Steady flow , Turbulent flow and Applications on the continuity equation

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

1. Imaginary lines do not intersect .
2. The tangent at any point along the streamline determines the direction of the instantaneous velocity of each particle of the liquid at that point .
3. The number of streamlines does not change as the cross-section area changes , while the streamlines density at a point changes as the cross-section area changes and expresses the flow velocity of the liquid at that point .
4. Therefore , streamlines cram up at points of high velocity ( its density increases ) and keep apart at points of low velocity ( its density decreases ) , This means that speed of fluid at any point inside the tube is directly proportional to the density of streamlines at that point .

Conditions of the steady flow

1. Liquid should fill the tube completely .
2. Speed of the liquid at a certain point in the tube is constant and does not change as the time passes .
3. Flow is irrotational , there is no vertex motion .
4. No frictional forces between the layers of the nonviscous liquid .
5. Flow such that the amount of liquid entering the tube equals that emerging out of it in the same period of time because the liquid is incompressible .

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 ( Q_v ) is the volume of fluid flowing through a certain area in one second , measuring unit is m³/s , When volume rate of a liquid = 0.05 m³/s , It means that volume of fluid flowing through a certain area in one second = 0.05 m³ .

Mass flow rate ( Q_m ) 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 ( ρ ) , volume ( V_ol ) and mass ( m ) flowing in speed ( v ) to move a distance ( Δx ) in time ( Δt ) through cross-sectional area of the tube ( A ) .

From the definition of the volume flow rate :

Q_v = ΔV_ol / Δt

ΔV_ol = A Δx = A v Δt     , where Δx = v Δt  

∴ Q_v = ( A v Δt ) / Δt

Q_v = A v

From the definition of the mas flow rate :

Q_m = Δm / Δt

Δm = ρ ΔV_ol

ΔV_ol = A Δx = A v Δt

Q_m = ( ρ A v Δt  ) / Δt

Q_m = ρ A v = ρ Q_v

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 :  Q_v = A₁ v₁  , The mass flow rate : Q_m = ρ A₁ v₁

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

The volume flow rate : Q_v = A₂ v₂ , the mass flow rate : Q_m = ρ A₂ v₂

The flow rate ( volume or mass ) is constant in case of steady flow .

 ρ A₁ v₁ = ρ A₂ v₂

A₁ v₁ = A₂ v₂

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

The continuity equation

The velocity of a fluid in a steady flow at any point is inversely proportional to cross-sectional area of the tube at that point .

Based on the previous relation ( A₁ v₁ = A₂ v₂ ) if :

The tube is cylindrical having two cross-sectional area one is wide and the other narrow .

A₁ v₁ = A₂ v₂

r²₁ v₁ = r²₂ v₂

The tube is branched into ( n ) branches of the same cross-sectional area .

A₁ v₁ = n A₂ v₂

r²₁ v₁ = n r²₂ v₂

The tube is branched into number of branches of different cross-sectional area 

A₁ v₁ = A₂ v₂ + A₃ v₃ + A₄ v₄

r²₁ v₁ = r²₂ v₂ + r²₃ v₃ + r²₄ v₄

Where : A = π r² , r = radius of the tube .

The speed is inversely proportional to the cross-sectional area ( v ∝ 1/A ) , so , the liquid flows slowly in the tube when its cross-sectional area is big and vice versa .

Applications on the continuity equation

Flow of blood is faster in the main artery than in the blood capillaries because the sum of cross-sectional areas of blood capillaries is greater than the cross-sectional area of the main artery and since ( v ∝ 1/A ) , so , speed of blood decreases in the blood capillaries to allow exchange of oxygen and carbon dioxide gases in the tissues to supply it with food .

Design of the gas opening in the stoves , Opening are small so that the gas rushes fast out of it in a high speed ( v ∝ 1/A ) .

Turbulent flow

The turbulent flow is the flow when the speed of the fluid exceeds a certain limit and is characterized by small eddy currents , The steady flow of a fluid ( liquid or gas ) becomes turbulent flow if :

1. The speed of the fluid exceeded a certain limit and is characterized by small eddy currents .
2. A gas transfers from small space to a wider space .
3. A gas becomes turbulent when it transfers from high pressure to low pressure .

Applications on the pressure at a point ( Connected vessels , U-shaped tube & Mercuric barometer )

Applications on pascal’s principle , Manometer types and uses

Factors affecting the force of viscosity and Applications on the viscosity