This is statement of the exercise: In this exercise we consider as example the case of a disk of radius R centered at the origin of coordinates immersed in a fluid of density σ and velocity field $u(x, y) = (ux, uy) = (Re\ {U (z)} Im\ {U (z)}).U (z)$ represents the velocity field and is determined as $U (z) = \overline{dW/dz}$, where we take the complex potential $W (z) = U_0[z + R^2/z + i2R \log(z)]$ ($U_0$ is a real constant). real constant). The torque $N$ that the fluid exerts on the disk is given by the Blasius theorem. the Blasius theorem: $N = -Re \{\int γ dz z (\overline{U (z)})^2σ/2 \}$ (the curve $γ$ represents the contour of the positively oriented object).From the Blasius theorem, calculate the torque $N$.
So I tried the following way, as $\overline{U (z)}=dW/dz$ and: $$dW/dz=Uo[1-R^2/z^2+i2R/z] \Rightarrow (dW/dz)^2=Uo^2\left[\frac{z^4+R^4-6R^2z^2+i4Rz^3-i4R^3z}{z^4}\right]=Uo^2f(z)/z^4$$ Applying Blasius Theorem I got the following integral that can be solved with Cauchy Formula $$Uo^2\sigma/2\int \frac{f(z)}{z^3}dz =Uo^2\sigma/2\times\pi i(-12R^2)$$ So by applying the Re function the result the torque will be 0 and I believe it is not correct. Where am I failing?