Given V = (3y)i + (6x)j + (x^2)k, calculate (a) the angular rotation, (b) the vorticity, and (c) the stress tensor.
Work:
(a) Calculate the angular roation W
W = Wx i + Wy j + Wz k, where i,j,k are direction vectors
Wx = (1/2)*(dw/dy - dv/dz), where d/dy and d/dz are partial derivatives
Wy = (1/2)*(du/dz - dw/dx), where d/dz and d/dx are partial derivatives
Wz = (1/2)*(dv/dx = du/dy), where d/dx and d/dy are partial derivatives
V = u i + v j + w k, where i,j,k are direction vectors
u = 3y; du/dz = 0; du/dy = 3
v = 6x; dv/dz = 0; dv/dx = 6
w = x^2; dw/dy = 0; dw/dx = 2x
Wx = (1/2)*(0-0) = 0
Wy = (1/2)*(0-2x) = -x
Wz = (1/2)*(6-3) = (3/2)
W = -x j + (3/2) k
W = -x j + (3/2) k, where j,k are direction vectors
(b) Calculate the vorticity VOR
VOR = (del) X V, where V = velocity vector
W = (1/2)*(del) X V, where W = angular rotation
VOR = 2W
VOR = 2*[-x j + (3/2) k]
Solution
VOR = 2*[-x j + (3/2) k]
(c) Calculate the stress tensor Pij
Pij = mu * [(du i/dx j) + (du j/dx i)], j not equal to i, and j,i are direction vectors
and -P + (2*mu)*(du i/dx i), j=i
where x1,x2,x3 = x,y,z, respectively and u1,u2,u3 = u,v,w, respectively