11.10  Image of a Fluid Flow

    We have already examined several two-dimensional fluid flows and have discovered that the image of a flow under a conformal transformation is a flow.   The conformal mapping ,  which is obtained by using the Schwarz-Christoffel formula, (see Section 11.9), allows us to find the streamlines for flows in domains in the w plane that are bounded by straight-line segments.

    The first technique is finding the image of a fluid flowing horizontally from left to right across the upper half plane  .  The image of the streamline    is a streamline given by the parametric equations

              

and is oriented in the positive sense (counterclockwise).  The streamline    for    is considered to be the boundary wall for a containing vessel for the fluid flow.

Example 11.29.  Show that the mapping    maps the upper half plane   onto the domain in the w-plane that lies above the boundary curve consisting of the rays   and the segment     (see Figure 11.87).

            Figure 11.87  (a) Flow over a step.                  (b) Flow around a blunt object.

Hint:  Set    and   .  In the w-plane , and the exterior angles are  ,  and the formula for the derivative is  

            

Integrate and get  

            

Solve for the coefficients A and B and obtain

            

It maps the upper half-plane   onto the domain in the w plane that lies above the boundary curve consisting of the rays   and the segment  .  Furthermore, the image of horizontal streamlines in the z plane are curves in the w plane given by the parametric equation

        

for  .  The new flow is that of a step in the bed of a deep stream and is illustrated in Figure 11.87(a).  The function is also defined for values of z in the lower half-plane, and the images of horizontal streamlines that lie above or below the x axis are mapped onto streamlines that flow past a long rectangular obstacle, which is illustrated in Figure 11.87(b).

Explore Solution 11.29.

Extra Example 1.  Use Figure 11.88 to find the flow over the vertical segment from .

            Figure 11.88  Flow over the vertical segment from .

Solution.  Set  .  In the w-plane , and the exterior angles are  ,  and the formula for the derivative is  


            

Integrate and get  

            

Solve for the coefficients A and B and obtain

            

Explore Extra Solution 1.

Extra Example 2.  Use Figure 11.89 to find the flow around an infinitely long rectangular barrier.

            Figure 11.89  Flow around an infinitely long rectangular barrier.

Solution.  Set  .  In the w-plane , and the exterior angles are  ,  and the formula for the derivative is  


            

Integrate and get  

              

Solve for the coefficients and and obtain

            

Explore Extra Solution 2.

Extra Example 3.  Use Figure 11.90 to find the flow around

            Figure 11.90  (a) Flow around one inclined segment in the upper half-plane.

                         (b) Flow around two inclined segments forming a "V" in the plane.

Solution.  Set  .  In the w-plane , and the exterior angles are  ,  and the formula for the derivative is  


            

Integrate and get  

            

Solve for the coefficients A and B and obtain

            

Explore Extra Solution 3.

Extra Example 4.  Use Figure 11.91 to find the flow over a dam.

            Figure 11.91  Flow over a dam.

Solution.  Set  .  In the w-plane , and the exterior angles are  ,  and the formula for the derivative is  

            

Integrate and get  

            

Solve for the coefficients A and B and obtain

            

Explore Extra Solution 4.

Extra Example 5.  Use Figure 11.92 to find the flow around

            Figure 11.92  (a) Flow up an inclined step                    (b) Flow around a pointed object.

Solution.  Set  .  In the w-plane , and the exterior angles are  ,  and the formula for the derivative is  


            

Integrate and get  

            

Solve for the coefficients A and B and obtain

            

Explore Extra Solution 5.