Nanoscale Flow Choking at the Zero Slip Length: Universal Benchmark Data for In Silico Experiments An Exact Prediction of the 3D Blockage Factor in Nanoscale Systems

The Sanal flow choking ( PMCID: PMC7267099) and the streamtube flow choking are new theoretical concepts applicable to both the continuum and non-continuum fluid flows. Once the streamlines compacted, the considerable pressure difference attains within the streamtube and the flow within the streamtube gets accelerated to the constricted section for satisfying the continuity condition set up the conservation law of nature, which leads to the Sanal flow choking and supersonic flow development at a critical-total-to-static pressure ratio (CPR) due to the convergent-divergent (CD) shape of the streamtube. As the pressure of the nanofluid /non-continuum-flows rises, average-mean-free-path diminishes and thus, the Knudsen number lowers heading to a zero-slip wall-boundary condition with compressible viscous (CV) flow regime. Sanal flow choking is a CV flow phenomenon creating a physical situation of the sonic-fluid-throat in a duct at a CPR. Herein, we presented a closed-form-analytical-Nature model, which is capable to predict exactly the three-dimensional boundary-layer-displacement-thickness of nanoscale diabatic fluid flow ( flow involves transfer of heat ) systems at the zero-slip-length. The innovation of Sanal flow choking model is established herein through the entropy relation, as it satisfies all the conservation laws of nature. The exact value of the 3D boundary-layer-displacement-thickness in the sonic-fluid-throat region presented herein for each gas is a universal benchmark data for performing high-fidelity in vitro and in silico experiments for the lucrative design optimization of nanoscale systems. The physical insight of the Sanal flow choking and streamtube flow choking presented in this letter sheds light on finding solutions for numerous unresolved scientific problems.


Nature
thrusters 11 operating at both gravity and microgravity environments where the flow field exhibit both the continuum and non-continuum fluid properties. In such physical situations multiscale and hybrid modeling approaches are encouraged 12  Nanofluid flow is a blend of nano-sized particles in a traditional operating fluid 12 , which obeys all the conservation laws of nature. The occurrence of slip in gas flows, due to the local thermodynamic non-equilibrium, was originally reported by Maxwell 14,15 and its scale varies on the extent of rarefaction of the gas. It describes in terms of the Knudsen number (Kn), which gives an explicit clue on the type of flow, viz., continuum or non-continuum. Note that numerous modeling efforts have been reported in the open literature for nanoscale flow simulation without authentic code verification using any benchmark data and/or any closed-form analytical solution [16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33][34][35] . The fact is that generating benchmark data from the nanoscale system is a challenging task or quite impractical using a conventional in vitro methods and it is anticipated that the classical assumptions on the hydrodynamic model will ride into hitches as the composite flow system reaches nanometer (nm) size. 19 Obviously the brilliant conclusions drawn using sophisticated models, by various investigators across the globe, viz., direct simulation Monte Nature 4 Carlo (DSMC), molecular dynamics (MD), Burnett equation and the hydrodynamic models, will not be endorsed by the high precision industries for the highly expensive nanoscale systems designs for practical applications without providing an exact solution for the data verification.
Cooper et al. 19 reported that in vitro data well matched with the predicted results using the  hydrodynamic Navier Stokes method with the first-order slip condition for the range of average pore diameters from 169-220 nm. Singh and Myong 36 reported that neither continuum models nor free-molecular models could be invoked for fluid flow cases when the Knudsen number falls in the intermediate range between the continuum (Kn < 0.01) and free-molecular flow regimes (Kn>10). When the Knudsen number becomes large (Kn > 0.01), the conventional assumptions of no-slip boundary condition, thermodynamic equilibrium, and linear stress-strain relationship Nature 5 fail. Admittedly, when the pressure of the nanofluid rises, the average-mean-free-path diminishes and thus, the Knudsen number lowers heading to a zero-slip wall-boundary condition with compressible viscous (CV) flow regime creating streamline pattern in the nanoscale fluid flow system. Therefore, the Sanal flow choking and supersonic flow development leading to shock-wave generation due to the fluid-throat effect at the zero-slip-length is a valid physical situation in continuum and non-continuum flows where CD shaped nanoscale streamtubes persists (see Fig.1(b)). Herein, we provide a proof of the concept of fluid-throat persuaded flow choking in the nanoscale diabatic fluid flow system. For establishing this concept authoritatively, we are presenting an infallible closed-form analytical model for predicting the three-dimensional (3D) boundary layer displacement thickness (defined herein as 3D blockage factor) at the sonicfluid-throat location where the slip length is zero, which will be a useful tool for the in vitro and in silico experiments in both the continuum and non-continuum flows with due consideration of heat transfer effects (real-world fluid flow effect). D.M.Holland et al. 37 reported that an in silico model enhanced with molecular-level information could accurately predict unsteady nanoscale flows in non-trivial geometries.
Authors presented that slip at the nanoscale fluid flow system could make a noteworthy effect on the optimum channel dimensions; and formulated an analytical equation which corrects the wellknown Murray's Law. 38 It was reported that for a given cavitation resistance throughout the xylem network, Murray's law should apply, which predicts the optimal taper of viscoelastic vessels. [38][39][40][41] While searching for singularities in the Navier-Stokes equations Terence Tao 42 reported that the partial differential equations that describe nonlinear effects of several cases are It is an admitted fact that all the fluids in nature are compressible because the specific heat at the constant-pressure (Cp) is always higher, ranging from a zero-plus (0+) value, than the specific heat at the constant-volume (Cv) of all real-world fluids. 1 The traditional assumption of taking water as an incompressible fluid 10-27 is patently not true as its specific volume (or density) does change with temperature and/or pressure. 1 The exact prediction of the 3D blockage factor throughout the flow field is a meaningful objective of any fluid flow system design from yocto to yotta scale and beyond, which is still an unresolved problem. Of late V.R.S.Kumar et al. 1,2 exactly predicted the 3D blockage factor in the sonic-fluid-throat region at the Sanal flow choking condition. The real scientific fact is that the Sanal flow choking is a compressible fluid flow effect, which occurs in any duct with uniform port geometry, due to the boundary layer blockage induced internal flow choking at a critical-total-to-static pressure ratio (CPR), as the boundary layer blockage factor will never be zero in any real-world flows. 45 The critical TSPR for flow choking of composite fluids would vary based on the lowest heat capacity ratio (HCR) of the evolved species at the constriction region (fluidthroat) of the streamtube (see Fig.1(b)). Note that the molecular dynamics condition in the composite fluid flow system could alter the streamline-pattern at different time and location. Therefore, pinpointing the exact location of streamtube flow choking is a challenging in vitro and in silico topic of great interest to the nano-microscale system designers.

RESULTS AND DISCUSSION
Herein, we presented an exact solution of the 3D blockage factor with molecular precision at In this letter analytical models are presented for establishing the causes and effects of the Sanal flow choking in an internal nanoscale fluid flow system with sudden expansion or divergent region. This was an unresolved world-wide scientific problem for more than a century.
Note that in general any fluid flow system from yocto to yotta scale and beyond with sudden expansion or divergent region (see Fig.1(a-b)) could create the physical situation of the Sanal flow choking at a CPR, which is regulated by HCR (γ) as dictated by Eq.1. The 3D nondimensional blockage factor (3DNBF) in an internal flow system with the cylindrical upstream-duct is derived from the compressible fluid flow theory 2, 46 and presented herein as Eq.2(a-b) for diabatic nanoscale flow systems with a desirable inflow condition (see Eq.3). At the Sanal flow choking condition, the 3D blockage factor (3DBF) at the sonic-fluid-throat (Maxial = 1) is derived and presented herein as Eq.2(b), which is the highest blockage factor in the internalmultispecies-choked-nanoscale fluid flow system. the HCR of gases decreases due to the decreases in CPR of the nanoscale flow system as dictated by Eq.1. The solution curve of Eq.3 is presented as Fig.2(a). While performing the in silico model verification and calibration, the average friction coefficient must be chosen in accordance with the Fanno flow choking condition. 2,46 Admittedly, at the sonic-fluid-throat of the nanoscale fluid flow system (see Fig.1(a-b)  The solution curve of Eq.4 is given in Fig.2(b) in the semi-log plot.
Note that the dominant species with the lowest HCR predisposes for an early Sanal flow choking at the sudden expansion or transition region of any internal flow system (see Fig.1(a-b)).
Note that Eqs. 1-5 are useful mathematical models for the high-performance aerospace chemical systems architects for predicting the limiting condition of deflagration to detonation transition    Vigneshwaran's Table (Table-1 The CPR value is an indication of the lower critical detonation index (LCDI) of the internal flow systems having accumulated with such types of gases. The 3DNBF is a very useful benchmark data for nanoscale in vitro experiments and in silico model verification, validation and calibration with credibility, which was an unresolved problem over centuries. The corresponding non-dimensional blockage factor for two-dimensional 2 case is also given in Table-I Table-1 & Fig.2(a)). The LCDI presented in Table-1 is a powerful indicator of knowing the detonation index of nanoscale chemical energy systems with sudden expansion or divergent port for prohibiting the catastrophic failures due to the Sanal flow choking and/or streamtube flow choking (see Fig.1(b)).

Fig. 2(e) Mach Number-Entropy chart of Fanno, Rayleigh and
Sanal flow models at the choked flow condition. It is pertinent to state that, as seen in Table- The self-explanatory equations (see Eq.7(a-c)), derived from the compressible flow theory, are highlighting herein for demonstrating the various influencing parameters and the conflicting requirements to prohibit the Sanal flow choking in the nanoscale fluid flow system. Eq.7(a) reveals that a disproportionate increase of the thermal conductivity of nanofluid increases the risk of Sanal flow choking leading to supersonic flow development followed by shock wave and pressure-overshoot in the nanoscale flow systems with sudden expansion or divergent region (see Fig.1(a-b)). Therefore, the condition set by Eq.7(a) must be satisfied while addition x . Please note that the coupled influence of the above-listed parameters will be controlled by a unique property of the nanofluid, which is stated herein with brevity as the heat capacity ratio (HCR) of nanofluid as dictated by Eq.1. This is a remarkable finding for the nanoscale systems development.
Viscosity variations are depending on the shear rate or shear rate history of the fluid, which could vary due to the variations in the thermophysical properties of nanomaterials and local effects too. The boundary-layer-blockage factors presented in Table-1 (the solution of Eq.2(b)) and the average friction coefficient given in Table- Table-1 and Table-2  inviting memory effects on the viscoelastic walls of the duct. This is a grey area, which needs to be examined in detail through fluid-structural interactive multiphase, multispecies in silico models, which is beyond the scope of this letter.

CONCLUSIONS
The Sanal flow choking model presented herein is useful for in vitro and in silico experiments of all real world fluid flow problems (continuum / non-continuum). Although the interdisciplinary science of nanotechnology has been advanced significantly over the last few decades there were no closed-form analytical models to predict the 3D blockage factor of diabatic nanoscale fluid flow system. The Sanal flow choking for the diabatic condition presented herein is valid for all the real-world fluid flow problems for designing various nanoscale fluid flow systems and sub systems due to the fact that the model is untied from empiricism and any types of errors of discretization. Using Eq.1 and Eq.2 the chemical propulsion system designers could easily predict the likelihoods of detonation with the given inlet flow Mach number and the lowest value of the HCR of the leading gas coming from the upstream port of the chemical system. 47 In a nutshell, the best choice of increasing the solid fuel loading in the nanoscale thruster design without inviting any undesirable detonation and catastrophic failures, is to increase the HCR of the working fluid. Further discussion on the nanoscale propulsion system design is beyond the scope of this letter.
We have established herein that, due to the evolving boundary layer and the corresponding area blockage in the upstream port of any internal nanofluid flow system with sudden expansion or divergent region, the creeping diabatic nanoflow (Mi << 1) originated from the upstream port of the system could accelerate to the supersonic flow leading to an undesirable phenomenon of pressure-overshoot due to shock wave generation as a result of the Sanal flow choking. Through the proposed mathematical methodology, we could disprove the general belief of the impossibilities of internal flow choking in such real-world nanoscale fluid flow systems at the creeping inflow conditions. There was a general belief in the scientific community over the centuries that the subsonic/creeping flow would not be augmented up to supersonic flow without shipping through a geometric throat, which we have disproved herein through the closed-form analytical model. Note that if the total-to-static pressure ratio at the fluid-throat is lower than the  Authors have no competing interests as defined by your publications or other interests that might be perceived to influence the results and/or discussion reported in this paper.

Data and materials availability:
All data needed to evaluate the conclusions of the paper are available in the manuscript.

Declarations of interest: None
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