id	author	title	date	pages	extension	mime	words	sentence	flesch	summary	cache	txt
american_scientific_journal-7932	Robin Ramlal; Karim Rahaman	Internal Newtonian Flow Due to a Cylinder Undergoing Longitudinal and Torsional Oscillations of Different Frequencies	2022	12	.pdf	application/pdf	3143	164	63	Hence, the velocity of the fluid takes the form, 𝑞 = v(𝑅, 𝑡)�̂� + 𝑤(𝑅, 𝑡) �̂� (2.2) = 𝜕v 𝜕𝑡 (3.2) 𝜈 ( 1 𝑅 𝜕𝑤 𝜕𝑅 + 𝜕2𝑤 𝜕𝑅2) = 𝜕𝑤 𝜕𝑡 (3.3) Assuming that the �̂� component of the velocity field takes the form, v(𝑅, 𝑡) = Re[𝑓(𝑅)𝑒𝑖Ω1𝑡] (3.4) where Re refers to the real part, along with the fact that the velocity must remain finite as 𝑅 → 0, one gets, on solving (3.2) and using the no-slip condition with (2.1), v(𝑅, 𝑡) = Re [𝑞0 cos 𝛽 𝐼1(√𝑖𝜎1𝑅) 𝐼1(√𝑖𝜎1𝑎) 𝑒𝑖Ω1𝑡] (3.5) where 𝜎1 = √ Ω1 𝜈 In a similar manner, the �̂� component of the velocity is, 𝑤(𝑅, 𝑡) = Re [𝑞0 sin 𝛽 𝐼0(√𝑖𝜎2𝑅) 𝐼0(√𝑖𝜎2𝑎) 𝑒𝑖Ω2𝑡] (3.6) where 𝜎2 = √ Ω2 𝜈 The tangential stresses on the wall of the cylinder are given by [10], 𝜏𝑅𝜃|𝑅=𝑎 = 𝜇 [ 𝜕v 𝜕𝑅 − v 𝑅 ] 𝑅=𝑎 (3.7) 𝜏𝑅𝑧|𝑅=𝑎 = 𝜇 [ 𝜕𝑤 𝜕𝑅 ] 𝑅=𝑎 (3.8) Substituting (3.5) into (3.7) gives for the torsional stress on the cylinder’s wall, 𝜏𝑅𝜃 = Re [𝜇 𝑞0 cos 𝛽 √𝑖𝜎1 𝐼0(√𝑖𝛼1)− 2 𝑎 𝐼1(√𝑖𝛼1) 𝐼1(√𝑖𝛼1) 𝑒𝑖Ω1𝑡] (3.9) American Academic Scientific Research Journal for Engineering, Technology, and Sciences (ASRJETS) (2022) Volume 89, No 1, pp229-240 232 where 𝛼1 = 𝜎1𝑎. Also, substituting (3.6) into (3.8) gives for the longitudinal stress on the cylinder’s wall, 𝜏𝑅𝑧 =	cache/american_scientific_journal-7932.pdf	txt/american_scientific_journal-7932.txt