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The influence of Rayleigh friction and chemical reaction on the onset of double‐diffusive convection in a Navier–Stokes–Voigt (NSV) fluid layer is investigated by conducting linear instability and nonlinear stability analyses. The fluid layer is subjected to isothermal conditions and chemical equilibrium at the boundaries. The solubility of the dissolved component exhibits a linear dependency on temperature. The analysis is conducted for two distinct cases: the fluid layer is heated and salted from the bottom (case‐1), and the fluid layer is heated from the bottom and salted from the top (case‐2). Analytical expressions for the thermal Rayleigh number are obtained for both linear and nonlinear theories, and these expressions depend on Kelvin–Voigt, Rayleigh friction, solutal Rayleigh, Lewis, Prandtl, and Damkohler numbers. Including the Rayleigh friction term in the NSV fluid model improves the stability of the system and hence instability occurs with less ease. For lower solutal Rayleigh numbers, convection commences in the stationary mode and subsequently transitions to the traveling wave mode occurred in case‐1. The Damkohler number plays a significant role in the linear instability thresholds. It is also found that the Kelvin–Voigt number acts as a stabilizing factor for oscillatory mode convection. The comparison between linear and nonlinear thresholds unveils the region characterized by subcritical instability. [ABSTRACT FROM AUTHOR]

A study of the rheological and heat transport characteristics in cone–disk systems finds relevance in many applications such as viscometry, conical diffusers, and medical devices. Therefore, a three‐dimensional axisymmetric flow with heat transport of a magnetized nanofluid in a cone–disk system subjected to Hall current and thermal radiation effects is investigated. The simplified Navier–Stokes (NS) equations for the cone–disk system given by Sdougos et al. [18] Journal of Fluid Mechanics, 138, 379–404 are solved by using the asymptotic expansion method for the four different models, such as rotating cone with static disk (Model I), rotating disk with static cone (Model II), co‐rotating cone and disk (Model III), and counter‐rotating cone and disk (Model IV). The Khanafer–Vafai–Lightstone (KVL) model along with experimental data‐based properties of 37 nm Al2O3–H2O nanofluid is considered. To obtain the transformations leading to self‐similar equations from the Navier–Stokes (NS) and energy conservation equations, the Lie group technique is used. The self‐similar nonlinear problem is solved numerically to examine the effects of physical parameters. There are critical values of the power exponent at which no heat transport from the disk surface occurs. Nanoparticles significantly enhance heat transport when both the cone and disk rotate in the same or opposite directions. The centrifugal force and thermal radiation improve the heat transport in cone–disk systems. [ABSTRACT FROM AUTHOR]

Linear stability and weakly nonlinear stability analyses are developed for Rayleigh–Bénard convection in water near 3.98 °C subject to isothermal boundary conditions. The density–temperature relationship (equation of state) is approximated by a cubic polynomial, including linear, quadratic, and cubic terms. The continuity equation, the Navier–Stokes momentum equation, the equation of state, and the energy equation constitute the governing system. Linear stability analysis is used to investigate how the maximum density property of water affects the onset of convective instability and the choice of unstable wave number for four different types of boundary conditions. Then, a weakly nonlinear stability study is done using the spectral Fourier method for isothermal tangential stress-free boundary conditions to quantify the heat transport of the system and demonstrate the transition from regular/periodic convection to chaotic convection. A Stuart-Ginzburg–Landau equation is obtained using the multiscale expansion method. Streamlines and isotherms are presented and analyzed. The influence of maximum density has been shown to delay the onset of instability and is, therefore, a stabilizing mechanism for thermal instability. Due to the maximum density, the onset of chaotic convection is also delayed. Among four different boundaries, the impermeable rigid boundaries require the highest Rayleigh number for instability to begin. Increasing boundary temperatures advance the onset of chaotic convection and improve the heat transport situation. [ABSTRACT FROM AUTHOR]

A fully developed laminar steady flow of an incompressible, viscous fluid in a horizontal cylindrical pipe is considered here. Flow patterns for an incompressible, viscous fluid for both Newtonian and non-Newtonian fluids such as shear-thinning, shear-thickening and Bingham plastic fluids are analyzed in this study. Assuming that the flow is only due to the wall shear stress and the pressure drop, the velocity component in the axial direction for these cases is derived. Computational results of the velocity profiles for various cases are obtained using MATLAB and presented in graphical forms. It is observed that the velocity profile is parabolic for Newtonian fluid whereas it is flatter for a shear-thinning fluid and sharper for a shear-thickening fluid. For a Bingham fluid, the velocity reaches a constant value known as the plug velocity in the central plug flow region, and it decreases gradually to zero at the pipe wall. [ABSTRACT FROM AUTHOR]

Anomalous diffusion problems are used to describe the evolution of particle's motion in crowded environments with many applications, such as modeling the intracellular transport and disordered media. In the present paper, we develop a Petrov–Galerkin spectral method for the fourth‐order anomalous fractional diffusion equations. For the dimension reduction, we use a discretization in time by the convolution quadrature. We then introduce the basis sets for the trial‐and‐test spaces using modal Bernstein basis functions with a presentation of the method in a weak spectral formulation along with a discussion of the structure of the resulting systems, the convergence, and stability of the proposed method. The theoretical results are supported by illustrating some numerical examples. [ABSTRACT FROM AUTHOR]

We consider mathematical modeling of blood glucose-insulin regulatory system with the additional effect of the secreted insulin by the pancreatic beta cells and in the presence of an external energy input to such system. Such modeling system is investigated to determine the time-dependent nonlinear dynamics that take place by the quantities, which represent the glucose and insulin concentrations in the blood, insulin action as well as in the absence or presence of secreted insulin due to the pancreatic beta cells. Using both analytical and numerical procedures, we determine such quantities versus time for both diabetes patients and normal human and for different values of the parameters. We find that the nonlinear effect of the dynamics of the investigated regulatory system increases the values of the insulin action and the glucose and insulin concentrations. In the absence of the beta cells effects, which can correspond to the case of severe type 1 diabetes, the plasma glucose is higher and the insulin action and the insulin concentration are less active than the corresponding ones for the case in the presence of beta cells, which is relevant for type 2 diabetes or moderate type 1 diabetes patients. For the present system, smaller values of the parameters of the model, which represent kinetics of the glucose and insulin action, insulin sensitivity, insulin secretion enhancement and the plasma insulin decay rate, can lead to notably lower values of the glucose concentration. In the presence of the secreted insulin by the pancreatic beta cells the insulin action and the insulin concentration are more effective to reduce the blood glucose, which can help to improve the diabetes patient's health. [ABSTRACT FROM AUTHOR]

Here we investigate the effect of the vertical rate of change in thermal diffusivity due to a hydrothermal convective flow in a horizontal porous medium. The continuity equation, the heat equation and the momentum-Darcy equation constitute the governing system for the flow in a porous medium. Assuming a vertically varying basic state, we derive the linear system and from this linear system, we compute the critical Rayleigh and wave numbers. Using fourth-order Runge-Kutta and shooting methods, we obtain the marginal stability curves and linear solutions to analyze the solution pattern for different diffusivity parameters. [ABSTRACT FROM AUTHOR]

Here we investigate the use of cubic B-spline functions in solving boundary value problems. First, we derive the linear, quadratic, and cubic B-spline functions. Then we use the cubic B-spline functions to solve second order linear boundary value problems. We consider constant coefficient and variable coefficient cases with non-homogeneous boundary conditions for ordinary differential equations. We also use this numerical method for the space variable to obtain solutions for second order linear partial differential equations. Numerical results for various cases are presented and compared with exact solutions. [ABSTRACT FROM AUTHOR]

We investigate the effect of hydro-thermal convection in an aquifer system. It is assumed that the aquifer is bounded below and above by impermeable boundaries and it is heated from below. The solution of the governing system is expressed in terms of the basic steady state solution and perturbed solution. We obtain the critical Rayleigh number and critical wavenumber using Runge-Kutta method in combination of shooting method and present the marginal stability curve. The amplitude equation is derived by introducing the adjoint system. After amplitude is obtained, we compute the linear solutions for super-critical and sub-critical cases. Numerical results for various spatial and time values are presented in graphical forms. [ABSTRACT FROM AUTHOR]