Why d'Alembert's paradox has zero drag? Why d'Alembert's paradox has zero drag?

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*Even if viscosity is zero why drag is zero if there is no stagnation pressure  at the back (right) side of ball?


*Why flow stay attached to ball surface,even at back? What force push fluid to follow surface? Pressure  gradient?
So viscosity trigger flow separation in real life?

 A: d'Alembert's Paradox arises because the theory of fluid dynamics (potential flow theory) does not allow for any energy dissipation in case of an inviscid fluid. However, on a microscopic level, this theory neglects the fact that even if the gas molecules do not interact with each other (i.e. if the gas is inviscid), they still collide with the object, which must result in a change of momentum and energy.
Consider a plate moving frontally through a gas with velocity $+v$. A molecule moving vertically towards the plate with velocity $-u$ in the lab frame has a velocity $-u-v$ in the reference frame of the plate and hence bounces back from the front side with the velocity $u+v$ (as the plate is much heavier than the molecule), i.e. in the lab frame the velocity of the molecule is now $u+2v$. Correspondingly, at the backside of the plate, a molecule with velocity $u$ in the lab frame has a velocity $u-v$ in the reference frame of the plate and thus after reflection a velocity $-u+v$,  i.e. in the lab frame the velocity is now $-u+2v$.
This means in the lab frame, the kinetic energy of the molecules being reflected from the front has increased from $$m/2\cdot u²$$ to $$m/2\cdot (u+2v)² = m/2\cdot u² +2\cdot m\cdot u\cdot v + 2\cdot m\cdot v²$$, i.e. an increase of $$+2\cdot m\cdot u\cdot v + 2\cdot m\cdot v²$$.
Molecules reflected from the back side on the other hand have changed their energy from
$$m/2\cdot u²$$
to
$$m/2\cdot (-u+2v)² = m/2\cdot u² -2\cdot m\cdot u\cdot v + 2\cdot m\cdot v²$$
i.e. a decrease of
$$-2\cdot m\cdot u\cdot v + 2\cdot m\cdot v²$$.
Adding and averaging the two contributions one obtains therefore the average increase of the kinetic energy of the molecules with mass $m$ hitting the plate with velocity $v$ as
$$ΔK = 2\cdot m\cdot v²$$
This energy must then be lost by the moving object.
(Note: this value will actually be smaller by about factor 1/2 as the molecules won't be reflected straight back but will be scattered into the whole half-space due to the roughness of the plate's surface).
One can derive this result also from the energy and momentum conservation equations applied to an elastic collision, with the same result if one assumes the plate mass as large compared to the molecular mass.
So there would be a drag force on an object even in an inviscid medium.
By the same microscopic considerations, the aerodynamic lift for instance would still exist for an inviscid medium as well.
A: The flow pattern is  left-right symmetric. As a consequence, by Bernoulli, the pressure at any point  on the left is the same as the corresponding point on the right. There is therefore no net force on the red object.
A: Sorry for my poor english. My native language is French.
At first sight, it is paradoxical because "zero viscosity" is equivalent to infinite Reynolds number and therefore surely instability and detachment.
The solution comes from the fact that there is an essential difference between "no viscosity" and "very low viscosity" i.e. very high Reynolds number. In the case of a real fluid, even with a very low viscosity, the fluid in contact with the solid has zero velocity: the famous no-slip condition. This condition is radically different from the one applied to a perfect fluid which can slide along the solid wall. The junction is made via the boundary layer which is very thin in the case of a high Reynolds number.
For a streamlined object, this boundary layer does not separate and we have almost everywhere a perfect flow. In this case, the D'Alembert's paradox leads to almost exact conclusions: the drag on a streamlined object is indeed extremely low.
For a non-streamlined object, there is a large separation and the upstream-downstream symmetry disappears. The drag becomes very large.
We can note that in a way, D'Alembert's paradox reappears in the drag crisis on the sphere: the turbulent boundary layer separates later, the wake is reduced and the flow is perfect on a wider domain: the drag is reduced, in accordance with the predictions of d'Alembert's paradox.
