The Continuity equation



Rating - 4/5
455 views

The mass of a moving fluid doesn’t change as it flows. This leads to an important relationship called the continuity equation. Consider a portion of a flow tube between two stationary cross sections with areas A1 and A2 (Fig. 12.21). The fluid speeds at these sections are v1 and v2, respectively. As we mentioned above, no fluid flows in or out across the side walls of such a tube. During a small time interval dt, the fluid at A1 moves a distance ds1 = v1 dt, so a cylinder of fluid with height v1 dt and volume dV1 = A1v1 dt flows into the tube across A1. During this same interval, a cylinder of volume dV2 = A2v2 dt flows out of the tube across A2.

 

Let’s first consider the case of an incompressible fluid so that the density r has the same value at all points. The mass dm1 flowing into the tube across A1 in time dt is dm1 = rA1v1 dt. Similarly, the mass dm2 that flows out across A2 in the same time is dm2 = rA2v2 dt. In steady flow the total mass in the tube is constant, so dm1 = dm2 and 

rA1v1 dt = rA2v2 dt or  

The Continuity equation

 

 

The product Av is the volume flow rate dV>dt, the rate at which volume crosses a section of the tube:

The Continuity equation

 

These Topics Are Also In Your Syllabus The Continuity equation
1 Types of Operating Systems - Batch operating system, Time-sharing systems, Distributed OS, Network OS, Real Time OS link
2 Turbulence link
You May Find Something Very Interesting Here. The Continuity equation link
3 SUMMARY OF FLUID MECHANISM link
4 Gravitation link
5 Types Of Systems link

 

The mass flow rate is the mass flow per unit time through a cross section. This is equal to the density r times the volume flow rate dV/dt.

The Continuity equation

 

 

 

 

 

 

 

These Topics Are Also In Your Syllabus The Continuity equation
1 Types of Operating Systems - Batch operating system, Time-sharing systems, Distributed OS, Network OS, Real Time OS link
2 Describing oscillation link
You May Find Something Very Interesting Here. The Continuity equation link
3 amplitude, Period, Frequency, and angular Frequency link
4 Simple Harmonic motion link
5 circular motion and the equations of SHM link

 

 

 

 

 

Equation (12.10) shows that the volume flow rate has the same value at all points along any flow tube (Fig. 12.22). When the cross section of a flow tube decreases, the speed increases, and vice versa. A broad, deep part of a river has a larger cross section and slower current than a narrow, shallow part, but the volume flow rates are the same in both. This is the essence of the familiar maxim, “Still waters run deep.” If a water pipe with 2-cm diameter is connected to a pipe with 1-cm diameter, the flow speed is four times as great in the 1-cm part as in the 2-cm part.

We can generalize Eq. (12.10) for the case in which the fluid is not incompressible. If r1 and r2 are the densities at sections 1 and 2, then

The Continuity equation

 

If the fluid is denser at point 2 than at point 1 1r2 7 r12, the volume flow rate at point 2 will be less than at point 1 1A2v2 6 A1v12. We leave the details to you. If the fluid is incompressible so that r1 and r2 are always equal, Eq. (12.12) reduces to Eq. (12.10)

These Topics Are Also In Your Syllabus The Continuity equation
1 Types of Operating Systems - Batch operating system, Time-sharing systems, Distributed OS, Network OS, Real Time OS link
2 Estimates and order of magnitudes link
You May Find Something Very Interesting Here. The Continuity equation link
3 Vectors and vector addition link
4 Equilibrium and Elasticity link
5 Types Of Systems link

The Continuity equation

 


Rating - 4/5
528 views

Advertisements