What does "His mouth is set in a grim line" mean?

Question

I've come across with the sentence below:

David’s mouth is set in a grim line.

The word grin means:

• making you feel worried or unhappy
• looking or sounding very serious
• very bad, ugly, or unpleasant(informal)
• ill(informal)[1]

But I don't understand what the phrase exactly means. Does it mean "David looks seriously"?

So, could you please expain it to me?

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Note: David is an attorney.

The fuller text:

Riley hears Gwen’s scream, and despite her own immediate fear, tears up the stairs. The others are close on her heels. She arrives at the open doorway to Candice’s room.The first thing she sees is Gwen with her face buried in David’s chest, to her right, and then, beyond them, the body on the floor. [...] Riley can’t bear to look at the body anymore; she turns her attention to the others instead. Bradley is staring at Candice as if he’s seen a ghost, grabbing the edge of the desk to steady himself. David’s mouth is set in a grim line. Gwen, beside him, has her hand pressed hard against her mouth, trying not to throw up. Ian mutters, “Dear God,” and stands flat-footed, as Lauren pushes past him to the body. She moves to pull the scarf loose, touching Candice’s neck. “Get back, everyone,” David commands harshly. “There’s nothing we can do for her.”

[1]https://www.ldoceonline.com/dictionary/grim

Answer 1

The shape of the closed mouth can be referred to as a line. When you are smiling the line of the mouth curves upwards, from the middle to both ends. When you are frowning the line of the mouth curves downwards, from the middle to both ends.

When someone has a grim face, their mouth is more likely to be relatively straight, not curving in either direction, with the lips curving slightly in on themselves and pressed firmly together. The eyes are usually focused directly in front and tend not to move around as much as normal, in fact the face as a whole shows little animation. This is caused by the facial muscles, and the small muscles around the eyes, being stiff and tense.

A grim face is usually a sign that someone has experienced something extremely upsetting and they are trying to keep themselves composed; or they have an unpleasant and difficult task to perform, which probably has a low chance of success, but the are determined to overcome the obstacles that they are facing and eventually succeed. Obviously, in this case, we are talking about the first situation.