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4. Force computation — the analytic model actually used

4. Force computation — the analytic model actually used

trapping_appendix.tex §sec:trapping_appendix:single_barrier_system. Both external forces are assumed constant along the pore (linear potential drop over L = 14 nm), i.e. spatially uniform 1-D scalars, not fields:

F_ep  = e · N_net · V_b / L            (eq:forceep,     N_net = N_body + N_tag)
F_eo  = e · N_eo  · V_b / L            (eq:osmoticforce)
E_ep(x) = −F_ep·x + b_ep ,  E_osm(x) = −F_osm·x + b_osm
ΔE^cis = −F·Δx_cis ,  ΔE^trans = +F·Δx_trans

N_eo = equivalent osmotic charge number — "the number of charges that must be added to DHFR to create an equal electrophoretic force"; asserted to be "an invariant related solely to the size and shape of the molecule". That is the entire force model: no Maxwell stress, no dielectric-gradient term, no pressure/viscous decomposition, no surface integral.

The general framework the author endorses (background ch. 2, nanopores.tex; adopted by the spec):

F_ele = ∮_Γ (T_E · n) dΓ ,   T_E = ε E E − ½ ε (E·E) I        (eq:nanopores_electric_force)
F_hyd = ∮_Γ (T_H · n) dΓ ,   T_H = p I − η[∇u + (∇u)ᵀ]        (eq:nanopores_hydrodynamic_force)

magnetic terms of T_E set to zero. The full-tensor form captures both EP and dielectrophoretic contributions (F_DEP = (p·∇)E) with no shape assumption; Stokes' law F_d = −6πηaU is only a first-order stand-in valid outside the pore. conclusion.tex (l. 180) makes it a finding: the 5–30 atm pressure hotspots mean the pressure term is non-negligible, "confirming the importance of using the full hydrodynamic stress tensor to calculate the force, rather than just the Stokes' drag". T_H is the same tensor as in the momentum equation (knowledge/01-physics-epnpns/02-governing-equations.md#23-navierstokes-variable-density-and-viscosity), so η is the corrected position-dependent viscosity — the integrand inherits the wall correction.

Sign, resolved [tested: tests/tier1/test_forces.py::test_num28_hydrodynamic_stress_check_values] [tested: tests/tier2/test_stokes_drag.py::test_ver19_the_three_routes_agree_on_the_drag]. T_H = p I − η[∇u + (∇u)ᵀ] as printed above is the negative of the Cauchy stress the momentum equation is written in, which is T_H = −p I + 2 η sym ∇u with ∇·T_H + f = Re ϱ(u·∇)u — a positive pressure pushing outward. NUM-28 and post/forces.hydrodynamic_stress use the momentum-equation convention, because the domain form is derived from that divergence and mixing the two flips the force. The two conventions give the same physics and opposite numbers, so the rule is: whichever is chosen, the tensor in the force integral and the tensor in the residual must be the same one. T_M = ε(E ⊗ E − ½|E|²I) has no such ambiguity — E appears twice, so the sign of E cannot leak into it.