An Integral Equation Formulation of the Equations of Motion of an Incompressible Fluid
NAVAL UNDERSEA WARFARE CENTER DIV NEWPORT RI
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A set of coupled integral equations is derived from the incompressible Navier-Stokes equations and the continuity equation. These equations are based on a vorticity-velocity-enthalpy formulation and are exact. The equations consist of a generalization of the Biot-Savart law for determining the velocity, an integral expression of the momentum equation for determining the vorticity, and a boundary integral equation for determining the stagnation enthalpy. The equations are linear in each independent variable, with the nonlinearities entering only through the cross terms of the vorticity and velocity. They possess a number of interesting properties, including the total absence of spatial derivatives and the fact that the stagnation enthalpy, or pressure, is required only on the boundary of the fluid domain. In addition, since the vorticity is present in all volume integrals, the domain of integration in this case is restricted to the region of nonzero vorticity. All boundary conditions, and in particular the farfield boundary condition, are naturally incorporated in the formulation.
- Numerical Mathematics
- Theoretical Mathematics