Venturi Vaccum increase in Momentum

[Timtam]

I was hoping to get a more complete understanding how a Venturi vacuum increases the Momentum of the fluid.

I know Momentum is an awful way to pose this question but it really is important that it can be explained by momentum instead of conservation - Hence the post.

An initial pressure difference Sp₁ -Sp₂ gives rise to fluid flow with a momentum p_x=m_x.v_x As the flow enters a constriction its flow rate v_x increases and a consequential fall in static pressure Sp₃ , this may even fall below Atmospheric Sp_A and this pressure difference Sp_A -Sp₃ will cause mass m₊ to flow into the pipe in the y direction.

As it exits the constriction reverting to original radius its velocity reduces to the mean flow rate v_x induced via the pressure difference Sp₁ -Sp₂ However this velocity is now carried by additional mass so overall p_x=(m_x+m₊).v_x increases

This increase in p_x is not paid for by the external x direction Sp₁ -Sp₂ pressure difference but by the y direction pressure difference Sp_A -Sp₃

I know this does but I want to understand how the y direction momentum p_y induced by the y direction pressure difference contributes to x direction momentum ?

Can someone please explain the mechanism is transferred ? as to me the y direction pressure difference (being random) has equal chance to reduce x momentum as increase it.

Bonus points for explaining the 15' requirement

 

[paisiello2]

Are you sure that v2 = v1?