If you want to do this in a single shot, you'll need to do so early in the morning or late in the evening so you can do a long exposure without blowing out the image. How long the exposure needs to be depends on ambient light, ISO setting, how fast the water is flowing, and your own personal preference on how silky you want it. Taking such photos during the day is harder.
One option would be to get a neutral density filter to dim the light hitting the sensor so you can do a long exposure in broad daylight. The downside to this is phone lens adaptors are often cheap and ruin image quality, and the good ones are almost prohibitively expensive.
The other option is photo stacking as you've seen, which can be done manually. I used this app to create the following image.
https://play.google.com/store/apps/details?id=com.iudesk.android.photo.editor
View attachment 328244
That was a combination of the following two separate images.
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To create it, I used the text/image insert function and set the options to the following.
View attachment 328248
For something like flowing water, this may take some experimenting with the blend modes, and would need to be repeated many times with a lot of exposures.
Whatever method you use, you will need something to prop up the phone. That could be anything from a tripod to simply leaning your phone against something solid. You need the phone to remain still during a long exposure to prevent blur from camera shake. If you do image stacking, this keeps everything lined up (ignore my watermark misalignment, as I placed those manually in their respective photos).
If you want to try and automate the photo stacking option, I have seen some light painting apps that appear to work like that. I haven't personally tried them because I don't need them (they are more for phones unable to do manual long exposures), so I can't vouch for their effectiveness or make any specific recommendation.