Boundary Conditions
Boundary Conditions Definition
A Boundary Condition is used when integrating a function over a domain. They can be used in microscopic differential balances to find the constants that are generated when integrating the differential equations that describe those balances (for example, the navier stokes equation). There are multiple types of boundary conditions.
Dirichlet Boundary Condition
The most basic type of boundary condition is a Dirichlet boundary condition [1]. A Dirichlet boundary condition is when the value of the function that is being derived in the balance is directly stated. For example, if the energy equation is being solved and one of the boundaries is a wall held at a constant temperature. After the differential balance is integrated twice and the temperature is defined as a function of position, the constant temperature of the wall can be plugged in with its corresponding position in its coordinate system, and one of the constants can be determined.
Neumann Boundary Condition
The other type of boundary condition commonly used in differential balances is a Neumann boundary condition [2]. This is a condition that is related to the derivative of the function being solved for at a boundary of the domain. The flux of the quantity is related to the derivative, so Neumann boundary conditions are often called flux conditions. A common flux related boundary condition is that there is no flux through the center of a domain in cylindrical coordinates when r=0 is included in the domain. These conditions can be plugged into the flux laws, which include the first derivative of the function being solved for. After integrating the differential balance once, a flux related boundary condition can be plugged into the flux law to find another constant of integration.
Common Boundary Conditions
No Slip Condition
The no slip condition is used in momentum transfer. The no slip condition says that the fluid that is in contact with a wall or boundary is not moving.
Pressure Related Conditions
Pressure related boundary conditions are used in momentum transfer. Oftentimes, pressure related conditions are found when one of the boundaries is the atmosphere, so pressure at that boundary is atmospheric pressure.
No Flux Through Center
In spherical and cylindrical coordinate systems, when the center of the domain is at a radius zero, there is no flux through that point. This condition applies to heat, mass, and momentum transfer.
Flux Through a Phase Change
When one of the boundaries is a phase change of solid to liquid, liquid to gas, or solid to gas, the flux through that boundary can often be modeled by multiplying a constant by the change over the boundary. An example of this is newtons law of cooling. This type of boundary condition is found in both heat and mass transfer.
References
1. Weisstein, Eric W. "Boundary Conditions." From MathWorld--A Wolfram Web Resource. http://mathworld.wolfram.com/BoundaryConditions.html