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.tracker890 Source h said:
Thank you for reminding me about the importance of organizing and reviewing information. I'll be more careful about it in the future.erobz said:
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.
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.
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.
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.
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.