The figure shows a viscous incompressible liquid flow- ng down a vertical surface. A boundary layer...

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The figure shows a viscous incompressible liquid flow- ng down a vertical surface. A boundary layer develops near the vertical surface and grows to approach the free surface. Assuming that the volume flow rate down the vall is Q (constant) formulate the problem as follows: Taking into account the force due to gravity, write down the boundary layer equations for this situation. Assume that the outer inviscid flow is one-di- mensional, i.e., U=U(x) and p=p(x), and show that U=[2gx] 1/2 and p=constant=0 by consider- ing the governing equations and boundary condi- tions for the inviscid region. From the equations show that the corresponding momentum integral for the boundary layer can be written as, 8(x) free 8(x) 2 u d SU an U dx [ ( ) Txy(x,0) VU d dn gd: 8 dn ; ( Consider the region x x* where x* is the position where the fluid layer depth eq boundary layer thickness. Assume that the velocity distribution can be approximate second order polynomial, i.e., u/U=a + b + cm where n = y/8 and determine the coe a,b, & c by enforcing reasonable conditions on this velocity distribution. Obtain an expression for the boundary layer thickness 8(x) for x x* . Hint: eval momentum integral and look for a solution of the form 8(x)=Cx. For xx* also use a second order polynomial for the velocity distribution and shov differential equation for the fluid layer thickness 8(x) is 6Q d8 dx 5g = 83 - 83 0 where S is a constant that you should determine. Hint: for x x* the velocity dist must satisfy u d = flow rate = constant = Q U The figure shows a viscous incompressible liquid flow- ng down a vertical surface. A boundary layer develops near the vertical surface and grows to approach the free surface. Assuming that the volume flow rate down the vall is Q (constant) formulate the problem as follows: Taking into account the force due to gravity, write down the boundary layer equations for this situation. Assume that the outer inviscid flow is one-di- mensional, i.e., U=U(x) and p=p(x), and show that U=[2gx] 1/2 and p=constant=0 by consider- ing the governing equations and boundary condi- tions for the inviscid region. From the equations show that the corresponding momentum integral for the boundary layer can be written as, 8(x) free 8(x) 2 u d SU an U dx [ ( ) Txy(x,0) VU d dn gd: 8 dn ; ( Consider the region x x* where x* is the position where the fluid layer depth eq boundary layer thickness. Assume that the velocity distribution can be approximate second order polynomial, i.e., u/U=a + b + cm where n = y/8 and determine the coe a,b, & c by enforcing reasonable conditions on this velocity distribution. Obtain an expression for the boundary layer thickness 8(x) for x x* . Hint: eval momentum integral and look for a solution of the form 8(x)=Cx. For xx* also use a second order polynomial for the velocity distribution and shov differential equation for the fluid layer thickness 8(x) is 6Q d8 dx 5g = 83 - 83 0 where S is a constant that you should determine. Hint: for x x* the velocity dist must satisfy u d = flow rate = constant = Q U