Thanks to projects like Google thousands and thousands of books are digitized and become available through the internet. Most of these books are collector items and not available to the common researcher. I just searched in four of these books for more information dealing with the Austrian navy. Below are the results.
Army and navy chronicle, vol 8-9 Benjamin Homans, p. 294, 9 May 1839, p. 294 “The Austrian navy consists of 8 ships of the line, 8 frigates, 4 corvettes, 6 brigs, 7 schooners, and several lighter vessels; in all 47 sail. Austria , besides, keeps up on the Danube , for the protection of her frontier on the Turkish side, a flotilla of 25 boats, mounting 131 guns, and manned by the battalion of Tschaikists.”
The American almanac and repository of useful knowledge for the year 1838,volumes 9-10, p. 267 “The Austrian Navy is under the management of a naval commandant at Venice . It consists of from 30 to 34 vessels of war, of which 3 are ships of the line in ordinary, 5 frigates, 5 sloops, 8 brigs , and 6 schooners.”
Penny encyclopediea of the Society for the Diffusion if Useful Knowledge, 1835, volumes 3, p. 147 “The Austrian navy is under the management of a naval commandant at Venice , who is accountable to the council of war. It consists of between thirty and thirty-four vessels of war; among which are three ships of the line in ordinary, five frigates, five sloops, eight brigs, and six schooners. The establishments attached to it are a corps of naval artillery and engineers, a cadet academy, and a corps of marines. The flag, which is borne also by the merchant vessels, is of a red ground, with a broad white stripe in the centre.”