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Fluid Statics and Fluid Dynamics
Published in Robert E. Masterson, Nuclear Reactor Thermal Hydraulics, 2019
Fluids in motion behave differently than fluids at rest. This is because the local pressure can be affected by the way in which a fluid moves. Sometimes fluids in motion move at the same speed everywhere, and sometimes they do not. When the velocity of a fluid changes, a force is applied to the fluid that causes it to accelerate or decelerate. This acceleration or deceleration can result in a pressure change that is interpreted as a pressure drop or a pressure gain. The magnitude of this change can affect the way the core of a reactor behaves. The motion of a fluid can be described by either Lagrangian or Eulerian fluid mechanics (see Section 14.17), and each of these descriptions of fluid mechanics leads to an important parameter called the material derivative. The material derivative is needed to understand the motion of fluids when area or direction changes occur.
Kinematics of Deformation and Motion
Published in G. Thomas Mase, Ronald E. Smelser, Jenn Stroud Rossmann, Continuum Mechanics for Engineers, 2020
G. Thomas Mase, Ronald E. Smelser, Jenn Stroud Rossmann
In this section let us consider any physical or kinematic property of a continuum body. It may be a scalar, vector or tensor property, and so we represent it by the general symbol Pij… with the understanding that it may be expressed in either the material description Pij…=Pij…(X,t), or in the spatial description Pij…=Pij…(x,t), The material derivative of any such property is the time rate of change of that property for a specific collection of particles (one or more) of the continuum body. This derivative can be thought of as the rate at which Pij… changes when measured by an observer attached to, and traveling with, the particle, or group of particles. We use the differential operator d/dt, or the superpositioned dot, to denote a material derivative, and note that velocity and acceleration as we have previously defined them are material derivatives.
Theory of Motion
Published in Prasun Kumar Nayak, Mijanur Rahaman Seikh, Continuum Mechanics, 2022
Prasun Kumar Nayak, Mijanur Rahaman Seikh
The most important among various time rates is the concept of the material derivative. The time rate of change of any property of a continuum (such as temperature or velocity or stress tensor) with respect to specific particles of the moving continuum is called the material derivative of that property. The material derivative (also known as the substantial, or convective derivative) may be thought of as the time rate of change that would be measured by an observer traveling with the specific particles under study. The instantaneous position xi of a particle is itself a property of the particle.
Nonlinear wind-drift ocean currents in arctic regions
Published in Geophysical & Astrophysical Fluid Dynamics, 2022
The governing equations in the f-plane approximation for arctic ocean flow in the deep Canadian, Makarov and Nansen basins are (see Talley et al.2011) the Navier-Stokes equations coupled with the equation of mass conservation where is time, is the velocity vector, is the pressure, is the constant gravitational acceleration at the Earth's surface, is the depth-dependent water density, is the Coriolis parameter (with the constant rate of rotation of Earth about its polar axis), and are the constant horizontal and vertical eddy viscosity coefficients, respectively, and is the material derivative.
Large deformation mixed finite elements for smart structures
Published in Mechanics of Advanced Materials and Structures, 2020
Let denote the body of interest in undeformed configuration. The motion, or deformation of is described by the mapping , where is the current or deformed configuration in space. Each material point is mapped to a spatial point via with the displacement. The deformation gradient is the material derivative of ,
Modelling supernova-driven turbulence
Published in Geophysical & Astrophysical Fluid Dynamics, 2020
F. A. Gent, M.-M. Mac Low, M. J. Käpylä, G. R. Sarson, J. F. Hollins
In the momentum equation, the shock capturing viscosity is applied as where denotes velocity, ρ gas density and is the material derivative. The viscous coefficient takes the form where Taking only positive values of and otherwise zero, at any point the maximum value within two zones in any direction is applied.3 This field is then smoothed using a seven-point smoothing polynomial with Gaussian weights [1,9,45,70,45,9,1]/180 to obtain . Hence, the artificial viscosity is applied only locally at the shocks and has quadratic dependence on the divergence. The dimensionless constant .