Determination of Two-Dimensional Solutions to the Navier-Stokes Equations

Abstract

The Navier-Stokes Equations describe the motion of a viscous fluid. Although they have a wide range of practical uses, finding an analytical solution is nearly impossible, except for under certain simplifications. We examine the problem of a constant unidirectional flow applied to a semi-in finite plate. Several theorems and properties, including Reynold's Transport Theorem and the Divergence Theorem, are used to derive the Navier-Stokes Equations from first principles. This is followed by a Nondimensionalization, which puts the equation in a dimensionless form, with no fi xed length or time scales. We use the Similarity Solution method to introduce a change of variables, which reduces the PDE to a soluble ODE, the Blasius Equation. We solve the ODE numerically after considering various boundary conditions and assumptions. Finally, the results of the numerical solution are transformed back into the original spatial variables and discussed in terms of Boundary Layer Theory.