Hermione could use her intelligence to outwit Bellatrix. She might pretend to be compliant and gradually gain Bellatrix's trust, all the while looking for an opportunity to escape. For instance, she could trick Bellatrix into thinking she has some important information that can only be retrieved in a certain location, and then make a run for it when they get there.
One possible motive could be Bellatrix's loyalty to Voldemort. She might think that by kidnapping Hermione, she can get information about Harry Potter and the Order of the Phoenix. Hermione is known to be very intelligent and Bellatrix could believe she has valuable knowledge. For example, Hermione might know the location of Potter or the Order's plans against Voldemort.
Perhaps Bellatrix kidnapped Hermione to use her as bait to lure Harry. Bellatrix has always been extremely loyal to Voldemort and might think that by having Hermione, she can draw Harry out into the open. Hermione, being the intelligent witch she is, could try to outwit Bellatrix while in captivity, maybe by finding a way to send messages to her friends or even by trying to escape on her own.
The Gryffindor house would be in an uproar. Her friends and classmates would be desperate to do something. Neville, for example, might step up more than ever, given his own growth in courage. They would likely start by informing the teachers, like McGonagall, who would then take the matter to Dumbledore's Army.
In Hermione and Bellatrix fanfiction, the power dynamic is often an interesting point. Bellatrix has a great deal of power as a Death Eater, but Hermione has her own kind of power in her knowledge and magic skills. Some fanfictions may show a battle of wills between them, where Hermione refuses to be intimidated by Bellatrix's threats. This can lead to some really intense and exciting storylines, like when Hermione stands up to Bellatrix in a duel - like situation in a magically - created arena within the fanfic's plot.
One possible aspect of Hermione and Bellatrix fanfiction could be the exploration of their opposing natures. Hermione is a studious, rule - following Gryffindor, while Bellatrix is a wild and chaotic Death Eater. Fanfiction might play with the idea of how they interact in different scenarios. For example, in a scenario where Hermione is captured by Bellatrix, their exchanges could be filled with tension and power plays. Hermione's intelligence could be pitted against Bellatrix's brute force and loyalty to Voldemort.
Yes, there are likely fanfictions out there with such a plot. The 'Harry Potter' fandom is vast and diverse, and some fanfiction authors explore dark and alternate storylines. However, these often deviate from the original positive and heroic themes of the series.
Living with Bellatrix would be a terrifying experience for Hermione. However, she would rely on her wits. She would probably keep her Gryffindor courage in check and act more like a Slytherin, being cunning and sly. She might start to understand Bellatrix's insecurities and use them to her benefit. For instance, if Bellatrix is insecure about her standing among the Death Eaters, Hermione could offer advice that seems helpful but actually undermines Bellatrix's loyalty to Voldemort. And she would constantly be on the lookout for any signs of kindness or humanity in Bellatrix, hoping to use it as a way to change the situation.
One reason could be pure hatred. Bellatrix despises Muggle - borns, and Hermione being a very intelligent and powerful Muggle - born would be a prime target for her wrath. Another could be jealousy. Hermione's skills and knowledge might make Bellatrix feel inferior, leading her to lash out.
At first, Hermione would probably be in shock and defiance. But she'd quickly start to strategize. She'd use her vast knowledge of magic to protect herself from Bellatrix's potential outbursts.