Question

5. Boundary Layer theory: a. In this problem we consider a two-dimensional boundary layer problem that is not adjacent to a solid boundary. Consider a semi-infinite fluid occupying the domain

x>0 to the right of a wall along the y-axis at x=0. At the origin, there is a small hole of height b through which we introduce a jet of uniform velocity u(0) across b. The two-dimensional mass flux pu(0)b is small since b is small. Naturally, for x>0, u depends on both xand y. However, if u is large enough, the momentum flux is not small and so this problem involves a momentum source that is a negligible mass source. Naturally, the further downstream inx that one looks, due to the fluid viscosity, the jet drags along ever more layers of fluid with it,and so the height of the fluid being dragged in the x direction by the jet, i.e., the boundary layer,grows with x. This is the region that we seek to describe in this problem. a. Write an expression for the momentum flux at x=0. (Recall the expression for th3e mass flux above.) Write an expression for the momentum flux, M, at any x>0. Find dM/dx? b. What is U, the potential flow very far (y→±0) from the jet's centerline? What is the result in boundary layer equation in terms of the stream function? c. Show that dM/dx=0 by integrating the x-component of the boundary layer equations in terms of u and v times p dy from -o to +0 and using a manipulation similar to the one that Adeyinka used in deriving the Karman Pohlhaussen method, i.e., adding and subtracting uðu/ðx and relating derivatives of u and v from the continuity equation. What does dM/dx =0 tell you about how the velocity at y=0 changes with x? d. Write out the boundary conditions as v=0, at x=0 and at v=±0. e. Non-dimensionalize the equation and do a scaling analysis to find the form of the similarity solution i.e find (x,y) (x.Y)along with the boundary conditions in terms of f(n).

Fig: 1

Fig: 2

Fig: 3

Fig: 4

Fig: 5

Fig: 6