11.7  Two-Dimensional Fluid Flow

    Suppose that a fluid flows over the complex plane and that the velocity at the point    is given by the velocity vector  

(11-30)        .  

We also require that the velocity does not depend on time and that the components and have continuous partial derivatives.  The divergence of the vector field is given by

              

and is a measure of the extent to which the velocity field diverges near the point.  We ill consider only fluid flows for which the divergence is zero.  This is more precisely characterized by requiring that the net flow through any simple closed contour be identically zero.  

    If we consider the flow out of the small rectangle shown in Figure 11.47, then the rate of outward flow equals the line integral of the exterior normal component of taken over the sides of the rectangle.  The exterior normal component is given by on the bottom edge, on the right edge,   on the top edge, and on the left edge.  Integrating and setting the resulting net flow to zero yield  

(11-31)        .  

Figure 11.47  A two-dimensional vector field.

    Both and are continuously differentiable, so we can use the mean value theorem to show that  

(11-32)        

where and .  Substitution of the expressions in Equation (11-32) into Equation (11-31) and subsequently dividing through by results in  

            .  

    We can use the mean value theorem for integrals with this equation to show that  

            ,

where and .  Letting and in this equation yields

(11-33)        ,

which is called the equation of continuity.

    The curl of the vector field in Equation (11-30) has magnitude

            

and is an indication of how the field swirls in the vicinity of a point.  Imagine that a "fluid element" at the point is suddenly frozen and then moves freely in the fluid.  The fluid element will rotate with an angular velocity given by

            .

    We consider only fluid flows for which the curl is zero.  Such fluid flows are called irrotational.  This condition is more precisely characterized by requiring that the line integral of the tangential component of along any simply closed contour be identically zero.  If we consider the rectangle in Figure 11.47, then the tangential component is given by on the bottom edge, on the right edge, on the top edge, and on the left edge.  Integrating and equating the resulting circulation integral to zero yields

            .  

    As before, we apply the mean value theorem and divide through by , and obtain the equation

            .  

    We can use the mean value for integrals with this equation to deduce that  

            .

Letting and yields

            .

    Equations (11-33) and this equation show that the complex function  

              

satisfies the Cauchy-Riemann equations and hence it is an analytic function.  If we let denote the antiderivative of , then

(11-34)        ,  

which is the complex potential of the flow and has the property

            

Since    and  , we also have

            ,  

so   is the velocity potential for the flow, and the curves  

              

are called equipotentials.  The function    is called the stream function.  The curves  

              

are called streamlines, and describe the paths of the fluid particles.  

    To demonstrate this result, we implicitly differentiate and find that the slope of a vector tangent is given by  

            .  

Using the fact that    and this equation, we find that the tangent vector to the curve is

            .  

The main idea of the preceding discussion is the conclusion that, if

(11-35)        ,  

is an analytic function, then the family of curves

            

represents the streamlines of a fluid flow.

    The boundary condition for an ideal fluid flow is that should be parallel to the boundary curve containing the fluid (the fluid flows parallel to the walls of a containing vessel).  In other words, if Equation (11-35) is the complex potential for the flow, then the boundary curve must be given by for some constant K;  that is, the boundary curve must be a streamline.

Theorem 11.5  (Invariance of Flow).  Let  

              

denote the complex potential for a fluid flow in a domain G in the w plane where the velocity is  

            .  

If the function      is a one-to-one conformal mapping from a domain D in the z plane onto G,  then the composite function  

              

is the complex potential for a fluid flow in D where the velocity is  

            .    

The situation is shown in Figure 11.48.

Figure 11.48  The image of a fluid flow under conformal mapping.  

 Proof.

Proof of Theorem 11.5 is in the book.
Complex Analysis for Mathematics and Engineering

    We note that the functions   

              

are the new velocity potential and stream function, respectively, for the flow in D.  A streamline or natural boundary curve

            
            
in the z plane is mapped onto a streamline or natural boundary curve

            


in the w plane by the transformation .  One method for finding a flow inside a domain D in the z plane is to conformally map D onto a domain G in the w plane in which the flow is known.

    For an ideal fluid with uniform density , the fluid pressure    and speed    are related by the following special case of Bernoulli's equation:  

            .  

Note that the pressure is greatest when the speed is least.

Example 11.22.  The complex potential    has the velocity potential and stream function of    

              and   ,  
            
respectively, and gives rise to the fluid flow defined in the entire complex plane that has a uniform parallel velocity of  

            .  

The streamlines are parallel lines given by the equation    and are inclined at an angle  ,  as indicated in Figure 11.49.

Figure 11.49  A uniform parallel flow.  

Solution 11.22.

Example 11.23.  Consider the complex potential    where A is a positive real number.  The velocity potential and stream function are given by

              

respectively.  

The streamlines    form a family of hyperbolas with asymptotes along the coordinate axes.  The velocity vector    indicates that in the upper half-plane , the fluid flows down along the streamlines and spreads out along the x axis, as against a wall, as depicted in Figure 11.50.

Figure 11.50  The fluid flow with complex potential  .  

Explore Solution 11.23.

Extra Example 1.  Consider the complex potential    where A is a positive real number.  The velocity potential and stream function are given by

              

respectively.  

The streamlines    form a family of hyperbolas with asymptotes along the coordinate axes.  The velocity vector    indicates that in the upper half-plane , the fluid flows down along the streamlines and spreads out along the positive x axis and negative y axis, as shown in Figure 11.55.

Explore Extra Solution 1.

Example 11.24.  Find the complex potential for an ideal fluid flowing from left to right across the complex plane and around the unit circle  .  

Solution.  We use the fact that the conformal mapping    maps the domain    one-to-one and onto the w plane slit along the segment  .  The complex potential for a uniform horizontal flow parallel to this slit in the w plane is  

            ,  

where A is a positive real number.  The stream function for the flow in the w plane is    so that the slit lies along the streamline  .

    The composite function    determines the fluid flow in the domain D, where the complex potential is

            ,  

where .  We can use polar coordinates to express as

            

The streamline     consists of the rays

               and   

along the x axis and the curve  ,  which is the unit circle  .  Thus the unit circle can be considered as a boundary curve for the fluid flow.

    The approximation    is valid for large values of z, so we can approximate the flow with a uniform horizontal flow having speed    at points that are distant from the origin.  The streamlines    and their images    under the mapping are illustrated in Figure 11.51.

Figure 11.51  Fluid flow around a circle.  

Explore Solution 11.24.

Example 11.25.  Find the complex potential for an ideal fluid flowing from left to right across the complex plane and around the segment from  .  

Solution.  We use the conformal mapping  

              

where the branch of the square root of     in each factor is  ,  where  ,  and  ,  where  .  The transformation    is a one-to-one conformal mapping of the domain D consisting of the z plane slit along the segment    onto the domain G consisting of the w plane slit along the segment  .  The complex potential for a uniform horizontal flow parallel to the slit in the w plane is given by , where for convenience we choose and where the slit lies along the streamline  .

    The composite function

            

is the complex potential for a fluid flow in the domain D.  The streamlines given by    for the flow in D are obtained by finding the preimage of the streamline    in G given by the parametric equations  

            .  

The corresponding streamline in D is found by solving the equation

            

for x and y in terms of t.  Squaring both sides of this equation yields  

              

Equating the real and imaginary parts leads to the system of equations  

               and   .  

To eliminate the parameter t in the last two equations, first use the equations  

            ,  
            
then get    which can be rewritten as  

            

and we can solve for y in terms of x to obtain

              

for streamlines in D.  For large values of x, this streamline approaches the asymptote    and approximates a horizontal flow, as shown in Figure 11.52.

Figure 11.52  Flow around a segment.  

Explore Solution 11.25.