13.17
Equation of Continuity
Fluid motion is represented by either velocity vectors or streamlines. The volume of a fluid flowing past a given location through an area during a pe…
The volume of fluid flowing through a point in an area in unit time gives the volume flow rate.
On substituting volume with area times distance, and using the relation between velocity and distance, volume flow rate equals area times velocity of the fluid.
Consider an incompressible fluid with the same density at all points flowing steadily through an irregular cross-sectional pipe. For a steady flow, the velocity and density of the fluid at a point remain constant with time.
The mass of the fluid passing through a point per unit time is termed as mass flow rate, and it equals density times the volume flow rate.
Since the pipe does not have any other source or sink, the mass flowing into the pipe must equal the mass leaving the pipe. This gives the general equation of continuity for fluids.
For incompressible fluids, density cancels out. Hence, the volume flow rate into the pipe equals the volume flow rate out of the pipe.
View the full transcript and gain access to JoVE Core videos
Q1: What is volume flow rate and how is it calculated?
Volume flow rate is the volume of fluid flowing through a point in an area per unit time. It is calculated by multiplying the cross-sectional area of the pipe by the velocity of the fluid. This relationship shows that flow rate depends on both the size of the conduit and how fast the fluid moves through it.
Q2: Why does fluid velocity increase when a pipe narrows?
When a pipe's cross-sectional area decreases, the fluid velocity must increase to maintain continuity of flow. Since the same amount of incompressible fluid must pass through any point in the pipe per unit time, a smaller area requires higher velocity to conserve the volume flow rate.
Q3: What is the equation of continuity for incompressible fluids?
The equation of continuity states that the volume flow rate entering a pipe equals the volume flow rate leaving it. For incompressible fluids, density cancels out, so the product of cross-sectional area and velocity remains constant throughout the pipe, ensuring mass conservation.
Q4: How does mass flow rate relate to volume flow rate?
Mass flow rate equals density multiplied by volume flow rate. It represents the mass of fluid passing through a point per unit time. For steady flow through a pipe with no sources or sinks, the mass flowing in must equal the mass flowing out.
Q5: What conditions must be met for steady flow in a pipe?
For steady flow, the velocity and density of the fluid at any point must remain constant over time. Additionally, the pipe must have no sources or sinks that add or remove fluid, ensuring that mass entering the pipe equals mass leaving it at all times.
Q6: Why is the equation of continuity valid for liquids but not always for gases?
Liquids are essentially incompressible, so their density remains constant, making the equation of continuity universally valid for all liquid flow. Gases are compressible, so the equation must be applied with caution when gases undergo compression or expansion, as density changes affect flow relationships significantly.
Q7: How does the equation of continuity relate to energy conservation in fluid flow?
The equation of continuity ensures mass conservation in steady flow by maintaining constant volume flow rate through varying pipe sections. This principle connects to energy conservation and bernoulli equation principles, where changes in velocity and pressure are governed by both mass continuity and energy conservation in fluid systems.