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Lecture: Boundary Value Problem Boundary Value Problem 1 The Blasius equation describes the non-dimensional velocity distribution in the laminar boundary layer over a flat plate. It describes the similarity solution of the fluid flow influenced by viscous effects . The Navier Stokes equation and the boundary conditions on the problem are u v 0; x y u u u 2u v 2 x y y (1) y 0 : u v 0; y : u U (2) where u and v are the velocity in the x and y direction. The problem is described by the Figure below where U is the free stream velocity. The pressure is assumed constant and is not solved in this simplification. U y U u (y) x Figure: Laminar flow over a flat plate Using a stream function the continuity equation, the first equation in Equation (1), is satisfied. Introducing the dimensionless coordinate η, and a dimensionless stream function f , the second equation in Equation ((1) is reduced to d3 f d2 f 0.5 f 0 d 3 d 2 The boundary conditions then translate to f (0) 0; df (0) 0; d df (6) 1; d (3) This equation, termed as the Blasius equation, is a third order nonlinear ordinary differential equation with boundary conditions at two points. Published solutions indicate that the infinity boundary condition is easily met at η = 6. The analytical solution was obtained by Blasius by a series expansion at η = 0 and asymptotic expansion for large η. It was solved numerically for the first time in 1912. A better solution was obtained by Howarth. Howarth’s solution can now be reproduced using a numerical boundary value problem solver. The Blasius problem is part of the set of problems defined by the Falkner-Skan equations, also appears to be a bench mark for comparing the different techniques for the solution to the NLBVP, particularly analytical ones. Boundary Value Problem 2 This example is also a fluid flow problem. The original problem definition is available in Schlichting [Boundary Layer Theory]. Figure below illustrates the features of the problem. A disk of radius R is rotating with the angular speed ω in still fluid. The flow is steady, incompressible, has constant property, and is axisymmetric. The fluid at the disk has to satisfy the no slip condition. The centrifugal effects cause the fluid to leave the disk radially near the disk. The flow above the disk must replace this airflow through a downward spiraling flow. A cylindrical coordinate system (r, θ, z) is used for description. Vr, Vθ, Vz, are the velocity components. p is the pressure, ν, the dynamic viscosity. The continuity and the Navier-Stokes equations are z Vz r Vr R V Figure: Flow over a rotating disk Vr Vr Vz 0 r r z V V 2 V Vr r Vz z r r z Vr V VrV V Vz r r z Vr Vz r Vz Vz z 2V 1 p V 2r r r r r r 2V 2 r V r r 2Vr 2 z 2V 2 z 2V 1 p 1 Vz 2Vz 2z 2 z r r z r (4 The boundary conditions are z 0 : Vr 0; V r ; Vz 0; z : Vr 0; V 0; (5) In Schlicting, it is derived that using a differential volume over the disk an approximate value for the boundary layer thickness can be found. This is used for scaling the z- coordinate. Next, scaling for the velocity and pressure are introduced so that dependence on r and z can be separated. This is an essential technique for reducing the PDE to ODE. The scaling relations for the various terms are z ; Vr r F (Z ); V r G(Z ); Vz H (Z ); p( z ) P(Z ); Zz (6) Substituting in Equations (4, 5), and simplifying with Z being the independent variable and primes representing derivatives with respect to Z, 2F H 0 2 2 F F H G F 0 2 FG HG G 0 P HH H 0 (7) The transformed boundary conditions are Z 0; F 0; G 1; H 0; P 0 Z ( 6); F 0; G 0 (8) The original Navier-Stokes equations and boundary conditions are now reduced to the set of ODE and corresponding boundary conditions in Equations (7, 8). Usually, the last differential equation for pressure is not solved with the first three. It can be obtained from the solution of F, G, and H. The solution to Equation (7, 8) was first obtained using a power series around Z =0, and an asymptotic series for large values of Z. The numerical solution was obtained by Sparrow and Gregg . Numerical solutions are currently accepted as alternate analytic solutions for this problem today. MATLAB is used to generate the numerical solutions used for comparison. Once again the final value of the residuals should provide reasonable confidence that the Bezier functions provide an excellent solution.