Capillary pressure equation defining

Capillary Pressure. At the surface of a single liquid-vapor interface, a capillary pressure difference, defined as (Pv - P,) or APc, exists. This capillary pressure difference can be described mathematically from the Laplace-Young equation,... 

When two immiscible fluids (or a fluid and a gas) are in contact, molecular attractions between similar molecules in each fluid are greater than the attractions between the different molecules of the two fluids and a clearly defined interface exists between them. The force that acts on this interface is called interfacial tension (or surface tension in case of a gas-fluid contact). As a result of this force, a pressure difference exists across the interface. This pressure difference is known as capillary pressure and is given by the following equation (Dake, 1978) ... 

The momentum equations in the whole field formulation can also be rewritten in a momentum conserving form [132] [183] [92], defining a 3D capillary pressure tensor... 

Equation(ll), the variable defines the capillary pressure. The parameter is Biot s modulus related to capillary pressure, which is a function of water saturation degree through the water retention curve... 

The capillary length is defined by equating the Laplace pressure yn with the hydrostatic pressure pgK, so that... 

The pressure difference, AP, between the two fluid phases that causes capillary rise is a function of the interfacial tension (surface free energy) and the mutually perpendicular radii of curvature, ri and r2, for the interface between the two fluids. The pressure difference is known as the capillary pressure. Pc, and is defined as the pressure in the oil phase, Pq, less the pressure in the water phase, Pw If the porous medium is regarded as a bundle of capillaries with an average radius, r, equation 5 can be expressed as... 

The integration of equation 15 is used to obtain the area under the individual curves (Ai for secondary drainage and A2 for forced imbibition). A limit of final saturation or capillary pressure must be chosen to provide consistent results. Wettability is defined as... 

FIGURE4.16 Example of equilibrium shape of bridging drop with perturbed and curved region of the air-water surface of radius adjacent to drop. Capillary pressure jumps are balanced as required by Equation 4.32 if equilibrium is to prevail. However, beyond the perturbed region, air-water surface is flat and film becomes plane parallel. Therefore, equilibrium between the region defined by radius and the remainder of foam film does not exist. (Reprinted with permission from Denkov, N.D. Langmuir, 15, 8530. Copyright 1999 American Chemical Society.)... 

One Fortran subtlety deserves elaboration. The 200 do-loop defines two separate difference equations for the flows left and right of the front, but the pressure updating in the 260 do-loop refers to a single pressure. So long as the front does not move more than one mesh in a time step, errors due to copying liquid pressure as gas pressure, or conversely, do not exist, assuming small capillary pressures. Pressure continuity assures that both blocks will contain identical pressures. 

The coefficients A = (1 -Aq)IBq (3 Cay, Aq, and Bq as functions of the capillary number according to Equation 3.237 and Equation 3.236 obtained from the numerical calculation and are presented in Figure 3.18. The isotherm of the disjoining pressure is defined by Equation 3.240, that is, the case of complete wetting is under consideration. Figure 3.18 shows that in this case at Ca 0, B 3/2, which corresponds to the prediction (3.249). At the same lime, Aq 1 - (3/2) He as shown above, and A . 

We defined the equation of motion as a general expression of Newton s second law applied to a volume element of fluid subject to forces arising from pressure, viscosity, and external mechanical sources. Although we shall not attempt to use this result in its most general sense, it is informative to consider the equation of motion as it applies to a specific problem the flow of liquid through a capillary. This consideration provides not only a better appreciation of the equation of... 

As noted above, it is possible that a different pair of radii of curvature applies at different locations on a surface. In this case the Laplace equation shows that Ap also varies with location. This is the reason for the variation of the pressure with z in the meniscus shown in the capillary in Figure 6.3b. As is often true of pressures, it is convenient to define pressure variations relative to some reference plane. 

In actual use for mobility control studies, the network might first be filled with oil and surfactant solution to give a porous medium with well-defined distributions of the fluids in the medium. This step can be performed according to well-developed procedures from network and percolation theory for nondispersion flow. The novel feature in the model, however, would be the presence of equations from single-capillary theory to describe the formation of lamellae at nodes where tubes of different radii meet and their subsequent flow, splitting at other pore throats, and destruction by film drainage. The result should be equations that meaningfully describe the droplet size population and flow rates as a function of pressure (both absolute and differential across the medium). 

The importance of this equation is that it demonstrates that 7 is a linear function of the test pressure P, as long as the transition pressure between diffusive flow and capillary flow is not reached or exceeded. Other variables that must be controlled in diffusion testing include (a) the filter membrane area, because it defines the effective area of pores or void fraction (b) the temperature, because it defines the solubility of gas in liquid and (c) the composition of the liquid phase, because the presence of solutes affects the solubility coefficient. 

There are two main corrections that have to be applied to the information obtained from the capillary rheometer. First, there is an entrance pressure drop when the molten polymer enters the capillary, which is taken into account through the entrance or Bagley correction. This pressure drop is related to elastic deformations of the melt at the entry of the capillary [15]. Secondly, the non-Newtonian shear rate is expressed in terms of an apparent viscosity (defined in terms of a Newtonian flow). The relationship between the non-Newtonian and Newtonian shear rates, expressed as in the following equation, is known as the Rabinowitch correction [13, 16] ... 

At given gas and liquid superficial velocities Jg and ji that are defined by the volumetric flow rate of the respective phase divided by the cross-sectional area of the microchannel, de Mas et cd. [30] solved the force balance for the two co-flowing fluid phases. A capillary with a diameter equal to the hydraulic diameter of the microreactor was considered (Figure 11.2). The annular flow was assumed to be axisym-metric, laminar and fully developed. Using a constant pressure gradient in the gas and liquid phases along the streamwise direction, the flow satisfies the equations... 

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