Numerical and analytical solutions of Stokes theory of COUPLE STRESS uid under the e ects of constant, space, and variable viscosity in the inclined channel are discussed in this paper. The considered COUPLE STRESS uid is described mathematically with the de nition of the STRESS tensor. The dimensional form of the boundary value problem is transformed into dimensionless form by de ning dimensionless quantities, and is then solved with the aid of the perturbation technique. The analytical expressions of velocity and temperature of all cases based on the viscosity of the COUPLE STRESS uid are presented. For the validity of the perturbation solution, the Pseudo-Spectral collocation method is employed for each case of the viscosity model including constant, space, and Vogel's models, respectively. The solution of the perturbation method and Pseudo-Spectral methods are shown together in the graphs. The e ects of COUPLE STRESS parameters on velocity and temperature distributions are also elaborated with physical reasoning in the results and discussion section. It is observed that the velocity and temperature of uid escalate via the pressure gradient parameter and Brinkman number, while decelerating via the COUPLE STRESS parameter.

The thermosolutal convection in a COUPLE-STRESS FLUID layer heated and soluted from below in porous medium is considered to include the effects of uniform vertical magnetic field and uniform vertical rotation. Following the linearized stability theory and normal mode analysis, the dispersion relation is obtained. For the case of stationary convection, the stable solute gradient and rotation have stabilizing effects on the system. In the presence of rotation, the medium permeability has a destabilizing (or stabilizing) effect whereas magnetic field and COUPLE-STRESS parameter have stabilizing (or destabilizing) effect on the system. On the other hand, in the absence of rotation, medium permeability has a destabilizing effect whereas magnetic field and COUPLE-STRESS parameter have a stabilizing effect. The dispersion relation is also analyzed numerically. The stable solute gradient, rotation and magnetic field introduce oscillatory modes in the system, which were nonexistent in their absence. A condition for the system to be stable is obtained by using Rayleigh-Ritz inequality.

In this paper the effects of COUPLE STRESS FLUID on the control of Rayleigh-Taylor instability at the interface between a dense FLUID accelerated by a lighter FLUID is studied. A simple theory, based on fully developed approximations, is used to derive the growth rate of Rayleigh-Taylor instability. The cutoff and maximum wave numbers and the corresponding maximum frequency are obtained. It is shown that the effect of COUPLE STRESS parameter reduces the growth rate considerably compare to the classical growth rate in the absence of COUPLE-STRESS and hence favorable to control the growth rate of surface instabilities in many practical problems.

The thermal instability of a COUPLE-STRESS FLUID heated from below is investigated. Following the linearized stability theory and normal mode analysis, the paper mathematically establishes that the onset of instability at marginal state, cannot manifest itself as stationary convection, if the thermal Rayleigh number R and the COUPLE-STRESS parameter F, satisfy the inequality R ≤ (2+ Ö1+3p2F) (3p2F+Ö1+3p2F -1)3/27F2 (-1+Ö(1+3p2F)), and when the COUPLE-STRESS parameter F is infinitesimally small, R≤27p4/4 {1+p2/2 F}, the result which also clearly mathematically established the stabilizing character of the COUPLE-STRESS.

In this study, a realistic model for dynamic instability of embedded single-walled nanotubes (SWCNTs) conveying pulsating FLUID is presented considering the viscoelastic property of the nanotubes using Kelvin-Voigt model. SWCNTs are placed in longitudinal magnetic fields and modeled by sinusoidal shear deformation beam theory (SSDBT) as well as modified COUPLE STRESS theory. The effect of slip boundary condition and small size effect of nano flow are considered using Knudsen number. The Gurtin-Murdoch elasticity theory is applied for incorporation the surface STRESS effects. The surrounding elastic medium is described by a visco-Pasternak foundation model, which accounts for normal, transverse shear and damping loads. The motion equations are derived based on the Hamilton's principle. The differential quadrature method (DQM) in conjunction with Bolotin method is used in order to calculate the dynamic instability region (DIR) of visco-SWCNTs induced by pulsating FLUID. The detailed parametric study is conducted, focusing on the combined effects of the nonlocal parameter, magnetic field, visco-Pasternak foundation, Knudsen number, surface STRESS and FLUID velocity on the dynamic instability of SWCNTs. The results depict that increasing magnetic field and considering surface effect shift DIR to right. The results presented in this paper would be helpful in design and manufacturing of nano/micro mechanical systems.

Activation energy is of considerable significance in diverse applications such as chemical kinetics, catalyst development, enzymes, semiconductors, and systems sensitive to temperature, such as chemical reactors and engines. The objective of this research is to investigate the influence of activation energy on a magnetized COUPLE STRESS FLUID over an inclined stretching permeable cylinder in a non-Darcy porous medium. The effects of cross-diffusion and stratified mixed convection are also considered in FLUID model. The boundary layer equations, which describe the flow, have been converted into dimensionless form through suitable transformable variables. Subsequently, these transformed equations are solved using fourth order Runge-Kutta mechanism along with the shooting technique. The outcomes comprise visual depictions and comprehensive explanations demonstrating the influence of relevant variables on thermal, concentration, and velocity fields. Observations reveal that the concentration profile is directly influenced by the Forchheimer number and activation energy parameter, whereas both temperature and concentration fields decrease with elevated thermal and solutal stratification parameters. Additionally, numerical outcomes for the skin-friction coefficient, Nusselt number, and Sherwood number are presented in tabular form.

The radiation and thermal diffusion effects on mixed convection flow of COUPLE STRESS FLUID through a channel are investigated. The governing non-linear partial differential equations are transformed into a system of ordinary differential equations using similarity transformations. The resulting equations are then solved using the Spectral Quasi-linearization Method (QLM). The results, which are discussed with the aid of the dimensionless parameters entering the problem, are seen to depend sensitively on the parameters.

Exact analysis of miscible dispersion of solute with interphase mass transfer in a poorly conducting COUPLE STRESS FLUID flowing through a rectangular channel bounded by porous layers is considered because of its application in many practical situations. The generalized dispersion model of Sankarasubramanian and Gill is used, which brings into focus the exchange coefficient, the convective coefficient and the dispersion coefficient. The exchange coefficient comes into picture due to the interphase mass transfer and independent of solvent FLUID viscosity. It is observed that the convective coefficient increases with an increase in the porous parameter while it decreases with an increase in the COUPLE STRESS parameter. The dispersion coefficient is plotted against wall reaction parameter for different values of porous parameter and COUPLE STRESS parameter. It is noted that the dispersion coefficient decreases with an increase in the value of COUPLE STRESS parameter but increases with porous parameter.

In the present research, the forced vibration of a truncated conical nanoshell in contact with FLUID under harmonic load has been investigated. Classical plate theory along with modified STRESS COUPLE theory has been used for shell analysis. The energy method and Hamilton's principle have been used to determine the governing equations. The applied pressure from the FLUID to the shell has been determined by using the FLUID potential function and using the velocity boundary conditions at the contact surface of the FLUID and the shell. Also, to determine the vibration response, the method of hypothetical modes along with the Galerkin technique has been used. Finally, for a sample nanoshell, numerical results have been determined with the help of MATLAB software, and the numerical results of forced vibration and the effect of various parameters such as aspect ratios, cone apex angle, external force parameters, and FLUID density on the vibration response of the system have been analyzed.

The effect of different forms of basic temperature gradients on the criterion for the onset of convection in a layer of an incompressible COUPLE-STRESS FLUID-saturated porous medium is investigated. It is shown that the principle of exchange of stability is valid, and the eigenvalue problem is solved numerically using the Galerkin technique. The parabolic and inverted parabolic basic temperature profiles have the same effect on the onset of convection and similar is the case between piecewise linear temperature profiles heating from below and above. Amongst the various basic temperature profiles, the linear temperature profile is found to be more stabilizing on the onset of instability. In addition, the influence of thermal depth on the criterion for the onset of convection is assessed in the case of piecewise linear temperature profiles. Moreover, an increase in the value of COUPLE-STRESS parameter is found to delay the onset of convection and to increase the width of convection cells. It is also noted that the critical wave numbers are slightly affected by the nature of basic temperature profiles.