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High School Physics: Tutoring Solution22 chapters | 266 lessons

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Instructor:
*Aaron Miller*

Aaron teaches physics and holds a doctorate in physics.

The subject of this lesson is how we quantify the amount of fluid flowing through a pipe, channel, or other container. We will see that it depends on both the dimensions of the container and the speed of the fluid.

What is the difference between a fluid and a solid? If you think about your everyday experiences with fluids (like water or air) and a solid (like a potato), you may have a sense that one major difference between these two categories of matter is in the way they move. Potatoes are solid because they (mostly) retain their shape when they are thrown. However, the air you are breathing does not retain its shape as it moves into your lungs, so we say it is non-solid. Matter that is non-solid is called **fluid**. Fluids include liquids and gases. When a fluid moves (for example, as air moves in and out of your lungs during respiration), the matter is said to **flow**. As a fluid moves, it may change shape and volume (for example, air may be compressed), but, like a moving potato, its total mass stays the same as it moves. Often, the motion of a fluid is quantified by its **flow rate** instead of its velocity. We will see that flow rate obeys an important law described near the end of this lesson, which is what makes it an important quantity.

One of the most useful examples where flow rate can be calculated easily is a moving fluid in an enclosed volume (like a pipe), so we will discuss this specific example at length. Many situations in scientific and engineering applications are like fluid flow in a pipe, with the fluid moving uniformly in one direction on average through a contained volume. Some examples are blood flow in your circulatory system and air flow in a building's ventilation system.

The figure shows a fluid moving in a pipe. Let's assume for simplicity that everywhere within the pipe the fluid has the same velocity (i.e., near the edges of the pipe, the fluid has the same speed and direction of flow as the fluid near the center of the pipe). The direction of the velocity is toward the right, and the speed of the fluid is *v*. Because the fluid is moving in the same way everywhere in the pipe, this is an example of what is called a **uniform flow**.

The **flow rate** within the pipe is defined as the volume of fluid each second that is passing through a cross-sectional slice of the pipe, which is shown in the figure as a dotted circle. It turns out that under the assumption of uniform flow within the pipe, the flow rate, which is often represented by the symbol *Q*, can be directly related to the fluid speed *v* and a measure of the size of the slice -- the pipe's **cross-sectional area** *A.* In terms of these variables, the flow rate *Q* can be expressed as

where the units of *Q* are volume per time, or cubic meters per second in Standard International (SI) units. When applying this equation, your answer will only be in SI units if your area and speed are expressed in SI units: square meters and meters per second, respectively.

This formula says the flow rate of a pipe is greater for fluid that is passing more quickly through the pipe, and it is also greater when a pipe's cross-sectional area is larger. In the diagram, the cross-sectional area *A* is the area of the circle you could measure if you sliced open the pipe vertically, which is perpendicular to the velocity of the fluid. The cross-sectional slice of the pipe does not have to be circular, but in this diagram it is. A pipe carrying fluid can be any shape in general, but the meaning of *A* in the formula is always the same - it is the area of a slice that the fluid is flowing through. It is important to remember the reason why flow rate depends on *A*: wider pipes can deliver larger volumes of fluid at a higher rate than narrower pipes.

In real-life applications, most of the time the uniform flow assumption we used in the previous example is not entirely true. For example, near the inner surface of the pipe friction forces exist which restrict the fluid's motion. Consequently, fluid near the center of the pipe, where friction forces are weaker, moves with higher speed compared to fluid near the outer edge, so the flow is said to be **non-uniform**. In a non-uniform flow, the speed of the fluid depends on where exactly in the pipe's cross-section you are measuring. Fortunately, the flow rate formula above still works for non-uniform flows when we use the average speed of the fluid over the cross-section of the pipe for the variable *v*.

Along similar lines, situations often occur where fluid flow is not quite constant over time - for example, when the heart is pumping blood through the body's circulatory system. This is an example of a **non-steady** flow -- a flow with speed that changes with time. However, the motion of blood speeds up and slows down rhythmically as the heart beats. Again, the flow rate can be calculated with the formula above when the speed *v* is the average speed of the blood over one period of the motion.

After this discussion, perhaps you are wondering: if we can measure the velocity of fluid in the pipe, why is it necessary to wade through the extra complication of calculating a flow rate? Why isn't velocity a good enough measure of the fluid's motion?

The answer to this question is that when a fluid is in motion, its velocity often changes as it moves along its path, but in many practical situations, its flow rate does not change. This can be written as an equation:

The left-hand side of this equation is the total rate of **inflow** to a volume (like a pipe or other container) and the right-hand side of the total rate of **outflow**.

The statement means the rate of fluid flowing into a region of space is equal to the rate of fluid flowing out. It follows from the more familiar statement of conservation of mass: as matter (solid or fluid) moves, its total mass does not change. The statement relating inflow and outflow is true for many fluids in practice, as long as the fluid is **incompressible**, i.e., the mass fluid has a constant volume and does not expand or contract. The form of this equation is more complicated for compressible fluids whose volumes can more easily change as they move. However, the basic idea is the same for both kinds of fluids: the total mass of fluid flowing in to a system is equal to the total mass flowing out.

Flowing fluids are characterized by a quantity called the flow rate, which is defined as the volume of fluid flowing through an area each second. In a pipe, or other enclosed region, the flow rate can be expressed in terms of the fluid speed and the cross-sectional area of the pipe. Flow rate is an important quantity because it is conserved for incompressible fluids as the total mass of the moving fluid cannot change.

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High School Physics: Tutoring Solution22 chapters | 266 lessons

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