Fluid-Flow-and-Equation

Fluid Flow and Equation of Continuity

Fluid Flow and Equation of Continuity

Moving fluids are of great value. To find out about the behaviour of the fluid in motion, we consider their flow through the pipes. When a fluid is in motion, its flow can be either streamline or turbulent.

The flow is stated to be streamlined or laminar, if every particle that passes a specific point, moves along exactly the very same path, as followed by particles which passed that points earlier.

In this case, each particle of the fluid moves along a smooth path called a streamline. The different streamlines cannot cross each other. This condition is called steady flow condition. The directions of the streamlines are the same as the direction of the velocity of the fluid at that point. Above a specific velocity of the fluid flow, the motion of the fluid ends up being unsteady and irregular.

Under this condition the velocity of the fluid changes suddenly. In this case, the exact path of the particles of the fluid cannot be anticipated. The irregular or unsteady flow of the fluid is called turbulent flow.

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We can understand many features of the fluid in motion by considering the behaviour of a fluid which fulfils the following conditions.

  • The fluid is non-viscous i.e., there is no internal frictional force in between adjacent layers of fluid.
  • The fluid is incompressible, i.e., its density is constant.
  • The fluid movement is consistent.
Equation of Continuity

Suppose about a fluid flowing through a pipe of non-uniform size. The particles in the fluid move along the streamlines in a consistent state flow. In a small time Δt, the fluid at the lower end of tube moves a range ΔXi1, with a velocity v 1. If A1is the area of cross-section of this end, then the mass of the fluid contained in the shaded area is:

Equation-of-Continuity

 

Δ m 1 = ρ1A 1ΔX1 = ρ1 A 1 V 1 x Δt

Where ρ1is the density of the fluid. Likewise, the fluid that moves with velocity v2 through the upper end of the pipe (area of sample A2) in the exact same time Δt has a mass

Δm 2 = ρ2A2v2 xΔ t

If the fluid is incompressible and the flow is steady, the mass of the fluid is conserved. That is the mass that flows into the bottom of the pipe through A 1 in a time Δt should be equal to the mass of the liquid that drains through A2 in the very same time. For that reason,

Δ m 1 = Δm2

or ρ1A1v1 = ρ2A2v2

This equation is called the equation of continuity. Considering that density is constant for the steady flow of incompressible fluid, the formula of continuity becomes

A 1v 1= A 2 v 2

The product of the cross-sectional area of the pipe and the fluid speed at any point along the pipe is constant. This constant equals the volume flow per second of the fluid or simply flow rate.