Mathematical Proceedings of the Cambridge Philosophical Society · 1937 · 493 citations · 4 references
Natural SciencesPhysicsComputational Fluid DynamicsDifferential EquationsFluid DynamicsBoundary ConditionsBoundary-layer TheoryBoundary-value ProblemsIterative SolversNumerical AnalysisNumerical Methods For Differential EquationsAnalytical Solutions For Boundary LayersBoundary Condition ApproximationBoundary Condition TreatmentBoundary Layer ApproximationsFalkner-skan Boundary LayersLaminar Boundary LayersNumerical Analysis Of Partial Differential EquationsNumerical Solutions Of Partial Differential Equations
The differential analyser has been used to evaluate solutions of the equation with boundary conditions y = y ′ = 0 at x = 0, y ′ → 1 as x → ∞, which occurs in Falkner and Skan's approximate treatment of the laminar boundary layer. A numerical iterative method has been used to improve the accuracy of the solutions, and the results show that the accuracy of the machine solutions is about 1 in 1000, or rather better. It is shown that the conditions are insufficient to specify a unique solution for negative values of β a discussion of this situation is given, and it is shown that for the application to be made of the solution the appropriate condition is that y ′ → 1 from below, and as rapidly as possible, as x → ∞. The condition that y ′ → 1 from below can be satisfied only for values of β 0 , greater than a limiting value β 0 , whose value is approximately − 0·199, and which is related to the point at which the laminar boundary layer breaks away from the boundary.
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