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Modeling of Flow Problems
Published in Krishnan Murugesan, Modeling and Simulation in Thermal and Fluids Engineering, 2023
In Stokes flow, only the force due to the viscosity of the fluid gets balanced with the force due to the pressure field of the flow field when the inertial force of the fluid is considered negligible. It is only the pressure force that drives the flow by overcoming the viscous force of the fluid. Fluid momentum equations always balance the rate of change of momentum of the fluid to the net forces acting on the fluid. Hence, in a strict sense, in the absence of inertial force, the Stokes flow does not have momentum equation. However, the approximate momentum equations can be written from the Navier-Stokes equations by neglecting the inertial terms. Thus, we get the following equations as the governing equations for two-dimensional Stokes flow.
Impregnation and resin flow analysis during compression process for thermoplastic composite production
Published in Advanced Composite Materials, 2021
Osuke Ishida, Junichi Kitada, Katsuhiko Nunotani, Kiyoshi Uzawa
The in-plane flow experiments have been conducted using a testing machine equipped with an open-edge mold. In these experiments, just the one-dimensional flows in opposite directions were created in the open-edge mold as shown in Figure 5. The flow can be presumed as Stokes flow because the Reynolds number is very low due to the low velocity and high viscosity. The flow behavior of the thin viscous layer between the two mold surfaces can be expressed by the Reynolds equation [31,32], which is derived from the Navier-Stokes equation with several assumptions such as thin film geometry, negligible inertial and body force. The one-dimensional flow can be expressed under the above assumptions as the following equation.
Topology optimization of heat and mass transfer problems in two fluids—one solid domains
Published in Numerical Heat Transfer, Part B: Fundamentals, 2019
Rony Tawk, Boutros Ghannam, Maroun Nemer
The application of topology optimization on fluid flow problems was first performed by Borrvall and Petersson [3] on Stoke flows using density method. Stokes flow is a type of fluid flow where the advective inertial forces are insignificant compared to viscous forces, thus having low Reynolds number (Re ≪ 1). The optimization problem consists of minimizing the dissipated power in a fluid domain; the total volume of fluid should not exceed a maximum value considered as a constraint. They used density method in their work, which was the first application of topology optimization technique on fluid flow problems. The interpolation scheme introduced in [3] was widely used afterwards in topology optimization of mechanical fluid problems. Aage et al. [4] took advantage of parallel computations methods to solve the same problem of Borrvall and Petersson [3]. Gresborh-Hansen et al. [5] considered outflow rate as optimization target. Guest et al. [6] used Darcy’s law of fluid flow in a porous medium, to deal with the presence of solid and fluid phases simultaneously in the optimization domain. The flow was then modeled by Darcy–Stokes equation. Wiker et al. [7] also treated Darcy–Stokes topology optimization problem. Their work included solution to an area to point drainage problem, where the goal was to transport all fluid out from the domain through a single part of the boundary with a minimal possible power consumption. They also studied the impact of many geometrical and mathematical parameters on the final solution. Contrary to [6], their results showed that regularization is needed in topology optimization of fluid problems to avoid numerical problems. Evgrafov et al. [8] stated that the problem of Darcy-Stokes flow in topology optimization admit solutions even if the limiting zero and infinite permeabilities of Darcy’s law are included in the design domain. Gersborg-Hansen et al. [9] extended the application of topology optimization on Navier–Stokes flows for low and moderate Reynolds number. Deng et al. [10] used density method to solve topology optimization problem for unsteady flow. Results showed that the final optimum design is influenced by the dynamic effect and Reynolds number. The works of Kreissl et al. [11] on unsteady flow showed that more the problem become unsteady, more the fluid channel architectures differ from those obtained in steady state problems. Deng et al. [12] solved the problem of topology optimization of flows driven by body forces. Their work included steady and unsteady flows. Results showed that in both types of flows, optimal design depends strongly on the type of body force.