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This study presents a new intelligent computing method based on the Levenberg–Marquardt back-propagation-based supervised neural network (LMB-SNN) for the flow of Oldroyd 4-constant fluid in the forward roll coating process (OFC-FRCP). The mathematical formulations for OFC-FRCP are converted into ordinary differential equations using lubrication approximation theory (LAT) and suitable dimensionless parameters. The analytic expressions for velocity distribution, pressure and pressure gradients have been obtained using Adomian’s Decomposition Method (ADM) for several situations. The fluid’s separation point and coating thickness are then calculated using integration. The reference dataset for LMB-SNN has been obtained by ADM for multiple scenarios by adjusting various physical variables. The neural network’s training, testing and validation are carried out in parallel to minimize the mean-squared error function (MSEF). The proposed LMB-SNN’s effectiveness was confirmed by analyzing the MSEF, error histograms and regression plots. The precision of the method is verified by the close agreement between numerical outputs and dataset values. The intelligent numerical results computed using ADM and LMB-SNN are presented in various graphs and tables. The method’s accuracy is demonstrated by the numerical outputs closest to the built and dataset values at levels between $10^{-9}$ and $10^{-12}$.

Roll coating is a popular method for applying a thin coating film to a continuous substrate, such as paper or foil. This study aims to investigate analytically the fluid flow through counter rotating rolls using a non-Newtonian model. The governing equation for the constitutive of the Oldroyd-B as a non-Newtonian fluid with momentum equation has been discovered. The nondimensionalizing of the fluid flow equations is done by using suitable dimensionless parameters. The equations are then simplified by using Lubrication Approximation Theory (LAT) to obtain the solution for various quantities. Analytical solutions for the velocity distribution, pressure and pressure gradients have been obtained by regular perturbation method. Numerical integration has been used to calculate separation point, coating thickness, separation force and power input of the fluid. The effect of involved parameter such as Weissenberg number We and viscosity on velocities, pressure gradient and pressure is presented in a variety of graphs and several other results are shown in tables. It has been observed that as the modified capillary number $N_{C{a}_2}$ increases, the separation point moves toward the nip region, whereas the coating thickness declines. It has been witnessed that the We serves as the controlling parameter for flow rate, pressure distribution and power input, whereas for the coating thickness, the separation force increases, which are important for coating industries.

We investigate the cross-diffusion on MHD mixed convective thermo-solutal transport in viscoelastic fluid from a nonlinear stretching sheet. The novelty of the work is the analytical evaluation of the nonlinear stretching sheet in a porous medium with a variable magnetic parameter and radiative flux. Also, thermo-solutal transport of a mixed convective flow is analyzed with Dufour and Soret effects in the presence of the chemical reaction, heat source/sink. The governing PDEs are reduced into ODEs by similarity variables. ODEs were tackled analytically using the homotopy analysis method (HAM). The analysis is carried out up to the 15th order of approximation. HAM contains an h-curve, allowing a flexible method of controlling the series solution convergence area. Graphs and tables are plotted to analyze the influence of different parameters on the fluid. Rates of heat and mass flux are depicted using tables. Our results indicate that the increasing viscoelastic flow enhances the boundary layer, and the thermal boundary layer reduces as radiation enhances. The numerical demonstrations of the wall drag coefficient, Nusselt number and Sherwood number have been tabulated. The growing parameters are taken as $0.5\le M\le 2.0$, $0.1\le \mathrm{\text{Ri}}\le 0.7$, $0.3\le K_0\le 0.9$, $0.5\le f_w\le 2.0$, $1\le \mathrm{\text{Rd}}\le 4$, $0.3\le \mathrm{\text{Pr}}\le 0.9$ and $0.5\le K_2\le 2.0.$

To study the dynamics of rigid body immersed in viscoelastic fluid, an Oldroyd-B fluid flow past an eccentrically situated, free rotating cylinder in a two-dimensional (2D) channel is simulated by a novel lattice Boltzmann method. Two distribution functions are employed, one of which is aimed to solve Navier–Stokes equation and the other to the constitutive equation, respectively. The unified interpolation bounce-back scheme is adopted to treat the moving curved boundary of cylinder, and the novel Galilean invariant momentum exchange method is utilized to obtain the hydrodynamic force and torque exerted on the cylinder. Results show that the center-fixed cylinder rotates inversely in the direction where a cylinder immersed in Newtonian fluid do, which generates a centerline-oriented lift force according to Magnus effect. The cylinder’s eccentricity, flow inertia, fluid elasticity and viscosity would affect the rotation of cylinder in different ways. The cylinder rotates more rapidly when located farther away from the centerline, and slows down when it is too close to the wall. The rotation frequency decreases with increasing Reynolds number, and larger rotation frequency responds to larger Weissenberg number and smaller viscosity ratio, indicating that the fluid elasticity and low solvent viscosity accelerates the flow-induced rotation of cylinder.

In this paper, we propose a novel, simple and accurate analytical study based on nonlocal elasticity theory to forecast small-scale effects on the radial vibration of anisotropic gold nanospheres submerged in viscoelastic fluid (VEF). Eringen’s model is used to determine the motion equation for anisotropic nanospheres, with the fluid assumed to be viscoelastic and compressible. The frequency equation is derived by imposing the fluid-nanosphere interface continuity conditions. A comparison with the literature results is conducted to demonstrate the validity and correctness of this analysis, which indicates a very good agreement. The importance of small-scale effects in the radial vibration, which need to be included in the nonlocal elasticity model of submerged nanospheres, is eventually revealed by numerical examples. It is discovered that the nanosphere size, nonlocal parameter, and glycerol–water mixture have a significant impact on the vibration behaviors. Our results show that the small scale is crucial for the radial vibration of gold nanoparticles when the gold nanosphere is smaller than $5.5\mathrm{\text{nm}}$. Thus, the resulting frequency equation is very useful to interpret experimental measurements of the vibration characteristics of submerged gold nanospheres in VEF.

The combined heat and mass transfer, the so-called thermosolutal convective problem, has become an attractive field of research in many diversified areas. In this paper, for the first time, oscillatory flow analysis has been carried out for triple diffusive viscoelastic fluid flow in a porous medium. A comprehensive model is developed for the modified Darcy–Brinkman–Oldroyd-B fluid, porous medium, Boussinesq approximation, heat and mass transfer across a finite temperature and concentration difference in the chemical potential of two salts. Triple diffusive viscoelastic fluid flows through porous media have grown significantly as this situation occurs in more than a few applications such as improved oil recovery filtration, liquid complex molding, solidification of liquid crystals, cooling of metallic plate in a bath, exotic lubricants and colloidal solutions, polymer processing, chemical and bioengineering industries, among others. The governing coupled nonlinear partial differential equations with boundary constraints represent the modeled flow problem. In addition, these equations are converted into non-dimensional form by employing suitable non-dimensionalizing quantities. The impacts of the pertinent parameters and related dimensionless numbers on the dimensionless velocity, temperature, concentrations, shear stress, heat and mass transfer are examined for both suction and injection cases. It has been found that when the injection level on the heated plate is increased, the shear rate increases for each channel plate. Furthermore, we recognized that the viscoelastic parameters exhibit an opposite kind of behavior on the velocity, temperature, and concentrations fields.

In this work we report theoretical and numerical results on convection in a viscoelastic binary mixture under rotation. Instability thresholds for stationary convection are calculated. We obtain explicit expressions of convective thresholds in terms of the control parameters of the system for oscillatory convection. Finally, we analyze the stabilizing effect of rotation on instability thresholds for aqueous DNA suspensions.

A systematic experimental study of the Faraday instability in viscoelastic fluid is presented. We have used a shear thinning polymer solution in which the elastic effects are predominant within our work range. We have analyzed the dependence of the threshold instability as a function of the depth in layer. Depending on the fluid layer depth and the driving frequency, harmonic or subharmonic regimes are developed. We have focused our work on the subharmonic region and temporal and spatial behaviors were analyzed. In addition, we have used the onset acceleration to estimate the rheological properties of the fluid. These predictions are supported by experimental measurements.

We report theoretical and numerical results on convection for a magnetic fluid in a viscoelastic carrier liquid. We focus on the stationary convection for idealized boundary conditions. We obtain explicit expressions of convective thresholds in terms of the control parameters of the system. Close to bifurcation, the coefficients of the corresponding amplitude equation are determined analytically. The effect of the rheology and Kelvin force are emphasized and, finally, the nature of the bifurcation is discussed.

We report a 6D nonlinear dynamical system for thermo-viscoelastic fluid by selecting higher modes of infinite Fourier series of flow quantities. This nonlinear system demonstrates overstable convective motion and some organized structures such as period-bubbling and Arnold tongue-like structures. Studies reveal that the stability of the conduction state does not alter for the new 6D system in comparison with the lowest order 4D system of Khayat [1995]. However, the stabilities of the convective state have some differences. The onset of unsteady convection in the 6D system is delayed for weak elasticity of the fluid. There exists a critical range of fluid elasticity where the 4D system exhibits subcritical Hopf bifurcation while the 6D system shows supercritical Hopf bifurcation, which ensures the increase of the domain of stability. In this range, catastrophic route to chaos occurs in the 4D system, whereas the 6D system exhibits intermittent onset of chaos. Comparing the two-parameter dependent dynamics for the two systems, the chaotic zones enclosed by periodic regions are suppressed in the 6D system, so the flow behaviors become more predictable. Owing to interacting thermal buoyancy and fluid elasticity, both the models exhibit period-bubbling transition to chaos, but the period-bubbling cascade in the 6D model occurs at lower Rayleigh number than the 4D model. The convergence rate of the period-bubbling process slows down compared to usual period-doubling and approaches the square root of the Feigenbaum constant asymptotically.

This study describes ciliary motion on the transport of fluids in human body with heat transfer. The mathematical model of the flow of a Jeffrey fluid in a tube of finite length is considered due to metachronal wave of cilia motion. Flow equations have been modeled and simplified using similarity variables. Exact solutions of the formulated problem have been obtained for velocity, temperature and pressure gradient and graphs for velocity, pressure rise, pressure gradient and temperature profile have been plotted and studied for different values of specific physical parameters. Trapping phenomena and isotherms are presented at the end of the paper.

We explain the relativistic theory of viscoelasticity which we have recently constructed on the basis of Onsager's linear nonequilibrium thermodynamics. This theory universally reduces to the standard relativistic Navier-Stokes fluid mechanics in the long time limit. Since effects of elasticity are taken into account, the dynamics at short time scales is modified from that given by the Navier-Stokes equations, so that acausal problems intrinsic to relativistic Navier-Stokes fluids are significantly remedied. We then present conformal higher-order viscoelastic fluid mechanics with strain allowed to take arbitrarily large values. We particularly show that a conformal second-order fluid with all possible parameters in the constitutive equations can be obtained without breaking the hypothesis of local thermodynamic equilibrium, if the conformal fluid is defined as the long time limit of a conformal second-order viscoelastic system.

In this paper a new approach is developed to simulate the extrudate swell of viscoelastic fluid. Lattice Boltzmann model is used to solve the complex Navier-Stokes equations and constitutive equations simultaneously at each time iteration. The type of cell is re-initialized by calculating the mass fluxes between neighboring cells after each iteration. The boundary conditions are treated with the non-equilibrium extrapolation method. Finally, by tracking the moving interface, the shape of the extrudate swell is acquired.