Eveline is part of James Joyce's Dubliners, a collection of short stories that document the contemporary lives of the inhabitants of Dublin while simultaneously depicting themes related to aging and identity.
The central conflict in Eveline is, superficially, Eveline's option to go away with Frank or to remain in Ireland. At this scale, we might think of Eveline as simply a childish coward, too afraid to take the steps necessary to free herself of the oppressive circumstances of her life. However, on a deeper level, the conflict is a matter of identity; Eveline represents both the archetypal youth coming into adulthood, and the Irish identity facing the rest of the world. Is the prospect of better things worth leaving behind your heritage? Eveline decides that the answer is no.
Doubt is almost certainly one of the reasons for her decision. This is most revealed in the quote;
All the seas of the world tumbled about her heart. He was drawing her into them: he would drown her.
We may presume that Eveline is paralyzed (a theme through Dubliners) because her doubts are powerful enough to make her future with Frank seem like an actual, physical threat, such as drowning. Her world, up to this point, has been so small and familiar that she has no confidence that things will really be better with Frank. She is stepping from an unhappy but quiet world into one composed of nothing but risks.
Her eyes gave him no sign of love or farewell or recognition.
This is an enigmatic ending; we might think that Eveline has been consumed by her doubts, and she is a small person, trapped in a small but familiar world, destined to end up like her mother. Her crippling doubts explain why poverty and poor life decisions persist across generations, and this ending is a tragic one. On the other hand, perhaps she feels that going with Frank is a betrayal of her family and heritage, and her doubts were merely her conscience eating at her resolve. Her lack of love or recognition may simply a cold recognition that Frank represents temptation and hedonism.