The Falkner-Skan differential equation is a third-order nonlinear ordinary differential equation that describes the steady, incompressible, laminar boundary layer flow over a wedge. It is a fundamental equation in fluid dynamics, particularly in the study of boundary layer theory.
Understanding the Equation
The equation is named after V. M. Falkner and S. W. Skan, who first derived it in 1931. It describes the velocity profile of a fluid flowing over a wedge with a certain angle. The equation is written as:
f'''(η) + f(η)f''(η) + β(1 - [f'(η)]^2) = 0where:
- f(η) is the dimensionless stream function,
- η is the similarity variable,
- β is the wedge angle parameter, which is related to the angle of the wedge.
Applications
The Falkner-Skan equation has numerous applications in various fields, including:
- Aerodynamics: Understanding flow over airfoils and wings.
- Fluid mechanics: Analyzing boundary layer flow in different geometries.
- Heat transfer: Studying heat transfer in boundary layer flows.
- Chemical engineering: Modeling flow in reactors and pipes.
Solutions
The Falkner-Skan equation does not have an analytical solution for all values of β. However, it can be solved numerically using different methods, such as shooting methods or finite difference methods.
Practical Insights
- The Falkner-Skan equation provides a simplified model for understanding the complex behavior of boundary layers.
- It helps in predicting the flow separation and transition to turbulence.
- It has significant applications in designing aircraft and other aerodynamic structures.
Examples
- Flat plate: When β = 0, the Falkner-Skan equation reduces to the Blasius equation, which describes the flow over a flat plate.
- Wedge flow: For different values of β, the equation describes flow over wedges with different angles.
