**EM DRIVE FORCE vs. TIME **We continue the program started with posts

http://forum.nasaspaceflight.com/index.php?topic=37642.msg1403629#msg1403629 http://forum.nasaspaceflight.com/index.php?topic=37642.msg1404000#msg1404000http://forum.nasaspaceflight.com/index.php?topic=37642.msg1404004#msg1404004http://forum.nasaspaceflight.com/index.php?topic=37642.msg1404005#msg1404005http://forum.nasaspaceflight.com/index.php?topic=37642.msg1404006#msg1404006http://forum.nasaspaceflight.com/index.php?topic=37642.msg1404754#msg1404754http://forum.nasaspaceflight.com/index.php?topic=37642.msg1404783#msg1404783http://forum.nasaspaceflight.com/index.php?topic=37642.msg1405604#msg1405604http://forum.nasaspaceflight.com/index.php?topic=37642.msg1405605#msg1405605showing the Force (Newtons) on the small and the big base vs. time and the net force ( (force on big base) - (force on small base)).

The forces on the bases are obtained by numerical integration of the stress tensor σ

_{xx} (*) component ( obtained

**using ***Wolfram Mathematica* (

http://www.wolfram.com/mathematica/ ) , post-processed from the transient Finite Difference (using Meep) solution for RF feed ON for an EM Drive of the geometry of @rfmwguy (Db=11.01 in, Ds=6.25 in, L=10.2 in).

The stress component σ

_{xx} is compressive at both bases. Therefore the force obviously points in different directions: at the big base it points in the direction from the small base to the big base while at the small base it points in the direction from the big base to the small base. From the interior to the surface in both cases. In other words, the electromagnetic field exerts a pressure on both surfaces and the force points from the interior to the bases.

It is also very interesting to point out that:

1) The compressive force against the big base is larger (due to the greater surface of the big base) than the compressive force against the small base, even though the maximum stress at the small base is higher.

2)

** the time at which the maximum force occurs at the big base is phase-shifted with respect to the time at which the maximum force occurs at the small base. The force at the small base leads, because the antenna is much closer to the small base than to the big base.**3) Notice that when the net force is at a minimum, it actually points for a very short amount of time towards the small base (positive direction). Most of the time it points towards the big base (negative direction).

4) We naturally expect that force on the overall copper itself should sum up to zero in order to satisfy the momentum equilibrium equation implied by Maxwell's equation. We expect that the imbalance in net force between the bases should be compensated by the electromagnetic pressure on the lateral surfaces, leading to a component on the direction of the small base to result in a net overall force of zero. We don't have access to the electromagnetic fields at the lateral surfaces computed by Meep, in order to calculate the stress tensor at the lateral surfaces and integrate it to get the force on the lateral surface.

5) It is still highly suggestive that there is a net force in the direction of the big base as a result of the imbalance between the big base and the small base, although the stress at the small base is higher, and although the stress distribution is completely different (as previously shown) at both bases. Also, the Poynting vector is strongly pointing towards the big base (as previously shown). If the EM Drive is not an experimental artifact, whatever is responsible for the acceleration must be using this net imbalance and phase shift between the forces at the bases (for example: plasma ions produced by microwave heating of the air inside the cavity, leaking out and producing an exhaust, or evanescent waves acting on air molecules, or electromagnetic pressure acting on axionic dark matter or on a degradable level of the Quantum Vacuum, etc.). The net force imbalance between the bases, pointing towards the big base, is entirely consistent with a reaction acceleration of the EM Drive in the opposite direction as a result of a recoil force.

6) A fitted model of the time variation of the force (with excellent R^2 = 0.999981), shows that the present Finite Difference model (from which the force has been computed at the last two cycles ending at 0.013 microseconds from the time at which the Microwave feed was turned on), would have to be marched forward for 1,000 times longer, to a total of 10 microseconds, for the force to be magnified by the calculated exponential growth to a value of 10 microNewtons (for an inputPower of 43 watts). Given the fact that the present Meep model takes an hour to run on a good PC modern computer, 1,000 hours of computer time represents over 41 days of computing time. Thus running the Meep model to steady state is impractical. Rather than using a supercomputer to perform such a computation, I suggest to use an implicit (unconditionally stable) Finite (*****) Difference model in time (rather than the explicit time difference model presently used that is subject to stability problems that limit the maximum finite difference time step). Such implicit finite difference models are well known (I developed a version of them in my PhD thesis 35 years ago) and can be run much faster than explicit FD models. There are also numerous alternative numerical schemes that are more accurate than Finite Differences.

______________________________

NOTES:

Stress calculation:

(*) (where we denote by σ

_{xx}= T

^{11} the contravariant component of the tensor acting along the longitudinal direction "x" of the EM Drive, normal to the the plane yz having normal x, where direction "1" is "x")

(**) For the copper diamagnetism is assumed such that the magnetization

**M** is assumed proportional to the applied magnetic field such that

for free space it is assumed that

**M** is zero in free space in the relationship

(***) The Stress calculations are for an Input Power of 43 Watts (similar to the value used by NASA in some of their runs). The Stress values are proportional to the Input Power, so for example, if the Input Power were 860 Watts, that means that the calculated values for Stress are 860 Watts/ 43 Watts = 20 times greater than shown in the plots. In other words, for 860 Watts InputPower, the values for Stress in the plots need to be multiplied by a factor of 20. Ditto for the force.

(****) The total time from start of the RF feed in the Meep response analysis to the very last step is:

320 ( time slices) * 4.082199*10^(-11) seconds/timeSlice =

=

**0.013063 microseconds**Each "time slice" step is

**4.082199*10^(-11)** seconds/timeSlice

Duration of the total of 13 time slice steps = 53.068 *10^(-11) seconds

Last time step is at 0.013063 microseconds from the start of the RF feed ON

___________________________________________________________

Conversion to get SI Units from the graphs and equations in Meep units:

**TIME: **Multiply Meep Time Slice "t" in the horizontal axis and in the formulae by the following factor:

((Total Meep Time)/(#Time Slices))*((Length Scale Factor)/(Speed of Light in Vacuum)) =

=((13.054)/(320))*((0.3)/(299792458))

=

**4.082199*10^(-11)** seconds/timeSlice

ASSUMPTIONS: the validity of the following data:

Number of time slices for the total run = 320

Number of Meep time units for the total run = 13.054

Meep Length Scale factor= 0.3 meters

Meep Current (Io) = 1

(*****) One of the earliest unconditionally stable, implicit time domain methods was developed by John Houbolt (

https://en.wikipedia.org/wiki/John_Houbolt ) at NACA (predecessor of NASA) in 1950:

.J. C. Houbolt, ‘A recurrence matrix solution for the dynamic response of elastic aircraft’, J. Aeronaut. Sci., 17,540-550, (1950).

, the same Houbolt who had the genius to create the Lunar orbit rendezvous method to land astronauts on the moon with a single Saturn V, and prevail over Von Braun's Nova rocket concept (eventually von Braun came around and supported Houbolt's proposal). It was due to Houbolt's concept that the US was able to reach the Moon by 1969.