Continuity Equation

 

Fluids: Continuity Equation

        As we may have already learned, a liquid is a nearly incompressible fluid, and if compression is present in a given situation, such compression is effectively negligible. On the other hand, gas can be compressed and should be appropriately accounted for. This difference in compression makes liquid a lot preferable compared to gas as we can easily neglect compression. Hence, the reason why most entry-level problems involving pressure only include liquids.

        The Continuity Equation is somewhat different from Archimedes' and Pascal's Principles since it deals with fluid in motion or fluid dynamics, while the previous ones deal with fluid in a static or hydrostatic environment. However, for the Continuity Equation to make sense, the involved fluid must have a Laminar Flow. This means that the fluid we're working on must be free-moving or must have no viscosity nor friction.

Volume Flow Rate

        Before I present the Continuity Equation, let's first understand the concept of Volumetric Flux, commonly referred to as Volume Flow Rate. It is defined as the volume of a fluid that passes through a certain cross-sectional area over a period of time. Volumetric Flux is commonly denoted with Q, however, you may find some references which define it as R.

Hence the equation,

This equation, however, is further broken down for versatility and practicality. Since we'll be dealing with variations of pipes, we can use the formula V = Ad, where Volume equates to Area(A) multiplied by the distance(d) or length of displacement of the fluid. Hence giving us the equation,

And since we're dealing with fluid dynamics, we'll be taking advantage of one of the kinetic equations. Which is v = d/t, hence resulting in an equation of

The letter v in this situation denotes the velocity of the fluid. Area(A) will vary depending on the shape of the pipe, however, since most pipes are cylindrical, we can use A = πr² 

Continuity Equation

        We know that liquids are nearly incompressible, therefore we have the advantage to neglect their compression. Such property is important for the continuity equation because the volume should remain the same unless subjected to intense or enormous amounts of pressure.

Here's an example

Refer to the illustration above.

Suppose that A1 and v1 are the inputs, while A2 and v2 are the outputs.

Based on the liquid's property to be nearly incompressible. We can deduce that the input volumetric flux should equate to that of the output volumetric flux. So simply,

Since we are aware that Q = V/t, hence

Given this equation, we can understand that the volume of the water

should remain the same given the same amount of time that elapsed.

Or if we're interested in finding the relationship between the Areas and velocities, then we can use this equation.

Since V = Ad and v = d/t

With this equation, we can deduce that if the cross-sectional area were to be changed, then the velocity should inversely compensate. This means that if the cross-sectional area were to be reduced, then the velocity should increase. Consequently, if the cross-sectional area increases, then the velocity should decrease.

        If you are asked to show the equation of continuity, then you can provide this equation. This one equation is widely known as the equation of continuity.

 

        However, if you are asked to find the equations of continuity (plural), then you must provide these three equations.

~kaku