"::You're both questioning whether my summary of the momentum theorem is correct and uses the term correctly, and whether it is fully supported.  First, let's look at the apparent contradiction between me and Emanuel, as raised by  

 :::""Emanuel describes the force as acting within the control volume, McLean as acting at its boundary."" 

 ::There's actually no contradiction here.  Note that Emanuel defines the RHS of eq 14.1 as ""the vector sum of the applied forces that act on the system, which here is the fluid inside V.""  The force acting on the boundary (the second integral on the RHS) is part of that sum, along with the body force acting throughout the interior.  So Emanuel considers the force acting at the boundary to be part of the total force acting ""on the fluid inside V"".  Then if you neglect gravity, the first integral on the RHS (the volume integral) is zero, and  only the second integral (the force acting at the boundary) remains.  So my description is identical to his when gravity is neglected. 

 ::True, Emanuel doesn't say that gravity can be neglected, and neither does Shapiro's section 1.5, but gravity is neglected in every application of the momentum theorem to aerodynamics that I've ever seen, including the ones I cite in the article.  Clancy's firehose analysis also neglects gravity.  And in the rest of Shapiro's book, when he uses Newton's second law, he leaves the gravity force out without explaining why.  So does every other aerodynamics book on my shelf, except Batchelor.  On p. 176, in a section headed ""Modification of the pressure to allow for the effect of the body force"", Batchelor shows that gravity does not affect the dynamics of a constant-density flow because the body force (the weight of a fluid parcel) is canceled by a background gravitational-hydrostatic pressure gradient that always accompanies it.  Thus neglecting both the body-force term and the gravitational-hydrostatic part of the pressure gradient is rigorously justified for constant-density flow. 

 ::So neglecting the effect of gravity on the flow is something everyone does in aerodynamics applications, and most take it for granted.  In my quick check of the sources, Batchelor is the only one who takes the time to justify it rigorously.  Neither Shapiro nor Emanuel mentions the simplified form of the momentum theorem with the gravity terms omitted, so I see now that it's misleading for me to cite them when I state the simplified form.  I'll propose a fix for this problem below. 

 ::Now to the matter of steady flow.   wrote: 

 :::""I don't see anything in Shapiro 1.5 which says that the first RHS term of Eq 1.15 is zero in steady flow"". 

 ::True, he doesn't say it in 1.5.  He defines steady flow in 1.2 as ""a condition where at each point in space there is no variation of any property with respect to time"", and writes it as the partial derivative of any property with respect to time equals zero, using density as an example.  So I'd say that Eq 1.15 with the first RHS term set to zero for the case of steady flow is fully supported by Shapiro. 

 ::On this same topic  also wrote: 

 :::""If I was mistaken to say, 'The idea that either of the relationships derived above for steady flow conditions are ""The momentum theorem"" is not consistent with any of the sources I have seen', I ask   to specify where the evidence is."" 

 ::I said that I didn't understand your basis for saying this because it seemed to me at the time, and it still does, that your relationship ∑F = ∮c.s. is identical to Shapiro's eq 1.15, for the case of steady flow, and is thus consistent with Shapiro.  This version of the equation requires only steady flow; it doesn't require neglecting gravity.  So I think that in addition to being consistent with Shapiro, it is fully supported, though as I point out above, you have to look in section 1.2 in addition to 1.5.  The short answer to your question is that the evidence regarding the unsteady flow term is in Shapiro.   

 ::Whether it is correct to refer to it as ""the momentum theorem"" is the next question.   wrote: 

 :::""There is also the question of whether it is correct to refer to a degenerate case of Shapiro's eq 1.15, with the first RHS term set to zero for steady flow, as ""the momentum theorem"". By way of comparison, if one described a relationship derived from Newton's second law where F = 0, it would look very much like Newton's first law."" 

 ::In general I think it's fair to use the name of a theorem or law to refer to a special case, provided you also make it clear what special case you're talking about.  To use your example, I see n