Pressure Differential in Fluid Dynamics: Why More on the '+s' Side?

[Tymofei]

Here i added a page from my fluid dynamics book where it shows particle model for deriving the equation. My question is why pressure is more at stream side aka 'positive "s" direction'.I would expected more pressure on the other side because for example when you trying to push a rigid object or some system like train vagon system,every differantial mass part/vagon parts should have same acceleration so inner force difference acting on differrantial mass part should be equilevant to dm * acceleration.And on the 'back' aka 'force applied side' of that particle there should be always more force applied on back compared to front side where difference is again dm * a = dF.Whats wrong with fluids does they perceive more pressure/force on their front side?So how they can accalerate then at positive 's' direction?Isnt that a contradiction ? I am confused..

 

[boneh3ad]

Where does it say that pressure is more on the downstream side? The $dP$ term is not positive definite.

 

[Tymofei]

Where does it say that pressure is more on the downstream side? The $dP$ term is not positive definite.

Yes but if we will take it negative,or replace it to other side and take it positive,after integrating(im assuming that fluid is non-compressible so all terms are exact-differantials) we will get exact solution but with negative pressure term.And it means that at constant height,result for pressure drop will be decreased velocity instead of increased velocity.So point where I am confusing is how we determine sign or location side for dP ? And what's logic lays behind it

 

[boneh3ad]

I guess I am just not following the confusion here, perhaps. $dP$ is generally taken as positive in the sense of increasing pressure in the direction of an increasing coordinate system, here along the streamline. A positive (adverse) pressure gradient would result in a force pushing in the upstream direction and a negative (favorable) pressure gradient would result in a force pushing in the downstream direction.

 

[Tymofei]

I guess I am just not following the confusion here, perhaps. $dP$ is generally taken as positive in the sense of increasing pressure in the direction of an increasing coordinate system, here along the streamline. A positive (adverse) pressure gradient would result in a force pushing in the upstream direction and a negative (favorable) pressure gradient would result in a force pushing in the downstream direction.

Ok i think i got it ,as i see pressure calculations based on pressure gradient which has opposite direction compared to pressure gradient-force.BUT by choosing positive pressure gradient direction toward positive s,we threat negative net force coming from it like positive,it doesn't make sense because its all based on force equation not 'pressure gradient' equation.My primary question was actually what retain us from switching P and (P + dP) sides.

 

[boneh3ad]

If you did that everything would be backward with positive forces pushing away from the positive direction. Think of what it means for $dp < 0$. That means that pressure is decreasing in the streamwise direction, corresponding to a positive force in that direction.