How to make a round-bottomed, round-ended groove - without a router

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I would do it with a scratch stock - you can make one from two small offcuts plus a bit of good steel ( I use broken pieces of a very large hacksaw blade, but any bit of good steel 1mm thick or so will do ) which you can grind to whatever shape you want. If you grind it square it will work in both directions.
I think that with careful use you could do the whole thing with that.
 
I suspect I would mostly chisel it out, begin by cutting a V groove and slowly paring out from there bevel down. It's the kind of thing where 3mm or even finer chisels really are your friend. Switch to gouges when you are somewhere close but in my case they wouldn't be to hand so I'd probably do the final rounding with paper. Something like this most of it only needs to be somewhere close, it's only the edges of the channel that will really catch your eye if they're not right.
 
If you have a mounting stand for your drill this may help.
 

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I have eight strips of wood that need "channels" cutting into the faces.
Well, others have suggested it already, but a router is the obvious choice for speed. You don't have one, fair enough, but maybe you have a woodworking acquaintance or friend that owns a router, or a woodworking business nearby you could approach with a willingness to pay. I'd guess it might take half an hour to an hour to rout all eight pieces ready for a bit of final sanding. Slainte.
 
I would do it with a scratch stock - you can make one from two small offcuts plus a bit of good steel ( I use broken pieces of a very large hacksaw blade, but any bit of good steel 1mm thick or so will do ) which you can grind to whatever shape you want. If you grind it square it will work in both directions.
I think that with careful use you could do the whole thing with that.
I like this idea. Easy to test/practice and I can envisage a simple jig/fence to keep the line and depth consistent. I might have the parts to hand too in the form of a few cabinet scrapers. Perhaps one of those would serve as stock. I'm not sure if they're hardened all over though, or only at the edge. I've seen small, curved scrapers for sale too which might actually be the best option anyway - because my "grinder" is a Dremel and/or a flat file. Not ideal...

It might be difficult to scrape the ends exactly "as drawn" (hemispherical) but a slightly longer, half-oval-shaped lead-in might look acceptable.

Thanks
 
I suspect I would mostly chisel it out, begin by cutting a V groove and slowly paring out from there bevel down. It's the kind of thing where 3mm or even finer chisels really are your friend. Switch to gouges when you are somewhere close but in my case they wouldn't be to hand so I'd probably do the final rounding with paper. Something like this most of it only needs to be somewhere close, it's only the edges of the channel that will really catch your eye if they're not right.
Definitely a good plan - which I'll combine with Peter's "scratch stock" suggestion. All sounds within my capabilities and actually quite promising. Ta.

PS. I suspect the spec might change from 5mm deep to 2mm deep once I start scraping the first one :)
 
If you have a drill stand (or can cobble together some fixed method of holding your drill vertically or horizontally) you could create a ‘table’ with a fence perpendicular to the drill and buy a 1/4” router bit.
I would try mounting the drill horizontally on a bench, add a bit of timber to get the chuck to the correct height for the groove, screw on a fence and two stops. You will effectively have made a one-off router table. It will be slow but should work.
 
If you have a drill stand (or can cobble together some fixed method of holding your drill vertically or horizontally) you could create a ‘table’ with a fence perpendicular to the drill and buy a 1/4” router bit.
I would try mounting the drill horizontally on a bench, add a bit of timber to get the chuck to the correct height for the groove, screw on a fence and two stops. You will effectively have made a one-off router table. It will be slow but should work.
 
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