Develop the similarity forms of the boundary layer momentum and energy equations

Question

Develop the similarity forms of the boundary layer momentum and energy equations for uniform flow past an inclined wall (Uinfinity (x) = Cxm) as shown in the following sketch: The exponent in the free-stream velocity m is related to the wedge total angle by m = beta/(2pi - beta). For the momentum equation in the boundary layer, using the Bernoulli's equation in the free stream, we obtain an estimate for the pressure gradient as 1/rho dPinfinity/dx = -Uinfinity dUinfinity/dx, where Uinfinity (x) = C xm. Thus, the momentum boundary layer effectively becomes Apply the similarity transformation and show that the above momentum equation simplifies to the following similarity equation (known as the Falkner-Skan equation, which is a generalization of the Blasius equation 2f''' + (m + 1)ff'' + 2m[1 - (f')2] = 0 Apply the same transformation to the energy equation and show that the corresponding similarity equation now becomes 2 theta '' + Pr(m + 1)f theta ' = 0 where theta = T - T0/T infinity - T0. Establish whether the angle of inclination has any effect on the boundary condition to be used in conjunction with the equation above.

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