Question 7.5.6: Streamlines Defined Parametrically. Construct a flow of an i......
Streamlines Defined Parametrically
Construct a flow of an ideal fluid in the domain D consisting of all points in the upper half-plane y > 0 excluding the points on the ray y = π, −∞ < x ≤ 0, shown in Figure 7.58.
In Example 4 of Section 7.3 we used the Schwarz-Christoffel formula to find a conformal mapping of the upper half-plane y > 0 onto the domain D.By replacing the symbol z with the symbol w in the solution from Example 4, we obtain the mapping
z=\Omega^{-1}(w)=w+\mathrm{Ln}(w)+1 (10)
of the upper half-plane v > 0 onto D. The inverse Ω of the mapping in (10) is a complex velocity potential of a flow of an ideal fluid in D, but we cannot solve for w to obtain an explicit formula for Ω. In order to describe the streamlines, we recall that the streamlines in D are the images of horizontal lines v = c_{2} in the upper half-plane v > 0 under the mapping z = w + Ln(w) + 1. Since a horizontal line can be described by w(t) = t + ic_{2}, −∞ < t < ∞, it follows that the streamlines in D are given parametrically by
z(t)=\Omega^{-1}(w(t))=w(t)+\mathrm{Ln}[w(t)]+1=t+i c_{2}+\mathrm{Ln}[t+i c_{2}]+1,or x(t) = t+ \frac{1}{2} log_{e}(t^{2} + c^{2}_{2})+1, y(t) = c_{2}+ Arg (t+ic_{2}), −\infty < t <\infty . Some streamlines for this flow have been plotted using Mathematica in Figure 7.58.