= Solution
\b[Orbital <angular momentum> carried by the wind.] Use a circular orbit of separation $a$, and write $M=M_1+M_2$ and $\Omega_o^2=GM/a^3$. The donor's distance from the center of mass is $a_1=aM_2/M$. With spins neglected, its specific orbital <angular momentum> is $j_1=a_1^2\Omega_o$, while the <circular-binary orbital angular momentum> is
$$
J_{\rm orb}=\frac{M_1M_2}{M}a^2\Omega_o=M_1M_2\sqrt{\frac{Ga}{M}}.
$$
A fast spherically symmetric <stellar wind> has no net extra <angular momentum> in the donor's rest frame and escapes with mean specific orbital <angular momentum> $j_1$. Thus <Jeans-mode mass loss> gives
$$
\boxed{\frac{\dot J_{\rm orb}}{J_{\rm orb}}=\frac{j_1\dot M}{J_{\rm orb}}=\frac{M_2\dot M}{M_1M}.}
$$
Initially take negligible wind capture by the companion, so $\dot M_1=\dot M$, $\dot M_2=0$ and the total mass derivative is $\dot M$. Logarithmic differentiation of $J_{\rm orb}$ gives
$$
\frac{\dot J_{\rm orb}}{J_{\rm orb}}=\frac{\dot M}{M_1}+\frac12\frac{\dot a}{a}-\frac12\frac{\dot M}{M}.
$$
Combining the two expressions yields $\dot a/a=-\dot M/M$, so $aM$ is constant. <Kepler's third law> then gives $\dot P/P=(3/2)\dot a/a-(1/2)\dot M/M=-2\dot M/M$. Hence
$$
\boxed{a\propto M^{-1},\qquad P\propto M^{-2}.}
$$
These are secular relations for slow changes of the orbit through nearly circular states. A wind being fast compared with orbital speed does not mean its mass-loss timescale is shorter than an orbital period; impulsive removal would require a different orbital calculation.
\b[Whether wind drives contact.] Ignore nuclear expansion on the stated short evolutionary timescale. With $R_1\propto M_1^{-n}$ and $R_L\propto a(M_1/M)^{1/3}$, wind alone gives
$$
\begin{aligned}
\frac{d}{dt}\ln\frac{R_1}{R_L}
&=-\left(n+\frac13\right)\frac{\dot M}{M_1}+\frac43\frac{\dot M}{M}\\
&=\frac{\dot M}{3M_1(1+q)}[3(1-n)q-(1+3n)],\qquad q=\frac{M_1}{M_2}.
\end{aligned}
$$
Because $\dot M<0$, the ratio increases precisely when
$$
\boxed{q<\frac{1+3n}{3(1-n)}.}
$$
This is <wind-driven Roche-lobe overflow>: the escaping wind makes the donor expand relative to its lobe and drives a separate transfer stream. It is not the slow-wind capture mechanism called <Wind Roche-lobe overflow>. If the inequality is reversed, the donor recedes inside its expanding lobe and detaches; wind loss alone does not maintain a Roche-lobe transfer stream. At equality it gives no leading-order change of overfill, so additional evolution or higher-order changes decide the outcome.
\b[Contact rate and its stability.] Now let the transfer stream be accreted conservatively by star 2 while the separate wind still escapes at rate $\dot M<0$. Denote the stream rate by $\dot M_2>0$. Then
$$
\dot M_1=\dot M-\dot M_2,\qquad\dot M_{\rm total}=\dot M.
$$
Only the wind removes orbital <angular momentum>; internal transfer redistributes it. Using the same wind-loss law and differentiating $J_{\rm orb}$ gives
$$
\frac{\dot a}{a}=-\frac{\dot M}{M}+2\dot M_2\left(\frac1{M_1}-\frac1{M_2}\right).
$$
Substitute this into the donor and lobe radius derivatives. The overfill evolves as
$$
\frac{d}{dt}\ln\frac{R_1}{R_L}
=-\frac{\dot M}{3M_1(1+q)}[1+3n-3(1-n)q]
-\frac{\dot M_2}{3M_1}[5-3n-6q].
$$
Maintaining contact sets the left side to zero and gives the <contact transfer rate with Jeans-mode wind loss>,
$$
\boxed{\dot M_2=-\frac{1+3n-3(1-n)q}{(1+q)(5-3n-6q)}\dot M.}
$$
When the wind-drive condition holds, its numerator is positive. If $6q<5-3n$, the denominator is also positive and the stream rate is positive; increasing transfer reduces overfill and can balance the wind driver.
If instead $6q>5-3n$, positive transfer increases overfill, which drives still more transfer. The formal contact formula would demand a negative stream rate and is not a physical steady solution. \b[The system develops unstable, runaway mass transfer] on the timescale appropriate to the assumed donor response. For a giant with an adiabatically responding envelope, this can lead to a <common envelope> and rapid orbital evolution, rather than the stable contact sequence; the eventual outcome requires physics beyond these radius laws. At $6q=5-3n$, transfer gives no linear restoring effect, and a nonzero wind driver cannot be balanced by a finite rate within this approximation.
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