This article objectively assesses, the hypothesis of the streamline’s shape theory and its formulated equation. The deduction of proof uses algebra rather than ﬁrst-order partial diﬀerential equations to address the speciﬁc hypothesis of "Streamline’s shape theory" from the fundamental perspective of applied mathematics and scientiﬁcally derives mathematical relations of the axioms and corollaries in the ﬁeld of ﬂuid dynamics. The algebraic methods employed provide progressively more distinct and precise solutions compared to ﬁrst-order partial diﬀerential equations. The foremost objective of this work is to evaluate if the formulations of the streamline’s shape theory can have solutions for inviscid-incompressible and viscid-compressible ﬂows of Newtonian ﬂuids and to identify their nature. Secondly, to understand how the topology of the body and the free-stream conditions aﬀect these solutions with due regards to the shape and size of the body interacting with the ﬂuid ﬂow. Finally, to explore the possibility of this theory to develop a CFD solver for streamline simulation to reduce the experimentation in the analysis of ﬂow-structure interactions of Newtonian ﬂuids and also to identify its scope of applications and limitations. 1 Introduction The existing theories concerning streamlines like Prandtl–Batchelor theorem and free stream- line theory were fundamental in the development of several models for the streamline simulation. Prandtl–Batchelor theorem is applicable for ﬂows with high Reynolds numbers for closed stream- line conditions and invalid within the boundary layer regime.[19][17] Freestream line theory, on the other hand, uses classical Helmholtz free boundary theory and Laplace transformations. Based on it, few promising mathematical models were developed in recent years for various com- plex cases in hydrodynamics and aerodynamics, such as wake models, cavity ﬂows, free-surface perturbations caused by viscosity, and turbulence.[12, 5, 20, 20, 15, 22, 7, 18] Historically, many theories use energy conservation/ dissipation and mass conservation/ dissipation approaches. These approaches give an aggregate of ﬂuid ﬂow within the considered boundary limits. How- ever, they present a challenge to specify local parameters within the ﬂuid ﬂow. Diﬀerent types 2010Mathematics subject classiﬁcation. 35Q35 ; 74F10 ; 76M99 Key words and phrases. Fluid-Structure interactions, Topological ﬂuid dynamics, General ﬂuid mechanics, Mathematical ﬂuid dynamics, Applied mathematics. 1 of solvers use diﬀerent methods and equations concerning various cases of the ﬂuid ﬂow and types of ﬂuids. Most of them employ diﬀerential equations and partial diﬀerential equations to evaluate local parameters that have their shortcomings. In recent years streamline simulation has acquired its prominence among researchers as a promising approach compared to traditional methods.[6]Moreover,the reliability of CFD results heavily depends on the computational meth- ods, basic equations, and processing capabilities of the computer, and thus experimental data is always required.[11] Though existing literature conveys that streamlines form as a result of shear stress and dynamic pressure distributions over the surface of the body in a ﬂow.[3] There eﬃciently is no mathematical model that can explain how and why streamlines are forming concerning the nature of ﬂuid, nature of the ﬂow, the shape, and the size of the body.[8, 2] "Streamline’s shape theory" isaproposedmathematicalmodelthatclaimstoexplainthe mechanism of streamlines of a Newtonian ﬂuid in an open circuit ﬂow over a body.[21] According to this theory, Hypothesis :Laminar ﬂow of a Newtonian ﬂuid over a rigid body will displace the ﬂuid par- ticleswhenthesolidbodyhassuﬃcientstrengthtowithstandthedeformationanddisplacement. The dynamic pressure of the ﬂuid and the shear stress distribution over the body contributes to the streamline formation.[9, 8] Equation (1) shows the force that is resultant of ﬂow over a rigid body, which contributes to change in the trajectory of ﬂuid particles forming streamlines. ~FAi= A1 2Ai~ vAi2+1 2Ai1~ vAi12cos(r) dA + A Ai@vAi @y+Ai1@vAi1 @ycos(r) dA (1) Here, equation (1) is the sum of dynamic pressure and shear stress at a local point and the impact of surrounding points of the topology. The1 2Ai~ vAi2is dynamic pressure at a local area AiandAi@vAi @yis the shear stress at Ai. Similarly, the cosine multiples in the equation (1.1), are dynamic pressure and shear stress at surrounding areas and ris the angle between free-stream vector and the local force vector. ris the angle at which the resultant force FAi1acts onFAi. Remark 1.1.The aforementioned mathematical formula claims to predict and explain the tra- jectory of ﬂuid particles in a laminar ﬂow of Newtonian ﬂuid and provides an ability to explain various phenomena in an open circuit ﬂow such as stalling, ﬂow separation, down-wash, Coand a eﬀect, eﬀects of shape and size of the body over which ﬂuid ﬂows. Though the theory is argu- mentatively not disapproved but its reliability in terms of mathematical precision in analyzing open circuit Newtonian ﬂow over a body is yet to be evaluated. That imposes certain questions like: Does this equation always have solutions? If so, what is the nature of the resultant solution? What is the range of applicability for this formula? 2 Methodology The article published under the title “Mechanics of streamlines in an open circuit ﬂuid ﬂow and its impact on the ﬂow”[8] is a mathematical conjecture as it doesn’t have any mathematical proof or experimental data to validate it. A brief mathematical inference for the stated formulation of “Streamline’s shape theory," is carried out in section 3. Then, the mathematical proof employing inductive logic and the direct method of proof derives mathematical inferential arguments in the ﬁeld of ﬂuid mechanics. The section 4 titled “mathematical validation" has both graphical method and analytical methods of 2 algebraic calculations that proves the formulation of the considered hypothesis of “Streamline’s Shape Theory." 3 Mathematical Interpretation Ai−1AiθτAi−1=µAi−1∂u ∂y 1 2ρAi−1/vector vAi−12/vectorFAi−1 1 2ρAi/vector vAi2τAi=µAi∂u ∂y/vectorFAi−1cos(θ) +1 2ρAi/vector vAi2+µAi∂u ∂y+ =/vectorFAi c Figure1: Relationshipbetweentwosuccessivepointsonthetopologyofthebodyandthevectors due to dynamic pressure and shear stress distribution The shear stress always acts tangential to the surface of the body for a Newtonian ﬂuid, and dynamic pressure acts normal to the surface of the body.[10] Equation (1) is a time-independent and sum of two parabolic functions and two linear functions. Here, dynamic pressure at Aiand resultant dynamic pressure from surrounding areas Ai1 are the parabolic functions as they are in the form: ax2= 2~ v2. The shear stress distribution is a linear function for a Newtonian ﬂuid within subsonic velocities. The @v @yis in the form of mx+ 0. As the ﬂuid stream passes over the topology of a body, the shear stress distribution and the dynamic pressure distribution change the velocity and trajectory of the ﬂuid particles. Therefore, a free-stream ﬂuid ﬂow, a deﬁned parabolic function, when passes over a body develop shear stress on the topology of that body, which is a linear function for the Newtonian ﬂuids. The sum of these two functions will have a resultant force vector, which eﬀects succeeding locus of points gradually. Figure.1 illustrates these vectors. The dynamic pressure due to free-stream at1 2Ai1~v2 Aiand the shear stress developed at Ai1isAi1@vAi1 @yand the resultant sum due to these two forces isFAi1. The force on the succeeding area Aialong the path ’c’ is a sum of FAi1, the force due to the dynamic pressure of free-stream, and the force due to the shear stress distribution. Here, Ai1is the stagnation point. So, the formation of streamline starts at the stagnation point and gradually changes along with the shape and size of the body. 4 Mathematical Validation The algebraic form of the equation (1) at any point on the surface of the body with arbitrary shape is:[1] 1 2Ai~ vAi2+Ai~ vAi y+1 2Ai1~ vAi12cos(r) +Ai1~ vAi1 ycos(r) +PAi= 0 (2) Here,PAiis the pressure at Aiandv yis the rate of shear deformation. 4.1 In-viscid & In-compressible ﬂow For an inviscid-incompressible ﬂow, andare constants. Since equation (2) is the sum of algebraic equations. Here, the assumption is that the dynamic viscosity is constant, and the 3 eﬀects of temperature on the ﬂow are negligible. Ignoring in formulation causes d’Alembert’s paradox or hydrodynamic paradox, which does not yield a realistic mathematical model[16][17]. Upon simpliﬁcation, 1 2 ~ vAi2+ y~ vAi+qPAi |{z} ax2 1+bx1+c+1 2 ~ vAi12cos(r) + y~ vAi1cos(r) +rPAi | {z } ax2 2+bx2+c= 0 (3) where, q+r =1 i.e. qPAi+rPAi=PAiandyis the length of the boundary layer formed. The~ v y is the rate of shear deformation. The sum of a parabolic function and linear function results in parabola shifting. Here, equation (3) is the sum of two algebraic functions.[13] Therefore, f(x1) +g(x2)is the resultant function. Figure.2 and ﬁgure.3 graphically illustrates the algebraic sum of these two functions. Here,1 2~ v2and1 2~ v2cos(r)are in the form of ax2and fora > 0 the parabola will open upwards and will have a vertex at (0,0). The vertex of the function 1 2~ v2+ yv+qPAiis ( y; y( 2y1)+qPAi) and the vertex of1 2v2 Ai1cos(r)+ yvAi1cos(r)+rPAi is ( y; ycosr( 2y1) +rPAi). The Y-intercept of the equation (3) is (0, PAi). As the negative sign indicates the only directional component of the velocity vector, the minimum velocity required for a ﬂuid particle to change its trajectory and deform at a surface isj=yj. Thus, the corollary of kinematic viscosity is satisﬁed. Corollary 4.1. For any change, in the ﬂuid stream over a body. The minimum velocity after the interaction of a body with the ﬂuid stream must be y. As Kinematic Viscosity is the inherent viscosity of Newtonian ﬂuids, that does not change with a change in applied force. yis the thickness of the velocity boundary layer or displacement thickness given by Prandtl’s Boundary layer theory. Muller’s method is a root-ﬁnding numerical method developed by David Muller that calcu- lates roots.The standard formula where the co-eﬃcient a, will be in the denominator and that leads to in-determinant solutions for the algebraic equation when a= 0. Muller’s formula can be deduced from the standard quadratic formula by Vieta’s formulas, and it will give solutions even whena= 0. The solution for f(x1) +g(x2)will have four roots. The discriminants iand i1of the functions f(x1)andg(x2)are given in the equation (4) and equation (5). Likewise, equation (6) gives the roots of the equation (3) by using Muller’s method.[14] The increments and decrements of the resultant algebraic function depends on the value of cosine value in g(x2). Asrchanges from 0to360the value of cosrvaries between 1 to -1. This indicates the eﬀects of the shape of the body on the ﬂow. Therefore, the nature of the two roots for the considered algebraic expression depends on the shape of the body. i= y2 2(qPAi) (4) i1= ycos (r)2 2(rPAi1) cos (r) (5) ~ vir=2qPAi   y pi; ~ vir1=2rPAi   y pi1(6) 4 (0,0) /vector vP(a)Plot for the Velocity and Pressure at a local point of the ﬂow for f( x1) µ∂v ∂y+1 2ρ/vector v2+rPAi µ∂v ∂y 1 2ρ/vector v2 (0,0) /vector vP(b)Plot for the Velocity and Pressure at a local point of the ﬂow for g( x2) µ∂v ∂ycos (θr) 1 2ρ/vector v2cos (θr) µ∂v ∂ycos (θr) +1 2ρ/vector v2cos (θr) +qPAiFigure 2: Graphical representation of the algebraic sum of dynamic pressure & shear stress of thef(x1)&g(x2) 5 (0,0) /vector vPPlot for the Velocity and Pressure at a local point of the ﬂow for f(x1)+g(x2) µ∂v ∂ycos (θr) +1 2ρ/vector v2cos (θr) +qPAi µ∂v ∂y+1 2ρ/vector v2+rPAi 1 2ρ/vector v2cos (θr) +µ∂v ∂ycos (θr) +µ∂v ∂y+1 2ρ/vector v2+PAiFigure 3: Graphical representation of algebraic sum of the f(x1)&g(x2)i.e. 4.2 Viscid- Compressible ﬂow For a viscid compressible ﬂow, andare not constants. Therefore, modifying equation (3) and introducing , kinematic viscosity. 1 2v2 Ai+ y~ vAi+qPAi Ai|{z} ax2 1+bx1+c+1 2v2 Ai1cos(r) + y~ vAi1cos(r) +rPAi Ai1| {z } ax2 2+bx2+c= 0 (7) Since=y=andis an inherent characteristic of the ﬂuid which doesn’t change with applied force. Equation(7)isintheformof ax12+bx1+c1+ax22+bx2+c2i.ef(x1)andg(x2). Similartothe method mentioned in the section 4.1, There will be a parabola shifting and change in the vertex ofthealgebraicexpression. Thevertexofthefunction1 2v2 Ai+ y~ vAi+qPAi Aiis( y,1:5(=y)2+qPAi Ai) and the vertex of1 2v2 Ai1cos(r)+ y~ vAi1cos(r)+rPAi Ai1is ( y,1:5(=y)2cosr+rPAi1 Ai1). Similar to the method mentioned in section 4.1, the discriminants and roots of equation (7) are agiven in equations (8),(9) and (10). Graphical representations in ﬁgure (2) and ﬁgure (3) are also valid for a viscid-compressible ﬂow. i= y2 2qPAi Ai(8) i1= ycos (r)2 2rPAi1 Ai1cos (r) (9) 6 ~ vir=2qPAi=Ai   y pi; ~ vir1=2rPAi=Ai1   y pi1(10) The streamline shape theory satisﬁes corollary 4.1 for viscid-compressible ﬂows. However, a signiﬁcant diﬀerence exists in the resolved roots of equations (6) and (10). The roots of equation (10),~ vir&~ vir1have pressure to density ratios in their numerators. Since the speed of sound of a ﬂuidcfluis the square root of the ratio of change in pressure and change in density as stated in equation (11). cflu=s dP d '~ vir=2   y piqPAi Ai '~ vir1=2   y pi1rPAi Ai1(11) As one of the roots obtain value closer to the speed of sound, the ﬂuid ﬂow doesn’t form streamlines along the surface of the body. Equation (11) shows the limitation of streamline’s shape theory. Speed of sound cflualso equals top rT. The eﬀect of temperature on the viscosity and simultaneously on the formation of streamlines are in-evident in the streamline’s shape theory. Therefore, this theory is only applicable to subsonic ﬂows. Thus, streamline’s shape theory satisﬁes the second corollary. Corollary 4.2. At higher velocities, shock waves are formed rather than streamlines causing an abrupt change in the pressure as the body travels with velocity closer to the speed of sound. 5 Results and Discussion The streamlines adopt the shape of the surface topology within the subsonic regime. These statements justly apply to only Newtonian ﬂuids in a laminar ﬂow. For ﬂuid particles to alter their typical trajectory, the minimum velocity at a local point precisely is j=yj. Here, the value ofdoesn’t change with the change in the applied force at the local point, and yrepresent the displacement thickness from the surface. Foraninviscid-incompressibleﬂow, thehypothesisisvalid, andfromthedimensionalanalysis of equations (4), and (5); can be interpreted as: the discriminants of the equation at any point over the topology is the change in the mass ﬂow rate per unit area due to the pressure at that point. Consider discriminants of inviscid-incompressible ﬂow mentioned in equations (4) and (5). Here, M,L,T are base quantities mass, length, and time respectively. M2 L2T212qL2 L2 =M L2M T1 T12qL2 L2 (12) Equation (12) emphasizes that a formulation is a form of a continuum of the ﬂuid with a correction factor of (12q(Ai)2)=(Ai)2). The parameters of the determinant are the surface density, mass ﬂow rate, and rate of shear strain deformation. In other words, the constant dynamic viscosity and a boundary layer thickness will have an inherent continuity of the ﬂuid as 7 a nature of ﬂuid itself. It is apparent in the equations (4) and (5) as (=y)2. At every point on the surface, this constant continuity of mass is aﬀected by the pressure and density of the ﬂuid. qP is a component of pressure due to shear stress and dynamic pressure at a point concerning its surroundings. Therefore, achangeinthiscontinuumofﬂowoccursbetweentwosuccessivepoints by a factor of (12(Ai)2)=Aiwith respect to the product of density, mass ﬂow rate, and rate of shear strain deformation. Similarly, equation (5) can be interpreted but it will have cos(r), which is merely the component of the pressure aﬀecting from the preceding point on the topology and rP is the component of the total pressure distribution P. Equation (6) indicates two speciﬁc parameters, the denominator =yas deﬁned earlier is the inherent continuum property of the ﬂuid, deﬁned as the ratio of dynamic viscosity and boundary layer thickness. The discriminant is the continuum of the ﬂuid due to ﬂow structure interaction and subsequent distribution of the shear stress and dynamic pressure concerning the shape of the body. Thus, we can state that the velocity at any point on the streamline is directly proportional to the pressure at that point and inversely proportional to the diﬀerence between the inherent continuum of the ﬂuid and the change in the continuum of ﬂuid ﬂow at that point. The ﬂuid stream interacts with a structured body at Ai1, att0when the impact of the surrounding area’s interaction is non-existent. Equation (5) and equation (8) shows that one of the i1is dependant on the value of r. This mathematically conﬁrms that when a solid body interacts with a ﬂuid stream. The leading edge where the ﬂuid particles bombard on the surfaceAi1att0, the ﬂuid particles parallel in motion to the bombarded ﬂuid particles haven’t reachedAi, which contributes to non-existent of r. However, Ai1have already interacted with the surface of the body forming rthe resultant force vectors of f(x1) +g(x2)will have the impact from Ai1att1. Thus ﬂuid particles tend to travel along the surface of the body. In other words, the shear stress inﬂuences cause the free stream to ﬂow around the walls of the body by contributing to the streamline formation. Equations (8),(9) and (10) shows that viscid-compressible ﬂow will have similar eﬀects like inviscid-incompressible ﬂows of a Newtonian ﬂuid, regarding the impact of ror the proﬁle of the body. However, there is a signiﬁcant diﬀerence between equations (4), (5), and (6) when compared to equations (8), (9), and (10). Consider a dimensional analysis of the discriminant of the viscid-compressible ﬂow: L2 T22qL4 T2 =L T2 12qL2 (13) Unlike, inviscid-incompressible ﬂow the deﬁning parameter iis in terms of velocity square. It implies that the ﬂuid ﬂow in viscid-compressible ﬂow has a wave nature. Equations (8) and (9) are just diﬀerence between the square of the inherent speed of sound of ﬂuid with the speed of sound propagation at the local point. rsimply relies on the relationship between two successive points.Therefore, the formulation conﬁrms that compressibility causes waves in ﬂuid ﬂow[4]. Here,t0is the time at which ﬂuid particles reach the leading edge of the body at Ai1.t1is the time at which ﬂuid particles reach the subsequent point on the surface of the body Ai. From ﬁgure 2, it is noticeable that the resultant function of the shear stress and dynamic pressure is a parabolic function with the change in the axis of symmetry. Here, the quadratic term’s coeﬃcient is 2, and(density) is always positive for a ﬂuid. As it is well-known if 2is greater than unity the parabola will be narrow, and the parabola shall open wider if the value is less than 1. As all viscous ﬂuids will have positive viscosity the co-eﬃcient of the linear term will also be positive. Thus, the axis of symmetry shifts to the left. Thus for non-Newtonian ﬂuids or ﬂuids with zero viscosity (super-ﬂuids) can’t be governed by the “Streamline’s shape theory". In ﬁgure 3, which is the sum of two algebraic expressions f(x1) +g(x2)the change in vertex. Figure 2 (a) is the graphical representation of the resultant function due to the ﬂow-structure 8 interaction at the stagnation point, where there is no eﬀect on the ﬂuid interactions of the surrounding area. Figure 2 (b) is the graphical representation of the resultant function due to the ﬂow-structure interaction in due consideration of the eﬀect of the ﬂuid interactions of the surrounding area alone. Thus, it indicates how the shape of the body, the density, and viscosity oftheﬂuidaﬀectstheﬂow-structureinteractionsbycontributingtothedecrementinthevelocity of the ﬂow at a particular point on the topology of the body. The change in vertex is due to a change in pressure, and these functions mentioned and illustrated in ﬁgure 2 causes a change in pressure. Thus, the vertex change is observable in ﬁgure 3. The narrowing of the parabola indicates a higher coeﬃcient of the quadratic term, an indication of the compressibility eﬀect. However, this model doesn’t consider the eﬀects of electromagnetic ﬁelds, thermal boundary layers, among many other aspects. The Corollary 4.1 states that this model is mathematically appropriate for all inviscid- incompressible ﬂows. In practice M<0.3 (Mach number) is mostly considered as an inviscid- incompressible ﬂow. But beyond M>0.3, ﬂow will be assumed as a viscous ﬂow. The Corollary 4.2 indicates the mathematical limitation of this model which is observed in experimental re- sults of existing literature. Tran-sonic ﬂows defy the laws of low speed ﬂuid dynamics. At higher velocities close to the speed of sound ﬂuids exhibit wave characteristics unlike low speed ﬂows.[2, 3] From section 4, it is evident that this theory distinguishes from other mathematical models in a few particular aspects. 1. Most of the earlier theories and their mathematical models based on experimental data emphasizedonBuckingham- theorem. Thisapproachenabledtheresearchersbyreducing the number of parameters to monitor during experimentation and made the process more convenient. However, this caused ignorance of a few parameters in the empirical equations when formulated. The shape of the body is one such parameter, which is often represented as correction in terms of coeﬃcients. 2. The introduction of angle , makes it possible to calculate the change in the ﬂow along with the topology of the surface. If the direction of ﬂow, shape of the body, and size of the body are known, then it will be possible to calculate the force at the stagnation point. Then from the stagnation point to the force vectors at succeeding locus of points are possible to be calculated at each corresponding time and space intervals. 3. In this formulation, the shape and size are inclusive in the fundamental equation. Trigono- metric relations and ﬁnite element methods, coupled with this model, could make it a more reliable CFD solver. 4. Although this model doesn’t consider Buckingham- theorem, it is not resistant to Pi parameters. Once velocity is calculated at each point the surface of the body, through the historical data of Mand Re (Reynolds number) various phenomena like ﬂow transition, ﬂow separation, the formation of shock-waves can be predictable on the topology of the body. 6 Conclusion Hence, the stated hypothesis of the principal argument is satisfactorily proven, and the stream- lines occur as a resultant of the shear stress and dynamic pressure distribution over the body. The formation of streamlines and their mechanisms are the fundamental aspects that don’t haveamathematicalmodelinﬂuiddynamics. Although, manyexpertsamongvariousdisciplines opine that this may not be of signiﬁcant importance. But to develop more sophisticated CFD 9 applications and methods of analysis, this is very critical. Several methods in the ﬁeld of ﬂuid dynamics are in practice often produce reasonable solutions to the modeled problems, and yet most of them are empirical and lack formal mathematical proof. This formulation and its proof conﬁrm various aspects of a Newtonian ﬂuid ﬂow, which are often explained and formulated by diﬀerent theories. For an inviscid-incompressible ﬂow, this formulation addresses continuity eﬀects, d‘Alembert’s paradox, and deﬁnes minimum criteria for velocity change at any local point on the ﬂuid ﬂow. Likewise, this formulation avoids the employment of coeﬃcients as it incorporates a geometric relationship between the ﬂuid ﬂow and the shape of the body. Most of all, this formulation unlike, the Navier-Stokes equation has a formal mathematical proof along with derivable solutions. The foremost advantage of Streamline’s shape theory is that it is a lucid mathematical for- mulation and solutions always exist. Streamline’s shape theory can evaluate various phenomena such as the Coand aeﬀect, ﬂow transition, ﬂow separation, stalling, etc. Although other math- ematical models can serve the same purpose, one needs diﬀerent solvers and has to perform iterative simulations to compute various parameters. Further course of possible research involves the evaluation of this theory utilizing experimen- tation methods and development of CFD solver based on the streamline’s shape theory. A study is necessary to perceive the behavior of the non-Newtonian ﬂuids concerning the topology of the body in ﬂow-structure interactions and the formation of streamlines. Abbreviations CFD Computational Fluid Dynamics M Mach number Re Reynolds number y length of boundary layer Nomenclature Discriminant of the quadratic equation @u=@y Rate of shear deformation Dynamic viscosity of the ﬂuid Kinematic viscosity of the ﬂuid Density of the ﬂuid rAngle of resultant force on Aifrom preceding area Ai1 ~FResultant force due to the ﬂuid motion ~ vVelocity of the ﬂuid cfluSpeed of sound of the ﬂuid d Change in the density dP Change in the pressure PPressure at a local point @u=@y rate of shear deformation Subscript AiLocal parameters such as ,,~Fand~ vat the point ( Ai) Ai1Local parameters such as ,,~Fand~ vat the preceding point ( Ai) ir&ir1Roots of the quadratic equations Acknowledgment I, Yuvaraj George, express my gratitude to all my colleagues from Moscow Aviation Institute, who have extended their moral support and encouragement to carry out this work. I would like 10 to dedicate this work to my friend and well-wisher Bakka Pavan Kumar Reddy, who has been phenomenal in motivating me to pursue this work. Funding This research received no speciﬁc grant from any funding agency in the public, commercial, or not-for-proﬁt sectors.”