1. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 The curve γ and an approximation of it by a Dirac chain. (a) The curve γ. (b) A depiction of A 3, which approximates γ. Figure Legend:

2. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 (a) Depiction of Δu(p;α). The arrow on the right-hand side illustrates –(p; α), the opposite of the original chain (p; α), and the arrow on the left-hand side illustrates (p+u;α), which is a translation of (p; α). (b) Depiction of ΔvΔu(p;α). The arrow on the upper right depicts the original chain (p; α) and the other arrows illustrate chains obtained from this one by translation and inversion. Figure Legend:

3. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 Convergence of the rate of change of area as given by the transport identity in the right-hand side of Eq. (104). (a) Plot of the evolution of A·TT(n) for different values of n, which correspond to the different approximations of the right-hand side of Eq. (104). (b) Convergence of the rate of change of the area A·TT for different snapshots in time. Notice the logarithmic abscissa in (b). Figure Legend:

4. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 Convergence of the rate of change of the area for different snapshots in time using (a) Riemann sums and (b) Simpson's rule. Notice the logarithmic abscissas in (a) and (b). Figure Legend:

5. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 Comparison of A·TT(8),A·Rie,c(8), and A·Sim,c(8) with A·TT(16). Figure Legend:

6. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 Graph of h˜ at various times. These graphs also appear in Seguin and Fried [10]. (a) Graph of h˜(·,1/4). (b) Graph of h˜(·,3/5). Figure Legend:

7. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 (a) Rate of change of the circulation computed with the transport identity (117) for different n with dimensionless integration time step Δτ = 0.001, dimensionless kinematic viscosity ν = 1, and dimensionless circulation Γ = 10. The inset in (a) shows a detailed view of the region around the peak of maximal rate of change of the circulation. (b) Original fractal curve at τ = 0 and deformed fractal curve at τ = 2 for n = 8 and Γ = 10 along with streamlines of the velocity field of the Lamb–Oseen [25, 26] vortex (dashed lines). Figure Legend:

8. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 Snapshots of a cross section of the domain D˜R in the (ξ, ζ)-plane at τ = 0.25. (a) The entire cross section [–1, 1] × [–1, 1]. (b) A detailed view [−0.25, −0.15] × [−0.25, −0.15] of the cross section showing the crack tip. Figure Legend:

9. Date of download: 7/8/2016 Copyright © ASME. All rights reserved. From: Extending the Transport Theorem to Rough Domains of Integration Appl. Mech. Rev. 2014;66(5):050802-050802-16. doi:10.1115/1.4026910 Comparison of ˜·ETT(n) over nondimensional time τ for different resolutions n Figure Legend: