## Explanation for moving fluids having lower pressure

 Quote by Aero51 Believe it or not there is not set theory on how lift is generated. We have a very good mathematical expression, circulation, which is essentially a measure of the rotationality of the air around the wing. If you were to subtract the freestream velocity (the speed of the wing) from the velocity vector field you would find that the vectors form a circular pattern: pointing upwards at the leading edge, backwards at the top (with the freestream flow), downward at the trailing edge, and forwards (against the freestream flow). Lift is therefore generated through a combination of the conservation of momentum and pressure gradients along the wing. This is not a very specific explanation, but I have yet to find a better one. Note that changes in pressure due to changes fluid velocity above the wing are insufficient to account for the total lift of a wing generated. Bernoulli's equation is thus not an accurate model of air moving over a wing. On the other hand, potential flow, which uses potential equations to describe flow patterns, is a model that uses Bernoulli's theorem for ideal (no viscosity, no heat transfer, all molecular kinetic energy is translational) fluid behavior.
I wouldn't say that there is no set theory - lift, at least in the incompressible case (and even to some degree in the compressible case) is fairly well understood, and the key is viscosity. Viscosity causes the air at the trailing edge to smoothly combine (the flow from the bottom cannot wrap around to the top and vice versa) due to the sharpness of the trailing edge. This condition (known as the kutta condition) is basically what forces the air to circulate in the first place - in the absence of circulation, the air on the bottom of the airfoil would wrap around the trailing edge to the top, and no lift would be generated.

The circulation imposed on the flow by this trailing edge, viscosity-driven condition is what causes the difference in flow speed between the upper and lower surface of the wing. As stated earlier in this thread, there's absolutely no reason a given parcel of air from ahead of the wing needs to recombine behind it (and in fact it does not in practice), so the common explanation is bogus. However, the result is the same - the flow on the upper surface is moving faster, while the flow on the lower surface is moving slower. Once this velocity distribution is obtained, the derived pressure distribution (from the Bernoulli relation) is both accurate and sufficient to explain the airfoil's lift. The key is in getting the correct velocity distribution in the first place, and potential flow (which you mentioned) does a fairly good job so long as the reynolds number is fairly high and the flow is effectively incompressible, both of which are true for aircraft flying below about mach 0.3 (so long as they aren't tiny UAVs).
 Cjl your first paragraph is completely wrong. The kutta condition is simply a mathematical boundary condition to solve a potential flow field. It has nothing to do with viscosity or imposing circulation directly, though it will affect the circulation magnitude. In addition, flows with viscosity will not leave the trailing edge of an airfoil smoothly; the total pressure will not be recovered. This is one reason why the Bernoulli equation does not describe lift. The kutta condition pertains to ideal flows only.

Gosh folks why so heavy?

This is what the OP last asked

 This is getting slightly more complex than I thought it would. What explains the fact that paper will fly upwards when you hold it up against your mouth and blow?
 To answer that question: the coanda effect. I cant think of a way to describe it simply other than the "stickyness" of air.

 Quote by Aero51 Cjl your first paragraph is completely wrong. The kutta condition is simply a mathematical boundary condition to solve a potential flow field. It has nothing to do with viscosity or imposing circulation directly, though it will affect the circulation magnitude. In addition, flows with viscosity will not leave the trailing edge of an airfoil smoothly; the total pressure will not be recovered. This is one reason why the Bernoulli equation does not describe lift. The kutta condition pertains to ideal flows only.
No, my first paragraph is completely correct. The kutta condition can be looked at a couple of ways - the way it is often used is to effectively "correct" an inviscid flowfield. However, the reason why the kutta condition holds is because of viscosity. In the (admittedly nonphysical) limit of an infinitely sharp trailing edge, in the absence of the kutta condition, the flow around the trailing edge causes an infinite shear. Of course, for a real airfoil, the trailing edge is not infinitely sharp, but so long as it is sufficiently sharp, an extremely high shear would still develop in this case. Any viscosity at all (even a very small viscosity) will cause large forces to act on the fluid due to this shear, and these forces are what actually cause the flow to come smoothly off both the top and bottom surfaces of the wing at the trailing edge. So, viscosity is the physical reason why the kutta condition holds.

As for the statement that flows with viscosity won't leave the trailing edge smoothly? Why do you think that? Flows with viscosity can flow smoothly off the trailing edge just fine. Depending on the details, the boundary layer will typically be turbulent for most aircraft, true, but this is on the scale of millimeters - the overall flow pattern is still smooth and attached.

I'm also not sure why you're bringing total pressure into this. It's true that there's energy loss in a boundary layer, but to a pretty good approximation, the flow outside of the boundary layer itself is inviscid and lossless (for aircraft). As a result, the Bernoulli relation can be used along with the air velocity just outside of the boundary layer to determine the pressure just outside the boundary layer (which can be found by running an inviscid simulation of the airfoil, or if you really want to be picky about it, you can replace the airfoil shape with a slightly modified shape that adds in the boundary layer displacement thickness). In boundary layers, the pressure gradient in the normal direction to the surface is minimal (and very frequently assumed to be zero), so knowing the pressure distribution just outside the boundary layer also tells you the pressure distribution inside the boundary layer, at the surface of the airfoil. If you know the pressure distribution at the surface of the airfoil, all you need to do is integrate around the airfoil to get the lift. Note that this entire process uses only potential flow (inviscid, incompressible, lossless) and the bernoulli relation, but it is very accurate at describing the lift generated by an airfoil at high reynolds number below mach 0.3.

To answer your final sentence, the Kutta condition applies to pretty much every flow around a non-stalled airfoil. It isn't purely a theoretical construct - if anything, it's more of an empirical observation describing the behavior of flows around objects with a sharp trailing edge. By itself, it doesn't actually serve much of a purpose, but when combined with other fluid dynamic principles, it can be quite useful in determining the properties of the flow around an object.

 The kutta condition can be looked at a couple of ways - the way it is often used is to effectively "correct" an inviscid flowfield. However, the reason why the kutta condition holds is because of viscosity. In the (admittedly nonphysical) limit of an infinitely sharp trailing edge, in the absence of the kutta condition, the flow around the trailing edge causes an infinite shear. Of course, for a real airfoil, the trailing edge is not infinitely sharp, but so long as it is sufficiently sharp, an extremely high shear would still develop in this case. Any viscosity at all (even a very small viscosity) will cause large forces to act on the fluid due to this shear, and these forces are what actually cause the flow to come smoothly off both the top and bottom surfaces of the wing at the trailing edge. So, viscosity is the physical reason why the kutta condition holds.
I agree that the kutta condition needs to be employed to model the effects of viscosity however, your explanation about viscosity allowing flow to leave the trailing edge smoothly is incorrect.

 As for the statement that flows with viscosity won't leave the trailing edge smoothly? Why do you think that? Flows with viscosity can flow smoothly off the trailing edge just fine. Depending on the details, the boundary layer will typically be turbulent for most aircraft, true, but this is on the scale of millimeters - the overall flow pattern is still smooth and attached.
Any airfoil at a non zero angle of attack (well this isnt 100% true but for most "normal" airfoils it is) will have some degree of separation at the trailing edge. Obviously this becomes more prevalent at higher angles. Take a look at some pictures of google of airfoils at different angles of attack, notice that even around 5 degrees the separation point can be around 80% of the chord. The flow, therefore cannot leave the trailing edge smoothly. If this were true then that would imply that there is a stagnation point at the TE and no separation will be present.

 I'm also not sure why you're bringing total pressure into this. It's true that there's energy loss in a boundary layer, but to a pretty good approximation, the flow outside of the boundary layer itself is inviscid and lossless (for aircraft). As a result, the Bernoulli relation can be used along with the air velocity just outside of the boundary layer to determine the pressure just outside the boundary layer (which can be found by running an inviscid simulation of the airfoil, or if you really want to be picky about it, you can replace the airfoil shape with a slightly modified shape that adds in the boundary layer displacement thickness). In boundary layers, the pressure gradient in the normal direction to the surface is minimal (and very frequently assumed to be zero), so knowing the pressure distribution just outside the boundary layer also tells you the pressure distribution inside the boundary layer, at the surface of the airfoil. If you know the pressure distribution at the surface of the airfoil, all you need to do is integrate around the airfoil to get the lift. Note that this entire process uses only potential flow (inviscid, incompressible, lossless) and the bernoulli relation, but it is very accurate at describing the lift generated by an airfoil at high reynolds number below mach 0.3.
I am not sure if you deliberateness did this - you basically described the algorithm used by XFOIL.

 To answer your final sentence, the Kutta condition applies to pretty much every flow around a non-stalled airfoil. It isn't purely a theoretical construct - if anything, it's more of an empirical observation describing the behavior of flows around objects with a sharp trailing edge. By itself, it doesn't actually serve much of a purpose, but when combined with other fluid dynamic principles, it can be quite useful in determining the properties of the flow around an object.
I answered your first sentence by noting that the Kutta condition doesn't apply to separated airfoils. The Kutta condition, to be exact, is a mathematical boundary condition based on empirical observation for nonviscus flows. Flows modeled with viscosity do not need to satisfy the Kutta condition because it does not need to be employed. The Kutta condition is not needed to solve the potential flow field, however it is needed for a nontrivial solution (IE the real world flow).

Also, I would like to show you why the Bernoulli equation does not correctly explain why lift is generated on a airfoil. Yes, it can be used as a mathematical approximation to a real flow, but it is a very misleading statement from a physical standpoint. Here is an example:

Assume that the lift generated by an airfoil can be described entirely by the differences in velocities on the upper and lower surface. Lets also assume that we are flying at sea level under normal conditions. The equation describing the lift due to the average velocity gradient will be given by

$1/2 \rho (V_{top}^2 - V_{bot}^2) = L/S$

ρ = air density
L = Lift generated by the body (lets say wing for simplification)
S = lifting surface area (wing area)
V = velocity

The equation will become with some substitution:
$V_{top}^2 - V_{bot}^2 = C_L V^2$

Assuming that the reference velocity for the lift coefficient (freestream) is equal to the velocity under the bottom half of the wing:

$V_{top} = \sqrt{C_L} V_{bot}$

It is clear that when the lift coefficient is low (low angle of attack, flying a high speed with large wing area, cruise conditions) that the bernoulli principle describes lift sufficiently well. However, lets consider take off or landing conditions when the lift coefficient is large. For a specific example, consider a large airliner landing at 160 mph (74 m/s) with a lift coefficient of 4. The average velocity above the wing will be:

$V_{top} = 2*74 = 148 [m/s]$

If the lift on this wing was due entirely to the differences in velocities above and below the wing, then the mach number at the top surface will be .43, and the bottom will be .215. It doesn't make physical sense that the mach number above the wing will be nearly twice the free-stream speed. Lift, therefore, cannot be described by the bernoulli principle alone. A better explanation is the conservation of momentum.

SIDE NOTE:I wish I could find better/specific numbers, but these are ball-parked estimates from the internet. I know that triple slotted flaps, used on some large airliners, can have maximum lift coefficients around 6. Specifics are proprietary information sadly.
 I had a dream I made a stupid mistake and I was right. The math under the radical should be C L +1 making the results from my calculation Vtop = √5 * 74 =165 m/s or a Mach number of. 48. Again, physically unrealistic.

 S = lifting surface area (wing area)
Perhaps you would like to clarify?

S is different for top and bottom except for bricks.

Also there is a significant variation of pressure over this area, both top and bottom. In fact the solution for pressure on the underside has a singularity without friction.

 Quote by Aero51 I agree that the kutta condition needs to be employed to model the effects of viscosity however, your explanation about viscosity allowing flow to leave the trailing edge smoothly is incorrect.
Aero51, his explanation is not incorrect. To put it another way, in an inviscid flowfield, for all cases except for the 0° angle of attack, the rear stagnation point is predicted to be somewhere other than the trailing edge. This would require the flow to completely turn around the trailing edge. In an inviscid flow field, this makes plenty of sense mathematically (and is as nonsensical physically as the assumption of invsicid flow). For a viscous flow, however, this represents a singularity at an infinitely sharp trailing edge and would result in an infinite velocity when turning around that trailing edge. Clearly that doesn't happen. Viscosity requires that the rear stagnation point be located at the trailing edge.

The Kutta condition, as mathematically expressed in airfoil theory, is the means of imposing mathematically this physical reality.

 Quote by Aero51 Any airfoil at a non zero angle of attack (well this isnt 100% true but for most "normal" airfoils it is) will have some degree of separation at the trailing edge.
This is an extreme case of overgeneralization. Far from every airfoil experiences separation, including those at modest angle of attack. It all just comes down the pressure gradient over the airfoil for the given angle of attack and the state of the boundary layer. You can't make generalizations like this.

 Quote by Aero51 Obviously this becomes more prevalent at higher angles. Take a look at some pictures of google of airfoils at different angles of attack, notice that even around 5 degrees the separation point can be around 80% of the chord. The flow, therefore cannot leave the trailing edge smoothly. If this were true then that would imply that there is a stagnation point at the TE and no separation will be present.
Because, of course, Googling images of airfoils turns up all possible airfoils.

 Quote by Aero51 I am not sure if you deliberateness did this - you basically described the algorithm used by XFOIL.
And thousands of other panel codes.

 Quote by Aero51 I answered your first sentence by noting that the Kutta condition doesn't apply to separated airfoils. The Kutta condition, to be exact, is a mathematical boundary condition based on empirical observation for nonviscus flows. Flows modeled with viscosity do not need to satisfy the Kutta condition because it does not need to be employed. The Kutta condition is not needed to solve the potential flow field, however it is needed for a nontrivial solution (IE the real world flow).
Rather than saying "nonviscous", you ought to use the term "invsicid". Anyway, this isn't true. The Kutta condition was created to rectify the predictions of inviscid theory with the observations of the real world. The real world, of course, is viscous, so the Kutta condition is based on empirical observation of viscous flows as cjl has correctly stated.

Flows modeled with viscosity do not need to "satisfy the Kutta condition" because they are, by default, already satisfied because a solution doesn't exist otherwise.

Some may argue that by definition, it is needed to solve the inviscid flow field because without it, the solutions are of course trivial and aren't actually the solutions in which we are interested. I would count myself among those who would argue this.

 Quote by Aero51 If the lift on this wing was due entirely to the differences in velocities above and below the wing, then the mach number at the top surface will be .43, and the bottom will be .215. It doesn't make physical sense that the mach number above the wing will be nearly twice the free-stream speed. Lift, therefore, cannot be described by the bernoulli principle alone. A better explanation is the conservation of momentum. SIDE NOTE:I wish I could find better/specific numbers, but these are ball-parked estimates from the internet. I know that triple slotted flaps, used on some large airliners, can have maximum lift coefficients around 6. Specifics are proprietary information sadly.
Not only does it make perfect physical sense to have the air moving twice as fast over the top of the wing than over the bottom, but it happens all the time. This is one of the fast ways to disprove the equal transit time fallacy that high school teachers love. You show an actual CFD simulation of the flow over the wing and you see the parcels of air leaving the upper surface long before those below the wing leave that side.

Recall the solution for the inviscid flow around a circular cylinder. The speed at the top and bottom of the cylinder (+/-90° from the forward stagnation point) is precisely twice the freestream velocity. This should give you some intuition into the fact that your example actually could happen. Even something as simple as a NACA 0012 at 0° AoA will accelerate the air to around u/U≈1.2 without even generating any lift. Put that at an angle of attack and that number will increase.

 Quote by Aero51 I had a dream I made a stupid mistake and I was right. The math under the radical should be C L +1 making the results from my calculation Vtop = √5 * 74 =165 m/s or a Mach number of. 48. Again, physically unrealistic.
I'm not going to go into a full response right now (I don't have time - I might give a more detailed response tomorrow), but I'd like to know on what basis you assume that a flow speed of ~2x freestream is unreasonable, given your incredibly high assumed Cl. That actually sounds in the correct ballpark to me, with a couple of caveats (namely, that bernoulli assumes no significant compressibility effects, and with an upper surface flow speed of >M0.3, that is no longer a valid assumption, as well as with the assumption of an unstalled airfoil).

Also, your Cl numbers seem too high to me. I'm not aware of any airfoil that can actually pull off a lift coefficient of 4, and 6 (your quoted number for a triple slotted) is unachievable by any normal means that I am aware of. One thing you might consider checking is the reference area - many aircraft use high lift systems that (sometimes dramatically) increase the area of the wing, so if you continue to use the clean wing area as the reference area, the Cl numbers will seem incredibly high (while in reality, the Cl might be a more reasonable 2-2.5 or so if the actual wing area with flaps extended were used).

Studiot: In general, the wing surface area is considered to be the area projected onto a plane defined by the wing's chord line. It's true that the top has more area, but the area used for computations is basically the area that the wing would have if it were zero thickness and flat.

 In general, the wing surface area is considered to be the area projected onto a plane defined by the wing's chord line. It's true that the top has more area, but the area used for computations is basically the area that the wing would have if it were zero thickness and flat.
Thank you - This is why one should always define one's terms, but isn't there more to it than that?

The interaction between the object and the flow produces two orthogonal forces the lift perpendicular to the flow and the drag parallel to the flow.
But the relative magnitude of these depends not only on the projected area but also on the angle between the plane of this area and the flow ie the angle of attack.
Whether you attribute this to two projected areas in two orthogonal planes or by separately accounting for the angle is a matter of choice, but this needs to be defined.

 Quote by Aero51 cjl At one point in time I had access to some information in which lift coefficients exceeding 5 were measured on certain flap configurations. I cannot disclose specifics (as this information may have accidentally fallen into my hands when it fell into my desk one evening), but a publicly available paper is: High-Lift Systems on Commercial Subsonic Airliners. If my Cl seems high consider safety requirements for emergency landings
Once again, I ask you: were the lift coefficients using the reference area of the clean wing? If so, those numbers do sound completely achievable. In many cases, the reference area is kept fixed for a given aircraft (to reduce the number of variables), and then the Cl is calculated based on the clean wing reference area, even in the presence of high-lift devices (even though high-lift devices frequently increase the wing's area, sometimes dramatically, as is the case with the Boeing 727 seen here or the Boeing 747 seen here). This method is completely mathematically valid, and even allows for slightly more intuitive treatment of the aircraft's behavior, but it also makes your derivation wrong (since the actual area the pressure is acting over is different than the reference area used for deriving the coefficient of lift).

As for safety requirements for emergency landings? Those don't necessarily require a high Cl - they require the right combination of Cl, wing area, and acceptable landing speed. Modern airliners are actually going towards simpler high lift devices, along with lower Clmax values because modern airfoils perform much better at high speed, allowing for a larger wing area, higher aspect ratio, and less sweep for a given cruise speed. This increase in wing area and aspect ratio allows for lower landing and takeoff speeds without needing dramatic high lift devices, which allows for a simpler mechanical design as well.

 Once again, I ask you: were the lift coefficients using the reference area of the clean wing? If so, those numbers do sound completely achievable. In many cases, the reference area is kept fixed for a given aircraft (to reduce the number of variables), and then the Cl is calculated based on the clean wing reference area, even in the presence of high-lift devices (even though high-lift devices frequently increase the wing's area, sometimes dramatically, as is the case with the Boeing 727 seen here or the Boeing 747 seen here). This method is completely mathematically valid, and even allows for slightly more intuitive treatment of the aircraft's behavior, but it also makes your derivation wrong (since the actual area the pressure is acting over is different than the reference area used for deriving the coefficient of lift).
They were calculated using the clean wing configuration as a reference area. You raise a good point noting that the wing area will change and thereby reduce the lift coefficient. In that case the velocities above and below the wing may not change so drastically and do seem plausible.

 As for safety requirements for emergency landings? Those don't necessarily require a high Cl - they require the right combination of Cl, wing area, and acceptable landing speed. Modern airliners are actually going towards simpler high lift devices, along with lower Clmax values because modern airfoils perform much better at high speed, allowing for a larger wing area, higher aspect ratio, and less sweep for a given cruise speed. This increase in wing area and aspect ratio allows for lower landing and takeoff speeds without needing dramatic high lift devices, which allows for a simpler mechanical design as well.
In the paper I mentioned above there is an illustration showing how much wing area would be required if an aircraft did not employ high lift devices. It seems counter intuitive that aerospace companies would want larger wings because this will result in an increase of material cost, drag and weight. If the wing area is reduced, on the other hand, all a company has to lose is fuel volume. Also, I believe it is safe to assume that safety requirements will call for a higher CL. For example, in a case when a full loaded airliner has to make an emergency landing the aircraft must reduce its speed significantly - the increase in wing area due to the flaps being deflected is not sufficient to account for the new speed. I would also argue that increasing the aspect ratio will increase the likelyhood of stall and weight. As you said, it comes down to design optimization

 Quote by Aero51 I also believe my statement about slight separation on most airfoils is a fair generalization (even if it is negligible for practical calculations) because total pressure is never fully recovered on a wing.
Total pressure if never recovered in any situation with a boundary layer or any other viscous phenomena so your point is moot. This has nothing necessarily to do with separation (though it can) since it happens even on an unseparated airfoil due to viscous dissipation.

 Quote by Aero51 A true stagnation point does not exist at the trailing edge. The argument you made about lift being dictated by the boundary layer, pressure distribution and angle of attack is somewhat trivial because those facts are true for anything that generates lift - even a brick.
It isn't trivial at all. You can't make such generalizations because the separation phenomenon is much more involved than the shape simply being an airfoil, which is effectively what you have suggested though not expressly stated. In fact, I routinely work with airfoils that do not experience separation until fairly large angles of attack. That alone disproves your assertion that any airfoil has some degree of separation. It doesn't.

 Quote by Aero51 The Kutta condition does not apply to real flows - it can neither be satisfied nor unsatisfied. In theory introducing viscosity into the model will accurately predict the flow pattern.
 Quote by Aero51 I already agreed with cjl by stating that the Kutta condition is a mathematical boundary condition based on observation.
However, you also said this:

 Quote by Aero51 The Kutta condition, to be exact, is a mathematical boundary condition based on empirical observation for nonviscus flows.
That is false. You can't even have empirical observations of an inviscid flow because such a flow does not exist in any situation that would lead to the development of the Kutta condition. The only truly and globally inviscid flow is a uniform free stream. The Kutta condition is, mathematically, a boundary condition used to rectify inviscid computations with the viscous reality. Of course, the Kutta condition itself applies to viscous flow because originally it was just the statement that "A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge." It is satisfied by default in a viscous flow and need not be independently enforced, but that does not mean that it does not apply to viscous flows, after all, without viscosity there would be no Kutta condition.

 Quote by Aero51 Also, your point about the velocity at the peak of the cylinder being twice the freestream flow is invalid because the discussion pertains to differences between the upper and lower surfaces of a wing. The difference in velocities for the case you described is 0 and no lift is generated. I used the freestream flow as the flow speed on the lower surface purely for simplicity.
If you didn't mean the free stream, then don't say the free stream. Regardless, the principle remains the same. Bernoulli does not physically explain lift, but that doesn't mean that you can't have the flow moving twice as fast over the upper surface of the wing than it is over the bottom surface. In fact, in just about 5 min of playing around with a simple Euler solver that NASA has available online can show situations where the flow moves 4 times as fast over the top compared to over the bottom. Camber the airfoil a bit and you can see the effect if you tell it to plot velocity.

Sorry for the belated response. I recently moved about 400 miles from my old home and it took me about week to get things settled. Anyway, on to "business".

 Total pressure if never recovered in any situation with a boundary layer or any other viscous phenomena so your point is moot. This has nothing necessarily to do with separation (though it can) since it happens even on an unseparated airfoil due to viscous dissipation.
You make a good point, total pressure is not recovered in stokes flow, but there is no separation. I believe in the context of your typical airfoil flow regime (RE>=10^6) there is always a little bit of separation at the trailing edge. I found a video on youtube showing this effect, though I cant find it now.

 That is false. You can't even have empirical observations of an inviscid flow because such a flow does not exist in any situation that would lead to the development of the Kutta condition. The only truly and globally inviscid flow is a uniform free stream. The Kutta condition is, mathematically, a boundary condition used to rectify inviscid computations with the viscous reality. Of course, the Kutta condition itself applies to viscous flow because originally it was just the statement that "A body with a sharp trailing edge which is moving through a fluid will create about itself a circulation of sufficient strength to hold the rear stagnation point at the trailing edge." It is satisfied by default in a viscous flow and need not be independently enforced, but that does not mean that it does not apply to viscous flows, after all, without viscosity there would be no Kutta condition.
Quite frankly I think this argument is becoming too philosophical and less scientific.
 Plus, I tried the Euler Solver. I got flow speeds about 4.5 x as fast between the top and the bottom surfaces. These points were inside the boundary layer though (the program uses Zhukovski transformations), so the difference between the top/bottom is probably slightly less. I also noticed that the flow below the surface was also less than the freestream velocity, invalidating one of the assumptions I made in the equation I wrote earlier.