Out-of-equilibrium soft matter is prevalent in nature due to large energetic barriers allowing for long-lived mestastable states [1] and commonly used for industrial applications where flows and external fields are used to design materials with desirable properties. Active matter is a perfect example of intrinsically out-of-equilibrium systems, as they consume energy from the embedding medium at the level of each particle to perform work, typically, self-propulsion. Examples are found in living (bacteria [2], flocs of birds etc) or inert matter (Janus particles, nanorobots [3]), across a vast range of length scales. The emergence of complex and out-of-equilibrium behaviour constitutes a modelling challenge for which novel numerical and theoretical tools are required [4,5,6].  

One of the main challenges is to understand the behaviour of dynamic and complex systems from simple modelling tools [7], such as motility-induced phase separation [8]. A now classical example of the emergence of collective motion in active particles is the Viczek model [9], capable of modelling the behaviour of aligning self-propelled particles. Increasingly, computational works have been devoted to active particles in complex media [10]. These include active particles under hard [11] and soft [12] confinement, active particles in the presence of passive obstacles [13,14] or with space-dependent activity [15]. Often, active particles are found at interfaces [16] and the interactions with the interfaces are crucial to understand their behaviour [17]. Active liquid crystals are an indicative example of out-of-equilibrium complex systems where defects are continuously created inducing complex flows [18].

Externally driven soft matter systems can produce extremely well-organised materials with desired properties. Block copolymer melts have intrinsic ordering but controlling their global orientation requires the use of external fields. Shear flows [19] and electric fields [20] have been used to obtain long-ranged alignment of block copolymers, while computational tools have been used to address the stability [21], as well as the mechanism of alignment [22] and phase transitions [23].

Flow is a prevalent mechanism for perturbing complex materials from their equilibrium state. From a modelling perspective, the key question is what equations we should use to describe this material under flow. Connecting micro-scale motion of individual molecules to the continuum-level flow dynamics is paramount; and these model equations must address the issues of not only physics, but chemistry and engineering too. There is a clear need for the development of constitutive models from both the continuum and microstructural levels [24,25]; this requires numerical methods that can solve the resulting system of partial differential equations, molecular simulations techniques capable of exploring rheology [26,27], and perhaps machine learning approaches to make connections across length scales. For non-Newtonian fluids, we must be prepared for instabilities [28], phase changes, and problems arising from transport phenomenon, such as mixing, and heat and mass transfer [29]. We may also have to consider unsteady, turbulent, and chaotic flow characteristics [30]. Flow may arise from metrology or processing, imparting challenges such as inhomogeneity, sources/sinks, and free surfaces; on the other hand, flow may be induced by the locomotion inherent to active matter [31].