Praise be to Allaah.
The man has to treat both of his wives fairly, and to fear Allaah with regard to them. The woman should resist her jealousy and strive to control herself (jihad al-nafs) and not cause trouble to her husband because he has another wife.
“The basic principle concerning divorce is that it is makrooh (disliked), and if we say that the basic principle is that it is forbidden, this is not far-fetched. This is indicated by the words of Allaah concerning those who take an oath not to have sexual relations with their wives. He said (interpretation of the meaning):
‘… then if they return (change their idea in this period), verily, Allaah is Oft-Forgiving, Most Merciful.
And if they decide upon divorce, then Allaah is All-Hearer, All-Knower’
The aayah ends with these two names, ‘All-Hearer, All-Knower’, if they decide upon divorce, to indicate that Allaah does not like that, because in the case of one who returns or comes back to his wife after swearing not to have intercourse with her, it says that ‘Allaah is Oft-Forgiving, Most Merciful.’
This makes it clear that Allaah likes the one who has sworn such an oath to go back (to his wife). In the case of one who has decided to divorce his wife, the use of the words ‘All-Hearer, All-Knower’ indicates that Allaah dislikes that. It was narrated that the Prophet (peace and blessings of Allaah be upon him) said: ‘The most hated of permissible things to Allaah is divorce.’ This hadeeth is not saheeh, but its meaning is saheeh. Allaah dislikes divorce, but He does not forbid it to His slaves, to make things easier for them. If there is a valid reason for divorce, in sharee’ah or otherwise, then that is permissible. Whether there is a valid reason depends on whether the woman’s remaining married will lead to some shar’i reservation that can only be alleviated by divorce, then he may divorce her, such as when the woman is lacking in religious commitment or chastity, and he is unable to reform her. In such cases, we say that it is better for him to divorce her. But if there is no such valid reason, whether shar’i or otherwise, then we say that it is better not to divorce her; indeed in such a case divorce would be makrooh.”
(As’ilat al-Baab al-Maftooh by Ibn ‘Uthaymeen, p. 113)
With regard to the woman mentioned in the question, if she is able to live with her husband in a good manner, and if each of them can put up with the other at times of short-lived anger, then this will be better for her, for him, for their children and for the family of each partner. If a good life cannot continue between them for some reason on the part of one or both of them, and it appears that separation is better for her or for him or for both of them, then Allaah says (interpretation of the meaning):
“But if they separate (by divorce), Allaah will provide abundance for everyone of them from His Bounty”
Allaah may provide her with a husband better than him, who will be more righteous and treat her more kindly. May Allaah help us all to do that which He loves and which pleases Him.