routing to ping automatically via the correct network

If I have a network namespace with two network adapters

eth0 192.168.11.10/24, gw 192.168.10.1

eth1 10.10.10.10/24


I can do successful pings to 192.168.10.x, 10.10.10.x and 192.168.10.1

But I cannot successfully ping 8.8.8.8. 

I can however successfully ping 8.8.8.8 with
 
ping -I eth0 8.8.8.8

Is there a way to redirect processes that 'bind' to the incorrect nic, 
to the correct nic? I have been trying a bit with custom routing 
tables/rules, but I do not seem to get this to work.