Firstly: 

Nifaas (postpartum bleeding) refers to the blood that comes out of the uterus because of giving birth. 

If the birth took place without any bleeding -- which is very rare -- then the woman is not regarded as being in nifaas and she is not subject to the same rulings as women who are in nifaas, because the ruling depends on the presence of blood. 

Ibn Qudaamah al-Maqdisi said: If she gives birth and does not see any blood, then she is pure (taahir) and not in nifaas, because nifaas is blood, and there is no blood in this case. End quote. Al-Mughni, 1/429 

Ibn Hajar al-Haytami said: If a woman gives birth and does not see any blood, then there is no nifaas for her at all, and if she does ghusl, she comes under the same rulings as women who are pure (taahir) in all aspects. End quote from al-Fataawa al-Fiqhiyyah al-Kubra, 1/358 

In al-Mawsoo‘ah al-Fiqhiyyah (41/15) it says: If the birth was free of blood or there was no bleeding, such as if the infant came out dry, then she is pure (taahir) and there is no nifaas for her, because nifaas is blood, and there is no blood in this case. End quote. 

The scholars of the Standing Committee for Issuing Fatwas were asked: Some women experience difficulties in giving birth and they have to give birth by means of a Caesarian section, in which case the child does not come out through the vagina. What is the ruling on these women in sharee‘ah, with regard to the bleeding of nifaas? 

They replied: 

They come under the same ruling as women who are in nifaas: if the woman sees blood, she should refrain (from prayer etc.) until she becomes pure, and if she does not see any blood, then she should fast and pray like all other women who are pure (taahir). End quote from Fataawa al-Lajnah al-Daa’imah (2/420) 

Secondly: 

The scholars differed as to whether ghusl is obligatory in this case. 

It was said that she does not have to do ghusl, because Islam only requires it of women who are in nifaas, and in this case the woman is not in nifaas in any way. 

This is the view of the Maalikis and Hanbalis. 

See: al-Mughni, 1/429; al-Mawsoo‘ah al-Fiqhiyyah, 14/51  

Shaykh Ibn ‘Uthaymeen (may Allah have mercy on him) favoured this view, and said: If a woman gives birth and does not see any blood, which is very rare, in this case she should not refrain (from prayer and so on) for the period of nifaas. If she gives birth at sunrise and the time for Zuhr begins and she did not see any blood, then she does not have to do ghusl; rather she should do wudoo’ and pray. End quote from al-Sharh al-Mumti‘, 1/281 

And it was said that she does have to do ghusl, because giving birth is usually a cause of nifaas which makes ghusl obligatory, so giving birth should make ghusl obligatory too. 

This is the view of the Shaafa‘is and is the view favoured by the scholars of the Standing Committee for Issuing Fatwas, who said: 

If a pregnant woman gives birth and no blood comes out, then she has to do ghusl and pray and fast, and her husband may have intercourse with her after she does ghusl, because what usually happens in the case of childbirth is that blood comes out, even if it is a little, with the baby or straight afterwards. End quote from Fataawa al-Lajnah al-Daa’imah, 5/421 

In order to be on the safe side, she should do ghusl so as to avoid an area in which there is a difference of opinion among the scholars. 

Thirdly: 

If she bleeds for several days, then the bleeding stops, she has to do ghusl and pray and fast, even if that is before forty days have passed since giving birth. There is scholarly consensus on this point, because there is no minimum limit for nifaas. This has been discussed in the answer to question number 50308. If the bleeding resumes within the 40 day period, then this is nifaas; and whatever goes beyond the 40 days is istihaadah (irregular bleeding) which does not prevent her from praying and fasting. 

For more information please see the answers questions number 7417 and 106464. 

And Allah knows best.