Consider a (p)rototype consisting in an incompressible and newtonian fluid flowing in a pipe of diameter $D_P$, studied by similarity in a (m)odel of the same fluid in another pipe of diameter $D_m<D_p$.
Equating the Reynolds numbers: $D_m v_m = D_p v_p$. That is
$$v_m = \frac{D_p}{D_m}v_p > v_p$$
so the fluid moves faster in the model which is smaller in order to archive 'some kind' of similarity.
Q1 Which kind of similarity?
The Reynolds number arises after adimiensionalize the viscosity using the density ($\rho$), velocity ($v$) and a lenght ($D$). Using the same variables to adimensionalize the forces $F$ and the powers $P$,
$$ F^* = \frac{F}{\rho v^2 D^2} \qquad P^* = \frac{P}{\rho v^3 D^2}$$
If the adimensional forces and powers are the same for the model and the prototype ($F^*_m = F^*_p$) and ($P^*_m = P^*_p$), it is ready found that
$$ \begin{align} F_p &= F_m\\ P_p &= \frac{D_m}{D_p} P_m \end{align}$$
which means that the forces are equal in the model and prototype, and the power of the prototype (relative to the model) is smaller than the model.
The conclusion above must to be false.
Q2 What is wrong here? I think that it is related to lack of time correspondence between the model and the prototype, but I am not sure about how to treat this.
Doing the same for energy, I found that $E_p = \frac{D_p}{D_m}E_m$, but I would expect that the proportionality is with volume or mass and not with diameter.
