The streamlines for an incompressible, inviscid, two-dimensional $$v_{\theta}=K r$$ where \(K\) is a constant. (a) For this rotational flow, determine, if possible, the stream function. (b) Can the pressure difference between the origin and any other point be determined from the Bernoulli equation? Explain.

The velocity is given as \(v_{\theta}=K r\). The stream function, \(\psi\), is related to the velocity in polar coordinates by \(v_{r} = \frac{1}{r} \frac{\partial \psi}{\partial \theta}\) and \(v_{\theta} = -\frac{\partial \psi}{\partial r}\). Since \(v_{r} = 0\) for the given problem, the first equation leads us to \(\psi = a \theta + b\), where \(a\) and \(b\) are constants. Using the second equation, and \(v_{\theta}=K r\), we can integrate this to find \(\psi = -K r^2/2 + a\theta + b\)

In general, the constants \(a\) and \(b\) will depend on the specifics of the flow, including the boundary conditions. Without any additional information (such as whether it is necessary to match a certain flow rate, or match a velocity at the edge of a pipe or a boundary in the space), we can't find these constants, so we would just leave \(\psi\) as \(\psi = -K r^2/2 + a\theta + b\)

The Bernoulli equation is \(P1 + 0.5*\rho*v1^2 + \rho*gh1 = P2 + 0.5*\rho*v2^2 + \rho*gh2\), where \(P\) represents pressure, \(\rho\) represents fluid density, \(v\) represents fluid speed, \(g\) represents gravity, and \(h\) represents height. From the given information, (inviscid, incompressible flow), we can use the Bernoulli equation except for the cases where the change in height is important, or where viscous or compressible effects come into play. Since it is two-dimensional flow and no height change is mentioned, we can ignore \(gh\) term in equation. Also, we can consider any other point in fluid to have pressure \(P2\) and speed \(v2\) while at origin pressure is \(P1\) and velocity is zero. After substituting these values Bernoulli equation appears as : \(P1 = P2 + 0.5*\rho*(Kr)^2\), hence pressure difference \(P1-P2 = 0.5*\rho*(Kr)^2 \), which gives the pressure difference between the origin and any point.