The relationship between the Mother of the Believers ‘Aa’ishah and the Prophet’s daughter Faatimah (may Allah be pleased with them both)
Question

what was the relationship bwtween fatimah and the prophert wife aisha
Praise be to Allah.The relationship between Faatimah (may Allah be pleased with her) and ‘Aa’ishah (may Allah be pleased with her) was based on good manners and a good attitude, a relationship of respect, upholding ties and love. We have never heard of any incident between them to suggest that there was any breach or cutting off of ties between them. 
‘Aa’ishah (may Allah be pleased with her) loved Faatimah (may Allah be pleased with her) and spoke highly of her, to such an extent that she likened her to the Messenger of Allah (blessings and peace of Allah be upon him). 
Abu Dawood (5217) and at-Tirmidhi (3872) narrated that ‘Aa’ishah (may Allah be pleased with her) said: I never saw anyone who resembled the Messenger of Allah (blessings and peace of Allah be upon him) more in dignity, calmness and pleasant disposition – al-Hasan said: in speaking and talking, and he did not mention dignity, calmness and pleasant disposition – than Faatimah (may Allah honour her face). When she entered upon him, he would stand up for her, take her by the hand, kiss her and make her sit where he was sitting, and when he entered upon her, she would stand up for him, take him by the hand, kiss him and make him sit where she was sitting.
Classed as saheeh by al-Albaani in Saheeh Abi Dawood. 
“Dignity” refers to good conduct in matters of religion, and also has to do having a balanced attitude. 
“Calmness”: Abu ‘Ubayd said: Calmness and pleasant disposition are similar in meaning; this may be said with regard to tranquillity and dignity, or with regard to appearance and attributes. 
“Pleasant disposition” this refers to elegance in walking, speaking and so on. 
Fath al-Baari (10/510) 
Al-Bukhaari (6285) and Muslim (2450) narrated that ‘Aa’ishah the Mother of the Believers said: We, the wives of the Prophet (blessings and peace of Allah be upon him) were all with him [on one occasion] and not one of us was absent. Then Faatimah (peace be upon her) came walking, and by Allah, her manner of walking was like that of the Messenger of Allah (blessings and peace of Allah be upon him). When he saw her, he welcomed her and said: “Welcome to my daughter,” then seated her to his right or his left. Then he whispered something to her and she wept greatly, and when he saw how sad she was, he whispered something else to her and she smiled. I, out of all his wives, said to her: The Messenger of Allah (blessings and peace of Allah be upon him) singled you out, from all of us, to whisper something to you, then you wept. When the Messenger of Allah left, I asked her: What did he whisper to you about? She said: I will not disclose the secret of the Messenger of Allah (blessings and peace of Allah be upon him). Then when he died, I said to her: I adjure you by virtue of the rights I have over you  that you should tell me. She said: Now, yes (I will tell you) – and she told me. She said: As for when he whispered to me the first time, he told me: “Jibreel used to review the Qur’an with me once every year, but this year he reviewed it with me twice, and I realized that my death has drawn near, so fear Allah and be patient, for what a good example I am to you.” She said: So I wept as you saw me weep. But when he saw how upset I was, he whispered to me a second time and said: “O Faatimah, does it not please you to be the leader of the believing women, or the leader of the women of this ummah?” 
Undoubtedly the fact that ‘Aa’ishah narrated a hadeeth like this, which praises Faatimah (may Allah be pleased with her) in the highest terms and highlights her high status before Allah, may He be exalted, and before His Messenger (blessings and peace of Allah be upon him) offers definitive proof of her great love for her and her desire to convey that praise of her which surpasses all other. 
Moreover, in the same hadeeth we see that she, among all the wives of the Prophet (blessings and peace of Allah be upon him) is the one who asked Faatimah, which indicates how close she was to her. Then she adjured her and confirmed her request by virtue of the rights that she had over Faatimah. Such a thing cannot be said unless there is a good and loving relationship. 
Al-Qaari (may Allah have mercy on him) said: The words “I adjure you by virtue of the rights I have over you” mean: by virtue of the established motherhood (as ‘Aa’ishah was a Mother of the Believers) or sisterhood or sincere love or prior friendship.
End quote from Mirqaat al-Mafaateeh (9/3964) 
With regard to Faatimah (may Allah be pleased with her), she was no less loving and loyal, especially as she was aware of how dear ‘Aa’ishah was to her father (blessings and peace of Allah be upon him). Al-Bukhaari (2581) and Muslim (2442) narrated that ‘Aa’ishah, the wife of the Prophet (blessings and peace of Allah be upon him), said: The wives of the Prophet (blessings and peace of Allah be upon him) sent Faatimah the daughter of the Messenger of Allah (blessings and peace of Allah be upon him) to the Messenger of Allah (blessings and peace of Allah be upon him). She asked permission to enter when he was lying down with me under my cover, and he gave her permission. She said: O Messenger of Allah, your wives have sent me to you to ask you to be just with regard to the daughter of Abu Quhaafah. I [‘Aa’ishah] kept quiet. The Messenger of Allah (blessings and peace of Allah be upon him) said to her: “O my daughter, do you not love that which I love?” She said: Yes. He said: “Then love this one.” Faatimah got up when she heard that from the Messenger of Allah (blessings and peace of Allah be upon him), and she went back to the wives of the Messenger of Allah (blessings and peace of Allah be upon him) and told them what she had said and what the Messenger of Allah (blessings and peace of Allah be upon him) had said to her. They said to her:  You have been of no avail for us. Go back to the Messenger of Allah (blessings and peace of Allah be upon him) and say to him: Your wives urge you to be just with regard to the daughter of Abu Quhaafah. Faatimah said: By Allah, I will never speak to him about her. 
Undoubtedly Faatimah (may Allah be pleased with her) heeded the advice of her father (blessings and peace of Allah be upon him) with regard to ‘Aa’ishah (may Allah be pleased with her), when he said to her: “O my daughter, do you not love that which I love?” She said: Yes. He said: “Then love this one.” 
Al-Bukhaari (5361) and Muslim (2727) narrated from ‘Ali (may Allah be pleased with him) that Faatimah (peace be upon her) came to the Prophet (blessings and peace of Allah be upon him) to complain about the pain caused to her hand by the mill, and she had heard that some prisoners had been brought to the Prophet (blessings and peace of Allah be upon him). But she not find him, so she told ‘Aa’ishah about that. When the Prophet (blessings and peace of Allah be upon him) came, ‘Aa’ishah told him (about Faatimah coming to her). He [‘Ali] said: (The Prophet (blessings and peace of Allah be upon him)) came to us, and we had gone to bed. We started to get up, but he said: “Stay where you are.” Then he came and sat between me and her, until I could feel the coolness of his foot on my stomach. Then he said: “Shall I not teach you something better than what you asked for? When you go to your bed, glorify Allah thirty-three times, praise Him thirty-three times and magnify Him thirty-four times. That is better for you than a servant.” 
The fact that she spoke to ‘Aa’ishah about such a matter is indicative of the good relationship between them, and highlights the fact that she trusted her and valued her opinion. Her relationship with the Mother of the Believers ‘Aa’ishah in that regard and in other ways was like the relationship of any married daughter with her mother, when she goes to her father’s house. 
The relationship between ‘Aa’ishah and Faatimah (may Allah be pleased with them both) continued after the death of the Messenger of Allah (blessings and peace of Allah be upon him), as ‘Aa’ishah asked her again about what the Prophet (blessings and peace of Allah be upon him) had whispered to her, and she told her what he had told her, as stated above in the hadeeth. 
For more information on the virtues of ‘Aa’ishah (may Allah be pleased with her), please see the answer to question no. 7878
For more information on the virtues of Faatimah (may Allah be pleased with her), please see the answer to question no. 11787
And Allah knows best.