Shear localisation with 2D Viscous Froth and its relation to the Continuum Model 15 the relative angle between the normal vector to the soap film and the shear direction. To relate the normal drag force to the actual drag force of the Continuum Model, we need to project v ⊥ in the shear direction (see Fig. 11(ii)), resulting in | v | = | v $^{⊥}$| cos θ = | v$_{app}$ | (cos θ ) $^{2}$. giving the (continuum) drag force the required dimen- sions of force per area. In Sec. 4, we showed how the viscous stress has a λV dependence using dimensional arguments (see eq. 7). Using these arguments, but rather defining the strain rate as a locally changing quantity (see eq. 20), we see that where the Herschel-Bulkley exponent a and the dimen- sionless quantity ˆ c$_{v}$ are free parameters. Using all of the approximations calculated in this sec- tion, we proceed to solve eq. 11 numerically, accepting solutions only if they obey the inequality given by eq. 19, as done in Sec. 5. The key difference here is that localisation length is a function of the product λV . The upper bound for the Continuum Model predic- tion is formulated in terms of λV by inserting eq. 24 and eq. 25 into eq. 16, resulting in The corresponding lower bound must be found numeri- cally. To simplify this calculation, we fix λ and allow V to vary. A comparison of the VF and Continuum Model re- sults can be seen in Fig. 12, where values of a = 0 . 3 and ˆ c$_{v}$ = 0 . 26 are chosen as they give a reasonable prediction for both upper and lower bounds (although Finally, we consider how the orientation of the soap films in our foam might affect the drag force. We assume that the foam is isotropic and proceed to average over all possible values of θ . As < (cos θ ) 2 > = 1 / 2, our final expression for the continuum drag force coefficient c$_{d}$ is Fig. 11 Two projections are necessary to relate the veloc- ity v$_{app}$ of a soap film segment in the VF model to the local velocity v of the Continuum Model: (i) projection of the av- erage velocity of a soap film segment v$_{app}$ in the direction of the normal to that segment, and (ii) projection of the normal velocity of the soap film segment v ⊥ back in the direction of shear. θ is the angle between the normal vector to the soap film segment and the shear direction. It is the magnitude of the vectors that is displayed in the figure. c$_{v}$ = ˆ c$_{v}$γ 1 − a ¯ A a − 1 / $^{2}$λ a (25) l $^{+}$( λV ) = $^{(}$2 a ˆ c$_{v}$σ 1 − a ¯ A a − 1 / 2 (1 + a ) ˆ c$_{d}$ ) 1 1+ a ( λV ) a − 1 1+ a . (26) c$_{d}$ = ˆ c$_{d}$λ = 1 2$^{√}$ 2 √ 3 ¯ A λ , (24) θ |v$_{app}$| | v Т |=|v app | co s θ |v|=|v Т |cosθ soap film (i) (ii) Direction of shear 