Viscous Fluid Model Research Articles

A dynamic solution for predicting resonant frequency of piston mode fluid oscillation in moonpools/narrow gaps

The objective of this study is to develop a simple tool for predicting the resonant frequency of the piston mode wave oscillation in a moonpool or a narrow gap between twin floating barges. Based on the dynamic analysis and the domain-decomposition, a linear harmonic oscillator model for the fluid oscillation in a confined space is formulated, which immediately leads to an explicit dynamic solution for the resonant frequency. In the model, it is assumed that the water depth is equal to or greater than a critical value dependent on the ratio of the draft to the breadth of the floating body. The predicted resonant frequencies are compared with the numerical results of the linear potential model and the viscous fluid model, the semi-analytical potential solutions and the experimental measurements. Good agreements are observed for a wide range of geometric parameters of the moonpool and the narrow gap of either sharp or round edge shapes. The present dynamic method is also applied to predict the resonant frequency of the piston mode oscillation in the gap between a floating body and a vertical wall with a reasonable success.

Influence of magnetic field on double convection problem of fractional viscous fluid over an exponentially moving vertical plate: New trends of Caputo time-fractional derivative model

In this article, the influence of a magnetic field is studied on a generalized viscous fluid model with double convection, due to simultaneous effects of heat and mass transfer induced by temperature and concentration gradients. The fluid is considered over an exponentially accelerated vertical plate with time-dependent boundary conditions. Additional effects of heat generation and chemical reaction are also considered. A generalized viscous fluid model consists of three partial differential equations of momentum, heat, and mass transfer with corresponding initial and boundary condition. The idea of non-integer order Caputo time-fractional derivatives is used, and exact solutions for velocity, temperature, and concentration in terms of Wright function and function of Lorenzo–Hartley are developed for ordinary cases. Graphical analysis of flow and fractional parameters is made by using computational software MathCad, and discussed. The results obtained are also in good agreement with the published results from the literature. As a result, it is found that temperature and fluid velocity can be enhanced for smaller values of fractional parameters.

A second order fully-discrete linear energy stable scheme for a binary compressible viscous fluid model

We present a linear, second order fully discrete numerical scheme on a staggered grid for a thermodynamically consistent hydrodynamic phase field model of binary compressible fluid flows, derived from the generalized Onsager Principle. The hydrodynamic model possesses not only the variational structure in its constitutive equation, but also warrants the mass, linear momentum conservation as well as energy dissipation. We first reformulate the model using the energy quadratization method into an equivalent form and then discretize the reformulated model to obtain a semidiscrete partial differential equation system using the Crank-Nicolson method in time. The semi-discrete numerical scheme preserves the mass conservation and energy dissipation law in time. Then, we discretize the semi-discrete PDE system on a staggered grid in space to arrive at a fully discrete scheme using 2nd order finite difference methods, which respects a discrete energy dissipation law. We prove the unique solvability of the linear system resulting from the fully discrete scheme. Mesh refinements and numerical examples on phase separation due to spinodal decomposition in binary polymeric fluids and interface evolution in the gas-liquid mixture are presented to show the convergence property and the usefulness of the new scheme in applications, respectively.

Group Analysis of Equations of Hydrostatic Model of Viscous Fluid

The group properties of three-dimensional equations of hydrostatic model of viscous fluid are studied. Several exact solutions are presented. The free surface of fluid and pressure on this surface are deter- mined

Modeling the technological process of tillage

The purpose of the research is to develop a model of the technological process of tillage, justify the initial and boundary conditions for the functioning of the model and determine the ratio of the physical and mechanical properties of the soil and the viscosity of the medium.As a model of the soil medium, it was proposed to use the rheological model of Newtonian viscous fluid, which allows using the methods of continuous media dynamics for modeling the technological process of soil treatment, and computational fluid dynamics for numerical implementation of the model. The initial and boundary conditions for the functioning of the model are substantiated and the ratio of the physical and mechanical properties of the soil and the viscosity of the medium are found. The results of determining the traction resistance of the tool and their comparison with the results obtained during laboratory experiments on the soil channel are presented. The proposed method of modeling the soil treatment allows you to analyze the power characteristics of the working bodies and the quality of the soil at the design stage of tillage machines. The obtained characteristics make it possible to optimize the structural and technological parameters of the working bodies of machines on the computer.

Hydroelastic response of three-layered beam resting on winkler foundation

The hydroelastic oscillations problem of the sandwich beam resting on elastic foundation and interacting with the pulsating fluid layer was investigated. The three-layered beam with an incompressible lightweight filler was considered. The broken normal hypothesis for the three-layered beam and the model of viscous incompressible fluid, as well as Winkler model for elastic foundation, was chosen to study hyroelastic problem. A mathematical model of the considered mechanical system consists of dynamic equations of the three-layered beam with incompressible lightweight filler and the Navier-Stokes equations with continuity equation for the pulsating viscous fluid layer. The no-slip conditions and pressure coincidence at the edges with the given pressure into edges cavities were selected as boundary ones. The plane problem of hydroelastic beam bending oscillations for the regime of steady harmonic ones were considered. The solution of the hydroelastic problem was carried out by the perturbations method using proposed small parameters of the hydroelastic problem. As a result, the laws of three-layered beam elastic deflections and the hydrodynamic parameters of pulsating viscous fluid layer were obtained.

Global attractors for a nonlinear one-dimensional compressible viscous micropolar fluid model

This paper considers the dynamical behavior of solutions of constitutive systems for 1D compressible viscous and heat-conducting micropolar fluids. With proper constraints on initial data, we prove the existence of global attractors in generalized Sobolev spaces $$H_{\delta }^{(1)}$$ and $$H_{\delta }^{(2)}$$ . These attractors are unique in corresponding phase spaces.

Asymptotic Stability of Combination of Contact Discontinuity with Rarefaction Waves for the One-Dimensional Viscous Micropolar Fluid Model

In this paper, we consider with the large time behavior of solutions of the Cauchy problem to the one-dimensional compressible micropolar fluid model, where the far field states are prescribed. When the corresponding Riemann problem for the Euler system admits the solution consisting of contact discontinuity and rarefaction waves, it is shown that the combination wave corresponding to the contact discontinuity, with rarefaction waves is asymptotically stable provided that the strength of the combination wave and the initial perturbation are suitably small. This result is proved by using elementary L2-energy methods.

Simulation of the flow of a two-layer liquid film on a cylindrical surface

The problem of a stationary joint flow of a two-layer liquid film and gas along the outer (or inner) surface of a circular cylinder of radius r0 is considered. It is assumed that the films are insoluble in one another, and there are no chemical reactions. The axis of the body is located vertically, and the films flow down from its top. The film is affected by gravity, as well as a gas stream directed upwards or downwards. A cylindrical coordinate system (r, θ, z) is introduced: the z coordinate is measured along the axis of the cylinder, r and θ are the polar coordinates in a plane perpendicular to the axis of the body. To describe the flow of a liquid film, a viscous incompressible fluid model is used, which is based on the equations of continuity and Navier-Stokes. The following boundary conditions are set on the interface surfaces: on the solid surface - draw off “sticking”, on the “liquid-liquid” and “liquid-gas” interfacial surfaces - the conditions of equilibrium of forces and continuity of speeds. To simplify these differential equations, the method of a small parameter, for which the relative thickness of the films is selected, is applied. Solutions of simplified equations (in a zero approximation) are obtained in analytical form. Functional dependences are obtained for calculating the optimal effect of the gas flow on the "working" film. In accordance with the described method, calculations of the flow of a two-layer film on the outer and inner surfaces of the circular cylinder are performed in cases where the gas stream is directed upwards, downwards , and also when the gas flow is absent. An increase in the relative thicknesses of the films δ1 and δ2 (with decreasing radius of the cylinder r0) leads to an increase in deviations from the case of a plane surface that corresponds to the limiting case δ1 = δ2 = 0. The results of calculations of the flow of a two-layer liquid film on the surface of a circular cylinder are presented. The analysis of the influence of physical parameters on the speed profiles is carried out. The results of calculations for determining the optimal effect of gas flow on a liquid film are presented, when the profile of the speed of the "working" film is the most uniform

Mathematical modeling of film flow of a liquid on a surface of a body of a rotation

The problem of the spatial nonwave stationary flow of the viscoplastic fluid on the surface of the body of rotation under the action of gravity is considered. It is assumed that the axis of the body is located at a certain angle to the vertical, and the film of liquid flows down from its top. A curvilinear orthogonal coordinate system (ξ, η, ζ) associated with the body surface is introduced: ξ is the coordinate along the generatrix of the body, η is the polar angle in the plane perpendicular to the axis of the body of revolution, ζ is the dis-tance along the normal to the surface. To describe the flow of a liquid film, a viscous in-compressible fluid model is used, which is based on partial differential equations - the equations of motion and continuity. The following boundary conditions are used: sticking conditions on the solid surface; on the surface separating liquid and gas, the conditions for continuity of stresses and normal component of the velocity vector. For the closure of a system of differential equations, the Schulman rheological model is used, which is a gener-alization of the Ostwald-de-Ville power model and the Shvedov-Bingham viscoplastic model. To simplify the system of differential equations, the small parameter method is used. The small parameter is the relative film thickness. It is assumed that the generalized Reynolds number has an order equal to one. The solution of the equations of continuity and motion (taking into account the principal terms of the expansion) was obtained in an analytical form. The obtained formulas for the components of the velocity and pressure vector generalize the known relations for flat surfaces. To determine the unknown film thickness, an initial-boundary value problem was formulated for a first-order partial differential equation. The solution to this problem is found with the help of the finite difference method. The results of calculations according to the proposed method for the circular cone located at a certain angle to the vertical are presented. Calculations show that the parameters of nonlinearity and plasticity of this rheological model of a liquid can significantly affect the speed profiles and the distribution of the thickness of the viscous layer on the surface of the body

An analysis of the semi-analytic solutions of a viscous fluid with old and new definitions of fractional derivatives

In this paper we present the natural convection flow of an incompressible viscous fluid subject to Newtonian heating and constant mass diffusion using a recently developed definition of the Caputo–Fabrizio fractional derivative. Boundary layer equations in dimensionless form are obtained by means of dimensionless variables. The expressions for the temperature, concentration and velocity fields are obtained in the Laplace transformed domain. The inverse Laplace transform for the temperature, concentration and velocity field are found numerically by means of Stehfest's and Tzou's algorithms. A comparative analysis has been carried between the Caputo–Fabrizio and the Caputo fractional model obtained by Vieru (2015) through graphical illustration. At the end, we can see the impact of the flow parameters, including the new fractional parameter, on the flow which is presented graphically. As a result, the fractional viscous fluid model with the Caputo–Fabrizio fractional derivative has a higher velocity than with the Caputo.

Viscous numerical examination of hydrodynamic forces on a submerged horizontal circular cylinder undergoing forced oscillation

As a classical topic, the hydrodynamic forces on a submerged horizontal cylinder undergoing forced oscillation have been widely studied based on potential flow theory. However, the fluid viscosity and the flow rotation may play an important role when the oscillation amplitude of the circular cylinder is large, and large discrepancy will occur between the potential flow simulation and the experimental results. This study focuses on the study of hydrodynamic forces on a submerged horizontal circular cylinder undergoing forced oscillation by means of a viscous fluid numerical wave tank (NWT) model. The accuracy of the numerical model is validated against available experimental data. The comparisons between the hydrodynamic forces on the circular cylinder predicted by the viscous fluid model and the potential flow model are conducted to show the viscous effects on the hydrodynamic forces. By the study on the flow fields, the mechanism of the viscous effects is explained by the vortex effect. The basic reason for the difference between the results based on the viscous fluid theory and the potential flow theory is revealed by analyzing the force components.

Mathematical model of viscous petroleum fluid for consequential migration in porous solid

Viscous fluid transport or crude natural petroleum migration in a form of the laminar flow of incompressible fluids and hydrocarbon classification are principal to study the mass flow rate according to Darcy’s law and Hagen-Poiseuille law. Natural petroleum classification is concerned for its migration from a reservoir to underground surface. Its chemical aspect can be categorized into hydrocarbon as paraffinic, intermediate paraffinic, aromatic, and asphaltic classes, in the ratio of 5:10:4:1, respectively. Its property signifies two types of paraffinic and intermediate classes. The volumetric flow rates implicate to petroleum migration which refers to hydrocarbon classifications. These indicate possibility of migration with the mass flow of hydrocarbon class. Correlation of their volumetric flow rates can appraise the configuration of the volumetric flow rates as a function of mass of flow rates with three classifications: typical crude oil, heavy crude oil, bitumen. Crude petroleum classes are bitumen to heavy crude oil, and can be anticipated a progress of the migration zones of heavy crude oil, heavy crude oil-bitumen, and bitumen zone. Consequently, assessment results expose that the slow volumetric flow rates associate to petroleum migration, pressure depths, and crude petroleum classification.

A Mathematical Model Governing Tornado Dynamics: An Exact Solution of a Generalized Model

Abstract We investigate in this paper the dynamics of tornadoes by considering that the real inflow radial velocity depends on both the radial and vertical coordinates. The formulation is based on the model for the radial velocity that has been deduced from an experimentally verified model of azimuthal velocity. We present an analytical model for steady, incompressible, and viscous fluids and try for exact solutions. Although all the three components depend on radial and axial coordinates, viscosity affects merely the azimuthal velocity and the pressure. It is observed that the magnitude of the radial velocity increases to the maximum at the core but reverses the trend beyond and vanishes as it reaches the centerline. The magnitude reduces linearly with axial distance as per the supposition. At the core, the larger the Reynolds number, the lower is the velocity for moderate Reynolds numbers. Insignificant impact is observed for very large Reynolds number. However, inside and outside the core, the trends are reversed. Radial pressure distributions for different axial positions are similar to theoretical, numerical, and experimental observations. As we move outward from the axis, pressure increases. The difference between the pressures at the axis and that in outward regions increases with height. Pressure falls with rising Reynolds number uniformly for all radial distances. This is an indication that quantitative difference in pressure is large between viscous and inviscid flows.

Viscous Effects on Wave Forces on A Submerged Horizontal Circular Cylinder

Numerical simulations are carried out for wave action on a submerged horizontal circular cylinder by means of a viscous fluid model, and it is focused on the examination of the discrepancies between the viscous fluid results and the potential flow solutions. It is found that the lift force resulted from rotational flow on the circular cylinder is always in anti-phase with the inertia force and induces the discrepancies between the results. The influence factors on the magnitude of the lift force, especially the correlation between the stagnation-point position and the wave amplitude, and the effect of the vortex shedding are investigated by further examination on the flow fields around the cylinder. The viscous numerical calculations at different wave frequencies showed that the wave frequency has also significant influence on the wave forces. Under higher frequency and larger amplitude wave action, vortex shedding from the circular cylinder will appear and influence the wave forces on the cylinder substantially.

Reducing Friction with a Liquid Film on the Body Surface

A flow of a thin layer of liquid is simulated on a flat surface of a body located in a stream of air. Liquid film on the surface of the body reduces frictional resistance and can be used as a boundary layer control element. The paper presents a mathematical model of the film flow on a half-plane, located at an angle to the horizon. The fluid flow is determined by the force of gravity and friction from the external air current. A model of an incompressible viscous fluid is used in the boundary-layer approximation. The terms of the motion equation are averaged over the film thickness according to the Leibniz rule. In the cross section of the film, a quadratic law is adopted for the distribution of the longitudinal velocity, taking into account friction on the film surface. An analytical solution of the problem is obtained in the form of series in powers of the small parameter for determining the film thickness and the average longitudinal velocity along the length of the plate. It is shown that the friction decreases with flow around a half-plane with a film of liquid on the surface.

Generalized Continuum Phenomenological Model of Viscous Fluid: Analysis of Local Mass Displacements

The concept of local mass displacements (as a measure of nonlocality of the state of a physically small element of a body) is used to formulate a complete system of relations of the gradient-type mathematical model of viscous compressible fluid. In this model, the relationship between the thermomechanical processes and the process of local mass displacements, its irreversibility, and the inertia properties are taken into account. It is shown that, in the stationary approximation, the obtained system of relations enables us to justify the appearance of wedging pressure in thin interlayers of fluid.

3-D flow of a compressible viscous micropolar fluid model with spherical symmetry: A brief survey and recent progress

We consider the non-stationary 3-D flow of a compressible viscous and heat-conducting micropolar fluid with the assumption of spherical symmetry. We analyze the flow between two concentric spheres that present solid thermo-insulated walls. The fluid is perfect and polytropic in the thermodynamical sense and the initial density and temperature are strictly positive. The corresponding problem has homogeneous boundary data. In this work, we present the described model and provide a brief overview of the progress in the mathematical analysis of the associated initial-boundary problem. We consider existence and uniqueness of the generalized solution, asymptotic behavior of the solution and regularity of the solution.

Theoretical and numerical investigations of wave resonance between two floating bodies in close proximity

A simple theoretical dynamic model with a linearized damping coefficient is proposed for the gap resonance problem, as often referred to as the piston mode wave motion in a narrow gap formed by floating bodies. The relationship among the resonant response amplitude and frequency, the reflection and transmission coefficients, the gap width, and the damping coefficient is obtained. A quantitative link between the damping coefficient of the theoretical dynamic model (ɛ) and that devised for the modified potential flow model (up) is established, namely, up = 3πɛω (where ωn is the natural frequency). This link clarifies the physical meaning of the damping term introduced into the modified potential flow model. A new explicit approach to determine the damping coefficient for the modified potential model is proposed, without resorting to numerically tuning the damping coefficient by trial and error tests. The effects of the body breadth ratio on the characteristics of the gap resonance are numerically investigated by using both the modified potential flow model and the viscous RNG turbulent model. It is found that the body breadth ratio has a significant nonlinear influence on the resonant wave amplitude and the resonant frequency. With the modified potential flow model with the explicit damping coefficient, reasonable predictions are made in good agreement with the numerical solutions of the viscous fluid model.

Global Existence of Weak Solutions to a Higher-Order Parabolic System

In this paper, we study the coupling of the non-Newtonian Navier–Stokes equation and the oil–water–surfactant equation which stand for a model of a multi-phase incompressible dipolar viscous non-Newtonian fluid under shear. Based on Galerkin approximation and Simon’s compactness results, we obtain the existence and uniqueness of weak solutions of the system.