# Capillary pressure

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In fluid statics, capillary pressure (${\displaystyle {p_{c}}}$) is the pressure between two immiscible fluids in a thin tube (see capillary action), resulting from the interactions of forces between the fluids and solid walls of the tube. Capillary pressure can serve as both an opposing or driving force for fluid transport and is a significant property for research and industrial purposes (namely microfluidic design and oil extraction from porous rock). It is also observed in natural phenomena.

## Definition

Capillary pressure is defined as:

${\displaystyle p_{c}=p_{\text{non-wetting phase}}-p_{\text{wetting phase}}}$

where:

${\displaystyle p_{\text{c}}}$is the capillary pressure
${\displaystyle p_{\text{non-wetting phase}}}$ is the pressure of the non-wetting phase
${\displaystyle p_{\text{wetting phase}}}$ is the pressure of the wetting phase

The wetting phase is identified by its ability to preferentially diffuse across the capillary walls before the non-wetting phase. The "wettability" of a fluid depends on its surface tension, the forces that drive a fluid's tendency to take up the minimal amount of space possible, and it is determined by the contact angle of the fluid. [1] A fluid's "wettability" can be controlled by varying capillary surface properties (e.g. roughness, hydrophilicity). However, in oil-water systems, water is typically the wetting phase, while for gas-oil systems, oil is typically the non-wetting phase. Regardless of the system, a pressure difference arises at the resulting curved interface between the two fluids. [2]

## Equations

Capillary pressure formulas are derived from the pressure relationship between two fluid phases in a capillary tube in equilibrium, which is that force up = force down. These forces are described as: [1]

${\displaystyle {\text{force up = interfacial tension of the fluid(s) acting along the perimeter of the capillary tube}}}$
${\displaystyle {\text{force down = (density gradient difference) x (cross-sectional area) x (height of the capillary rise in the tube)}}}$

These forces can be described by the interfacial tension and contact angle of the fluids, and the radius of the capillary tube. An interesting phenomena, capillary rise of water (as pictured to the right) provides a good example of how these properties come together to drive flow through a capillary tube and how these properties are measured in a system. There are two general equations that describe the force up and force down relationship of two fluids in equilibrium.

The Young–Laplace equation is the force up description of capillary pressure, and the most commonly used variation of the capillary pressure equation: [2] [1]

${\displaystyle p_{c}={\frac {2\gamma \cos \theta }{r_{c}}}}$

where:

${\displaystyle \gamma }$ is the interfacial tension
${\displaystyle r}$ is the effective radius of the interface
${\displaystyle \theta }$ is the wetting angle of the liquid on the surface of the capillary

The force down formula for capillary pressure is seen as: [1]

${\displaystyle p_{c}={\frac {\pi r^{2}h(\Gamma _{w}-\Gamma _{nw})}{\pi r^{2}}}=h(\Gamma _{w}-\Gamma _{nw})}$

where:

${\displaystyle h}$ is the height of the capillary rise
${\displaystyle \Gamma _{w}}$ is the density gradient of the wetting phase
${\displaystyle \Gamma _{nw}}$ is the density gradient of the non-wetting phase

## Applications

### Microfluidics

Microfluidics is the study and design of the control or transport of small volumes of fluid flow through porous material or narrow channels for a variety of applications (e.g. mixing, separations). Capillary pressure is one of many geometry-related characteristics that can be altered in a microfluidic device to optimize a certain process. For instance, as the capillary pressure increases, a wettable surface in a channel will pull the liquid through the conduit. This eliminates the need for a pump in the system, and can make the desired process completely autonomous. Capillary pressure can also be utilized to block fluid flow in a microfluidic device.

The capillary pressure in a microchannel can be described as:

${\displaystyle p_{c}=-\gamma \left({\frac {cos\theta _{b}+cos\theta _{t}}{d}}+{\frac {cos\theta _{l}+cos\theta _{r}}{w}}\right)}$

where:

${\displaystyle {\gamma }}$ is the surface tension of the liquid
${\displaystyle {\theta _{b}}}$ is the contact angle at the bottom
${\displaystyle {\theta _{t}}}$ is the contact angle at the top
${\displaystyle {\theta _{l}}}$ is the contact angle at the left side of the channel
${\displaystyle {\theta _{r}}}$ is the contact angles at the right side of the channel
${\displaystyle {d}}$ is the depth
${\displaystyle {w}}$ is the width

Thus, the capillary pressure can be altered by changing the surface tension of the fluid, contact angles of the fluid, or the depth and width of the device channels. To change the surface tension, one can apply a surfactant to the capillary walls. The contact angles vary by sudden expansion or contraction within the device channels. A positive capillary pressure represents a valve on the fluid flow while a negative pressure represents the fluid being pulled into the microchannel. [3]

#### Measurement Methods

Methods for taking physical measurements of capillary pressure in a microchannel have not been thoroughly studied, despite the need for accurate pressure measurements in microfluidics. The primary issue with measuring the pressure in microfluidic devices is that the volume of fluid is too small to be used in standard pressure measurement tools. Some studies have presented the use of microballoons, which are size-changing pressure sensors. Servo-nulling, which is historically used for measuring blood pressure, has also been demonstrated to provide pressure information in microfluidic channels with the assistance of a LabVIEW control system. Essentially, a micropipette is immersed in the microchannel fluid and is programmed to respond to changes in the fluid meniscus. A displacement in the meniscus of the fluid in the micropipette induces a voltage drop, which triggers a pump to restore the original position of the meniscus. The pressure exerted by the pump is interpreted as the pressure within the microchannel. [4]

#### Examples

Current research in microfluidics is focused on developing point-of-care diagnostics and cell sorting techniques (see lab-on-a-chip), and understanding cell behavior (e.g. cell growth, cell aging). In the field of diagnostics, the lateral flow test is a common microfluidic device platform that utilizes capillary forces to drive fluid transport through a porous membrane. The most famous lateral flow test is the take home pregnancy test, in which bodily fluid initially wets and then flows through the porous membrane, often cellulose or glass fiber, upon reaching a capture line to indicate a positive or negative signal. An advantage to this design, and several other microfluidic devices, is its simplicity (for example, its lack of human intervention during operation) and low cost. However, a disadvantage to these tests is that capillary action cannot be controlled after it has started, so the test time cannot be sped up or slowed down (which could pose an issue if certain time-dependent processes are to take place during the fluid flow). [5]

Another example of point-of-care work involving a capillary pressure-related design component is the separation of plasma from whole blood by filtration through porous membrane. Efficient and high-volume separation of plasma from whole blood is often necessary for infectious disease diagnostics, like the HIV viral load test. However, this task is often performed through centrifugation, which is limited to clinical laboratory settings. An example of this point-of-care filtration device is a packed-bed filter, which has demonstrated the ability to separate plasma and whole blood by utilizing asymmetric capillary forces within the membrane pores. [6]

### Petrochemical industry

Capillary pressure plays a vital role in extracting sub-surface hydrocarbons (such as petroleum or natural gas) from underneath porous reservoir rocks. Its measurements are utilized to predict reservoir fluid saturations and cap-rock seal capacity, and for assessing relative permeability (the ability of a fluid to be transported in the presence of a second immiscible fluid) data. [7] Additionally, capillary pressure in porous rocks has been shown to affect phase behavior of the reservoir fluids, thus influencing extraction methods and recovery. [8] It is crucial to understand these geological properties of the reservoir for its development, production, and management (e.g. how easy it is to extract the hydrocarbons).

[ dubious ]The Deepwater Horizon oil spill is an example of why capillary pressure is significant to the petrochemical industry. It is believed that upon the Deepwater Horizon oil rig’s explosion in the Gulf of Mexico in 2010, methane gas had broken through a recently implemented seal, and expanded up and out of the rig. Although capillary pressure studies (or potentially a lack thereof) do not necessarily sit at the root of this particular oil spill, capillary pressure measurements yield crucial information for understanding reservoir properties that could have influenced the engineering decisions made in the Deepwater Horizon event. [9]

Capillary pressure, as seen in petroleum engineering, is often modeled in a laboratory where it is recorded as the pressure required to displace some wetting phase by a non-wetting phase to establish equilibrium. [10] For reference, capillary pressures between air and brine (which is a significant system in the petrochemical industry) have been shown to range between 0.67 and 9.5 MPa. [11] There are various ways to predict, measure, or calculate capillary pressure relationships in the oil and gas industry. These include the following: [7]

#### Leverett J-function

The Leverett J-function serves to provide a relationship between the capillary pressure and the pore structure (see Leverett J-function).

#### Mercury Injection

This method is well suited to irregular rock samples (e.g. those found in drill cuttings) and is typically used to understand the relationship between capillary pressure and the porous structure of the sample. [12] In this method, the pores of the sample rock are evacuated, followed by mercury filling the pores with increasing pressure. Meanwhile, the volume of mercury at each given pressure is recorded and given as a pore size distribution, or converted to relevant oil/gas data. One pitfall to this method is that it does not account for fluid-surface interactions. However, the entire process of injecting mercury and collecting data occurs rapidly in comparison to other methods. [7]

#### Porous Plate Method

The Porous Plate Method is an accurate way to understand capillary pressure relationships in fluid-air systems. In this process, a sample saturated with water is placed on a flat plate, also saturated with water, inside a gas chamber. Gas is injected at increasing pressures, thus displacing the water through the plate. The pressure of the gas represents the capillary pressure, and the amount of water ejected from the porous plate is correlated to the water saturation of the sample. [7]

#### Centrifuge Method

The centrifuge method relies on the following relationship between capillary pressure and gravity: [7]

${\displaystyle p_{c}=hg(\rho _{w}-\rho _{nw})}$

where:

${\displaystyle h}$ is the height of the capillary rise
${\displaystyle g}$ is gravity
${\displaystyle \rho _{w}}$ is the density of the wetting phase
${\displaystyle \rho _{nw}}$ is the density of the non-wetting phase

The centrifugal force essentially serves as an applied capillary pressure for small test plugs, often composed of brine and oil. During the centrifugation process, a given amount of brine is expelled from the plug at certain centrifugal rates of rotation. A glass vial measures the amount of fluid as it is being expelled, and these readings result in a curve that relates rotation speeds with drainage amounts. The rotation speed is correlated to capillary pressure by the following equation:

${\displaystyle p_{c}=7.9e^{-8}(\rho _{1}-\rho _{2})\omega ^{2}(r_{b}^{2}-r_{t}^{2})}$

where:

${\displaystyle r_{b}}$ is the radius of rotation of the bottom of the core sample
${\displaystyle r_{t}}$ is the radius of rotation of the top of the core sample
${\displaystyle \omega }$ is the rotational speed

The primary benefits to this method are that it's rapid (producing curves in a matter of hours) and is not restricted to being performed at certain temperatures. [13]

Other methods include the Vapor Pressure Method, Gravity-Equilibrium Method, Dynamic Method, Semi-dynamic Method, and the Transient Method.

#### Correlations

In addition to measuring the capillary pressure in a laboratory setting to model that of an oil/natural gas reservoir, there exist several relationships to describe the capillary pressure given specific rock and extraction conditions. For example, R. H. Brooks and A. T. Corey developed a relationship for capillary pressure during the drainage of oil from an oil-saturated porous medium experiencing a gas invasion: [14]

${\displaystyle p_{cgo}=p_{t}({\frac {1-S_{or}}{S_{o}-S_{or}}})^{(1/\lambda )}}$

where:

${\displaystyle P_{cgo}}$ is the capillary pressure between oil and gas phases
${\displaystyle S_{o}}$ is the oil saturation
${\displaystyle S_{or}}$ is the residual oil saturation that remains trapped in the pore at high capillary pressure
${\displaystyle P_{t}}$ is the threshold pressure (the pressure at which the gas phase is allowed to flow)
${\displaystyle \lambda }$ is a parameter that is related to the distribution of pore sizes
${\displaystyle \lambda >2}$ for narrow distributions
${\displaystyle \lambda <2}$ for wide distributions

Additionally, R. G. Bentsen and J. Anli developed a correlation for the capillary pressure during the drainage from a porous rock sample in which an oil phase displaces saturated water: [15]

${\displaystyle p_{cow}=p_{t}-p_{cs}ln({\frac {S_{w}-S_{wi}}{1-S_{wi}}})}$

where:

${\displaystyle P_{cow}}$ is the capillary pressure between oil and water phases
${\displaystyle P_{cs}}$ is a parameter that controls the shape of the capillary pressure function
${\displaystyle ({\frac {S_{w}-S_{wi}}{1-S_{wi}}})}$ is the normalized wetting-phase saturation
${\displaystyle S_{w}}$ is the saturation of the wetting phase
${\displaystyle S_{wi}}$ is the irreducible wetting-phase saturation

## Averaging capillary pressure vs. water saturation curves

It has been shown that as reservoir simulators use the primary drainage capillary pressure data for saturation-height modeling calculations, primary drainage capillary pressure data should be averaged in the same manner that water saturations are averaged. Also, as reservoir simulators use the imbibition and secondary drainage capillary pressure data for fluids displacement calculations, these capillary pressures should not be averaged like primary drainage capillary pressure data. These can be averaged by Leverett J-function. The averaging equations are as follows [16]

### averaging primary drainage capillary pressure vs. normalized saturation data

${\displaystyle {\text{average Pc}}={\frac {\sum _{i=1}^{n}\left({\phi V_{\mathit {b}}P_{\mathit {c}}}\right)_{i}}{\sum _{i=1}^{n}\left({\phi V_{\mathit {b}}}\right)_{i}}}}$

in which ${\displaystyle n}$ is the number of core samples, ${\displaystyle \phi }$ is the effective porosity, ${\displaystyle Vb}$ is the bulk volume of sample, and ${\displaystyle Pc}$ is the primary drainage capillary pressure data vs. normalized water saturation.

### averaging imbibition and secondary drainage capillary pressure vs. normalized saturation data

${\displaystyle {\text{average Pc}}={\frac {\sum _{i=1}^{n}\left({\frac {P_{\mathit {c}}{\sqrt {k/\phi }}}{\gamma }}\right)_{i}}{\sum _{i=1}^{n}\left({\frac {\sqrt {k/\phi }}{\gamma }}\right)_{i}}}}$

in which ${\displaystyle n}$ is the number of core samples, ${\displaystyle \phi }$ is the effective porosity, ${\displaystyle k}$ is the absolute permeability, ${\displaystyle \gamma }$ is the interfacial tension or IFT, and ${\displaystyle Pc}$ is the imbibition or secondary drainage capilalry pressure data vs. normalized water saturation.

## In nature

### Needle ice

In addition to being manipulated for medical and energy applications, capillary pressure is the cause behind various natural phenomena as well. For example, needle ice, seen in cold soil, occurs via capillary action. The first major contributions to the study of needle ice, or simply, frost heaving were made by Stephen Taber (1929) and Gunnar Beskow (1935), who independently aimed to understand soil freezing. Taber’s initial work was related to understanding how the size of pores within the ground influenced the amount of frost heave. He also discovered that frost heave is favorable for crystal growth and that a gradient of soil moisture tension drives water upward toward the freezing front near the top of the ground. [17] In Beskow’s studies, he defined this soil moisture tension as “capillary pressure” (and soil water as “capillary water”). Beskow determined that the soil type and effective stress on the soil particles influenced frost heave, where effective stress is the sum of pressure from above ground and the capillary pressure. [18]

In 1961, D.H. Everett elaborated on Taber and Beskow’s studies to understand why pore spaces filled with ice continue to experience ice growth. He utilized thermodynamic equilibrium principles, a piston cylinder model for ice growth and the following equation to understand the freezing of water in porous media (directly applicable to the formation of needle ice):

${\displaystyle P_{s}-P_{l}=\Psi _{sl}{\frac {dA_{r}}{dV}}=\Psi _{sl}{\tilde {K}}}$

where:

${\displaystyle {P_{s}}}$ is the pressure of the solid crystal
${\displaystyle {P_{l}}}$ is the pressure in the surrounding liquid
${\displaystyle {\Psi _{sl}}}$ is the interfacial tension between the solid and the liquid
${\displaystyle {A_{r}}}$ is the surface area of the phase boundary
${\displaystyle {V}}$ is the volume of the crystal
${\displaystyle {\tilde {K}}}$ is the mean curvature of the solid/liquid interface

With this equation and model, Everett noted the behavior of water and ice given different pressure conditions at the solid-liquid interface. Everett determined that if the pressure of the ice is equal to the pressure of the liquid underneath the surface, ice growth is unable to continue into the capillary. Thus, with additional heat loss, it is most favorable for water to travel up the capillary and freeze in the top cylinder (as needle ice continues to grow atop itself above the soil surface). As the pressure of the ice increases, a curved interface between the solid and liquid arises and the ice will either melt, or equilibrium will be reestablished so that further heat loss again leads to ice formation. Overall, Everett determined that frost heaving (analogous to the development of needle ice) occurs as a function of the pore size in the soil and the energy at the interface of ice and water. Unfortunately, the downside to Everett's model is that he did not consider soil particle effects on the surface. [19] [20]

### Circulatory system

Capillaries in the circulatory system are vital to providing nutrients and excreting waste throughout the body. There exist pressure gradients (due to hydrostatic and oncotic pressures) in the capillaries that control blood flow at the capillary level, and ultimately influence the capillary exchange processes (e.g. fluid flux). [21] Due to limitations in technology and bodily structure, most studies of capillary activity are done in the retina, lip and skin, historically through cannulation or a servo-nulling system. Capillaroscopy has been used to visualize capillaries in the skin in 2D, and has been reported to observe an average range of capillary pressure of 10.5 to 22.5 mmHg in humans, and an increase in pressure among people with type 1 diabetes and hypertension. Relative to other components of the circulatory system, capillary pressure is low, as to avoid rupturing, but sufficient for facilitating capillary functions. [22]

## Related Research Articles

Surface tension is the tendency of liquid surfaces at rest to shrink into the minimum surface area possible. Surface tension is what allows objects with a higher density than water such as razor blades and insects to float on a water surface without becoming even partly submerged.

Electrowetting is the modification of the wetting properties of a surface with an applied electric field.

In physics, Washburn's equation describes capillary flow in a bundle of parallel cylindrical tubes; it is extended with some issues also to imbibition into porous materials. The equation is named after Edward Wight Washburn; also known as Lucas–Washburn equation, considering that Richard Lucas wrote a similar paper three years earlier, or the Bell-Cameron-Lucas-Washburn equation, considering J.M. Bell and F.K. Cameron's discovery of the form of the equation in 1906.

Wetting is the ability of a liquid to maintain contact with a solid surface, resulting from intermolecular interactions when the two are brought together. This happens in presence of a gaseous phase or another liquid phase not miscible with the first one. The degree of wetting (wettability) is determined by a force balance between adhesive and cohesive forces.

The contact angle is the angle, conventionally measured through the liquid, where a liquid–vapor interface meets a solid surface. It quantifies the wettability of a solid surface by a liquid via the Young equation. A given system of solid, liquid, and vapor at a given temperature and pressure has a unique equilibrium contact angle. However, in practice a dynamic phenomenon of contact angle hysteresis is often observed, ranging from the advancing (maximal) contact angle to the receding (minimal) contact angle. The equilibrium contact is within those values, and can be calculated from them. The equilibrium contact angle reflects the relative strength of the liquid, solid, and vapour molecular interaction.

Cassie's law, or the Cassie equation, describes the effective contact angle θc for a liquid on a chemically heterogeneous surface, i.e. the surface of a composite material consisting of different chemistries, that is non uniform throughout. Contact angles are important as they quantify a surface's wettability, the nature of solid-fluid intermolecular interactions. Cassie's law is reserved for when a liquid completely covers both smooth and rough heterogeneous surfaces.

In fluid dynamics, the Buckley–Leverett equation is a conservation equation used to model two-phase flow in porous media. The Buckley–Leverett equation or the Buckley–Leverett displacement describes an immiscible displacement process, such as the displacement of oil by water, in a one-dimensional or quasi-one-dimensional reservoir. This equation can be derived from the mass conservation equations of two-phase flow, under the assumptions listed below.

In petroleum engineering, the Leverett J-function is a dimensionless function of water saturation describing the capillary pressure,

In physics, the Young–Laplace equation is a nonlinear partial differential equation that describes the capillary pressure difference sustained across the interface between two static fluids, such as water and air, due to the phenomenon of surface tension or wall tension, although use of the latter is only applicable if assuming that the wall is very thin. The Young–Laplace equation relates the pressure difference to the shape of the surface or wall and it is fundamentally important in the study of static capillary surfaces. It is a statement of normal stress balance for static fluids meeting at an interface, where the interface is treated as a surface :

The capillary length or capillary constant, is a length scaling factor that relates gravity and surface tension. It is a fundamental physical property that governs the behavior of menisci, and is found when body forces (gravity) and surface forces are in equilibrium.

HydroGeoSphere (HGS) is a 3D control-volume finite element groundwater model, and is based on a rigorous conceptualization of the hydrologic system consisting of surface and subsurface flow regimes. The model is designed to take into account all key components of the hydrologic cycle. For each time step, the model solves surface and subsurface flow, solute and energy transport equations simultaneously, and provides a complete water and solute balance.

Capillary condensation is the "process by which multilayer adsorption from the vapor [phase] into a porous medium proceeds to the point at which pore spaces become filled with condensed liquid from the vapor [phase]." The unique aspect of capillary condensation is that vapor condensation occurs below the saturation vapor pressure, Psat, of the pure liquid. This result is due to an increased number of van der Waals interactions between vapor phase molecules inside the confined space of a capillary. Once condensation has occurred, a meniscus immediately forms at the liquid-vapor interface which allows for equilibrium below the saturation vapor pressure. Meniscus formation is dependent on the surface tension of the liquid and the shape of the capillary, as shown by the Young-Laplace equation. As with any liquid-vapor interface involving a meniscus, the Kelvin equation provides a relation for the difference between the equilibrium vapor pressure and the saturation vapor pressure. A capillary does not necessarily have to be a tubular, closed shape, but can be any confined space with respect to its surroundings.

The Amott test is one of the most widely used empirical wettability measurements for reservoir cores in petroleum engineering. The method combines two spontaneous imbibition measurements and two forced displacement measurements. This test defines two different indices: the Amott water index and the Amott oil index.

Jurin's law, or capillary rise, is the simplest analysis of capillary action—the induced motion of liquids in small channels—and states that the maximum height of a liquid in a capillary tube is inversely proportional to the tube's diameter. Capillary action is one of the most common fluid mechanical effects explored in the field of microfluidics. Jurin's law is named after James Jurin, who discovered it between 1718 and 1719. His quantitative law suggests that the maximum height of liquid in a capillary tube is inversely proportional to the tube's diameter. The difference in height between the surroundings of the tube and the inside, as well as the shape of the meniscus, are caused by capillary action. The mathematical expression of this law can be derived directly from hydrostatic principles and the Young–Laplace equation. Jurin's law allows the measurement of the surface tension of a liquid and can be used to derive the capillary length.

Elasto-capillarity is the ability of capillary force to deform an elastic material. From the viewpoint of mechanics, elastocapillarity phenomena essentially involve competition between the elastic strain energy in the bulk and the energy on the surfaces/interfaces. In the modeling of these phenomena, some challenging issues are, among others, the exact characterization of energies at the micro scale, the solution of strongly nonlinear problems of structures with large deformation and moving boundary conditions, and instability of either solid structures or droplets/films.The capillary forces are generally negligible in the analysis of macroscopic structures but often play a significant role in many phenomena at small scales.

The finite water-content vadose zone flux method represents a one-dimensional alternative to the numerical solution of Richards' equation for simulating the movement of water in unsaturated soils. The finite water-content method solves the advection-like term of the Soil Moisture Velocity Equation, which is an ordinary differential equation alternative to the Richards partial differential equation. The Richards equation is difficult to approximate in general because it does not have a closed-form analytical solution except in a few cases. The finite water-content method, is perhaps the first generic replacement for the numerical solution of the Richards' equation. The finite water-content solution has several advantages over the Richards equation solution. First, as an ordinary differential equation it is explicit, guaranteed to converge and computationally inexpensive to solve. Second, using a finite volume solution methodology it is guaranteed to conserve mass. The finite water content method readily simulates sharp wetting fronts, something that the Richards solution struggles with. The main limiting assumption required to use the finite water-content method is that the soil be homogeneous in layers.

The rise in core (RIC) method is an alternate reservoir wettability characterization method described by S. Ghedan and C. H. Canbaz in 2014. The method enables estimation of all wetting regions such as strongly water wet, intermediate water, oil wet and strongly oil wet regions in relatively quick and accurate measurements in terms of Contact angle rather than wettability index.

Morris Muskat et al. developed the governing equations for multiphase flow in porous media as a generalisation of Darcy's equation for water flow in porous media. The porous media are usually sedimentary rocks such as clastic rocks or carbonate rocks.

In the theory of capillarity, Bosanquet equation is an improved modification of the simpler Lucas–Washburn theory for the motion of a liquid in a thin capillary tube or a porous material that can be approximated as a large collection of capillaries. In the Lucas–Washburn model, the inertia of the fluid is ignored, leading to the assumption that flow is continuous under constant viscous laminar Poiseuille flow conditions without considering effects of mass transport undergoing acceleration occurring at the start of flow and at points of changing internal capillary geometry. The Bosanquet equation is a differential equation that is second-order in the time derivative, similar to Newton's Second Law, and therefore takes into account the fluid inertia. Equations of motion, like the Washburn's equation, that attempt to explain a velocity as proportional to a driving force are often described with the term Aristotelian mechanics.

Liquid phase sintering is a sintering technique that uses a liquid phase to accelerate the interparticle bonding of the solid phase. In addition to rapid initial particle rearrangement due to capillary forces, mass transport through liquid is generally orders of magnitude faster than through solid, enhancing the diffusional mechanisms that drive densification. The liquid phase can be obtained either through mixing different powders—melting one component or forming a eutectic—or by sintering at a temperature between the liquidus and solidus. Additionally, since the softer phase is generally the first to melt, the resulting microstructure typically consists of hard particles in a ductile matrix, increasing the toughness of an otherwise brittle component. However, liquid phase sintering is inherently less predictable than solid phase sintering due to the complexity added by the presence of additional phases and rapid solidification rates. Activated sintering is the solid-state analog to the process of liquid phase sintering.

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