After time $t$, thickness of liquid will remain $\left(\frac{d}{3}-v t\right)$.
Now, time constant as function of time

$$\begin{aligned} \tau_c & =C R \\ & \left.=\frac{\varepsilon_0(1) \cdot R}{\left(d-\frac{d}{3}+v t\right)+\frac{d / 3-v t}{2}} \quad \quad \text { Applying } C=\frac{\varepsilon_0 A}{d-t+\frac{t}{k}}\right) \\ & =\frac{6 \varepsilon_0 R}{5 d+3 v t} \end{aligned}$$

$\therefore$ correct option is (a).