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Steady flow , Turbulent flow and Applications on the continuity equation

From Online Sciences - November 5, 2017

We can distinguish between two types of flow in fluids which are Steady flow and Turbulent flow , When a liquid moves such that its adjacent layers slide smoothly with respect to each other , we describe this motion as a laminar flow or a streamline ( steady ) flow , Every small amount of the liquid follow continuous path called streamline .

Steady flow

Steady flow is the flow in low speed such that its adjacent layers slide smoothly with respect to each other , Streamline is an imaginary line shows the path of any part of the fluid during its steady flow inside the tube ,The density of the streamlines at a point is the number of streamlines crossing perpendicular a unit area point .

Characteristics of the streamlines

Conditions of the steady flow

Flow rate is the quantity of liquid flowing through a certain cross-sectional area of a tube in one second , Flow rate could be volume flow rate and mass flow rate .

Volume flow rate ( Qv ) is the volume of fluid flowing through a certain area in one second , measuring unit is m/s , When volume rate of a liquid = 0.05 m/s , It means that volume of fluid flowing through a certain area in one second = 0.05 m .

Mass flow rate ( Qm ) is the mass of fluid flowing through a certain area in one second , measuring unit is kg/s , when mass flow rate of a liquid = 3 kg/s , It means that mass of fluid flowing through a certain area in one second = 3 kg .

Calculating the flow rate at any cross-sectional area :

Considering a quantity of liquid of density () , volume ( Vol ) and mass ( m ) flowing in speed ( v ) to move a distance ( x ) in time ( t ) through cross-sectional area of the tube ( A ) .

From the definition of the volume flow rate :

Qv = Vol / t

Vol = A x = A v t , where x = v t

Qv = ( A v t ) / t

Qv = A v

From the definition of the mas flow rate :

Qm = m / t

m =Vol

Vol = A x = A v t

Qm = (A v t ) / t

Qm =A v =Qv

The amount of liquid entering the tube = that emerging out of it in the same period of time .

Flow rate ( volume or mass ) is constant at any cross-sectional area and this is called law of conservation of mass that leads to the continuity equation .

Deduction of the continuity equation ( relation between flow speed of liquid and cross-sectional area of the tube )

Imagine that a tube has a fluid in a steady flow where the previous conditions of steady flow are verified .

Consider two-cross sectional areas ( A1 , A2 ) perpendicular to the streamlines :

At first cross-sectional area ( A1 ) , the speed of liquid through it ( v1 ) then :

The volume flow rate : Qv= A1 v1 , The mass flow rate : Qm =A1 v1

At second cross-sectional area ( A2 ) , the speed of liquid through it ( v2 ) then :

v1 / v2 = A2 / A1 , this relation is called the continuity equation

The continuity equation

Applications on the continuity equation

Turbulent flow

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