= Solution
The first equation is <mass conservation> for an <incompressible flow> whose area may change: local area storage balances the <divergence> of the volume flux $\alpha u$. The two left-hand momentum terms are local acceleration and <convective acceleration>, while $-p_x$ is the driving <pressure gradient> and $-R(\alpha)u$ models viscous drag per unit mass. The <pressure-area law of a collapsible tube> expresses transverse wall balance using internal minus prescribed external <pressure>.
For example, choose a reference area $\mathcal A_*$, length $L$ and <velocity> $U_*$. If dimensional variables are $\mathcal A,s,t_d,v,P$, take
$$
\alpha=\mathcal A/\mathcal A_*,\quad x=s/L,\quad t=U_*t_d/L,\quad
u=v/U_*,\quad p=P/(\rho U_*^2).
$$
The dimensionless external <pressure> uses the same scale. A dimensional friction rate $r_f(\mathcal A)$ becomes $R(\alpha)=Lr_f(\mathcal A_*\alpha)/U_*$, and the dimensional wall <pressure> law is divided by $\rho U_*^2$. These choices make both inertia and the <pressure>-gradient coefficient unity; other equivalent reference scales are possible. On the mechanically stable wall branch, increasing area requires increased transmural <pressure>, whereas an enlarged tube reduces resistance. Thus \b[$\boxed{\widetilde P'(\alpha)>0,\qquad R'(\alpha)<0}$].
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