Derive the Continuity Equation for Incompressible Fluids One Dimensional Flow

General Concept of Fluid Flow

One way of describing the motion of a fluid is to divide the fluid into infinitesimal volume elements,which we may call fluid particles,and to follow the motion of each particle. If we knew the forces acting on each fluid particle,we could then solve for the coordinates and velocities of each particle as functions of the time. This procedure,which is a direct generalization of particle mechanics, was first developed by Joseph Louis Lagrange (1736 – 1813). Because the number of fluid particles is generally very large,using this method is a formidable task.

There is a different treatment, developed by Leonhard Euler (1707 – 1783),that is more convenient for most purposes. In it we give up the attempt to specify the history of each fluid particle and instead specify the density and the velocity of the fluid at each point in space at each instant of time. This is the method we shall use. We describe the motion of the fluid by specifying the densityρ(x,y,z,t)and the velocityv( x,y,z,t) at the point x,y,z at the time t. We thus focus our attention on what is happening at a particular point in space at a particular time, rather than on which is happening to a particular fluid particle. Any quantity used in describing the state of the fluid,for example, the pressure p, will have a definite value at each point in space and at each instant of time. Although this description of fluid motion focuses attention on a point in space rather than on a fluid particle, we cannot avoid following the fluid particles themselves,at least for short time intervals dt.After all, the laws of mechanics apply to particles and not to points in space.

We first consider some general characteristics of fluid flow.

• Fluid flow can be steady or non-steady.We describe the flow in terms of the values of such variables at pressure,density,and flow velocity at every point of the fluid. If these variables are constant in time,the flow is said to be steady. The values of these variables will generally change from one point to another, but they do not change with time at any particular point. The condition can often be achieved at low flow speeds; a gently flowing stream is an example. In non-steady flow, as in a tidal bore, the velocities v are functions of the time. In the case of turbulent flow, such as rapids waterfall, the velocities vary erratically from point to point as well as from time to time.
• Fluid flow can be compressible or in-compressible.If the densityρof a fluid is a constant,independent of x,y,z, and t,its flow is called in-compressible flow. Liquids can usually be considered as following in-compressible. But even for a highly compressible gas the variation in density may be insignificant,and for practical purposes we can consider its flow to be in-compressible. For example,in flights at speeds much lower than the speed of sound in air (described by subsonic aerodynamics), the flow of the air over the wings is nearly in-compressible.

•  Viscosity in fluid motion is the analogue of friction in the motion of solids. When a fluid flows such that no energy is dissipated through viscous forces, the flow is said to be non-viscous. In many cases,such as in lubrication problems,viscosity is extremely important;motor oils,for example,are rated according to their viscosity and its vibration with temperature. In other cases,viscosity may be relatively unimportant, and by neglecting it we can use a simple description in terms of non-viscous flow.
•  If an element of the moving fluid does not rotate about an exis through the center of mass of the element, the flow is said to be irrotational. We can imagine a small paddle wheel immersed in the moving fluid. If the wheel moves without rotating,the motion is irrotational;otherwise it is rotational. Note that a particular element of fluid can move in a circular path and still experience irrotational flow;an analogy is the motion of the hanging cars in a Ferris wheel – even though the wheel rotates;the people in the cars do not rotate about their center of mass. The vortex formed when water flows around a bathtub drain is an example of this kind of irrotational flow.
• The ideal fluid moves without turbulence.This implies that each element of the fluid has zero angular velocity about its centre.

To simplify the mathematical description of fluid motion,we confine our discussion of fluid dynamics for the most part of steady, in-compressible,non-viscous,irrotational flow. We run the danger,however,of making so many simplifying assumptions that we are no longer talking about a real fluid. Furthermore, it is sometimes difficult to decide whether a given property of a fluid- its viscosity,say – can be neglected in a particular situation. In spite of all this,the restricted analysis that we are going to give has wide application in practice,as we shall see.

What is Continuity equation in fluid mechanics?

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

Mathematically it is represented as:

Continuity equation formula

Av = Constant

Continuity equation derivation

Consider a fluid flowing through a pipe of non uniform size.The particles in the fluid move along the same lines in a steady flow.

If we consider the flow for a short interval of time Δt,the fluid at the lower end of the pipe covers a distance Δx₁with a velocity v₁,then:

Distance covered by the fluid = Δx₁= v₁Δt

Let A₁be the area of cross section of the lower end then volume of the fluid that flows into the pipe at the lower end =V= A₁Δx₁= A₁ v₁ Δt

If ρ is the density of the fluid,then the mass of the fluid contained in the shaded region of lower end of the pipe is:

Δm₁=Density × volume

Δm₁= ρ₁A₁v₁Δt ——–(1)

Now the mass flux defined as the mass of the fluid per unit time passing through any cross section at lower end is:

Δm_{1/Δt =}ρ₁A₁v₁

Mass flux at lower end =ρ₁A₁v _{1 ———————(2)}

If the fluid moves with velocity v₂through the upper end of pipe having cross sectional area A₂in time Δt,then the mass flux at the upper end is given by:

Δm_2/Δt= ρ₂A₂v₂

Mass flux at upper end =ρ₂A₂v _{2 ———————–(3)}

Since the flow is steady,so the density of the fluid between the lower and upper end of the pipe does not change with time.Thus the mass flux at the lower end must be equal to the mass flux at the upper end so:

ρ₁A₁v_{1 =}ρ ₂ A ₂ v _{2  ———————-(4)}

In more general form we can write :

ρ A v =constant

This relation describes the law of conservation of mass in fluid dynamics.If the fluid is in compressible ,then density is constant for steady flow of in compressible fluid so

ρ₁=ρ₂

Now equation (4) can be written as:

A₁v_{1 =}A₂v₂

In general:

A v = constant

If R is the volume flow rate then :

R = A v = constant

Watch also:

Read also:
Bernoulli's equation
Viscosity
Terminal velocity

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Source: https://oxscience.com/continuity-equation/

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