Dependence on capillary pressure

Also, the vegetable world is known to be dependent on capillary pressure (and osmotic pressure) to bring water up to the higher parts of plants. Using these forces, some trees succeed in bringing the essential liquid (water) up to 120 m above the ground. 

In equation 3, p stands for water density, pi for liquid dynamic viscosity and for relative conductivity. The liquid conductivity is associated to darcean liquid flow, in water mass conservation equation. This term is non-linear because water relative conductivity depends on capillary pressure, which is the main variable associated with water mass conservation equation. Furthermore it is coupled to thermal effects because liquid dynamic viscosity depends on temperature, which is the main variable associated with energy conservation equation. 

Conventionally, the sample is initially saturated with one fluid phase, perhaps including the other phase at the irreducible saturation. The second fluid phase is injected at a constant flow rate. The pressure drop and cumulative production are measured. A relatively high flow velocity is used to try to negate capillary pressure effects, so as to simplify the associated estimation problem. However, as relative permeability functions depend on capillary number, these functions should be determined under the conditions characteristic of reservoir or aquifer conditions [33]. Under these conditions, capillary pressure effects are important, and should be included within the mathematical model of the experiment used to obtain property estimates. 

Cf., Poiseuille s equation, according to which the fluid velocity through a capillary is linearly dependent on the pressure gradient along it. 

Brain perfusion refers to the microcirculation of the brain (see Chap. 6). Microcirculation comprises the blood circulation in capillary networks and the exchange of oxygen and nutrients between the blood and the brain tissue. The effectiveness of brain perfusion depends on blood pressure, blood velocity, characteristics of the capillary network, capillary wall permeability, and diffusion rates of oxygen and nutrients. In the healthy brain, perfusion is symmetrical, and is substantially higher in the CM than in the WM. It has been approximated that the CBF is about 80 ml/100 g/min in the CM, and 20 ml/100 g/ min in the WM. Brain perfusion is usually quantified in terms of ml/100 g (cerebral blood volume,... 

There are static and dynamic methods. The static methods measure the tension of practically stationary surfaces which have been formed for an appreciable time, and depend on one of two principles. The most accurate depend on the pressure difference set up on the two sides of a curved surface possessing surface tension (Chap. I, 10), and are often only devices for the determination of hydrostatic pressure at a prescribed curvature of the liquid these include the capillary height method, with its numerous variants, the maximum bubble pressure method, the drop-weight method, and the method of sessile drops. The second principle, less accurate, but very often convenient because of its rapidity, is the formation of a film of the liquid and its extension by means of a support caused to adhere to the liquid temporarily methods in this class include the detachment of a ring or plate from the surface of any liquid, and the measurement of the tension of soap solutions by extending a film. 

When h r a the contribution of vertexes to the liquid volume fraction of a foam estimated from Eq. (4.4) is smaller compared to that of films, Eq. (4.2) and borders, Eq. (4.3). The relative contribution of borders and films depends on capillary and disjoining pressures. In a foam with thin (black) films at capillary pressures up to 5.103 Pa the border volume is... 

Effects of Capillary Number, Capillary Pressure, and the Porous Medium. Since the mechanisms of leave-behind, snap-off, lamella division and coalescence have been observed in several types of porous media, it may be supposed that they all play roles in the various combinations of oil-bearing rocks and types of dispersion-based mobility control (35,37,39-41). However, the relative importance of these mechanisms depends on the porous medium and other physico-chemical conditions. Hence, it is important to understand quantitatively how the various mechanisms depend on capillary number, capillary pressure, interfacial properties, and other parameters. 

The Cartesian manostat consists of a diver which moves up and down depending on the pressure of the distillation system, a means of setting the pressure in the diver, and a capillary which engages the short rubber stopper at the top of the diver. The body of the mano-... 

The substitution of the above equation into the generalized Laplace equation (with the dependence of capillary pressure on the z coordinate accounted for) yields the Laplace equation in the differential form, the numerical integration of which leads to the exact mathematical description of the drop or bubble surface shape in the gravitational field [6,14]. The exact description of the equilibrium surface shape is of importance in the evaluation of surface tension from the experimental data at interfaces with high mobility, such as liquid-gas and liquid-liquid ones (See Chapter 1,4). 

Here, 7j is the liquid surface tension, Sp>0 is a scaling constant, which includes both contact angle effects and the substantially larger scaling for capillary pressure in hydrophobic media. For the dependence of capillary pressure on liquid volume fraction, we use a simplified van Genuchten function see [14],... 

Let us consider the schematic diagram of the experimental setup (Fig. 11.6). Let Pi denote the pressure in the inner chamber and P2 denote the pressure in the outer chamber containing a liquid of lower density. Fluid flow across the capillary is mainly responsible for generating oscillations. The upward flow is opposed by the viscosity of heavier fluid in the inner vessel, while the downward flow is opposed by buoyancy. The buoyancy depends on the density difference Pi — Pf where Pi and P2 are the densities of heavier liquid and lighter liquid, respectively. This is so since the magnitude of the buoyancy force would depend on the difference of vertical components of fluid force on the upper and lower sides of the capillary. This is turn would depend on the pressure difference sP = Pi-P2. The density of the denser fluid in the capillary due to partial mixing would depend on the pressure difference. 

The optimum linear velocity for a capillary column depends on the pressure in the column because Wopt is proportional to the average diffusion coefficient, which varies inversely with pressure. Operation of a short wide-bore column at vacuum outlet conditions results in a significantly faster analysis than would occur if the same column was used under atmospheric outlet pressures. Mass spectrometry (MS) has made vacuum GC very easy to implement, since the mass spectrometer provides both detection and a source of vacuum. Vacuum GC can be achieved practically by incorporating a restriction at the inlet end of a wide-bore capillary column, and interfacing the terminal end of the column directly into the MS. The function of the restriction is to deliver an optimal helium flow for the mass spectrometer, and it can be as simple as a short section of 20 pm i.d. capillary (or a longer section of 100-150 pm i.d. capillary). An optimal carrier gas velocity of 90-100cms can be expected for a 10 m x 50 pm column with a restriction at the inlet, and a speed gain of a factor of 3-5 times can easily be obtained. 

FIG. 2 Calibration of a thin quartz capillary by measuring rates of meniscus motion v in dependence on applied pressure difference AP. The length of water column in the capillary was equal to / = 9.28 cm (curve 1) and 5.28 cm (curve 2) t = 19°C. 

Rates of meniscus back-and-forth movements are measured within a small portion of the capillary. A/ < 1 mm, much smaller than either /i or l2-According to Eq. (10), the rate of displacement must Knearly depend on the pressure drop AP when the dynamic capillary pressure P is constant. In this case, the lengths l and I2 and viscosity values and /2 influence oifly the slope of the linear v(AF) graphs. 

Convection is a process by which a substance is dragged along by the flow of fluid hence the term solvent drag is used to describe this type of transport. The flow is powered by osmotic or hydrostatic pressure gradients which exist across tissue boundaries. The kidney is an example of an organ which depends on hydrostatic pressure-driven convection for filtration of substances by the glomerulus and osmotic-pressure driven convection for solute reabsorption in the proximal tubule. Filtration and reabsorption by blood capillaries depends on Starling s relationship ... 

Capillary condensation on a fractal object Let us assume that a colloidal aggregate is exposed to a slightly unsaturated vapor at pressure p. The grain becomes surrounded by a "cocoon of liquid of volume Q (,p) dependent on the pressure. We construct the scaling form of Q (p) in section III. 

Figure 9.5a shows a portion of a cylindrical capillary of radius R and length 1. We measure the general distance from the center axis of the liquid in the capillary in terms of the variable r and consider specifically the cylindrical shell of thickness dr designated by the broken line in Fig. 9.5a. In general, gravitational, pressure, and viscous forces act on such a volume element, with the viscous forces depending on the velocity gradient in the liquid. Our first task, then, is to examine how the velocity of flow in a cylindrical shell such as this varies with the radius of the shell.

The basic design is that of the Ostwald viscometer a U-tube with two reservoir bulbs separated by a capillary, as shown in Figure 24a. The Hquid is added to the viscometer, pulled into the upper reservoir by suction, and then allowed to drain by gravity back into the lower reservoir. The time that it takes for the Hquid to pass between two etched marks, one above and one below the upper reservoir, is a measure of the viscosity. In U-tube viscometers, the effective pressure head and therefore the flow time depend on the volume of Hquid in the instmment. Hence, the conditions must be the same for each measurement. 

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