In this paper, a two-phase thermomechanical theory for granular suspensions is presented. Our approach is based on a mixture-theoretic formalism and is coupled with a nonlinear representation for the granular viscous stresses so as to capture the complex non-Newtonian behaviour of the suspensions of interest. This representation has a number of interesting properties: it is thermodynamically consistent, it is non-singular and vanishes at equilibrium and it predicts non-zero granular bulk viscosity and shear-rate-dependent normal viscous stresses. Another feature of the theory is that the resulting model incorporates a rate equation for the evolution of the volume fraction of the granular phase. As a result, the velocity fields of both the granular material and the carrier fluid are divergent even for constant-density flows. Further, in this article we present the incompressible limit of our model which is derived via low-Mach-number asymptotics. The reduced equations for the important special case of constant-density flows are also presented and discussed. Finally, we apply the proposed model to two test cases, namely, steady shear flow of a homogeneous suspension and fully developed pressure-driven channel flow, and compare its predictions with available experimental and numerical results.

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a b s t r a c tIn this paper, a numerical analysis of two velocity-two pressure models for flows of solid particles and fluids is presented. First, a formal exploitation of the weak formulation of such models asserts that they are amenable to integration via projection methods. The challenging issues in the algorithm development for these models are then documented and suitable numerical methodologies for their remedy are devised. Subsequently, an algorithm for the integration of the models of interest is proposed. This is a two-phase projection method on collocated grids that utilizes a fractional-step time-marching scheme. It is further endowed with an interface detection-and-treatment methodology to properly account for the stiffness induced by the presence of moving and deforming material interfaces. The efficiency and robustness of the proposed numerical method are assessed in a series of numerical experiments that are delineated in the last part of this paper.

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<div class="section abstract"><div class="htmlview paragraph">Vehicle electrification is bringing new challenges to the design of components for the automotive sector. New system requirements and functions are forcing either the development of new components or a complete redesign of the existing ones. In the absence of detailed pre-existing knowledge on operating conditions for these components, conservative requirements tend to result in overengineering. System modeling at vehicle level is a valuable approach in these circumstances, which can be used to efficiently estimate such conditions. With modeling, it is possible to define performance targets for components at an early development stage and to verify the impact of component design choices on vehicle performance. In this work we construct a full-vehicle model, which we use to frame the development of coolant distributor valves for electric powertrains.</div><div class="htmlview paragraph">In the first part of the work, we define the topology of the coolant circuit and the relevant interconnected systems (e.g. electrical power network, HVAC) based on vehicle teardown data. We identify representative operating conditions (e.g. driving cycle, ambient conditions). We combine the various systems into a vehicle global energy model.</div><div class="htmlview paragraph">In the second part of the work, we assess the influence of key design parameters for coolant distributor valves, such as the internal leakage, on global vehicle performance. The vehicle model includes a fluid-dynamic model of the valve calibrated on test measurements, and a simple control logic to define valve behavior as a function of the vehicle status (e.g. battery temperature, cabin requirements). We perform a parametric analysis for the internal leakage of the valve. With this analysis we can determine a leakage threshold up to which the energy efficiency of the vehicle and the quality of thermal management - expressed as time required to reach a temperature target - is not significantly affected. In turn, we show how realistic design constraints can be determined early in the development cycle of the system, avoiding overengineering and accelerating the development process.</div></div>