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Physics - Classical Mechanics - Introduction to Fluid Dynamics

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Introduction

Hey it's a me again @drifter1!

Today we continue with Physics, and more specifically the branch of "Classical Mechanics", in order to make an Introduction to Fluid Dynamics.

So, without further ado, let's get straight into it!

Ideal Fluid

Up to this point, we only studied fluids at rest. Fluids in motion are more complex, but there are some concepts that can be explained through the use of idealized models.

First of all, real fluids have internal friction, which is known as Viscosity (The concepts of viscosity and turbulence will be covered more thoroughly in a later part) and the density in compressed fluids is not constant. For that reason, models are based upon ideal fluids. An ideal fluid is basically an uncompressed fluid with negligible viscosity.

In the case of liquids, the assumption of no compression tends to be a good approximation. For gases no compression only makes sense when the difference in pressure between the various regions is not big.

Viscosity (internal friction) causes shear stress as the fluid is in motion. In some cases, the shear forces are negligible as they are much smaller than gravity and pressure. So, ignoring viscosity is sometimes feasible even in real occasions.

Flow Characteristics

The motion of fluids can be represented through streamlines. A streamline represents the path that a small volume of fluid "follows" as it flows. The velocity of the fluid is tangent to these streamlines.

If the flow can be represented by smooth, and mostly parallel, streamlines and doesn't change much over time, we are talking about steady or laminar flow. On the other hand, if the streamlines are irregular and change over time, the flow is known as turbulent.

Volume Flow Rate

When talking about fluids, velocity isn't that useful. For example, rapid mountain streams carry less water then wide and deep rivers. So, it makes sense to talk about the volume (or even the mass) that flows by a given location through an area during a period of time. This is known as volume flow rate and denoted by Q, or mathematically:

The volume of a cylinder is equal to A dx, where A is the area of the cross-section and dx the width of the portion of fluid. So, considering a cylinder the previous equation can be re-written as:

which relates the volume flow rate, cross-sectional area and velocity.

The S.I. unit of flow rate is the m3/sec, but its also common to use liters per minute (L/min).

Flow Continuity

Let's consider a fluid pipe with decreasing radius. We are talking about a real fluid which is incompressible, and so the same amount of fluid must pass by any point in the tube. In order to ensure flow continuity (no mass is added or removed) the same mass must flow in and flow out at any given point. When the density is the same, we can extend our logic towards volume. So, as the cross-sectional area decreases the velocity must increase. In other words, the volume flow rate must be constant.

Therefore, for any two points in a pipe, we define:

Liquids are considered incompressible and so this equation of continuity is valid for all liquids. For gases, however, which are compressible by nature, any compression or expansion of the gas must be considered as well.

Mass Flow Rate

Another way of describing the flow rate of fluids is by the mass flow rate. Mass can always be determined from the density and volume, and so the mass flow rate is defined as:

Applying the concept of continuity, the mass entering must be equal to the mass leaving the pipe, or any given point in the pipe, and so:

which is the general form of the continuity equation.

For an ideal fluid the densities cancel each other out, yielding the equation mentioned earlier. This general form makes only sense in the case of real fluids.

RESOURCES:

References

  1. https://courses.lumenlearning.com/suny-osuniversityphysics/chapter/14-5-fluid-dynamics/
  2. https://www.khanacademy.org/science/physics/fluids/fluid-dynamics/a/what-is-volume-flow-rate

Images

  1. https://pxhere.com/en/photo/1045542

Mathematical equations used in this article, where made using quicklatex.

Visualizations were made using draw.io.

Previous articles of the series

Rectlinear motion

Plane motion

Newton's laws and Applications

Work and Energy

Momentum and Impulse

Angular Motion

Equilibrium and Elasticity

Gravity

Periodic Motion

Fluid Mechanics

Final words | Next up

And this is actually it for today's post!

Next time we will talk about Bernoulli's Equation...

See ya!

Keep on drifting!