Figure 11.

(a) Plot showing ${\widehat{f}}_{\text{c}}({\alpha }_{\text{c}})$ for the choice (11) with parameters as given in (14) and (15) except that ${\widehat{k}}_{\text{w}}=20$ instead of ${\widehat{k}}_{\text{w}}=1$ signaling a reduced tissue conductivity. This change in fluid–ECM resistance force implies that the net effect of the fluid–ECM, cell–ECM, and cell–fluid interactions as expressed by ${\widehat{\zeta }}_{\text{w}}$, ${\widehat{\zeta }}_{\text{c}}$, and $\widehat{\zeta }$ (where the two latter are unchanged) removes the negative part of the fractional flow function ${\widehat{f}}_{\text{c}}({\alpha }_{\text{c}})$. On, the other hand, a 20 fold reduction in conductivity also gives a corresponding reduction in total velocity ${\mathbf{U}}_{\text{T}}$ such that the net effect on ${\mathbf{U}}_{\text{T}}{\widehat{f}}_{\text{c}}({\alpha }_{\text{c}})$ is adjusted correspondingly. (b) Plot showing ${\widehat{f}}_{\text{c}}({\alpha }_{\text{c}})$ for the choice (11) with parameters as given in (14) and (15) except that ${\widehat{k}}_{\text{w}}=20$ and $\widehat{k}=15$. This adjustment of the cell–fluid interaction effect through $\widehat{k}$ is sufficient to maintain a net upstream cell migration through the cell velocity component ${\mathbf{u}}_{c,\text{fluid}-\text{stress}}$. Zoomed version reveals a positive part reflecting downstream fluid–stress generated migration through ${\mathbf{u}}_{c,\text{fluid}-\text{stress}}$ for very small ${\alpha }_{\text{c}}$ ($<0.035$).