Compare the solution for Creeping flow to a very high Reynolds number flow

To contrast the solution for no inertia, we write down the average velocity for the inviscid (meaning viscous terms are not considered) flow case from a standard reference

The code is as the shading gets darker, the velocity is slower.  We see that coming in the y=0 axis from x=-Infinity, the velocity is decreasing to agree with the: no flow through the surface boundary condition.  Looking at the x=0 axis, coming in from y=∞, the velocity is increasing reflecting the deflection of fluid around the sphere.  This results in a Bernoulli effect with local fluid acceleration.  Below we check some numbers to verify these statements.  

The same question can be asked about the pressure field.  We will find that the pressure is always at or below the far away value for Re=0.  Substitute for the pressure field equation,

Pressure field for high Reynolds number:

In contrast, if a high velocity fluid is impinging on a surface, there will be a pressure increase.  For an inviscid flow, from Bernoulli's equation, we find that dp ~ (U0^2-u^2)/2, where we have already plotted u^2

So that the pressure is highest at y=0, x=1,-1 (i.e. the stagnation point) and decreases out to infinity.  The pressure is lowest, consistent with the Bernoulli effect, at y=1,-1,x=0, and increases out to y=infinity.    

A stream function, which is a function that is always tangent to the velocity field, is another way to view the solution of this problem.  The Re = 0 solution can be used to write the stream function which is, -1/2f[r] .  

We can make a plot of this if we transform from polar to Cartesian coordinates

Note that at y=4, the contours are still deformed, the effect of the sphere is felt very far away!  

Now compare the inviscid flow case:

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