is there any way to know, who are children of the specific process ? for example those children which their parent ID is foo ?
You are looking for the pstree command.
pstree by itself will list all the processes in a tree form (like lsblk does). You can use the -p flag to get the PIDs listed as well, and the -s to show parent process as well:
$ pstree -p 602
udisksd(602)-+-{cleanup}(607)
|-{gdbus}(605)
|-{gmain}(603)
`-{probing-thread}(606)
A (probably) POSIX-compliant way of getting the child PIDs (that I'd mentioned in the comments elsewhere):
ps -o ppid= -o pid= -A | awk '$1 == <some pid>{print $2}'
This tells ps to write the parent PID and PID of all processes (without headings), and then uses awk to see which lines have the given PID in the first field (the parent PID), and prints the corresponding second field (the child PID).
