Derivation of continuity equation

[makethings]

Homework Statement

Hi. I have a problem in fluid mechanics that is asking me to derive the conservation of mass equation using an infinitesimal control volume.


My problem is I do not know if I should be treating this problem as a fixed element or if the element is a parcel and its particles are tracked through together in time (a Langrangian method). Which would you assume?

I can solve the former, but if there is supposed to be such a thing for the latter I am not sure how to go about it. Would it be similar to a langangrian finite control volume?

 

[jbsteele]

Homework Equations The conservation of mass equation is ρ_out * V_out * u_out = ρ_in * V_in * u_inwhere ρ is the density, V is the volume of the control volume, and u is the velocity. The Attempt at a SolutionIf the element is a fixed infinitesimal element, then the conservation of mass equation can be derived by considering the mass flux through the element. The mass flux is defined as the product of the density and velocity of the fluid, which is then multiplied by the area of the element to find the total mass per unit time that is crossing the element. The conservation of mass equation is then given by: ρ_out * A_out * u_out = ρ_in * A_in * u_inwhere A is the area of the element. If the element is a parcel, then the conservation of mass equation can be derived using a Langrangian approach. In this approach, we track the particles of the fluid and the change in the total mass of the parcel is given by the difference between the mass entering and leaving the parcel. The conservation of mass equation is then given by: Δm_parcel = m_in - m_out where m is the mass of the fluid crossing the element.

 

[baba89]

I would provide the following response to this content:

The derivation of the continuity equation is a fundamental concept in fluid mechanics, and it is important to understand the assumptions and considerations that go into it. In terms of your question about whether to treat the problem as a fixed element or a parcel, it ultimately depends on the specific scenario you are studying and the assumptions that are appropriate for that situation.

If you are dealing with a system where the flow is steady and there is no change in the fluid properties over time, then it may be appropriate to treat the element as fixed and use the Eulerian method. However, if you are studying a system where the flow is unsteady and the fluid properties are changing over time, then the Lagrangian method may be more applicable.

In either case, the derivation of the continuity equation would involve considering an infinitesimal control volume and tracking the mass flow in and out of that volume. The main difference between the Eulerian and Lagrangian methods lies in how the control volume is defined and how the mass flow is calculated.

In summary, the approach you take for deriving the continuity equation should be based on the specific scenario you are studying and the assumptions that are appropriate for that situation. I recommend carefully considering the properties of your system and the assumptions you are making before proceeding with the derivation.