# 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**

**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 ( 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 ( A _{1} , A_{2} ) perpendicular to the streamlines :**

**At first cross-sectional area ( A _{1} ) , the speed of liquid through it ( v_{1 }) then :**

**The volume flow rate : Q _{v} = A_{1} v_{1} , The mass flow rate : Q_{m} = ρ A_{1} v_{1}**

**At second cross-sectional area ( A _{2 }) , the speed of liquid through it ( v_{2} ) then :**

** The volume flow rate : Q _{v} = A_{2} v_{2} , the mass flow rate : Q_{m} = ρ A_{2} v_{2}**

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

** ρ A _{1} v_{1} = ρ A_{2} v_{2}**

**A _{1} v_{1} = A_{2} v_{2}**

** v _{1} / v_{2} = A_{2} / A_{1} , 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 ( A _{1} v_{1} = A_{2} v_{2} ) if :**

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

**A _{1} v_{1} = A_{2} v_{2}**

**r² _{1} v_{1} = r²_{2} v_{2}**

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

**A _{1} v_{1} = n A_{2} v_{2}**

**r² _{1} v_{1} = n r²_{2} v_{2}**

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

**A _{1} v_{1}** =

**A**+

_{2}v_{2}**A**+

_{3}v_{3 }**A**

_{4}v_{4}**r² _{1} v_{1}** =

**r²**+

_{2}v_{2}

**r²**+

_{3}v_{3}**r²**

_{4}v_{4}**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 :**

**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 .**

**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**