Is an expiatory offering (kaffarah) for breaking an oath obligatory upon my brother's wife if she swore she would not stay in his house if he traveled alone, bearing in mind that my brother already paid the expiation on her behalf and traveled before she expiated, and she has not returned after three months? And is the expiation merely feeding ten needy people?

Your brother's wife must expiate her oath. Her husband may pay the expiation on her behalf with her permission. The expiation is: freeing a slave, or feeding ten poor persons, or clothing them. And whoever cannot find [the means] - then a fast of three days, as stated in verse 89 of Surah Al-Ma'idah. For feeding, it is sufficient to provide lunch or dinner for ten poor persons, or to give each of them one and a half kilograms of food. For clothing, it is a garment in which prayer is valid.