Lagrangian reference frame

The substantial derivative, also called the material derivative, is the rate of change in a Lagrangian reference frame, that is, following a material particle. In vector notation the continuity equation may oe expressed as... 

A three-dimensional simulation method was used to simulate this extrusion process and others presented in this book. For this method, an FDM technique was used to solve the momentum equations Eqs. 7.43 to 7.45. The channel geometry used for this method was essentially identical to that of the unwound channel. That is, the width of the channel at the screw root was smaller than that at the barrel wall as forced by geometric constraints provided by Fig. 7.1. The Lagrangian reference frame transformation was used for all calculations, and thermal effects were included. The thermal effects were based on screw rotation. This three-dimensional simulation method was previously proven to predict accurately the simulation of pressures, temperatures, and rates for extruders of different diameters, screw designs, and resin types. 

A convenient way of determining the deformations experienced by fluid elements, in a particular flow situation, is the use of the Lagrangian reference frame. In such reference frame, the position of a material element is described as a function of time and the initial position of the material element, as indicated in Equation 22.24. From the latter equation, the deformation tensor, which, as stated by Ottino [60], is the basic measure of deformation with respect to the reference conflguration, X, can be obtained. The relationship between x, X, and the deformation tensor has been written as Equation 22.25. In general, the solution of the flow field is determined numerically and, then, the deformation tensor can be calculated as the fluid element... 

The trajectory of a dispersed-phase bubble can be predicted by integrating the force balance written in the Lagrangian reference frame [33, 38], This force balance equates the bubble inertia with the forces acting on the bubble and can be written as... 

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