To see what processes are accessing the disk partition you can use either lsof or fuser.
lsof /tmp
mysqld 13103 mysql 6u REG 8,6 0 12 /tmp/ibisfIna (deleted)
mysqld 13103 mysql 7u REG 8,6 504 13 /tmp/ibjWhwnX (deleted)
mysqld 13103 mysql 8u REG 8,6 0 14 /tmp/ibi5qTnK (deleted)
mysqld 13103 mysql 12u REG 8,6 0 15 /tmp/ibhsV78x (deleted)
java 18852 tomcat4 mem REG 8,6 16384 96580 /tmp/hsperfdata_tomcat4/18852
# fuser -m /dev/sda6
/dev/sda6: 13103 18852
# ps aux | grep 13103
mysql 13103 2.7 1.6 143892 34252 ? Sl Oct11 2692:12 /usr/libexec/mysqld
# ps aux | grep 18852
tomcat4 18852 0.0 1.8 283192 37928 ? Sl Dec16 1:28 /usr/java/j2sdk1.4.2/bin/java
You can then gracefully shutdown the daemon using the disk partion and unmount it.