A liquid drug, with the viscosity and density of water, is to be administered through a hypodermic needle. The inside diameter of the needle is 0.27 mm and its length is 50 mm. Determine (a) the maximum volume flow rate for which the flow will be laminar (Re < 2300), (b) the pressure drop required to deliver the maximum flow rate, and (c) the corresponding wall shear stress

(a) The maximum volume flow rate for which the flow will be laminar is 0.0190 cubic meter per second

(b) The pressure drop required to deliver the maximum flow rate is 148962.96 Pascal

(c) The corresponding wall shear stress is 7600 Pascal

Explanation:

Reynolds number = 2299, density of water = 1000kg/m^3, diameter of needle = 0.27mm = 0.00027m, Length of needle = 50mm = 0.05m, viscosity of water = 0.00089kg/ms, area = 0.05m * 0.05m = 0.0025m^2, coefficient of friction = 64 : Reynolds number = 64 : 2299 = 0.028

Velocity = (Reynolds number * viscosity) : (density * diameter) = (2299 * 0.00089) : (1000 * 0.00027) = 2.046 : 0.27 = 7.58m/s

(a) Maximum volume flow rate = velocity * area of needle = 7.58 * 0.0025 = 0.0190 cubic meter per second

(b) Pressure drop = (coefficient of friction * length * density * velocity^2) : (2 * diameter) = (0.028 * 0.05 * 1000 * 7.58^2) : (2 * 0.00027) = 80.44 : 0.00054 = 148962.96 Pascal

(c) Wall shear stress = (density * volume flow rate) : area = (1000 * 0.0190) : 0.0025 = 7600 Pascal