10. In $\triangle A B C$, $A B=\sqrt{2}, A C=\sqrt{3}, \angle B A C=30^{\circ}, P$ is any point on the plane of $\triangle A B C$, then $\mu=\overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A}$ has the minimum value of $\qquad$.
$$
\mu=\frac{1}{2}\left[(\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C})^{2}-\left(\overrightarrow{P A}^{2}+\overrightarrow{P B}^{2}+\overrightarrow{P C}^{2}\right)\right] \geqslant-\frac{1}{2}\left(\overrightarrow{P A}^{2}+\overrightarrow{P B}^{2}+\overrightarrow{P C}^{2}\right) \text {, }
$$

Equality holds when $\overrightarrow{P A}+\overrightarrow{P B}+\overrightarrow{P C}=\mathbf{0} \Rightarrow P$ is the centroid of $\triangle A B C$. By the cosine rule $a^{2}=b^{2}+c^{2}-2 b c \cos A=2+3-2 \sqrt{6} \cdot \frac{\sqrt{3}}{2}=5-3 \sqrt{2}$,
Let $D$ be the midpoint of $B C$, by $A D^{2}+\frac{1}{4} a^{2}=\frac{1}{2}\left(b^{2}+c^{2}\right) \Rightarrow \overrightarrow{P A}^{2}=\frac{1}{9}\left(2 b^{2}+2 c^{2}-a^{2}\right)$
$$
\Rightarrow \overrightarrow{P A}^{2}+\overrightarrow{P B}^{2}+\overrightarrow{P C}^{2}=\frac{1}{3}\left(a^{2}+b^{2}+c^{2}\right)=\frac{1}{3}(10-3 \sqrt{2})=\frac{10}{3}-\sqrt{2} \text {. }
$$

Therefore, the minimum value of $\mu=\overrightarrow{P A} \cdot \overrightarrow{P B}+\overrightarrow{P B} \cdot \overrightarrow{P C}+\overrightarrow{P C} \cdot \overrightarrow{P A}$ is $\frac{\sqrt{2}}{2}-\frac{5}{3}$.