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, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
First of all, you should know, dear brother, that the Sunnah for a person performing ablution is to take fresh water for wiping over his head and not wipe with the water that remained after washing his hands (arms). Abdullaah Ibn Zayd said: “I saw the Prophet performing ablution and he wiped over his head with water other than the extra water that remained in his hands (after washing his arms).” [Ahmad, Abu Daawood and At-Tirmithi]
It is for this reason that the majority of the scholars are of the view that the ablution of a person who wiped over his head with the water that remained (on his hands) after washing his arms is not valid. Imaam Ash-Shaafi’i said in his book entitled Al-Umm: “If a person wipes over his head with the water that remained after washing his arms in ablution, or that he wipes over his head with the water that remained in his beard, then this is not sufficient for him; it would only be sufficient for him if he takes fresh water again.”
Moreover, Ibn Rushd said: “The majority of the scholars are of the view that it is an obligation to take fresh water for wiping over one’s head by analogy to all other organs of ablution.”
On the other hand, some scholars are of the view that it is permissible to wipe over the head with the water that remained after washing the hands (arms) as Ar-Rubayyi’ said that the Prophet wiped over his head with some extra water remaining in his hands. [Abu Daawood]
Besides, Ad-Daaraqutni reported the same Hadeeth with the wording: “He [the Prophet ] performed ablution and wiped over his head with the moisture that remained in his hands.” Also, in another narration by him [Ad-Daaraqutni], she [Ar-Rubayyi’] said: “The Prophet used to come to us and he would perform ablution and he would wipe over his head with the water remaining in his hands.”
However, the view that it is permissible to wipe over the head with the water that remained after washing the hands (arms) is the opinion of Imaam Al-Awzaa’i, Al-Hasan Al-Basri and ‘Urwa Ibn Az-Zubayr and this is the famous opinion of the Hanafi School of jurisprudence.
The Indian Fataawa (a Hanafi book) reads: “If there is some moisture in one's palm and he wiped (over his head) with it, it is sufficient for him, whether he took the water from the container or that he washed his arms and some moisture remained in his palm; this is the correct opinion.”
Contrary to the correct and famous opinion, there is another opinion that prohibits this. Radd Al-Muhtaar (the famous Hanafi book) reads: “There is another view that contradicts the famous opinion, which is the view of Al-Haakim that prohibits it, and most the Shaykhs (scholars of the Hanafi school of jurisprudence) refuted him, but the prominent jurist Ibn Al-Kamaal, supported the view of Al-Haakim and said: “The correct opinion is what Al-Haakim said.” Moreover, in his book entitled Al-Jaami’ Al-Kabeer, Al-Karkhi quoted the narration of Abu Haneefah and Abu Yusuf that if one wipes over his head with the water remaining (in his palms) after washing his arms, then this is not sufficient unless he takes fresh water because it (the remaining water) has been used once in purification...”
Finally, according to the view that it is permissible, there is no need for the questioner to ask whether or not rubbing the arms with the hands is considered as wiping because passing the hands over them is considered as washing and not wiping. The principle according to the Hanafi School is that it is permissible to wipe over the organ with the water remaining after washing while the water remaining after wiping is not permissible to use again.
Al-Bahr Ar-Raa’iq reads: “Wiping over one quarter of the head: it (wiping) linguistically means passing the hand over something, and in the Sharee’ah it means passing wet hands over the organ (of ablution) even with some moisture remaining after washing but not after wiping....”
The same thing is mentioned in Radd Al-Muhtaar: “It is permissible to wipe over the head with some moisture remaining after washing but not after wiping.”
Allaah Knows best.