Insights Blog
-- Browse All Articles --
Physics Articles
Physics Tutorials
Physics Guides
Physics FAQ
Math Articles
Math Tutorials
Math Guides
Math FAQ
Education Articles
Education Guides
Bio/Chem Articles
Technology Guides
Computer Science Tutorials
Forums
Trending
Featured Threads
Log in
Register
What's new
Search
Search
Search titles only
By:
Menu
Log in
Register
Navigation
More options
Contact us
Close Menu
JavaScript is disabled. For a better experience, please enable JavaScript in your browser before proceeding.
You are using an out of date browser. It may not display this or other websites correctly.
You should upgrade or use an
alternative browser
.
Forums
Physics
Classical Physics
Mechanics
Inertial Force in Fluid Mechanics
Reply to thread
Message
[QUOTE="pasmith, post: 6396734, member: 415692"] The reference frame is generally the frame in which the spatial coordinates are defined. But there are two choices for relating those coordinates to the flow field, and [URL="https://www.physicsforums.com/insights/partial-differentiation-without-tears/"]partial differentiation[/URL] with respect to time means different things in each of them. In the Eulerian description the flow field is [itex]\mathbf(\mathbf{x},t)[/itex] and the spatial coordinates [itex]\mathbf{x}[/itex] are fixed in space and fluid parcels move past them. Partial differentiation with respect to time is differentiation at a fixed point in space, and so [itex]\left(\dfrac{\partial \mathbf{u}}{\partial t}\right)_{\mathbf{x}}[/itex] doesn't give he acceleration of a particular fluid parcel. In the Lagrangian description, the flow field is [itex]\mathbf{v}(\mathbf{X},t)[/itex] and the spatial coordinates [itex]\mathbf{X}[/itex] labels the particular fluid particle which was initially at that position. Thus [itex]\left(\dfrac{\partial \mathbf{v}}{\partial t}\right)_{\mathbf{X}}[/itex] does give the acceleration of a fluid parcel, and if the frame of reference is inertial then there are no "ficticious forces". But [URL="https://www.physicsforums.com/insights/partial-differentiation-without-tears/"]partial differentiation[/URL] with respect to time doesn't tell you about a fixed point in space, because again the fluid parcel has moved. The two descriptions are related by noting that the fluid parcel [itex]\mathbf{X}[/itex] is at time [itex]t[/itex] at position [tex] \mathbf{x} = \mathbf{X} + \int_0^t \mathbf{v}(\mathbf{X},t)\,dt[/tex] and by the chain rule [tex] \left(\frac{\partial}{\partial t}\right)_{\mathbf{X}} = \left(\frac{\partial}{\partial t}\right)_{\mathbf{x}} + \mathbf{u} \cdot \nabla[/tex] which is where the "inertial force" in the Eulerian description comes from. Alternatively, in the Eulerian description you can consider conservation of momentum within a fixed volume. This momentum changes both due to forces acting on the volume and due to fluid parcels moving into or out of the volume. Now the flux of the [itex]i[/itex] component of mementum is [itex]\rho u_i \mathbf{u}[/itex] and taking the divergence and simplifying using the mass conservation equation yields [tex] \frac{\partial}{\partial t}(\rho u_i) + \nabla \cdot (\rho u_i \mathbf{u}) = \rho \left(\frac{\partial u_i}{\partial t} + \mathbf{u} \cdot \nabla u_i\right).[/tex] Does that make the [itex]\mathbf{u} \cdot \nabla \mathbf{u}[/itex] term a ficticious force? Depending on how you derive it it's either a component of acceleration or a term representing momentum flux, and is present in inertial frames. [/QUOTE]
Insert quotes…
Post reply
Forums
Physics
Classical Physics
Mechanics
Inertial Force in Fluid Mechanics
Back
Top