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Thursday, 16 April 2020

POTENTIAL FLOW THEORY

INTRODUCTION:

This particular flow is a ideal flow which is not a real flow.This particular flow may not be there in physically possible flows due to viscosity.we consider potential flow as the non-viscous flow i.e neglecting viscosity.This potential flow will help us to understand the lift force on an object.But doesn't talk about the drag force.


continuity equation:

       velocity vector⇒ v=uî+vĴ
       then,
              ටu/ ටx   +   ටv/ ටy  = 0
             This is the continuity equation.
*The velocity vectors which satisfies the above equation such flow fields are physically possible flow  fields.That particular flow field will have a stream function (ψ) or else no streamm function.

Irrotationality:

   Rotational means rotating itself on its own axis.not rotating about a common axis.
              ටv/ ටx   -  ටu/ ටy  = 0
            This is irrotationality condition.

 DERIVATION FOR POTENTIAL FLOW EQUATIONS:

>>stream line is a mathematical line which flow velocity is tangential.
eq of stream line:
         udy -vdx =0
     
    u= ටψ/ටy      v=  - ටψ/ටx

   tanθ=dy/dx   =v/u.

eq of stream line:
>> udy -vdx =0
>>(ටψ/ටy)dy  -  (- ටψ/ටx)dx =0
>> (ටψ/ටx)dx + (ටψ/ටy)dy =0
      This is the total derivative of  ψ.
>>d ψ=0
      integrating on both sides
>> ψ=const.

*This says that along any stram line, stream function is constant.
*Another point is no two stream lines have the same stream function  ψ.

From  irrotationality condition:
>>  ටv/ ටx   -  ටu/ ටy  = 0
>>ට(ටФ/ටy)/ ටx   -  ට(ටФ/ටx)/ ටy  = 0

*If a flow field is possible and irrotational it can have only  ψ.
*If the flow field is not possible and irrotational it can have Ф.
*suppose the flow field is possible,irrotational such flow field can be represented by ψ and Ф.

equations of potential flow:

If any flow have ψ and Ф.Then,it is called potential flow and it satisfies the below equations



and

   Фxx +Фyy =0.

     

         

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