In fluid statics, capillary pressure () 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:
where:
is the capillary pressure
is the pressure of the non-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. 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 wetting phase. Regardless of the system, a pressure difference arises at the resulting curved interface between the two fluids.
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:
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:
where:
is the interfacial tension
is the effective radius of the interface
is the wetting angle of the liquid on the surface of the capillary
The force down formula for capillary pressure is seen as:
where:
is the height of the capillary rise
is the density gradient of the wetting phase
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:
where:
is the surface tension of the liquid
is the contact angle at the bottom
is the contact angle at the top
is the contact angle at the left side of the channel
is the contact angles at the right side of the channel
is the depth
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.