Solution 3.

See text and/or instructor's solution manual.

Answer.   The transformation  ,  of Exercise 9 in Section 11.9, is known to map a horizontal flow in the upper half plane    

onto flow the upper half plane    that lies above the inclined segment from  .    

The derivative is   ,   integrate and get   ,  

then use the conditions     and     and obtain   .

Furthermore, the image of horizontal streamlines in the z-plane are curves in the w-plane given by the parametric equation

            ,    for    .  

Solution.   Along the x-axis use the points    and in the w-plane use  , ,   respectively.  

The exterior angles are   ,  

and the formula for the derivative is  given by the Schwarz-Christoffel formula  

                      

Integrate and get

                       

        The images of   ,   are   ,   respectively.

Use    and  ,  and obtain the system of equations

                      

Which simplifies to be   

                      

The values    are solutions for this system of equations.
            
Therefore,   

                    .  

We are done.   

Aside.  We can let Mathematica double check our work.

Remark.  The term    is undefined, but by convention we know that   .  

We are really done.   

        We can let Mathematica graph some of the streamlines.

        The flow around one inclined segment in the upper half-plane, as shown in Figure 11.90 (a).  

                    The streamlines in the w-plane given by the parametric equation

                    ,    for    .  

We are really really done.   

Aside.  We can extend the flow into the third and fourth quadrants using symmetry.

        The flow around two inclined segments forming a "V" in the plane, as shown in Figure 11.90 (b).  

                    The streamlines in the w-plane given by the parametric equation

                    ,    for    .  

We are really really really done.   

Aside.  We can use Mathematica to graph   .  

                    The image of the upper half plane    under the mapping   .  

We are really really really really done.   

Aside.  We can extend the mapping into the third and fourth quadrants using symmetry.

                      The image of the complex plane under the mapping   .  

This solution is complements of the authors.

(c) 2008 John H. Mathews, Russell W. Howell