Calculate the pressure force, or drag (per unit length), on an (infinite) circular cylinder of radius $a$ due to a uniform stream of speed $U_0$ of an incompressible, irrotational fluid flowing past it.
I have that $$\underline{u} = u_r\underline{\hat{R}} + v_\theta\underline{\hat{\phi}} = \frac{\partial \Phi}{\partial R}\underline{\hat{R}} + \frac{1}{R}\frac{\partial \Phi}{\partial \phi}\underline{\hat{\phi}}$$ Where $R=a$, $\underline{\hat{n}} = \underline{\hat{R}}$, the following must satisfy: $\underline{u}\cdot\underline{\hat{n}} = 0$. $$\underline{u}\cdot\underline{\hat{n}} = u_R = \frac{\partial \Phi}{\partial R} = 0 \Rightarrow \Psi = 0\quad \text{at}\quad R=a$$ But I am completely lost after this. How is a problem like this done?
Put the origin of the coordinate system at the center of the circular cross-section of the cylinder. With a uniform stream away from the cylinder, the velocity field and potential $\Phi$ is symmetric with respect to reflection across the $x-$ axis, i.e. $\Phi(x,y) = \Phi(x,-y)$. We also have fore-aft symmetry and $\Phi(x,y) = \Phi(-x,y).$
From Bernoulli's equation, $\frac{p}{\rho} = \text{const.} - \frac{1}{2}\left|\nabla \Phi \right|^2$, we see that the pressure at the surface of the cylinder has the same symmetry property.
Let $C = C_1 \cup C_2$ be the circular contour of the cylinder where $C_1$ is the upper half where $y > 0$ and $C_2$ is the lower half where $y < 0$. The drag force per unit length is obtained by integrating $-p \,\mathbb{n}$ around the $C$ to obtain
$$\mathbf{F} = -\int_Cp \,\mathbf{n}\,dl = - \int_{C_1}p \,\mathbf{n}\,dl - \int_{C_2}p \,\mathbf{n}\,dl . $$
Since $p(x,y) = p(x,-y)$ and $\mathbf{n}(x,y)\cdot \mathbf{j} = - \mathbf{n}(x,-y)\cdot \mathbf{j}$, the contributions from $C_1$ and $C_2$ cancel and we have
$$\mathbf{F} = \mathbf{0}.$$
Here we have D'Alembert's paradox which states that in steady, two-dimensional, potential flow, the net force on a body is zero. It is particularly easy to prove, as in this case, where the body is symmetric and placed in a unidirectional streaming flow.