Putting the next batch on the trub is a great way to have a major mess. The ferment takes off, fast! Great for heavy gravity IPAs, but messy. You could pour the trub through a sanitized sieve and use about 1/2 cup.
Keeping a liquid yeast culture going requires feeding it every two-three weeks, just like a sourdough starter. Pour off the top half of the water, add a teaspoon of DME and top off with cooled boiled water. Changing the water keeps the concentration of wastes down.