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Ordinary Differential Equations
THE LAMINAR BOUNDARY LAYER ALONG A FLAT PLATE[pic 1]The boundary layer along a flat plate at zero incidenceMomentum TransferÂ (Blasiusâ solution)The boundary layer equations for momentum transfer are as followings:[pic 2][pic 3]with the boundary conditions: at y = 0, u = v = 0; and at y = â, u = uâSimilarity Transformation: Transforming the independent variables x and y to Îˇ, and the dependent variable Î¨Â (stream function) to f(Îˇ) as the following definitions:[pic 4]Â and [pic 5]Then, the boundary layer equations become a single ordinary differential equation as:[pic 6]Â (Blasiusâs equation)with the boundary conditions as: atÎˇ= 0, f = fâ = 0 and atÎˇ= â, fâ = 1.The differential equation is solved numerically and the solution is [pic 7]Velocity distribution in the boundary layer along a flat plate, after Blasius [2]The function [pic 8]for the boundary layer a flat plate at zero incidence, after L. Howarth [16][pic 9]ff[pic 10]fâ0.0000.332060.20.006640.66410.331990.40.026560.132770.331470.60.059740.198940.330080.80.106110.264710.327391.00.165570.329790.323011.20.237950.393780.316591.40.322980.456270.307871.60.420320.516760.296671.80.529520.574770.282932.00.650030.629770.266752.20.781200.671320.248352.40.922300.728990.228092.61.072520.772460.206462.81.230990.811520.184013.01.396820.846050.161363.21.569110.876090.139133.41.746960.901990.117883.61.929540.923330.098093.82.116050.941120.080134.02.305760.955520.064244.22.498060.966960.050524.42.692380.975870.038974.62.888260.982690.029484.83.085340.987790.021875.03.283290.991550.015915.23.481890.994250.011345.43.680940.996160.007935.63.880310.997480.005435.84.079900.998380.003656.04.279640.998980.002406.24.479480.999370.001556.44.679380.999610.000986.64.879310.999770.000616.85.079280.999870.000377.05.279260.999920.000227.25.479250.999960.000137.45.679240.999980.000077.65.879240.999990.000047.86.079231.000000.000028.06.279231.00000.000018.26.479231.000000.000018.46.679231.000000.000008.66.879231.000000.000008.87.079231.000000.00000Boundary layer thickness Î´: The boundary layer thickness is defined as the distance from surface at u/uâÂ = 0.99. Â From the above table, it is atÎˇ= 5; thus,