continuity equation is derived by taking point approach, how can we justify that it is valid for a finite control volume analysis

[Manoj Dash]

continuity equation is derived by taking point approach, how can we justify that it is valid for a finite control volume analysis

[manav...@gmail.com]

On Thursday, March 19, 2015 at 6:23:27 AM UTC-7, Manoj Dash wrote:
> continuity equation is derived by taking point approach, how can we justify that it is valid for a finite control volume analysis

the governing equation are the differential form and those are actually derived from considering the integral over the control volume and we converge /shrink the volume to a point in space and write the differential equation saying that the volume integral at piont is approaching to zero ,so it is actually from the finite volume analysis that we approach to a continuity at a point equation

since it is difficult to compute a integration of governing equations on whole volume mesh in a computer it ,and due to limitation of computational cost and limit ,we calculate it as a discrete point by using algebraic equations over complete grid pionts
i hope that i have cleared your doubt

[jagdeep....@gmail.com]

no doubt the continuity equation is derived by taking point approach but actually it is in macroscopic level that is approaches to zero means it have sufficient amount of molecules within the point so that we can apply continuum hypothesis.

[Omkar Manav]

On Friday, April 3, 2015 at 11:01:32 PM UTC-7, jagdeep....@gmail.com wrote:

no doubt the continuity equation is derived by taking point approach but actually it is in macroscopic level that is approaches to zero means it have sufficient amount of molecules within the point so that we can apply continuum hypothesis.  

sir 

i guess that the continuity n momentum  equation is derived in such a way to capture both lagrangian and euler frame  simultaneously ,  to combine  analysis of both the frames we evalaute the continuity eqaution to a point wise analysis  this is my understanding correct me if i am wrong ?

[jagdeep....@gmail.com]

First of all, lagrangian and eulerian framework are different.these are framework to predict flow field.
If the laws of physics are written for a fixed region of space, so that different fluid particles occupy this
region at different times, then the frame of reference is said to be Eulerian. However, if the laws govern the same fluid particles in a particular region that moves with the fluid, the laws are written in Lagrangian reference frame. Basic conservation laws can be applied more easily to an arbitrary collection of matter of fixed identity (a system composed of the same quantity of matter at all times) than to a volume fixed in space. Applying conservation laws to matter of fixed identity gives rise to Lagrangian reference frame and the associated substantial derivatives of volume integrals. However, control volumes of fixed shape is preferred (Eulerian reference frame) for the following reasons of difficulty with the Lagrangian reference frame.
1. The fluid media is capable of continuous distortion and deformation, since it is often extremely difficult to identify and follow the same mass of fluid at all times as must be done in Lagrangian reference frame.
2. Also, our primary interest often is not in the motion of a given mass of fluid, but rather in the effect
of the overall fluid motion on some device or structure.