Classification of boundary conditions

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For any problem to be well defined, there are boundary/initial conditions that must be applied. There are three basic boundary conditions for second order PDEs. These boundary conditions are the Dirichlet [\Phi  = f\left( {\eta ,\zeta } \right)], the Neumann [\partial \Phi /\partial \eta  = f\left( \zeta  \right), or \partial \Phi /\partial \zeta  = f\left( \eta  \right)], and the mixed [a\,(\partial \Phi /\partial \eta ) + b\Phi = f(\zeta )
, or a(\partial \Phi /\partial \zeta ) + b\Phi  = f(\eta ) ] type.

The number of boundary conditions that must be applied for each independent variable is equal to the highest order of that variable. For example, if the PDE is second order in both η and ζ, then two boundary conditions are needed for each independent variable for a total of four boundary conditions. If ζ is time and first order, then an initial Dirichlet-type boundary condition is needed, and if it is second order, then both a Dirichlet and Neumann boundary condition are needed.

There are implications to experimental measurements and numerical analysis based on the classification of the governing PDE. For example, experimental measurements in an incompressible flow field with moderate to low Reynolds numbers are very difficult, because the governing equations are highly elliptic. The elliptic nature means that disturbances downstream greatly affect the upstream flow field. Therefore, any measurements that require a device in the flow field may change the nature of that flow field, and are inherently inaccurate. Numerical simulations are very reliable for these cases when the flow is in the laminar regime. In the laminar flow regime, the full Navier-Stokes equations can be directly solved as an elliptic problem with no approximations.

In the compressible flow regime with high Reynolds numbers, the characteristic of the flow is parabolic or hyperbolic, depending on the Mach number. In these cases, disturbances produced downstream do not affect the upstream flow field; therefore measurement devices in the flow will give an accurate depiction of the flow field without the device. In numerical simulations, it is computationally efficient to solve a parabolic flow field, because disturbances propagate in one direction. However, a flow field that is truly parabolic usually has a high Reynolds number and therefore is turbulent, which means it is three-dimensional and transient. Turbulence modeling either involves a very fine mesh, which is very computationally expensive, or an averaging technique, which requires a closure problem. The classification of a particular problem will help reduce the time it takes to get reliable results.

References

Faghri, A., and Zhang, Y., 2006, Transport Phenomena in Multiphase Systems, Elsevier, Burlington, MA
Faghri, A., Zhang, Y., and Howell, J. R., 2010, Advanced Heat and Mass Transfer, Global Digital Press, Columbia, MO.

Further Reading

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