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This study investigates the stability analysis of Rayleigh-Bénard configuration for a viscoelastic fluid subject to thermorheological effects, using the D 2 -Chebyshev- τ method. The fluid is modeled as a third-order viscoelastic fluid. This study accentuates how salting the fluid layer affects the thresholds for the onset of instability in a fluid of third order encompassing physically realistic rigid boundaries. The dynamic model incorporates advection-diffusion of temperature and solute concentration and a modified Navier–Stokes equation. We determine instability thresholds for the complex non-Newtonian fluid by analyzing the linear stability of the steady-state conduction solution. Our analysis proves the strong form of the principle of exchange of stabilities, demonstrating that convective motions can only occur through stationary motion. Additionally, a nonlinear stability analysis using the energy method is performed, deriving an unconditional nonlinear stability criterion. The results provide a comprehensive understanding of how variable viscosity and viscoelasticity impact system stability. Both the viscosity parameter and the third-grade fluid parameter exhibit stabilizing effects. Notably, we observe a discrepancy between the linear and global nonlinear stability results, indicating the presence of a subcritical instability region. This study contributes to the understanding of complex fluid dynamics in non-linear mechanical systems, with potential applications in various industrial and natural processes. • A system of differential equations models the double-diffusive convection in viscoelastic fluid. • The effect of temperature-dependent viscosity on the onset of convection is examined. • The strong form of principle of exchange of stabilities is proved. • Linear analysis determines thresholds above which steady solutions become unstable. • Nonlinear analysis proves the total perturbed energy of the system decays asymptotically. [ABSTRACT FROM AUTHOR]

The stability of Darcy–Brinkman–Bénard model for Kelvin–Voigt fluids is investigated using a local thermal non-equilibrium (LTNE) configuration. The study considers physically realistic rigid surfaces, with the lower surface heated from below. The effects of spatially varying gravitational force and constant magnitude Lorentz force on the onset of convection are investigated. A systematic analysis is conducted on two porous materials: Sander sandstone and aluminum metallic foam (A L 1050). Our mathematical model comprises the Kelvin–Voigt–Navier–Stokes equation, coupled with a two-equation, thermal non-equilibrium model of energy. The Fourier modes technique is used for linear instability analysis, while the energy method is applied for nonlinear stability analysis. The differential eigenvalue problems arising from linear and nonlinear theories are solved using the Spectral Collocation-QZ method. We determine the critical conditions for the onset of convection, examining the impact of the Darcy number, Darcy–Hartmann number, and inter-phase heat transport parameter. The results reveal a notable disparity between linear and nonlinear stability thresholds, with the latter being consistently lower. This discrepancy indicates the existence of a subcritical instability region. Our findings demonstrate that increased magnitudes of Lorentz and gravitational forces postpone the initiation of convective motion, thereby stabilizing the fluid system. This stabilization is reflected in higher Darcy–Rayleigh numbers, which potentially contribute to enhanced heat transfer rates. Similarly, an increase in the Darcy number leads to elevated Darcy–Rayleigh numbers, potentially improving thermal stability and heat transfer efficiency. Furthermore, we found that the subcritical instability regime expands significantly when both porosity and gravity field intensity reaches sufficiently high levels. • Mathematical model is developed for LTNE convection in Kelvin–Voigt fluid. • Global nonlinear stability analysis of conduction solution has been performed. • Conditions for the onset of convection are provided. • Existence of subcritical instability convection is demonstrated. • Differential eigenvalue problems are solved using the QZ-algorithm. [ABSTRACT FROM AUTHOR]

The linear and nonlinear stability analyses of thermosolutal convection in a non-Newtonian Navier-Stokes-Voigt fluid, considering Soret and Ekman damping effects, are conducted analytically. Instability thresholds are determined for thermosolutal convection within a viscoelastic fluid of the Kelvin-Voigt type, wherein a dissolved salt field exists. Two scenarios are examined: one where the fluid layer is heated from the bottom and concurrently salted from the bottom, and the other where the fluid layer is heated from the bottom and concurrently salted from the top. The governing partial differential equations system includes conservation laws of mass, momentum, energy, and salt concentration. Using the energy method, the disturbances to the fluid system are shown to decay exponentially. Analytical expressions are developed for the eigenvalue as a function of Soret, Lewis, Prandtl, Kelvin-Voigt, and Rayleigh friction numbers. The study illustrates the shift from a stationary mode of convection to an oscillatory mode and provides thresholds that indicate these transitions. It is found that the viscoelastic property of the fluid acts as a stabilizing agent for oscillatory mode convection. Rayleigh friction substantially controls the convection threshold. Upon comparing threshold values between linear and nonlinear theories, a subcritical instability region is observed in the heating bottom-salting bottom case (case-1), whereas such a region is absent in the heating bottom-salting top case (case-2). • Influence of Ekman damping on the convection in Kelvin-Voight fluid is highlighted. • Global nonlinear stability analysis of conduction solution has been performed. • Analytical expressions for the eigenvalue are obtained. • Conditions for the onset of convection are provided. • Transition from stationary convection to oscillatory convection is demonstrated. [ABSTRACT FROM AUTHOR]

Abstract: Computations of the hydrodynamic coefficients, displacement-amplitude ratios and loadings on floating vertical circular cylinder due to diffraction and radiation are presented here. The boundary value problem (BVP) is solved in terms of diffraction potential and three potentials due to radiation, two translational motions about x-axis (surge) and about z-axis (heave), one rotational motion about y-axis (pitch). The analytical expressions for the hydrodynamic coefficients, displacement-amplitude ratios and loadings for this case were obtained previously by Bhatta and Rahman . In this paper, we present the computational aspects of those analytical results for different depth to radius and draft to radius ratios. JMSL (Java Mathematical and Statistical Library) is used to compute special functions and solve complex matrix equations. [Copyright &y& Elsevier]

Abstract: The problem of weakly nonlinear convective flow in a mushy layer, with a permeable mush–liquid interface and constant permeability, is studied under operating conditions for an experiment. A Landau type nonlinear evolution equation for the amplitude of the secondary solutions, which is based on the Landau theory and formulation for the Rayleigh, , number close to its critical value, , is developed. Using numerical and analytical methods, the solutions to the evolution equation are calculated for both supercritical and subcritical conditions. We found, in particular, that for , the amplitude of the secondary solutions decays with time. For , the tendency for chimney formation in the mushy layer increases with time. In addition, in such a supercritical regime, the basic flow is linearly unstable and we see the presence of steady flow for large values of time. These results suggest a possible slow transition to turbulence in such a flow system. [ABSTRACT FROM AUTHOR]

The classical Boussinesq approximation (linear density-temperature variation) is valid for small temperature differences in the system. However, density varies with temperature nonlinearly in various applications, such as heat exchangers, solar collectors, and nuclear reactors. Therefore, unsteady nonlinear mixed convection (quadratic density-temperature variation) in a nanofluid on a vertical plate due to impulsive motion is investigated. Modified Nield and Kuznetsov nanofluid model is incorporated that considers physical realistic boundary conditions. Unsteady flow is governed by the conservation of mass, momentum, energy, and the continuity equation of nanoparticles. Williams-Rhyne transformations are used to parameterize and reduce the infinite time domain to a finite time domain. Subsequent nonlinear differential equations system is solved by using the Finite Difference Method (FDM) based routine. Rayleigh and Hiemenz type equations are obtained respectively in an initial-unsteady and final-steady state. The Response Surface Method (RSM) and Central Composite Design (CCD) are used for the simultaneous optimization of the friction factor and the rate of heat transport. It is found that the quadratic density-temperature variation significantly affects the flow fields. The friction factor has the highest and positive sensitivity toward the thermal buoyancy number. The heat transport rate has the highest and negative sensitivity toward the thermo-migration of nanoparticles. • Unsteady nonlinear convective flow of nanoliquid over a plate is proposed. • Heat transport rate and friction factor are optimized using the response surface method. • The quadratic density-temperature variation is examined. • Heat transport rate is enhanced due to nanoparticles. [ABSTRACT FROM AUTHOR]

Heat and mass transfer rates play a significant role in achieving a high-efficiency, low-cost wire coating process. Therefore, the flow analysis of molten polymer (polyvinyl chloride) carrying nanoparticles inside a pressure-type die is presented. The third-grade liquid model is used for the constitutive equation of the polyvinyl chloride (PVC), while the non-homogeneous biphasic model is used for nanoparticles. The properties of PVC are temperature-dependent. The melt flow is governed by the modified Navier-Stokes equation for the third-grade fluid, energy conservation, and nanoparticles' continuity equation. The finite difference method-based routine is applied to solve the nonlinear differential equations that include nine physical parameters. The Response Surface Method (RSM) is implemented to optimize the heat/mass transfer rate coated wire's eminence depends on the coating material's rheological characteristics of PVC. Full quadratic correlation models for heat/mass transfer rate of melt are proposed through central-composite-design (CCD). The optimal level of third-grade fluid factor and nanofluid factors was determined to achieve an optimum heat/mass transfer rate of the melt. The rheological characteristics of PVC have been improved due to the temperature-dependent viscosity and shear thickening/thinning property of the melt. The nanofluid factors improve the thermal field and subsequently reduce the Nusselt number. Maximum heat transfer occurs for a low level of Brownian factor, a high level of thermophoresis factor, and a third-grade fluid factor of 0.2177, also reaching the maximum mass transfer simultaneously. • Flow analysis of polyvinyl chloride carrying nanoparticles inside a die is proposed. • Heat and mass transfer rates are optimized using the response surface method. • The temperature-dependent properties of polyvinyl chloride are examined. • Heat transport rate is enhanced in the melt flow in a die due to nanoparticles. [ABSTRACT FROM AUTHOR]

Abstract: Numerical solution of Korteweg-de Veries–Burgers (KdVB) equation is presented using modified Bernstein polynomials (B-polynomials). Over the spatial domain, B-polynomials are used to expand the desired solution requiring discretization with only the time variable. Galerkin method is used to determine the expansion coefficients to construct initial trial functions. We use fourth-order Runge–Kutta method to solve the system of equations for the time variable. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. Numerical results obtained using this method are compared with existing analytical results. Excellent agreement is found between the exact solution and approximate solution obtained by this method. [Copyright &y& Elsevier]

Abstract: Here we present an algorithm for approximating numerical solution of Korteweg–de Vries (KdV) equation in a modified B-polynomial basis. A set of continuous polynomials over the spatial domain is used to expand the desired solution requiring discretization with only the time variable. Galerkin method is used to determine the expansion coefficients to construct initial trial functions. For the time variable, the system of equations is solved using fourth-order Runge–Kutta method. The accuracy of the solutions is dependent on the size of the B-polynomial basis set. We have presented our numerical result with an exact analytical result. Excellent agreement is found between exact and approximate solutions. This procedure has a potential to be used in more complex system of differential equations where no exact solution is available. [Copyright &y& Elsevier]