Q: I have a medium length beard with empty patches on the sides. Do I have to rub it with water during wudu’ and ghusl, or it is enough to just wash it once by a fist of water while washing my face, just as the rest of the facial hair (eyebrows, moustache, and soul patch)?
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A: Rubbing is mustahabb in wudoo’ and ghusl according to most of the scholars, except for the Maalikis.
Al-Nawawi (may Allaah have mercy on him) said in al-Majmoo’ (2/214): 
Our view is that rubbing the parts of the body when doing ghusl and wudoo’ is Sunnah and is not obligatory. If (a person) pours water over himself and makes it reach (the skin) without his hands touching it, or he immerses himself in a large amount of water, or stands beneath a downspout or in the rain with the intention of doing ghusl, and it reaches his hair and skin, that is sufficient for wudoo’ and ghusl. This is the view of all the scholars except Maalik and al-Muzani, who stipulated that (rubbing is essential) in order for ghusl and wudoo’ to be valid. End quote.
Facial hair – including the beard, moustache, tuft of hair beneath the lower lip (“soul patch”) and eyebrows – is subject to further discussion.
If it is thin and the skin can clearly be seen beneath it, then it is obligatory to make the water reach the bottom of it. If it is thick and the skin cannot be seen through it, the outer surface only has to be washed, but it is mustahabb to run the fingers through it. This applies in the case of wudoo’; when it comes to ghusl, both the outer surface and lower layer must be washed.
It says in Zaad al-Mustaqni’: And he should wash his face from the hairline to the lower edge of the beard and chin lengthwise, and from one ear to the other crosswise, along with whatever thin hair may be present, and the outer surface of thick hair and however far it extends. End quote.
Shaykh Ibn ‘Uthaymeen (may Allaah have mercy on him) said in his Sharh:
The words “along with whatever thin hair may be present, and the outer surface of thick hair” mean: that through which the skin is visible. Thick hair is that through which nothing is visible.
In the case of thin hair, both it and what is beneath it must be washed. With regard to thick hair, the outer surface, but not the inner layer, must be washed. 
Similarly it is essential to wash whatever hair there is on the face, such as the moustache, “soul patch”, eyelashes, eyebrows and sideburns.
It is mustahabb to run the fingers through thick hair, because the Messenger of Allaah (peace and blessings of Allaah be upon him) used to run his fingers through his beard when doing wudoo’.
End quote from al-Sharh al-Mumti’ (1/212).
It says in Fataawa al-Lajnah al-Daa’imah (5/206): It is essential to wash the outer surface of a thick beard, but it is not obligatory to wash the inner layers or the skin beneath it, but it is prescribed to run the fingers through it. Al-Nawawi (may Allaah have mercy on him) said: There is no difference of opinion that it is obligatory to wash the thick beard, but it is not obligatory to wash the inner layer of the beard or the skin beneath it, according to scholarly consensus. This is the view of all the scholars among the Sahaabah, Taabi’een and those who came after them. Ibn Rushd said: This is something that is too well known for there to be any dispute. End quote.
As for a thin beard through which the skin may be seen, both the inner layer and outer surface must be washed. End quote.