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 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
- Imaginary lines do not intersect .
- The tangent at any point along the streamline determines the direction of the instantaneous velocity of each particle of the liquid at that point .
- 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 .
- 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
- Liquid should fill the tube completely .
- Speed of the liquid at a certain point in the tube is constant and does not change as the time passes .
- Flow is irrotational , there is no vertex motion .
- No frictional forces between the layers of the nonviscous liquid .
- 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 ( Qv ) 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 ( 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 ( ρ ) , volume ( Vol ) 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 :
Qv = ΔVol / Δt
ΔVol = A Δx = A v Δt , where Δx = v Δt
∴ Qv = ( A v Δt ) / Δt
Qv = A v
From the definition of the mas flow rate :
Qm = Δm / Δt
Δm = ρ ΔVol
ΔVol = A Δx = A v Δt
Qm = ( ρ A v Δt ) / Δt
Qm = ρ A v = ρ 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 = ρ A1 v1
At second cross-sectional area ( A2 ) , the speed of liquid through it ( v2 ) then :
The volume flow rate : Qv = A2 v2 , the mass flow rate : Qm = 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 ) .
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 :
- The speed of the fluid exceeded a certain limit and is characterized by small eddy currents .
- A gas transfers from small space to a wider space .
- A gas becomes turbulent when it transfers from high pressure to low pressure .