum across the interface, and the presence of a monolayer at the interface will affect it in two ways. Firstly, the surface viscosity of the monolayer may cause a dissipation of energy and momentum at the surface, so that the drop behaves rather more as a solid than as a liquid, i.e., the internal circulation is reduced. Secondly, momentum transfer across the surface is reduced by the incompressibility of the film, which the moving stream of gas will tend to sweep to the rear of the drop (Fig. 14b) whence, by its back-spreading pressure n, it resists further compression and so damps the movement of the surface and hence the transfer of momentum into the drop. This is discussed quantitatively below, where Eq. (32) should apply equally well to drops of liquid in a gas. 

When a drop (water) falls to a flat interface (benzene-water) the entire drop does not always join the pool (water). Sometimes a small droplet is left behind and the entire process, called partial coalescence, is repeated. This can happen several times in succession. High-speed motion pictures, taken at about 2000 frames per second, have revealed the details of the action (W3). The film (benzene) ruptures at the critical film thickness and the hole expands rapidly. Surface and gravitational forces then tend to drag the drop into the main pool (water). But the inertia of the high column of incompressible liquid above the drop tends to resist this pull. The result is a horizontal contraction of the drop into a pillar of liquid above the interface. Further pull will cause the column to be pinched through, leaving a small droplet behind. Charles and Mason (C2) have observed that two pinches and two droplets occurred in a few cases. The entire series of events required about 0.20 sec. for aniline drops at an aniline-water interface (C2, W3). 

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