For laminar flow

(10)

    
and in turbulent flow region .

(11)

For gases at low speeds  and isothermal conditions we may use eq. 8 as well. At higher velocity of gases, density of the gas is not constant and Bernouli equation is to be written in a differential form i.e.

(12)

                                                                         
For the compressible fluid flowing under adiabatic conditions and assuming ideal gas law, the relation between P and  is 

(13)

                                                                
Where  is the isentropic exponent of the gas. Its value is 1.3 for monoatomic, 1.4 for diatomic and 1.669 for He and argon.

By 12 and 13 we get.

(14)

References:
D. R. Poirier and G.H.Geiger: Transport: Transport Phenomena is materials processing 1994.
R. Schuhmann: metallurgical Engineering, vol .1 Engineering principles