All the answers that I can see here, including the currently "accepted" one, is actually wrong in that they set:
scrollTop = scrollHeight
Whereas the correct approach is to set:
scrollTop = scrollHeight - clientHeight
In other words:
$('#div1').scrollTop($('#div1')[0].scrollHeight - $('#div1')[0].clientHeight);
Or animated:
$("#div1").animate({
scrollTop: $('#div1')[0].scrollHeight - $('#div1')[0].clientHeight
}, 1000);