Okay. So you want to clear the cookie and not the session. Think about this. Even the big socializing site (facebook for example) don't have that kind of security. So honestly its hard to give an idea on how to do that.
If you have a login process, then you need to have a timed process that checks if an account is still active/online and if they aren't then you automatically log them out. You could do this every 15 minutes and any account that is logged-in but inactive would be logged out. Obviously, this also means that you need the ability to track activity. Unless you are doing financial transactions or some other very sensitive process, this isn't a big issue. If you have provided the ability to logout and the user chooses not to do it, then he/she can still access your site from that machine & browser without logging in again. In most cases, this is a "so what" situation.
What are you currently setting the cookies' expiration date to? If you don't set it to anything, then they will expire automatically when the browser closes.
If you're trying to automatically log the client out after, say, 15 minutes of inactivity, then you will need a system to monitor the user's actions and log when the last action (like loading a new page) took place.