Confusion in using the continuity equation here

In summary, the discussion revolves around the challenges and misunderstandings associated with applying the continuity equation in certain contexts. It highlights the importance of correctly interpreting the equation's variables and conditions to avoid miscalculations and ensure accurate representation of fluid dynamics.
  • #1
tracker890 Source h
90
11
Homework Statement
I feel that the mass within the system changes over time, but this perception contradicts the solution.
Relevant Equations
continuity equation
Q： Why does assuming "Properties in the tank are uniform, but time-dependent" lead to the validity of
(DmDt)sys=0? Doesn't the mass within the system change over time?
reference.
1700656597283.png
 
Physics news on Phys.org
  • #2
tracker890 Source h said:
Homework Statement: I feel that the mass within the system changes over time, but this perception contradicts the solution.
Relevant Equations: continuity equation

Q： Why does assuming "Properties in the tank are uniform, but time-dependent" lead to the validity of
(DmDt)sys=0? Doesn't the mass within the system change over time?
reference.
View attachment 335945
The mass of the system is the total mass, i.e. what’s inside and what’s outside the control volume at a particular time. It is invariant (at least in classical physics?). at ##t=0## all of the system is inside the control volume, as time progresses some portion of the system is outside. That ∫ on the left (unsteady) represents what portion of the system is inside (only) the control volume at a particular time.

Summarizing: The system is not the control volume. The system is the stuff (matter) inside the control volume, on its way into the control volume, or what has left the control volume.
 
Last edited:
  • Like
Likes tracker890 Source h and TSny
  • #3
To add a little to @erobz, the first integral ##\int_{\small CV}\rho dV## is the instantaneous mass within the control volume. This mass changes with time. So, ##\frac{\partial}{\partial t}\int_{\small CV}\rho dV## represents the rate of change of mass within the tank. The second integral represents the rate at which mass is flowing out through the neck of the container.
 
  • Like
Likes tracker890 Source h, Chestermiller and erobz
  • #5

FAQ: Confusion in using the continuity equation here

What is the continuity equation?

The continuity equation is a mathematical expression that describes the transport of some quantity, such as mass or energy, within a physical system. In fluid dynamics, it is often used to express the conservation of mass, stating that the rate of mass entering a system is equal to the rate of mass leaving the system plus the rate of accumulation of mass within the system.

How do you apply the continuity equation to fluid flow?

To apply the continuity equation to fluid flow, you need to ensure that the mass flow rate is conserved across different sections of a flow system. For an incompressible fluid, this can be expressed as A1V1 = A2V2, where A is the cross-sectional area and V is the velocity of the fluid at different points (1 and 2) in the system. This equation helps in understanding how changes in area affect the velocity of the fluid.

What are common mistakes when using the continuity equation?

Common mistakes include assuming the fluid is incompressible when it is not, neglecting changes in density for compressible fluids, and misapplying the equation to scenarios where the flow is not steady. Additionally, errors can occur if the cross-sectional areas or velocities are not accurately measured or if the flow is not uniform across the section.

Can the continuity equation be used for compressible fluids?

Yes, the continuity equation can be used for compressible fluids, but it requires accounting for changes in density. The general form of the continuity equation for compressible flow is ∂(ρA)/∂t + ∂(ρAV)/∂x = 0, where ρ is the fluid density, A is the cross-sectional area, V is the velocity, t is time, and x is the spatial coordinate. This form ensures mass conservation by incorporating density variations.

How does the continuity equation relate to other conservation laws?

The continuity equation is a specific case of the more general conservation laws, which include the conservation of mass, energy, and momentum. It specifically deals with the conservation of mass in a fluid system. When combined with the Navier-Stokes equations (for momentum) and the energy equation, it provides a comprehensive framework for analyzing fluid flow and other physical processes.

Similar threads

Why can a fluid that satisfies the continuity equation cross streamlines?
Replies
14
Views
918
I Static sphere with gravitating fluid
Replies
6
Views
896
Choosing Reference Points in Bernoulli's Equation for Fluid Flow
Replies
2
Views
2K
I Momentum of a Water Jet Impacting Plate
Replies
1
Views
2K
Fluid dynamics: Knowledge continuity equation
Replies
2
Views
2K
Floating block with objects that are thrown in a liquid
Replies
8
Views
2K
Why does this solution assume it's a steady-flow system?
Replies
6
Views
2K
Velocity of a Cart accelerated by a Jet stream
Replies
10
Views
2K
I Calculating Water Pool Depth: Factors to Consider in Fluid Dynamics
Replies
13
Views
1K
Pressure Differential in Fluid Dynamics: Why More on the '+s' Side?
Replies
5
Views
1K

Hot Threads

Recent Insights

Back
Top