[Europan Ocean (OP)]

Looking at the Milky Way, it looks like the stars are not orbiting but swirling down into the black hole at the centre of the Milky Way. It seems to me that our solar system is orbiting down closer and closer to the centre of the galaxy rather than remaining at the same distance. It takes millions of years for one solar orbit.

Is this true that the stars are falling, swirling down into the increasingly strong gravity of the black hole in the centre? As it absorbs more stars?

Could this mean the solar year is getting shorter?

[Bored chemist]

it looks like the stars are not orbiting but swirling down into the black hole at the centre of the Milky Way.

To whom does it look that way?
Ah; I see...

It seems to me that our solar system is orbiting down closer and closer to the centre of the galaxy

You seem to be mistaking your imagination for evidence.

[Europan Ocean (OP)]

it looks like the stars are not orbiting but swirling down into the black hole at the centre of the Milky Way.

To whom does it look that way?
Ah; I see...

It seems to me that our solar system is orbiting down closer and closer to the centre of the galaxy

You seem to be mistaking your imagination for evidence.

The portrayals of our galaxy do not look like Saturn's rings, but the stars look like they are going into the centre rather than  simply orbiting the centre with a circle or elliptical rotation.
https://www.universetoday.com/106062/what-is-the-milky-way-2/

[Kryptid]

but the stars look like they are going into the centre

They're generally not (other than through gravitational radiation, which is such a slow process for regular stars that it might as well not be happening).

[Halc]

The portrayals of our galaxy do not look like Saturn's rings, but the stars look like they are going into the centre rather than  simply orbiting the centre with a circle or elliptical rotation.

You seem to be interpreting the portrayals as a stationary pattern along which the stars move. Both of these assumptions are wrong. The spiral pattern rotates, and at a rate considerably faster than once every 200 million years, so if the stars kept in their respective arms, they'd be traveling outward and would exit the galaxy after less than half a billion years. But no, the stars follow a generally circular path around the galaxy and move from one arm to the next as each overtakes our solar system in turn. The spaces between the arms are about as filled with stars of our magnitude as is the spaces within the arms, but it has far fewer short-lived bright stars that make the arms so visible.

All that said,

Looking at the Milky Way, it looks like the stars are not orbiting but swirling down into the black hole at the centre of the Milky Way.

There are some stars (a hundred or so?) that actually orbit the central black hole (Sgr-A), numbered S1, S2, ...
They're all quite large and I think we just cannot see the smaller ones. We're not one of those stars. The sum of our kinetic energy relative to Sgr-A plus the potential energy relative to Sgr-A is positive, and it must be negative for us to be in orbit about it. That's pretty much the definition of an orbital relationship, at least with only two objects. Only those S1... stars have negative total orbital energy relative to Sgr-A, which is actually a very small black hole for a galaxy our size, notorious for low rate of incoming new material.

It seems to me that our solar system is orbiting down closer and closer to the centre of the galaxy rather than remaining at the same distance.

It's actually getting further away slowly, for the same main reason Earth is moving away from the sun: The mass of the galaxy is decreasing as it consumes matter via fusion reactions. This raises our potential energy which correspondingly increases our orbital distance. That effect is greater than the other four processes (friction, random interaction with nearby masses, tides, gravitational waves) that also change that distance.

It takes millions of years for one solar orbit.

Between 200 and 250 million years in a not-particularly circular or even elliptical path.

Is this true that the stars are falling, swirling down into the increasingly strong gravity of the black hole in the centre?

The gravitational pull of our galaxy is decreasing, not increasing. Dropping a star into the black hole would probably increase our orbit a tiny bit just like dropping Mercury into our sun would likely result in Earth orbiting a tiny bit further away.

Could this mean the solar year is getting shorter?

Solar year is one trip of Earth around the sun. You mean the galactic year, and it is probably sufficiently variable from one orbit to the next that these cumulative changes would be overwhelmed by the noise.

[Janus]

To build on what Halc said about the spiral arms.  They are produced by compression waves moving through the galaxy. This induces a greater rate of new star formation within them.  So they tend to have a higher percentage of young stars.  The brighter the star, the faster it goes through its fuel, and the shorter its lifetime.  By the time a newly born really bright star leaves the spiral arm, it has passed its lifetime as a main sequence star and is a husk of its former self.  The stars that tend to live long enough to leave the spiral arm it was born in are the medium to dim stars.
In other words, the spiral arms look bright not because they are made up of a significantly higher concentration of stars or total material, but because they contain a higher ratio of young, bright stars.

[Bored chemist]

The black hole has a mass of about 4.154±0.014 million solar masses.
The milky way has something like  100 thousand million stars
So, if the sun is a "typical" star only about 1 in 25000 of the mass of the milky way  is due to the BH in the middle.
If that BH disappeared it wouldn't make much difference to the motion of the sun.
Mainly the MW is in orbit around itself.

[evan_au]

Mainly the MW is in orbit around itself.

And the mass of the Milky way is dominated by the (so far) invisible Dark Matter halo.

it looks like the stars are swirling down into the black hole

Out in the fringes (where we live), the stars are well-separated, most of the stars are in a fairly thin disk, have fairly circular orbits, and and have roughly the same angular velocity. So they won't interact very strongly with each other.

However, in the central bulge, stars are closely spaced, they have wildly different orbital planes, and the ones we can see near the central black hole have rather elliptical orbits, so the stars will transfer angular momentum between each other.

Some of these gravitational interactions will "cancel" angular momentum (if the stars have different orbital axes).
I expect that this will result in the central bulge flattening out over time - but the average distance from the central black hole will then be less than it is now. When stars are moving in a disk, they will have fairly stable orbits.

...that is, until the central black hole merges with another black hole; the incoming black hole will add angular momentum into the system on an entirely different axis, and once again send the stars off in all directions.
https://en.wikipedia.org/wiki/Sagittarius_A*_cluster

it looks like the stars are swirling down into the black hole

You inferred this from looking at a still image, and imagining the galaxy as a drain emptying in a whirlpool. This is not what "Whirlpool Galaxy" means.
https://www.nasa.gov/feature/goddard/2017/messier-51-the-whirlpool-galaxy

The Gaia space probe has been able to measure the velocity of stars in the Large Magellanic Cloud, and it is showing the stretching associated with being spaghettified by its close approach to the Milky Way. But the orbits are still roughly elliptical, rather than converging on the central black hole (if it has one...)
https://www.esa.int/ESA_Multimedia/Images/2018/04/Rotation_of_the_Large_Magellanic_Cloud#.Y4PLHulNzqU.link

[Janus]

Mainly the MW is in orbit around itself.

And the mass of the Milky way is dominated by the (so far) invisible Dark Matter halo.

Though a great bit of that halo extends beyond the Galactic disk, and thus, due to the shell theorem, has no effect on the stars' orbits around the Sun.  Only that DM at a lesser distance from the center than the star is question would.
 So for the Sun, DM has an effect on its orbit, but doesn't dominate, as the DM closer to the center than we are, while a significant fraction of the total mass affecting our orbit, it isn't the majority of it.
If you look at the predicted and observed rotation curves here:
https://www.astronomy.ohio-state.edu/thompson.1847/1101/RotCurve2.gif
Note that close in to the center, you see little to no difference, as the mass of visible matter heavily dominates there. You don't see the deviation until you get out of the central bulge region and into the disk.

[Zer0]

The gravitational pull of our galaxy is decreasing, not increasing. Dropping a star into the black hole would probably increase our orbit a tiny bit just like dropping Mercury into our sun would likely result in Earth orbiting a tiny bit further away.

At first glance, what you said seemed Incorrect.
But after reading it Slowly & Steadily 4 times, & applying Alot of thought, i Finally Understood it.

P.S. - Thanks!