By inhibiting the motion of dislocations by impurities in a solid solutions, is a strengthening mechanism. In solid solutions it is atomic level strengthening resulting from resistance to dislocation motion. Hence, the strength of the alloys can differ with respect to the precipitate's property. Example, the precipitate is stronger (ability to an obstacle to the dislocation motion) than the matrix and it shows an improvement of strength.
<h2> We now focus on purely two-dimensional flows, in which the velocity takes the form
</h2><h2>u(x, y, t) = u(x, y, t)i + v(x, y, t)j. (2.1)
</h2><h2>With the velocity given by (2.1), the vorticity takes the form
</h2><h2>ω = ∇ × u =
</h2><h2></h2><h2>∂v
</h2><h2>∂x −
</h2><h2>∂u
</h2><h2>∂y
</h2><h2>k. (2.2)
</h2><h2>We assume throughout that the flow is irrotational, i.e. that ∇ × u ≡ 0 and hence
</h2><h2>∂v
</h2><h2>∂x −
</h2><h2>∂u
</h2><h2>∂y = 0. (2.3)
</h2><h2>We have already shown in Section 1 that this condition implies the existence of a velocity
</h2><h2>potential φ such that u ≡ ∇φ, that is
</h2><h2>u =
</h2><h2>∂φ
</h2><h2>∂x, v =
</h2><h2>∂φ
</h2><h2>∂y . (2.4)
</h2><h2>We also recall the definition of φ as
</h2><h2>φ(x, y, t) = φ0(t) + Z x
</h2><h2>0
</h2><h2>u · dx = φ0(t) + Z x
</h2><h2>0
</h2><h2>(u dx + v dy), (2.5)
</h2><h2>where the scalar function φ0(t) is arbitrary, and the value of φ(x, y, t) is independent
</h2><h2>of the integration path chosen to join the origin 0 to the point x = (x, y). This fact is
</h2><h2>even easier to establish when we restrict our attention to two dimensions. If we consider
</h2><h2>two alternative paths, whose union forms a simple closed contour C in the (x, y)-plane,
</h2><h2>Green’s Theorem implies that
</h2><h2></h2><h2></h2><h2></h2><h2></h2><h2></h2><h2></h2><h2></h2>
