All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah and that Muhammad is His slave and Messenger.
The husband is obliged to spend on his wife, even if she is rich, as we have already highlighted in fataawa 277414 and 31645.
On the other hand, the wife is obliged to obey her husband in what is reasonable and permissible. Abu Hurayrah narrated that the Prophet said, "If I would have ordered anyone to prostrate to someone other than Allaah, then I would have ordered the wife to prostrate to her husband." [At-Tirmithi]
Therefore, if you order your wife to come back to you and she refuses, then she is recalcitrant and you are not obliged to spend on her in this case. Please refer to fatwa 90507.
Also, if she travels for her need with your permission, then you are not obliged to spend on her as well. For more benefit, please refer to fatwa 306811.
Allaah knows best.