Background
Job can be divided into seven primary sections:
1. Introduction (1:1-1:5)
2. Job's first test (1:6-1:22)
3. Job's second test (2:1-2:10)
4. Job, his friends, and Elihu (2:11-37:24)
5. YHVH's first answer (38:1-40:5)
6. YHVH's second answer (40:6-42:6)
7. Epilogue (42:7-42:17)
Both of Job's tests have three elements: (1) Heavenly scene (2) Earthly scene (3) Concluding statement from the narrator regarding Job and sin.1
The main section, consists of dialogue where each person offers opinions about Job, each other, man in general, and God. There is also mention of past events. For example, in Chapter 29 Job describes his life before the trials and gives details which agree with YHVH's statement about Job:
Then the LORD said to Satan, “Have you considered My servant Job, that there is none like him on the earth, a blameless and upright man, one who fears God and shuns evil?” (Job 1:8 NKJV)
From the discussions it is obvious none of the people, including Job and Elihu are aware of the heavenly component to Job's tests. In relation to the real-time narrative of the events, the two heavenly scenes are like a prequel to the narrative. In other words, the heavenly scenes are correctly placed in the chronology of events, but the information in these scenes was unknown to anyone at the time the events took place. In essence, the reader has a more complete understanding of what is happening, then the humans who are involved.
With respect to Satan (literally, Ha-Satan, the accuser), it is also clear none of the humans involved are aware there is a spiritual entity who challenges and defies YHVH. Everyone, Job included, sees the afflictions as coming directly from YHVH. Here too the reader knows otherwise:
6 And the LORD said to Satan, “Behold, he is in your hand, but spare his life.” 7 So Satan went out from the presence of the LORD, and struck Job with painful boils from the sole of his foot to the crown of his head (Job 2)
Essentially the book of Job is mankind's "introduction" to Ha-Satan. Without the benefit of the heavenly scenes, we would be unaware there is a spiritual entity who is actively opposing YHVH.
Answer
Quite simply, the reason Satan is not mentioned after 2:7, is none of the humans involved were aware of his existence, at the time in history. This lack of knowledge persists throughout YHVH's "answers" to Job. No where in what YHVH says does He mention Ha-Satan, the accusations made against Job, or Ha-Satan's role in afflicting Job. When YHVH finishes speaking, Job has no more direct knowledge of Ha-Satan than before. In addition, it is not just Job who is uninformed:
Then all his brothers, all his sisters, and all those who had been his acquaintances before, came to him and ate food with him in his house; and they consoled him and comforted him for all the adversity that the LORD had brought upon him... (Job 42:11)
Even after Job is restored to health, every one of Job's family members and acquaintances continues in the belief, YHVH brought about his afflictions. Everyone retains the conviction YHVH, not Ha-Satan was the direct cause of Job's afflictions.
In any war there are two types of battles, tactical and strategic. Strategic battles are essential to achieve the goal, but tactical battles may or may not contribute to the strategic success. Spiritual warfare is no different. Salvation by the death and resurrection of our Lord Jesus is the strategic outcome; Job describes a tactical conflict. Remove the Book of Job from the Bible and the strategy of salvation is unaffected; however, without Job, man lacks vital information about the nature, character, and abilities of Ha-Satan.
Therefore, YHVH allows Ha-Satan to "test" Job but limits his ability to take his life. YHVH knows eventually Job will make a request YHVH will answer bringing a tactical victory:
Who would grant me a hearer, that the Almighty may hear my desire; and that he himself that judgeth would write a book, (Job 31:35 DRA)
The reader learns of Ha-Satan because Job demanded the charges against him be written.
The key consideration is direct knowledge. Obviously Job has a change of heart after hearing YHVH's second answer, but this change corresponds to Job's response after his first test: in all this Job did not sin nor charge God with wrong. After the second answer Job finally "gets it:" YHVH was not the one who afflicted him.
While Job understands he was "collateral damage" (it is really YHVH who is under attack), neither he or anyone else in the events know the identity of the one who is opposed to YHVH. That information is found in the written account Job demanded.
1. The chiastic structure "pairs" elements from the beginning with those in the ending: (1) Introduction/(7) Epilogue; (2) Job's first test/(6) YHVH's second answer; (3) Job's second test/ (5) YHVH's first answer. Because of the structure, it is possible there is an implied connection to Ha-Satan in YHVH's responses. However, the obvious parallels are with Job's responses to the tests and to the "answers" he receives. That is, (2) Job 1:22 to (6) 42:1-6 and (3) Job 2:10 to (5) 40:3-5. It is important to recognize YHVH, who obviously knew of Ha-Satan and the charges made, never addresses them with Job; in this regard, Job never receives a full explanation for what happened.