Dimensionless Problem Research Articles

A free boundary problem of elasticity in an angle and a vector Riemann–Hilbert problem on a torus

An inverse problem of elasticity of reconstructing the shape of an inclusion placed in a wedge, when the whole inclusion is subjected to a prescribed antiplane uniform stress, is considered. The doubly connected physical domain is conformally mapped onto a parametric plane cut along two finite segments on the real axis. Determination of such a map requires solving a vector Riemann–Hilbert problem of the theory of holomorphic functions on a symmetric genus-1 Riemann surface. The main steps of the method include factorization of a discontinuous function on a torus, solution of the associated Jacobi inversion problem, representation of the unknown vector-functions in terms of Weierstrass integrals, and reduction of the vector Riemann–Hilbert problem to a single integral kernel of the second kind with a discontinuous kernel on the elliptic surface. An integral representation of the conformal map in terms of the solution to the integral equation is given. In addition to four dimensionless problem parameters, the conformal map possesses five free parameters. Numerical results, which reconstruct the inclusion shape for sample sets of the parameters, are reported and discussed.

Verification of Dual Solutions for Water and Kerosene-Based Carbon Nanotubes over a Moving Slender Needle

This article focuses on the boundary layer for an axisymmetric flow and heat transfer of a nanofluid past a moving slender needle with single-walled carbon nanotubes (SWCNTs) and multi-walled carbon nanotubes (MWCNTs). In this study, the streamlines of the flow are symmetrically located along the needle’s surface. Water and kerosene are two types of base fluids that are considered in this study. This analysis is presented with needle thickness, the ratio of velocity, nanoparticle volume fraction, and Prandtl number. The partial differential equations (PDEs) are transformed into dimensionless ordinary differential equations (ODEs) by adopting relevant similarity transformations. The bvp4c package is implemented in MATLAB R2018a to solve the governing dimensionless problems numerically. The behaviors of various sundry variables on the flow and heat transfer are observed and elaborated further. The magnitude of the skin friction, heat transfer rate, as well as velocity and temperature distributions are demonstrated in graphical form and discussed. It is worth mentioning that kerosene-based CNTs have the largest skin friction coefficient and heat transfer rate compared to water-based CNTs. The thin wall of the needle and the single-walled carbon nanotubes also contributes to high drag force and heat transfer rate on the surface. It is revealed from the stability analysis that the first solution exhibits a stable flow. Obtained results are also matched with the present data in the restricting situation, and excellent agreement is noticed.

Computational study of magneto-convective non-Newtonian nanofluid slip flow over a stretching/shrinking sheet embedded in a porous medium

A Steady flow of two-dimensional magnetohydrodynamic non-Newtonian fluid over a stretching/shrinking sheet in the presence of nanoparticles is exemplified theoretically and numerically. In this problem, we have considered the thermal radiation and adjust the hot fluid along with the lower surface of the wall namely convective boundary-layer slip. To the best of the authors' knowledge, this parameter was here incorporated for the first time in such field of magnetofluid dynamic characteristics of conducting bio-nanofluids embedded in a porous medium. The solution of governing dimensionless problem is executed by Legendre-based collocation method (LBCM). It is vital to remark that the account for the velocity slip in the boundary conditions increases the velocity component. Also, the liquid acts as a Newtonian fluid when the Casson parameter increases. Consequently, those parameters contribute to the cooling plate, while others have the opposite effect. Thus, by selecting the appropriate fluid model and adjusting the governing parameters, the cooling/heating mechanism can be created. These results will assist the engineers in designing applications that require high-temperature nanomaterials processing operations.

A wavelet‐based novel approximation to investigate the sensitivities of various path‐independent binary options

We present a novel idea of a wavelet‐based approximation technique using a multi‐resolution analysis to investigate the sensitivities of various path‐independent binary options under the Black–Scholes environment. The final value problem is transformed into a dimensionless initial value problem; also, to avoid the large truncation error, the infinite domain is truncated into a finite domain. A noteworthy observation is that the proposed Haar wavelet scheme is effective and easy to implement to analyze the different physical and numerical aspects of the options' Greeks. It explicitly provides the numerical approximation of all the derivatives of the solution function. Also, the non‐smooth payoff functions are approximated well with the Haar wavelet approximation technique of estimating the spiked functions, so there is no need to deal with the discontinuity separately. We prove the consistency and stability of the proposed method and show that the proposed method is the first‐ and second‐order accurate in the temporal and spatial directions, respectively. A variety of test examples conclusively demonstrates the computational proficiency and the theoretical results of the proposed scheme. Different attributes of the Greeks of distinct binary options are analyzed graphically. The motivational work of the study of various Greeks of different binary options significantly impacts the hedging strategies used by different financial institutes.

Numerical study for entropy optimized radiative unsteady flow of Prandtl liquid

Entropy analysis for magnetohydrodynamic (MHD) unsteady flow of Prandtl fluid by a stretchable surface is addressed. Thermal transport with radiation, magnetic field and dissipation is taken. Physical behaviors of Soret and Dufour impacts are examined. Features of entropy have been deliberated. Mass transfer with first order chemical reaction is addressed. Dimensionless problems are simulated by finite difference method. Graphical illustrations show outcomes of fluid motion, thermal and mass transport and entropy. An intensification in Reynold number improves the velocity. Higher magnetic variable lead to enhance thermal field and entropy rate. Temperature enhancement is observed through radiation while reverse trend holds for Prandtl number. Higher Soret number rise the concentration while opposite holds for Schmidt number. Higher estimation of radiation lead to improve both entropy rate and thermal field.

Analysis of hydromagnetically modulated multiple slips motion of hybrid-nanofluid through a converging/diverging moving channel

The utility of convergent/divergent channel driven flow to improve the effectiveness of heat transport rate in industrial and engineering systems is diverse. This motivates us to disclose hybrid nanofluid flow features through non-parallel walls under hydro-magnetic aspect. The modified Darcian (Darcy–Forchheimer) expression is utilized for formulation. Reflection of improved Darcian form modifies the expression of velocity via square of velocity term. The effects of temperature jump and viscous dissipation are implemented in energy expression. Additionally, the slip flow phenomenon under the stretching/shrinking characteristics is studied. The analysis is carried out under the theory of boundary layer. Significant variables are implemented to acquire the dimensionless mathematical expressions. Dimensionless problem is tackled through a well-known homotopy technique. To observe the upshots of numerous pertinent parameters upon non-dimensional profiles of velocity and temperature, the graphs are plotted for both convergent/divergent channels. The heat transfer rate as well as drag force is also analyzed. In this study, it is concluded that temperature field rises in both divergent/convergent channels for dominant thermal slip parameter. Moreover, inertia parameter effects are seen weaker in converging channel for the velocity profile, while opposite trend is observed for diverging channel.

Stratified heat transfer of magneto-tangent hyperbolic bio-nanofluid flow with gyrotactic microorganisms: Keller-Box solution technique

Abstract The purpose of the present investigation is to examine the heat, mass and microorganism concentration transfer rates in the magnetohydrodynamics (MHD) stratified boundary layer flow of tangent hyperbolic nanofluid past a linearly, uniform stretching surface comprising gyrotactic microorganisms as well as nanoparticles. The governing PDEs with relevant end point conditions are molded into a non-dimensional ordinary differential equation (ODE) form by means of the similarity transformation. The numerical solution of dimensionless problem is acquired within the frame of robust Keller-Box technique. The velocity, temperature, mass and motile microorganism density are investigated graphically within the context of different significant parameters. Numerical results have been inspected via plots and table (namely as the local Nusselt number, the local wall mass flux and the local microorganisms wall flux). This article proves that the energy, concentration and motile microorganism density reduce with increase in thermal, solutal and motile density stratification parameters. The asserted outcomes are beneficial to enhance the cooling and heating processes, energy generation, thermal machines, solar energy systems, industrial processes etc.

Entropy generation analysis for peristalsis of magneto Jeffrey materials

This article communicates peristalsis of Jeffrey material in curved geometry. Here, material has temperature-dependent thermal conductivity and viscosity. Mathematical modeling of an inclined magnetic field in curved configuration has been presented in this article. Irreversibility effects have been analyzed through entropy generation. Slip conditions are entertained both for velocity and thermal fields. Problem is first reduced in wave frame and then lubrication approach has been utilized. Numerical solution of dimensionless problem is obtained and important parameters of curiosity are examined. It is noticed that velocity enhances for higher viscosity whereas temperature decreases for higher thermal conductivity coefficient. Velocity of the flow is maximum for inclination of magnetic field to be zero and it is minimum for [Formula: see text] Heat transfer parameter enhances both for thermal conductivity parameter and Hartmann number. Temperature is high for curved configuration when compared with straight channel. It is observed that entropy remains unchanged in center of the channel and it is maximum near the channel walls. Entropy generation decays near the channel walls by higher viscosity and thermal conductivity parameters. However, entropy is more for higher inclination of magnetic field.

Bioconvection flow in accelerated couple stress nanoparticles with activation energy: bio-fuel applications

On the account of significance of bioconvection in biotechnology and several biological systems, valuable contributions have been performed by scientists in current decade. In current framework, a theoretical bioconvection model is constituted to examine the analyzed the thermally developed magnetized couple stress nanoparticles flow by involving narrative flow characteristics namely activation energy, chemical reaction and radiation features. The accelerated flow is organized on the periodically porous stretched configuration. The heat performances are evaluated via famous Buongiorno’s model which successfully reflects the important features of thermophoretic and Brownian motion. The composed fluid model is based on the governing equations of momentum, energy, nanoparticles concentration and motile microorganisms. The dimensionless problem has been solved analytically via homotopic procedure where the convergence of results is carefully examined. The interesting graphical description for the distribution of velocity, heat transfer of nanoparticles, concentration pattern and gyrotactic microorganism significance are presented with relevant physical significance. The variation in wall shear stress is also graphically underlined which shows an interesting periodic oscillation near the flow domain. The numerical interpretation for examining the heat mass and motile density transfer rate is presented in tubular form.

Darcy-Forchheimer flow with nonlinear mixed convection

An analysis of the mixed convective flow of viscous fluids induced by a nonlinear inclined stretching surface is addressed. Heat and mass transfer phenomena are analyzed with additional effects of heat generation/absorption and activation energy, respectively. The nonlinear Darcy-Forchheimer relation is deliberated. The dimensionless problem is obtained through appropriate transformations. Convergent series solutions are obtained by utilizing an optimal homotopic analysis method (OHAM). Graphs depicting the consequence of influential variables on physical quantities are presented. Enhancement in the velocity is observed through the local mixed convection parameter while an opposite trend of the concentration field is noted for the chemical reaction rate parameter.

Haar‐wavelet based approximation for pricing American options under linear complementarity formulations

AbstractIn this manuscript, we present a novel and highly accurate wavelet‐based approximation technique to explore the sensitivities and value of American options diagnosed by linear complementarity problems. For a detailed analysis of such financially relevant problems, we transform the actual final value problem into a dimensionless initial value problem. To avoid the unacceptable large truncation error, the unbounded domain is trimmed into a bounded domain. A remarkable observation is that to investigate the various physical and numerical aspects of the options' sensitivities; the proposed scheme is efficient as it explicitly provides the numerical approximation of all the derivatives of the solution function. The multiresolution technique of the wavelets and the convergence of the proposed wavelet scheme are comprehensively analyzed. The wavelet analysis is accompanied by illustrative examples to demonstrate the proficiency and robustness of the present method coupled with graphical representations. It has been shown that the present method is efficient to solve free boundary problems. It is worthy to note that the highly accurate and promising computational results are enough to confirm the performance of the proposed method. The simulated results of options' Greeks analyzed and discussed have vast applications in different financial institutes and trading markets.

Heat transfer analysis of an inclined porous fin using Differential Transform Method

ABSTRACT The thermal attribute of an inclined porous fin of a longitudinal profile has been determined in the presence of convection and radiation effect. The Darcy’s model is utilised here. The governing equation, nonlinear second-order ordinary differential equation, has been derived. An effective analytical method, Differential Transform Method, has been implemented to find the solution of the dimensionless problem. The obtained outcome has been compared with that of the Adomian Decomposition Method as well as Runge–Kutta–Fehlberg method. The efficiency of the fin for different values of dimensionless parameters has been investigated. The influence of significant parameters has been illustrated with graphical interpretation. The thermo-geometric parameter and radiative parameter support in heat transfer rate from fin surface.

Heat transfer and Helmholtz-Smoluchowski velocity in Bingham fluid flow

A mathematical study is developed for the electro-osmotic flow of a non-Newtonian fluid in a wavy microchannel in which a Bingham viscoplastic fluid model is considered. For electric potential distributions, a Poisson-Boltzmann equation is employed in the presence of an electrical double layer (EDL). The analytical solutions of dimensionless boundary value problems are obtained with the Debye-Huckel theory, the lubrication theory, and the long wavelength approximations. The effects of the Debyelength parameter, the plug flow width, the Helmholtz-Smoluchowski velocity, and the Joule heating on the normalized temperature, the velocity, the pressure gradient, the volumetric flow rate, and the Nusselt number for heat transfer are evaluated in detail using graphs. The analysis provides important findings regarding heat transfer in electroosmotic flows through a wavy microchannel.

Microvascular blood flow with heat transfer in a wavy channel having electroosmotic effects.

The present paper addresses microvascular blood flow with heat and mass transfer in complex wavy microchannel modulated by electroosmosis. Investigation is carried out with joule heating and chemical reaction effects. Further, viscous dissipation is also considered. Using Debye-Huckel, lubrication theory, and long wavelength approximations, analytical solutions of dimensionless boundary value problems are obtained. The impacts of different parameters are examined for temperature and concentration profile. Furthermore, nature of pressure rise is also investigated to analyze the pumping characteristics. Important results of flow phenomena are explored by means of graphs.

MHD Natural Convection of a Fe3O4–Water Nanofluid within an Inside Round Diagonal Corner Square Cavity with Existence of Magnetic Source

This study concerns a numerical investigation of a magnetohydrodynamic (MHD) natural convection of a Fe3O4–water nanofluid filled within a round diagonal corner square cavity. The cavity was subjected to imposed temperatures (hot and cold walls) and one magnetic source. The nanofluid flow and heat transfer problem was mathematically modeled and its dimensionless problem was established. The finite element method was implemented in order to solve the MHD problem. The effects of the Rayleigh number, Hartmann number and round corner radius on the nanofluid flow (streamlines and velocity magnitude) and heat transfer (isotherms and temperature distribution) were evaluated. Heat transfer was assessed when the convection or the conduction dominates with regard to the nature of the flow.

A proposed unsteady bioconvection model for transient thin film flow of rate-type nanoparticles configured by rotating disk

The revolution in nanotechnology has made new developments toward enhancement of energy resources which play a substantial role in the growth of industries and technologies. The aim of this novel investigation is to develop a mathematical model for unsteady volatile finite flow of the upper film transmitted by rate-type nanofluid due to the horizontal spinning disk in the presence of gyrotactic motile microorganisms. The most preferable subclass of rate-type fluid, namely Maxwell fluid, is utilized to predict the exclusive rheological parameters. The additional unique features like activation energy, thermal radiation and Joule heating are also included for the current flow situation. The indicated partial differential system is transformed into nonlinear ordinary differential equations using appropriate self-similar transformation. The governing equations are translated to ordinary differential equations, and later on, shooting procedure is followed to construct the numerical arrangement of the transmuted dimensionless flow problem. The value of physical limitations is shown in the plots and table tests. In contrast, wider Brownian motion and thermophoresis parameters have different intensity field features. The graphical and numerical results are drawn against different prominent parameters over velocity profile, temperature distribution, volumetric fraction, nanoparticle, density of motile microorganisms. The results revealed that the increase in buoyancy ratio parameter and Rayleigh number reduced the axial velocity, radial velocity and azimuthal distribution. A progressive temperature distribution is noted with increasing Rayleigh number and buoyancy ratio constant. It is further observed that change in unsteady parameter enhanced the concentration and motile microorganism distribution effectively.

An improved Darcian analysis for chemically reacted Maxwell liquid toward convectively heated moving surface with magnetohydrodynamics

Heat transportation analysis subjected to porous medium consideration has increased because of industrial and engineering utilization. Typical examples comprise crude oil production, porous insulation, groundwater pollution, grain storage, mastic transport modeling, nuclear waste discarding and numerous other. The above-mentioned effectiveness of porous medium motivates us to model rate-type Maxwell material under hydromagnetic aspects. We utilized improved Darcian (Darcy–Forchheimer) expression for formulation. Consideration of improved Darcian expression modifies velocity expression through square velocity term. In addition, we accounted Robin’s condition and chemical reaction characteristics. Boundary-layer theory is employed for simplification of mathematical expressions. Dimensionless problems are treated analytically through homotopy scheme. Their significant attributes are further explored in detail. We noticed an increment in solutal and thermal fields when Biot numbers (thermal and solutal) are enhanced.

РЕАЛЬНИЙ ВАРІАНТ ОДНОВИМІРНОЇ ЗАДАЧІ ТЕПЛОПРОВІДНОСТІ ДЛЯ СУЦІЛЬНОГО НЕОБМЕЖЕНОГО СТАЛЕВОГО ЦИЛІНДРА

Сучасний етап розвитку методів проведення наукових досліджень характеризується широким використанням засобів обчислювальної техніки та чисельного моделювання. Ці потужні обчислювальні засоби суттєво збільшують можливості нової методології наукових досліджень, а саме, встановлюють більш тісну взаємодію експериментальних та теоретичних досліджень. В умовах промислового виробництва особливо велике значення має скорочення матеріальних та часових ресурсів при розробці нових технологій та освоєнні нового сортаменту продукції та нових сталей. Для термічної обробки металовиробів ці питання можна вирішити за допомогою сучасних методів моделювання та розрахунків, які при мінімальних витратах часу та матеріальних засобів дозволяють досліджувати різні технологічні процеси, проводити їх розробку та оптимізацію для різних матеріалів з отриманням конкретних рішень.

Onset of Darcy–Bénard convection under throughflow of a shear-thinning fluid

We present an investigation on the onset of Darcy–Benard instability in a two-dimensional porous medium saturated with a non-Newtonian fluid and heated from below in the presence of a uniform horizontal pressure gradient. The fluid is taken to be of power-law nature with constant rheological index and temperature-dependent consistency index . A two-dimensional linear stability analysis in the vertical plane yields the critical wavenumber and the generalised critical Rayleigh number as functions of dimensionless problem parameters, with a non-monotonic dependence from and with maxima/minima at given values of , a parameter representing the effects of consistency index variations due to temperature. A series of experiments are conducted in a Hele-Shaw cell of aspect ratio to provide a verification of the theory. Xanthan Gum mixtures (nominal concentration from 0.10 % to 0.20 %) are employed as working fluids with a parameter range and . The experimental critical wavenumber corresponding to incipient instability of the convective cells is derived via image analysis for different values of the imposed horizontal velocity. Theoretical results for critical wavenumber favourably compare with experiments, systematically underestimating their experimental counterparts by 10 % at most. The discrepancy between experiments and theory is more relevant for the critical Rayleigh number, with theory overestimating the experiments by a maximum factor less than two. Discrepancies are attributable to a combination of factors: nonlinear phenomena, possible subcritical bifurcations, and unaccounted-for disturbing effects such as approximations in the rheological model, wall slip, ageing and degradation of the fluid properties.

A Theoretical Analysis for Mixed Convection Flow of Maxwell Fluid between Two Infinite Isothermal Stretching Disks with Heat Source/Sink

The aim of this current contribution is to examine the rheological significance of Maxwell fluid configured between two isothermal stretching disks. The energy equation is also extended by evaluating the heat source and sink features. The governing partial differential equations (PDEs) are converted into the ordinary differential equations (ODEs) by using appropriate variables. An analytically-based technique is adopted to compute the series solution of the dimensionless flow problem. The convergence of this series solution is carefully ensured. The physical interpretation of important physical parameters like the Hartmann number, Prandtl number, Archimedes number, Eckert number, heat source/sink parameter and the activation energy parameter are presented for velocity, pressure and temperature profiles. The numerical values of different involved parameters for skin friction coefficient and local Nusselt number are expressed in tabular and graphical forms. Moreover, the significance of an important parameter, namely Frank-Kamenetskii, is presented both in tabular and graphical form. This particular study reveals that both axial and radial velocity components decrease by increasing the Frank–Kamenetskii number and stretching the ratio parameter. The pressure distribution is enhanced with an increasing Frank–Kamenetskii number and stretching ratio parameter. It is also observed that thetemperature distribution increases with the increasing Hartmann number, Eckert number and Archimedes number.