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## 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**

**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 ( Q _{v} ) 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 ( 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 () , volume ( V _{ol} ) 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 :**

**Q _{v} **=

**V**

_{ol}/**t**

**V _{ol} = A x = A v t , where x = v t**

**Q _{v} = ( A v t ) / t
**

**Q**_{v} = A v

_{v}= A v

**From the definition of the mas flow rate :**

**Q _{m} = m / t **

**m =V _{ol}
**

**V _{ol} = A x = A v t**

**Q _{m} = (A v t ) / t**

**Q**_{m} =A v =Q_{v}

_{m}=A v =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} =A_{1} v_{1}**

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

**v**_{1} / v_{2} = A_{2} / A_{1} , this relation is called the continuity equation

_{1}/ v

_{2}= A

_{2}/ A

_{1}, this relation is called the continuity equation

**The continuity equation**

**Applications on the continuity equation**

** Turbulent flow **

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