# Vorticity Versus Strain Rate in 2-D

Consider a rectangular particle of initial size (Ax, Ay) near the wall y = 0. On the upper edge the velocity is u = Au. Equivalently, this corresponds to moving with a particle located at (0, 0) in a velocity field where v = 0. After a short time At the particle is sheared and takes the shape of a parallelogram. The strain rate is defined as є1і2 = limAy^0 Ay = &y. Similarly, considering a velocity field with u = 0 and upon superposition with the previous flow field, one finds the following strain rate, Fig.8.3

du dv

£1,2 = £2,1 = +

dy dx

 (a) y Fig. 8.3 Shear a in x-direction and b in ^-direction

 (b) y

 x

 Further, we define the following strain rates du. d v £1,1 = 2— and £2,2 = 2— dx dy

 (8.3)

 Vorticity is defined as

 du dv d y + dx

 (8.4)

 and the angular velocity of the particle as

 1 1 ( du dv Q = w = — + 2 2 dy dx

 (8.5)

 u(x + Ax, y + Ay, t) = u(x, y, t) + Au(x, y, t) +———– v(x + Ax, y + Ay, t) = v(x, y, t) + Av(x, y, t) + •••

 (8.6)

 where

 Au(x, y, t) = du Ax + Ц Ay + Av(x, y, t) = dvAx + dv Ay + ■

 (8.7)

 or, in matrix form

 (8.8) Note: any square matrix A can be decomposed uniquely into the sum of a sym­metric and antisymmetric matrices A = 1 (A + A*) + 1 (A – A)

where A* is the transposed of A. Hence 1 / £1,1 ё 1,2 2 £1,2 £2,2 1 ( 0 – из = 2 U 0

 (8.10)

 (8.11)  Therefore

u(x + Ax, y + Ay, t)
v(x + Ax, y + Ay, t)
general 2 – D motion (8.12) 