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I have a world where certain species of dragon live along the coasts, eating off the fish from the sea.
I like the idea that they can fly through the air and swim in pursuit of fish but I'm not sure that would be practical because the wings would cause drag. I'm also not sure whether they would even need them, because once they've got the sea who needs the air?
As you pointed out, the large wings needed to sustain flight in air would become a problem in water, causing excessive drag. This however doesn't make it impossible.
I think you have two choices:
The real world has plenty of examples of birds that ca swim brilliantly, so obviously wings aren't a disadvantage per se. However, there are a few caveats:
However, besides the mechanical, there's also a biological aspect to consider. A bird's wings are mostly feather. Feathers are not living tissue, do not require blood supply, and are very good insulators. This allows swimming birds to extend their wings more or less with impunity underwater, and to retain their body heat while swimming. A dragon's wings, however, are batlike, meaning that they are mostly skin, and a dragon is covered with scales, which do not trap air like feathers. This means that a dragon will rapidly lose body heat when submerged, and the moment they extend their wings, they start to act like radiators and the dragon starts losing heat even faster. Given that dragons are lizard-like, you may decide that they are cold-blooded, which means that heat loss is a serious problem. It's not an insurmountable problem - there are aquatic reptiles in the real world, such as swimming snakes and the marine iguanas of the Galapagos Islands - but it does limit you to warmer waters only.
Alternatively, you could say that because dragons breathe fire, they have an internal source of heat, but even so, they probably wouldn't enjoy this rapid cooling. On the other hand, you could say that a dragon's internal fire causes them to overheat, and they use their dives into the sea to cool off so that they don't expire (or combust!).
This could also answer your secondary question: why can swimming dragons still fly? A dragon who breathes fire might need to dip in the sea to cool off periodically, but their fire doesn't work so well underwater, so they never become fully aquatic. If they nest above water, then they're at risk from predators and egg thieves. Nesting on cliffs or high places makes it harder for most things to reach them, which gives them a reason to retain the ability to fly. At that point their main threat is birds, who use feathers, which fire is quite effective against. In fact, fire may be more effective at nest defence than at hunting; provided the cliffs (or wherever the nest is) are largely non-flammable (so that the dragon doesn't catch itself in a forest fire), breathing fire into the air from a static, grounded position is likely easier than trying to breathe fire on a grounded target while flying over it - not to mention that using fire to hunt may result in a meal that is either unreachable due to being in the middle of a brushfire, or inedible by virtue of being mostly charcoal and ash.
Thus, you have a creature that uses fire to defend their nest against the birds who are able to reach its inaccessible location, while frequently taking dives into the sea to cool off using its large, radiator-like wings to avoid self-inflicted heatstroke or self-ignition.
It depends on the dragon. Western mythological dragons look like pterosaurs, while Eastern (asian) mythological dragons are much more snake-like, and they don't even have wings. After all, they fly because of magic so, why should they need wings at all?
An asian style of dragon is an excellent swimmer, just like snakes or eels are, and there's no problem with wings.
Look for analogy.
There are birds. Some of them fly, some swim. They are like dragons, really. Except it takes a lot of work to make dragon flight possible.
We have birds that can't fly at all. We have some that can only on the surface, wings above waterline. Then we have wing propelled and foot propelled diving birds. You just need to read more about them and you'll see how it affects wings.
Generally birds that use wings in water evolve smaller wings and no longer can, nor need to fly. Just as you stated in your question. The ones that use foot can keep wings close to the body and avoid drag. Some can fly all right. Some gave up the sky. You can feasibly apply any of these to your dragons. You can even have few varieties, why not.
Another possibility to consider is the carapace, like some insects (mostly in the beetle family) have. Essentially, the flying wings fold up and are stored under the hard carapace, which is split and moves out of the way when the wings are to be extended for flying. If the carapace is shaped such that your dragon's body is hydrodynamic when the flying wings are not in use, you can then have a different propulsion mechanism for in-water movement.
The downside is that beetles aren't really efficient flyers, even among insects - and there's probably a point where the mass of the carapace makes it unreasonable, meaning that you're quite unlikely to have large dragons.
If wings cause drag, then just fold the wings up when underwater.
(Note: This may work better for short periods of time.)
Proof? See this.
