Hydrodynamic Stability Research Articles

Numerical study of flame stability within inert porous media with variable void area

This numerical study brings a comparison of a cylindrical porous burner with graded porosity and a conical porous burner with uniform porosity. Both burners have the same void area, creating a similar hydrodynamic stabilization mechanism. The study was performed for a laminar premixed CH4/air flame for different equivalence ratios with the FGM reduction technique employed to model the chemical kinetics. The results demonstrate that the cylinder with graded porosity presents a wider range of stable flames compared with the conical porous burner because the graded porosity leads to a larger heat recirculation as the flame approaches the outlet surface, which is mainly due to an increase of the solid-phase radiative heat transfer. The two-dimensional structure of the flame front was investigated in detail showing that the non-alignment of flow streamlines and solid-phase heat flux lines creates regions of sub-adiabatic and super-adiabatic temperatures along the flame front. Flame area increases with the inlet flow rate with the graded porosity burner presenting a smaller area and larger local consumption velocity than conical burner.

Mathematical and information models of decision support systems for explosion protection

This paper is dedicated to the issue of mathematical and information modeling of the combustion-to-explosion transition that makes it possible to create an adequate mathematical and information support for decision support systems (DSS) for automatedcontrol of explosive objects.A simple mathematical model for the transition of combustion to explosion is constructed. This model is based on solving mathematical problems of the hydrodynamic stability of flames and detonation waves. These problems are reduced to solving eigenvalue problems for linearized differential equations of gas dynamics. Mathematical model is universal enough. It provides opportunities for making simple analytical estimates for the explosive induction distance and the time of the shock wave formation. The possibilities of the transition of slow combustion to both a deflagration explosion and a detonation wave are considered. Theoretical estimates of the explosive induction distance and the time of the combustion-to-explosion transition are obtained. These estimates are expressed by algebraic a formula, the use of which save computer resources and does not require significant computer time. The application of fuzzy logic makes it possible to use the proposed mathematical model of the combustion-to-explosion transition for real potentially explosive objects in industry and transport.Mathematical models of potentially explosive objects are based on combination of the fuzzy logic and classical mathematical methods. These models give possibilities for creating corresponding information models. Thus mathematical and information support of DSS for automated control systems of explosive objects is developed. The main advantage of these DSS is that it makes it possible for decision makers to do without experts. In particular, developed mathematical and information models create the base forsoftware of DSS for explosion safety of grain elevators. Appropriate software is developed and some calculations are performed. These calculations are useful not only from the point of view of testing the proposed method of mathematical modeling of a grain elevator as a potentially explosive object or testing the software itself, but also from the point of view of the grain elevator designing.

Ionic Charge Emission Into Fog From a Remotely Piloted Aircraft

AbstractCharge influences the properties of liquid droplets, such as evaporation rates, hydrodynamic stability, and sticking probabilities. Modifying droplet charge therefore provides a possible method of influencing fogs or clouds. An instrumented, remotely piloted aircraft has been equipped with positive and negative corona emitters to cause droplet charging. With the aircraft circling at 20 m altitude, effects of ion release were compared in clear air and natural fog. In clear air, the surface atmospheric electrical field changed whenever the emitters were activated, but without significant differences in the short‐wave radiation as the aircraft passed over previously ionized air. In fog, radiation fluctuations showed a Gaussian distribution before either emitter operated or when both emitters were operating, but with different distributions during unipolar ion emission. Introducing unipolar ions led to a maximum change in fog reflectivity of ∼2%, about 25 s later.

Design of the Bengawan Unmanned Vehicle (UV) Roboboat: Mandakini Neo

Mandakini Neo is an autonomous vehicle that was designed and built by the students of the Universitas Sebelas Maret, which was included in the Bengawan Unmanned Vehicle (UV) Roboboat Team to compete in the annual international Roboboat competition of 2021. This competition requires participants to complete several missions; one of the main missions is to move through two gates made from four poles using full automatic navigation, in order to continue on with the other missions. To complete the course, we used Pixhawk and GPS to allow the ship to run automatically, while minimizing the ship’s movement tolerance. The use of Mission Planner software for monitoring, and also for color and shape image processing to help with the reading of objects, along with a sensor fitted on the ship, allowed the mission to be completed. Mandakini Neo was made with the capacity, speed, and comfort of the ship in mind, as well as the ship’s hydrodynamic performance, stability, volume, structural integrity, and construction cost. Following its development we conducted tests of stability, maneuverability, and seakeeping in order to achieve the smallest possible resistance rate; for this purpose, we used the Savitsky method. The manufacture of the ship also required the choosing of the material, the use of the sensor, and also selection of an appropriate system. Finally, the design that we developed was a ship with a catamaran hull type, for which the dimensions had already been calculated, and the proper materials decided, and simple electrical components were able to be obtained for the sensor and the system.

Understanding the role of flow dynamics in thermoacoustic combustion instability

Thermoacoustic combustion instability is one of the most challenging operational issues in several high-performance, low-emissions combustion technologies, including gas turbines, aircraft engines, rockets, and industrial boilers. Driven by the coupling between combustor acoustics and flame heat release rate fluctuations, thermoacoustic combustion instability can lead to reduced operability, increased emissions, and, in the most extreme cases, catastrophic failure of combustor components. The feedback loop between acoustics and combustion is often facilitated by fluid mechanic oscillations, referred to as “velocity coupling,” whereby acoustic oscillations drive flow fluctuations, which in turn create fluctuations in the flame. The character of these fluid mechanic oscillations is highly dependent on the structure of the flow field and the receptivity of the flow to external excitation. Combustor flow fields use features like fluid recirculation and shear to enhance flame holding and reduce emissions, but these are also the same features that can make the flow receptive to acoustic excitation or even drive self-excited oscillations. In this paper, we discuss the basics of thermoacoustic instability with a focus on the role of hydrodynamic oscillations in typical combustor flows. To facilitate this discussion, we explore the hydrodynamic instability characteristics of several key combustor unit flows (wakes, swirling jets, etc.) and show how the hydrodynamic stability of a flow is an important consideration in determining a combustor’s propensity for thermoacoustic oscillations. Several examples of coupling between hydrodynamics and thermoacoustics are discussed to illustrate this important link. The paper concludes by discussing the potential for designing flow fields that are thermoacoustic instability resistant, either through a reduction in the receptivity of the flow or through nonlinear coupling mechanisms by which self-excited flow instabilities can suppress velocity-coupled combustion oscillations.

Prandtl number effects on the hydrodynamic stability of compressible boundary layers: flow–thermodynamics interactions

Hydrodynamic stability of compressible boundary layers is strongly influenced by the Mach number ( $M$ ), Prandtl number ( $Pr$ ) and thermal wall boundary condition. These effects manifest on the flow stability via the flow–thermodynamics interactions. Comprehensive understanding of stability flow physics is of fundamental interest and important for developing predictive tools and closure models for integrated transition-to-turbulence computations. The flow–thermodynamics interactions are examined using linear analysis and direct numerical simulations in the following parameter regime: $0.5 \leq M \leq 8$ ; and $0.5 \leq Pr \leq 1.3$ . For the adiabatic wall boundary condition, increasing Prandtl number has a destabilizing effect. In this work, we characterize the behaviour of production, pressure–strain correlation and pressure dilatation as functions of the Mach and Prandtl numbers. First and second instability modes exhibit similar stability trends but the underlying flow physics is shown to be diametrically opposite. The Prandtl number influence on instability is explicated in terms of the base flow profile with respect to the different perturbation mode shapes.

On the Short Wave Instability of the Liquid/Gas Contact Surface in Porous Media

We consider a problem of hydrodynamic stability of the liquid displacement by gas in a porous medium in the case when a light gas is located above the liquid. The onset of instability and the evolution of the small shortwave perturbations are investigated. We show that when using the Darcy filtration law, the onset of instability may take place at an infinitely large wavenumber when the normal modes method is inapplicable. The results of numerical simulation of the nonlinear problem indicate that the anomalous growth of the amplitude of shortwave small perturbations persists, but the growth rate of amplitude decreases significantly compared to the results of linear analysis. An analysis of the stability of the gas/liquid interface is also carried out using a network model of a porous medium. It is shown that the results of surface evolution calculations obtained using the network model are in qualitative agreement with the results of the continual approach, but the continual model predicts a higher velocity of the interfacial surfaces in the capillaries. The growth rate of perturbations in the network model also increases with decreasing perturbation wavelength at a constant amplitude.

The onset of instability in a hydromagnetic channel flow of Casson fluid: the accurate solutions

We study the temporal stability of linear two-dimensional disturbances of plane Poiseuille flow of a Casson fluid in a rigid parallel channel under the influence of a uniform magnetic field. When the disturbance is taken in normal mode it transforms the perturbed equations to a fourth-order eigenvalue problem which is then solved by the spectral method. We first obtain sufficient conditions for the hydrodynamic stability in the channel by analyzing the nature of the growth rate and the Reynolds number. For the chosen set of parameters, the eigenspectra of the flow exhibit the familiar Y-shaped structure with three distinct modes depending on the phase speed approach. The eigenfunctions corresponding to the least stable and most unstable eigenmodes have the symmetric and antisymmetric modes. For a certain range of Casson parameter β and the magnetic number M, instability always appears on the wall mode for which the eigenfunction is antisymmetric about the channel center-line. The neutral stability curves that correspond to various values of β and M are found to be an extension of the continuation of the Tollmien-Schlichting instability that is noticed for a Newtonian channel flow. The regions of stability are enlarged for small β and large M.

Experimental studies on the hydrodynamic properties of a deposit in rapid filters during physicochemical removal of iron from groundwater

Abstract This article presents the results of experimental studies for information support of the hydraulic compartment of the mathematical model of physicochemical iron removal from groundwater. It is experimentally established that the bound water in the deposit significantly exceeds the mineral component (chemical iron) for any waters with the different levels of iron oxidation. The deposit dehydration takes place at its boundary with the liquid phase. As a result, ferrous iron is gradually oxidized and ferric iron transforms from hydroxide forms to oxide forms, which leads to a decrease in the deposit volume and a increase in its density and strength. Further, the coefficient of the deposit hydrodynamic stability is proposed to use to take into account changes in the properties of the deposit that accumulates in the granular medium. It is experimentally determined that the iron amount remaining in the medium after backwashing increases simultaneously with the deposit's age. It will lead to a gradual reduction of the operational dirt capacity of the granular medium and a need to have measures of the filtration properties restoration or to replace the filter medium entirely.

Blowoff of bluff body flames: Transient dynamics and three dimensional effects

This work examines the dynamics of confined bluff body flames in the process of lean blowoff (LBO) using simultaneous stereo-PIV (particle image velocimetry), OH PLIF (planar laser induced fluorescence) and formaldehyde (CH2O) PLIF. Flames at high density ratios blow off in at least two distinct stages: stage 1, where intermittent extinction occurs along the flame front, but the flame and flow remain qualitatively similar to stable conditions, and stage 2, where there is permanent downstream flame extinction and large-scale changes in dynamic flow characteristics. This paper particularly focuses on stage 2 processes, with the goal of understanding what ultimately leads to irrecoverable flame blowoff. A test facility was developed with the flexibility to achieve two goals: (1) approach LBO by keeping the parameters that influence its hydrodynamic stability approximately constant, particularly flow velocity (ubulk) and gas expansion ratio (σ), and (2) compare near-LBO dynamics under conditions where, well away from blowoff, the flame is globally stable (high σ case) and globally unstable (low σ case, where the Bénard-Von Karman, BVK, instability of the flow is present). The latter case was of particular interest as most prior detailed diagnostic studies of LBO have been performed at high σ, BVK-suppressed conditions. We find that the transient blowoff process remains largely unchanged in the high and low σ cases, apparently due to the fact that the BVK instability reappears in either case under conditions very close to LBO. In all cases, blowoff is preceded by permanent downstream extinction that moves progressively closer to the bluff body as LBO is approached. We also find that near-LBO dynamics are intrinsically three dimensional, due to both secondary instabilities of the shear layer and large out-of-plane motions believed to be due to confinement effects associated with bluff body-wall interactions. These three-dimensional structures often manifest themselves as burning reactant fingers which are caught in the backflow of the recirculation zone; under very near LBO conditions they impinge on the back of the bluff body and extinguish as well. At the very edge of blowoff, the recirculation zone is no longer composed of hot products and is unable to autoignite the oncoming reactant flow, leading to global extinction. The characteristic time associated with this feedback between downstream extinction and wake structure alteration that leads to blowoff is about two orders of magnitude larger than the characteristic flow time, D/ubulk. We also discuss several implications of these results on computations of LBO - in particular, LBO's intrinsically three-dimensionality and the need for many flow through times to capture it.

Resurrecting extinct cephalopods with biomimetic robots to explore hydrodynamic stability, maneuverability, and physical constraints on life habits

Externally shelled cephalopods with coiled, planispiral conchs were ecologically successful for hundreds of millions of years. These animals displayed remarkable morphological disparity, reflecting comparable differences in physical properties that would have constrained their life habits and ecological roles. To investigate these constraints, self-propelling, neutrally buoyant, biomimetic robots were 3D-printed for four disparate morphologies. These robots were engineered to assume orientations computed from virtual hydrostatic simulations while producing Nautilus-like thrusts. Compressed morphotypes had improved hydrodynamic stability (coasting efficiency) and experienced lower drag while jetting backwards. However, inflated morphotypes had improved maneuverability while rotating about the vertical axis. These differences highlight an inescapable physical tradeoff between hydrodynamic stability and yaw maneuverability, illuminating different functional advantages and life-habit constraints across the cephalopod morphospace. This tradeoff reveals there is no single optimum conch morphology, and elucidates the success and iterative evolution of disparate morphologies through deep time, including non-streamlined forms.

Production and stability of food-grade liposomes containing microbial carotenoids from Rhodotorula mucilaginosa

We demonstrated the technical feasibility of obtaining carotenoids-loaded liposomes from soybean lecithins using an ethanol injection technique. We evaluated the influence of three lecithin-types with different phosphatidylcholine content and microbial carotenoids produced by yeast Rhodotorula mucilaginosa on liposome properties. The liposomal systems were characterized by average hydrodynamic diameter, polydispersity index, ζ-potential, and stability under different pH and heat temperature conditions commonly observed in food processing. The lecithin type and concentration used to produce liposomes significantly influence the hydrodynamic diameter, polydispersity index, and ζ-potential. In general, we observed the presence of liposomes with diameters ranging from 150 to 221 nm and a polydispersity index of approximately 0.300 when Lipoid S45 was used. The incorporation of microbial carotenoids promoted a significant increase in hydrodynamic diameter and ζ-potential, while carotenoid-loaded liposomes showed a lower polydispersity index. The carotenoid-loaded liposomes submitted to pH 3 presented a visual phase separation, while liposomes submitted at 70 °C for 15 min showed degradation of carotenoids equal to control assay (without treatment). We concluded that it is possible to incorporate microbial carotenoids in food-grade liposomal systems, allowing for further studies aimed at their application as an encapsulating and/or delivery system to be used in aqueous food products.

Well-Mixed Boundary Layer–Top Entrainment Instability: Hydrodynamic Analysis

Abstract The present study shows by a linear hydrodynamic stability analysis that an unstable mixed-layer deep circulation can be generated in the dry convective well-mixed layer by the entrainment from the top. The newly identified instability arises under the two competing processes induced by the top entrainment: the destabilization by generating thermal perturbations and the damping by mechanical mixing. The former and the latter, respectively, dominate over the other in the limits of large and small scales. As a result, the instability is realized at the horizontal scales larger than the order of the mixed-layer depth (ca. 1 km), and the time scale for the growth is about 1 day. This study has been motivated from a question of whether the cloud-top entrainment instability (CTEI) can induce a transition of the stratocumulus-topped well-mixed boundary layer into trade cumulus. The present study intends to extend the previous studies based on the local parcel analyses to a full analysis based on the hydrodynamics. Unfortunately, being based on a dry formulation, the present result does not apply directly to the CTEI problem. Especially, the evaporative cooling is totally neglected. Nevertheless, the present result can still be applied to moist systems, to some extent, by redefining certain terms in the formulation.

Baroclinic instability of spatially-periodic flows in a discrete surface quasi-geostrophic model with two levels

Within the framework of a discrete surface quasi-geostrophic (SQG) model with two levels, the hydrodynamic stability of a spatially periodic flow with sinusoidal velocity profile (Kolmogorov flow) is investigated. Such spatially quasi-periodic flows are often observed in the atmospheres of the giant planets and in the Southern Ocean. To describe the dynamics of perturbations, the Galerkin method with three basis trigonometric functions was used. It provides a tool to study both the linear and nonlinear perturbation dynamics. Analytical expressions are obtained for the growth rates of perturbations in the linear theory of stability, and it is shown that the fastest growing perturbations have a wavelength on the order of the spatial period of the main flow. Using the Galerkin method, it is also shown that the exponential growth of perturbations at the linear stage of development is replaced by the stage of stable nonlinear oscillations. Such oscillations are analogous to the nonlinear fluctuations, or vacillations, found in R. Hide's laboratory experiments. In terms of elliptic integrals, an analytical expression for the period of nonlinear oscillations is obtained, and it is shown that for typical values of the parameters relevant to the Earth’s atmosphere, the period of oscillations is about one month. The paper also proposes a long-wavelength version of the discrete SQG model, which describes the dynamics of perturbations with a horizontal scale much larger than the baroclinic Rossby radius of deformation. It is shown that the use of this version simplifies the study of stability and leads to asymptotically correct results.

Abstract 294: Exploiting mutant PPM1D-induced metabolic defects with nanoparticle-encapsulated NAMPT inhibitors

Abstract Diffuse Intrinsic Pontine Glioma (DIPG) is a leading cause of death in pediatric cancer, with an abysmal <1% 5-year survival rate due to lack of effective treatment options. A significant effort is needed to understand the genetic landscape of this disease to develop novel therapeutic strategies and modalities. Recently, the Bindra laboratory found that a truncation mutation in the gene PPM1D, found in ~30% of DIPG cases, causes global epigenetic alterations that lead to therapeutic vulnerabilities. They found that mutant PPM1D activity results in loss of the NAD+ biogenesis protein NAPRT that can be therapeutically targeted by inhibitors of another protein in the NAD+ biogenesis pathway NAMPT (NAMPTi), providing a viable therapeutic strategy for these cancers. Some NAMPTis have been FDA-approved for clinical usage, but present with systemic and retinal toxicity. Moreover, while there are few NAMPTis that can pass the blood brain barrier through systemic delivery, studies show diminished concentrations at the target site. Together, current studies show that the use of NAMPTis for precision targeting of CNS cancers such as DIPG with mutant PPM1D status is a promising therapeutic strategy, but impractical given the limitations of these drugs. However, recent developments in drug delivery systems offer a chance to overcome these issues. Tools such as nanoparticle (NP) drug delivery and unique injection set-ups such as convection-enhanced delivery (CED) allow for drugs such as NAMPTis to be reconsidered for clinical usage. To circumvent the challenges presented by these drugs, we have developed and characterized nanoparticles that encapsulate NAMPTis (NAMPTi-NP) and use CED for sustained intratumoral delivery. Thus far, we have fabricated and optimized PLA-PEG copolymeric nanoparticles capable of encapsulating the NAMPTi, GMX-1778. We characterized these nanoparticles based on (1) hydrodynamic diameter, (2) zeta potential, and (3) stability within an artificial cerebrospinal fluid in vitro solution. We have performed both in vitro and in vivo assays to determine the functionality of the NAMPTi-NPs through (1) cellular uptake studies via immunofluorescence and flow cytometry, (2) functional analysis via NAD+ quantification, (3) short- and long-term cell viability assays to determine sensitivity to the NAMPTi-NP, and (4) in vivo biodistribution studies to assess sustained retention of NAMPTi-NP with CED intracranial injections. We find that NAMPTi-NPs have immediate and sustained cellular uptake, loss of NAD+ post-treatment indicating effective targeting, and enhanced sensitivity in long-term viability studies in mutant PPM1D models. Lastly, these NAMPTi-NPs display prolonged retention in brain tissue compared to free drug injection over time. With further in vivo validation, this NP-based strategy will be a powerful tool for targeting mutant PPM1D DIPG and other cancers. Citation Format: Matthew Murray, Yazhe Wang, Ranjini K. Sundaram, Jason Beckta, W. Mark Saltzman, Ranjit S. Bindra. Exploiting mutant PPM1D-induced metabolic defects with nanoparticle-encapsulated NAMPT inhibitors [abstract]. In: Proceedings of the American Association for Cancer Research Annual Meeting 2022; 2022 Apr 8-13. Philadelphia (PA): AACR; Cancer Res 2022;82(12_Suppl):Abstract nr 294.

Investigation of collector flow for 1:30 scale solar chimney power plant model

Investigations of the flow field in scaled solar chimney power plant model were carried out using a two pronged approach: Experiments were carried out for a 1:30 scale model of the Manzanares solar chimney power and accompanying numerical simulations. Emphasis was on investigating the hydrodynamic stability of developing inward radial Rayleigh-Bénard-Poiseuille flow under the collector of the 1:30 scale model. Temperature measurements under the collector of the model provided the reference data for comparison with results from a direct numerical simulation. The simulation results reveal that both steady longitudinal (streamwise) modes and unsteady oblique disturbance waves are amplified. Steady longitudinal modes with an azimuthal wavelength of approximately 1.5 times half-height of the collector exhibit the strongest spatial growth. Towards the collector outflow (or inflow to the chimney), the steady modes appear to merge and the growth rates are attenuated as a result of the strong streamwise flow acceleration. A scaling analysis suggests that the present results are relevant for the design and operation of realistic large-scale solar chimney power plants. Based on the present investigation, buoyancy-driven instability and coherent longitudinal flow structures are to be expected for full-size solar chimney power plants. The coherent flow structures will affect the heat transfer and total pressure losses in the collector.

Critical-layer instability of shallow-water magnetohydrodynamic shear flows

In this paper, the instability of shallow-water shear flow with a sheared parallel magnetic field is studied. Waves propagating in such magnetic shear flows encounter critical levels where the phase velocity relative to the basic flow,$c-U(y)$, matches the Alfvén wave velocities$\pm B(y)/\sqrt {\mu \rho }$, based on the local magnetic field$B(y)$, the magnetic permeability$\mu$, and the mass density of the fluid$\rho$. It is shown that when the two critical levels are close to each other, the critical layer can generate an instability. The instability problem is solved, combining asymptotic solutions at large wavenumbers and numerical solutions, and the mechanism of instability explained using the conservation of momentum. For the shallow-water magnetohydrodynamic system, the paper gives the general form of the local differential equation governing such coalescing critical layers for any generic field and flow profiles, and determines precisely how the magnetic field modifies the purely hydrodynamic stability criterion based on the potential vorticity gradient in the critical layer. The curvature of the magnetic field profile, or equivalently the electric current gradient$J' = - B''/\mu$in the critical layer, is found to play a complementary role in the instability.

Experimental study of the effects of Vee-shaped steps on the hydrodynamic performance of planing hulls

Speed and stability are two essential criteria for assessment of planing hull performance. Accordingly, the effects of step shape on these vessels’ hydrodynamic performance and stability are experimentally investigated. Five different models are studied which are based on a high-speed recreational craft and are primarily the same in form and dimensions. One of these models does not have a step, two of them have transverse steps, and two other models have Vee-shaped steps. These five models all have a length of 2.64 m, a beam of 0.55 m, a deadrise angle of 24°, and weigh 86 kg. As the parent model, that is, the non-stepped model is unstable at high speeds, this study is aimed to figure out how different types of steps could prevent this kind of instability. The trim angle, resistance, and lengths of the wetted keel and wetted chine of all these models are reported at beam Froude numbers ranging from 0.43 to 3.44. First, the non-step model is investigated in towing tank tests. The optimum position for the center of gravity of this model is determined to be at a distance of 30% of the model’s length to the transom, which leads to the lowest resistance while the model still remains stable. The models with the aft-wise Vee type and fore-wise Vee type experience the lowest and highest resistance among all studied models, respectively. The steps’ performance appears to be directly related to the ventilation condition at the bottom of stepped models. It is also observed that aft-wise Vee-shaped step leads to the most severe flow separation, and the model with this type of step has the least wetted surface area compared to other studied models.

Stability of yield stress fluid flows, Part 1

<h2>Abstract</h2> Yield stress fluids are idealized simple models (e.g., Bingham, Casson, Herschel-Bulkley), describing fluids for which there is no deformation wherever the deviatoric stress lies below the yield stress. This simple rheological change results in hydrodynamic stability features that are quite different from those of purely viscous fluids. We can classify the flows into those that have a non-zero base flow <b>U</b>, and those for which <b>U</b> = 0. Unlike purely viscous fluids, flows with <b>U</b> = 0 have meaningful applications beyond hydrostatics. It is these flows that we explore here, focusing on general features of stability, control and determining the critical limits.

Cubic spline solutions of the ninth order linear and non-linear boundary value problems

A lot of numerical formulations of physical phenomena contain 9th-order BVPs. The presented probe intends to consider the spline solutions of the 9th-order boundary value problems using Cubic B Spline(CBS). Ninth order boundary value problems arise in the study of laminar viscous flow in a semi-porous channel, astrophysics, hydrodynamic & hydro-magnetic stability. The derived technique is exceptionally useful and is appropriate for such kinds of linear and non-linear boundary value problems. Cubic B Spline(CBS) is successfully applied to mathematical models. The outcomes are contrasted to those presented in the literature, revealing that the introduced technique leads to an advisable estimation of the exact solution.