Exercise 3.   Consider the ideal fluid flow in the channel bounded by the hyperbolas      and      in the first quadrant,

where the complex potential is given by      and    is a positive real number.  

3 (a).   Find the formula for speed.   Find the point on the boundary at which the speed attains a minimum value.

3 (b).   Where is the pressure greatest?

Solution 3.

See text and/or instructor's solution manual.

Solution 3 (a).   The velocity vector is   

                    .  

Hence we obtain  

                    .  

The minimum is will occur at the point in the channel that is closest to the origin, i. e. at the point  :  

                    .  

Solution 3 (b).

      From part (a), the minimum speed occurs at the point in the channel that is closest to the origin, i. e. at the point  .

The pressure is greatest where the speed is least, which occurs at the point  .  

We are done.   

Aside.  We can let Mathematica double check our work.

We are really done.   

Aside. We can let Mathematica find the stream function.

Enter the complex potential and determine the stream function.  

Aside.  We can make a plot of the stream function   .  For illustration purposes, we choose  .  

                    The streamlines   ,     

                    for   .   

                    The contour graph   ,

                    for the constants   .   

We are really really done.   

        The inverse of      is   

                    .    

                      The conformal mapping   .  

This solution is complements of the authors.

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