Final Flashcards
Assumptions for Cartesian N-S
Unidirectional [v = v_x], fully developed [v_x =/= v_x(x)], steady [v_x =/= v_x(t)], third dimension doesn’t matter [v_x =/= v_x(z)]
v = v_x(y) only
Assumptions for Pipe/Annulus Flow N-S
Steady, unidirectional, fully developed, symmetry wrt/theta
v = v_z(r) only
Assumptions for Stirred/Rotating Tank Flow N-S
Steady, unidirectional, tank/stir bar is tall in z, symmetry wrt/theta (and fully developed)
Only:
-v_theta^2 / r = - dP/dr
0 = mu d/dr (1/r d/dr(r v_theta))
BCs for Cartesian N-S
Slip or no-slip at top (y=h) and bottom (y=0) plates
Shear stress = 0 or is specified
BCs for Pipe Flow N-S
Given at r=0 and r=R
BCs for Annulus N-S
Given at r=r_i and r=r_o
BCs for Rotating Tank N-S
v_theta = 0 at r=0 (v_theta(0) is a minimum); no-slip at r=R
BCs for Stirred Tank N-S
no-slip at r=R; v_theta = 0 at r=infinity
BCs for Marangoni Flow N-S
v_x=0 at y=0, Tau_xy at y=H is K (Marangoni stress, defined as gradient of surface tension)
Pressure stress
Re «_space;1, pi ~ mu U / L
Re»_space; 1, pi ~ rho U^2
Wall stress from friction factor
Tau = (1/8) * f_D * rho * U^2
Laminar friction factor
64 / Re
Pipe head loss
s = (1/2) * f_D * U^2 * (L/D) * (1/g) = delta P / (rho*g)
Pipe pressure drop
delta P = (1/2) * f_D * rho * U^2 * (L/D)
Laminar c_D
Sphere: 24/Re
Flat Plate: 1.328/sqrt(Re)
Boundary layer on a flat plate
delta = r sqrt(mu x / rho U), where x = distance from leading edge
Capillary number
mu U / sigma
Pipe U and L
U = average velocity, L = diameter
Sphere U and L
U = free stream velocity, L = diameter
Flat plate U and L
U = free stream velocity, L = distance from leading edge
What does Reynolds number compare
Inertial effects / viscous effects
