Downstream of gene expression, effectors such as the actomyosin contractile machinery drive embryo morphogenesis. During $Drosophila$ embryonic axis extension, actomyosin has a specific planar-polarised organisation, which is responsible for oriented cell intercalation. In addition to these cell rearrangements, cell shape changes also contribute to tissue deformation. While cell-autonomous dynamics are well described, understanding the tissue-scale behaviour challenges us to solve the corresponding mechanical problem at the scale of the whole embryo, since mechanical resistance of all neighbouring epithelia will feedback on individual cells. Here we propose a novel numerical approach to compute the whole-embryo dynamics of the actomyosin-rich apical epithelial surface. We input in the model specific patterns of actomyosin contractility, such as the planar-polarisation of actomyosin in defined ventro-lateral regions of the embryo. Tissue strain rates and displacements are then predicted over the whole embryo surface according to the global balance of stresses and the material behaviour of the epithelium. Epithelia are modelled using a rheological law that relates the rate of deformation to the local stresses and actomyosin anisotropic contractility. Predicted flow patterns are consistent with the cell flows observed when imaging $Drosophila$ axis extension $in\; toto$, using light sheet microscopy. The agreement between model and experimental data indicates that the anisotropic contractility of planar-polarised actomyosin in the ventro-lateral germband tissue can directly cause the tissue-scale deformations of the whole embryo. The three-dimensional mechanical balance is dependent on the geometry of the embryo, whose curved surface is taken into account in the simulations. Importantly, we find that to reproduce experimental flows, the model requires the presence of the cephalic furrow, a fold located anteriorly of the extending tissues. The presence of this geometric feature, through the global mechanical balance, guides the flow and orients extension towards the posterior end.