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In continuum mechanics, the material derivative〔 describes the time rate of change of some physical quantity (like heat or momentum) for a material element subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation. For example, in fluid dynamics, take the case that the velocity field under consideration is the flow velocity itself, and the quantity of interest is the temperature of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as it is being moved along its pathline (trajectory) while following the fluid flow. ==Names== There are many other names for the material derivative, including: *advective derivative *convective derivative *derivative following the motion〔 *hydrodynamic derivative〔 *Lagrangian derivative *particle derivative *substantial derivative *substantive derivative *Stokes derivative〔 *total derivative〔 ==Definition== The material derivative is defined for any tensor field ''y'' that is ''macroscopic'', with the sense that it depends only on position and time coordinates (y=y( x, ''t'' ) ): : where is the covariant derivative of the tensor, and u( x, ''t'' ) is the flow velocity. Generally the convective derivative of the field u•∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u•(∇y), or as involving the streamline directional derivative of the field (u•∇) y, leading to the same result. Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent by the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative ''D/Dt'', instead for only the spatial term, u•∇.,〔 which is also a redundant nomenclature. In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. The effect of the time independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection. 抄文引用元・出典: フリー百科事典『 u•∇., which is also a redundant nomenclature. In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. The effect of the time independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.">ウィキペディア（Wikipedia）』 ■u•∇., which is also a redundant nomenclature. In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. The effect of the time independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.">ウィキペディアで「In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) for a material element subjected to a space-and-time-dependent macroscopic velocity field. The material derivative can serve as a link between Eulerian and Lagrangian descriptions of continuum deformation.For example, in fluid dynamics, take the case that the velocity field under consideration is the flow velocity itself, and the quantity of interest is the temperature of the fluid. Then the material derivative describes the temperature evolution of a certain fluid parcel in time, as it is being moved along its pathline (trajectory) while following the fluid flow.==Names== There are many other names for the material derivative, including:*advective derivative*convective derivative*derivative following the motion*hydrodynamic derivative*Lagrangian derivative*particle derivative*substantial derivative*substantive derivative*Stokes derivative*total derivative==Definition==The material derivative is defined for any tensor field ''y'' that is ''macroscopic'', with the sense that it depends only on position and time coordinates (y=y( x, ''t'' ) )::\frac \equiv \frac + \mathbf\cdot\nabla y,where \nabla y is the covariant derivative of the tensor, and u( x, ''t'' ) is the flow velocity. Generally the convective derivative of the field u•∇y, the one that contains the covariant derivative of the field, can be interpreted both as involving the streamline tensor derivative of the field u•(∇y), or as involving the streamline directional derivative of the field (u•∇) y, leading to the same result. Only this spatial term containing the flow velocity describes the transport of the field in the flow, while the other describes the intrinsic variation of the field, independent by the presence of any flow. Confusingly, sometimes the name "convective derivative" is used for the whole material derivative ''D/Dt'', instead for only the spatial term, u•∇., which is also a redundant nomenclature. In the nonredundant nomenclature the material derivative only equals the convective derivative for absent flows. The effect of the time independent terms in the definitions are for the scalar and tensor case respectively known as advection and convection.」の詳細全文を読む スポンサード リンク
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