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\frac{d Q^{\downarrow}_{AQ}}{dt} = q^{\downarrow}_{AQ}(t)

q^{\downarrow}_{AQ}(t)= C_a \cdot \frac{\partial p_a(t,r)}{\partial r} \bigg|_{r=r_e}

p_a(t, r)= p(0) + \int_0^t p_1 \left(\frac{(t-\tau) \cdot r_e^2}{\chi}(t-\tau), \frac{r}{r_e} \right) \dot p(\tau) d\tau

\frac{\partial p_1}{\partial t_D} =  \frac{\partial^2 p_1}{\partial r_D^2} + \frac{1}{r_D}\cdot \frac{\partial p_1}{\partial r_D}

p_1(t = 0, r_D)= 0

\frac{\partial p_1}{\partial r_D} \bigg|_ {r_D=r_a/r_e} = 0