How can I process a large CSV file line by line?

I've successfully written a script that downloads a CSV file with the use of cURL and then parses the CSV into a array as follows:

$rows = array_map(function($a) {
    return str_getcsv($a, $delimiter);
}, explode("\n", $result));

Then I iterate over $rows using foreach to save something into the database.

The script works fine, but when using a larger CSV file (>10.000 lines), the script becomes rather slow and gives a lot more errors.

I would like to cut the CSV file into pieces, so not the whole file will be imported into a variable. I found the following solution, but that still processes the whole file at once.

Is there a method to cut the CSV into pieces and run the database function several times? Or is there a better way to process a large CSV file like this?

I'm relatively new to processing large files, so please be kind!