Question

When small aerosol particles or microorganism move through air or water, the Reynolds number is very small (Re << 1). Such flows are called creeping flows. The aerodynamic drag force, Fp, on an object in creeping flow is a function only of its speed V, some characteristic length scale L of the object, and fluid viscosity u (see Fig. 4). Use the method of repeating variables to generate a relationship for Fp as a function of the independent variables. Draw a sketch to illustrate the problem solution and show all details of your work. (6 points) M V FD -L- Figure 4

Answer 1

Method of i M, L, t repeating Variables and Buckingham poethesten Relevant parameters are Fo=f(V, L, M it four variables are listed in the functional form n 4 Number of fundamental dimensions Sepscented man this are M= 3 Buckingham No. of n-tams pi-theorem Combate the repeating parameters into products with each of the Searning parameters T = va Lb me to [MO LOT] = (wTt] a [L] ² [M2 "7-1] • [ML 7-e] Equate enponents TO=-a-c-2 ho=atb-c+1 M = 0 = c + 1 nom 4-3= C = -1 a+b=C-1 a + C = -2 (2) atb -0) a a=-2-(1) Solve for a -1 b=-2-(-1) 1 Substitute in LI vih Fo a Fe Fo= CONSTANT MUL VLM

Ž a object Drag force on 12 Gaping flow Le Assumptions (i) Recc kh (1) Gravitational effects are irrelevant Fo= Constant Mvh Dimensional analysis cannot predict the value of constant, since it depends the shape and orrentation of body in flow field. flow, For - Fe Fo= (3A) MVD E Sphone is cruping 11117 D