Viscous Incompressible Fluid Research Articles

Effects of temperature dependent viscosity and thermal conductivity on natural convection flow along a curved surface in the presence of exothermic catalytic chemical reaction.

The natural convection boundary layer flow of a viscous incompressible fluid with temperature dependent viscosity and thermal conductivity in the presence of exothermic catalytic chemical reaction along a curved surface has been investigated. The governing non dimensional form of equations is solved numerically by using finite difference scheme. The numerical results of velocity profile, temperature distribution and mass concentration as well as for skin friction, heat transfer rate and mass transfer rate are presented graphically and in tabular form for various values of dimensionless parameters those are generated in flow model during dimensionalization. From the obtained results, it is concluded that the exothermic catalytic chemical reactions is associated with temperature dependent viscosity and thermal conductivity. Further, it is concluded that the body shape parameter also plays an important quantitative role for change in velocity profile, temperature field and mass concentration behavior in the presence of exothermic catalytic chemical reaction.

An inverse problem for generalized Kelvin–Voigt equation with p-Laplacian and damping term

In this paper, we consider the nonlinear inverse problem for generalized Kelvin–Voigt equations with the p-Laplace diffusion and damping term, describing the motion of incompressible viscous fluids. We assume that the damping term in the momentum equation depends on whether its signal is positive or negative, which may realizes the presence of a source or a sink within the system. The investigated inverse problem consists of finding a coefficient f(t) of the right-hand side of the momentum equation, a vector of velocity field v, and a pressure π. An additional information on a solution of the inverse problem is given as integral overdetermination condition. Under several assumptions on the exponents p, m, the coefficients μ, κ, γ, the dimension of the space d, and specified initial data, we prove the existence and uniqueness of the weak solution of the problem.

On the Use of Probability-Based Methods for Estimating the Aerodynamic Boundary-Layer Thickness

In this paper, known probabilistic methods for estimating the thickness of the boundary layer of a two-dimensional laminar flow of viscous incompressible fluid are extended to three-dimensional laminar flows of a viscous compressible medium. Their applicability to the problems of boundary-layer stability is studied with the LOTRAN3 software package, which allows us to compute the position of laminar-turbulent transition in three-dimensional aerodynamic configurations.

Exact Solutions to the Navier–Stokes Equations with Couple Stresses

This article discusses the possibility of using the Lin–Sidorov–Aristov class of exact solutions and its modifications to describe the flows of a fluid with microstructure (with couple stresses). The presence of couple shear stresses is a consequence of taking into account the rotational degrees of freedom for an elementary volume of a micropolar liquid. Thus, the Cauchy stress tensor is not symmetric. The article presents exact solutions for describing unidirectional (layered), shear and three-dimensional flows of a micropolar viscous incompressible fluid. New statements of boundary value problems are formulated to describe generalized classical Couette, Stokes and Poiseuille flows. These flows are created by non-uniform shear stresses and velocities. A study of isobaric shear flows of a micropolar viscous incompressible fluid is presented. Isobaric shear flows are described by an overdetermined system of nonlinear partial differential equations (system of Navier–Stokes equations and incompressibility equation). A condition for the solvability of the overdetermined system of equations is provided. A class of nontrivial solutions of an overdetermined system of partial differential equations for describing isobaric fluid flows is constructed. The exact solutions announced in this article are described by polynomials with respect to two coordinates. The coefficients of the polynomials depend on the third coordinate and time.

Hydrodynamic model of blood flow in major arteries pulsing in various modes.

Traditionally, the arteries in mammals are viewed as the tubes with elastic wall, whose elasticity could be slowly (during a minute) tuned to change its diameter thereby regulating regional blood supply. Recent findings showed that an artery is a much more sophisticated organ, which can change elasticity of vascular wall within a fraction of a second during a cardiac cycle due to activation of its smooth muscles manifested by generation of arterial action potentials. The rapid variations in elasticity of vascular wall resulted in three basic modes of arterial pulsing: passive, active, and intermediate. The latter is characterized by counteraction of dilation force of arterial pressure and the contractile one of smooth muscle in arterial wall, which can result in seemingly "rigid" artery of constant diameter. The prevalence of any of these forces results in active or passive pulsing modes. Existence of various pulsing modes raises the question of their effect on the main function of blood vessels, i.e., the transport of blood. The aim of this study is to assess the effect of various modes of arterial pulsing on hydraulic impedance of major arteries. The linearized Navier-Stokes equation was employed to develop a model of pulsatile flow of viscous incompressible fluid at small velocity via a conduit artery with the walls of variable or constant elasticity. An essential feature of the developed model is the shape of variable pressure drop applied to the ends of arterial segment, which simulates the real changes in arterial pressure during the heartbeat. Here, it is modeled by periodic (systolic) positive bell-shaped impulses with maxima corresponding to systolic arterial pressure, while the minimal plateau level refers to diastolic arterial pressure. The model assesses the changes in arterial hydraulic impedance during a cardiac cycle relatively to the stable level corresponding to constant blood flow driven by persistent pressure drop. Within intermediate variety of pulsing modes between the active and passive ones, the approximation of rigid arterial segment with infinite elasticity of arterial wall showed that hydraulic impedance in rigid artery is not constant due to inertial properties of the flowing blood. In passive pulsing mode characterized with constant elasticity of arterial wall, the diameter of artery changes in parallel with systolic pressure applied to the ends of arterial segment. At this, the overall change of hydraulic resistance is negative. In active pulsing mode, elasticity of arterial wall varies at different phase shifts relative to arterial pressure due to periodic contractions and relaxation of the smooth muscles in arterial wall. An important feature of active mode is possibility to decrease the hydraulic impedance during the front of arterial pressure. Various experimental modes of artery pulsing can be mathematically simulated. The passive and active modes of pulsing as well as a broad variety of intermediate pulsing modes with various phase shifts between arterial pressure and its diameter result in potency of the arteries to tune its performance in order to meet the regional circulatory requirements. The model showed that active arterial pulsing can diminish the arterial hydraulic impedance and contribute to the work needed for circulation thereby helping the pumping action of the heart.

Incompressible viscous fluids in R2 and SPDEs on graphs, in presence of fast advection and non smooth noise

Le comportement asymptotique d’une classe d’equations stochastiques de reaction-diffusion-advection dans le plan est etudie. Nous montrons qu’a mesure que le terme d’advection sans divergence devient de plus en plus grand, les solutions de telles equations convergent vers la solution d’une EDP stochastique appropriee definie sur le graphe associe a l’Hamiltonien. Tout d’abord, nous traitons le cas ou la perturbation stochastique est donnee par un processus de Wiener spatialement homogene singulier prenant des valeurs dans l’espace des distributions de Schwartz. Comme dans les travaux precedents, nous supposons ici que la derivee de la periode du mouvement sur les level sets de l’Hamiltonien ne s’annule pas. Puis, dans la seconde partie, sans supposer cette condition sur la derivee de la periode, nous etudions un type de convergence plus faible pour les solutions d’une classe appropriee de EDPS lineaires.

Existence of dual solutions and three-dimensional instability in helical pipe flow

Three-dimensional (3D) direct numerical simulations (DNS) of the viscous incompressible fluid flow through a helical pipe with circular cross section were performed. The flow is governed by three parameters: the Dean number (or the Reynolds number), curvature, and torsion. First, we obtained steady solutions by steady 3D calculations, where dual solutions were found, one was uniform in the pipe axial direction and the other varied very slowly, if torsion exceeded a critical value. Then, the instability of the steady solutions obtained was studied by unsteady 3D calculations. We obtained critical Reynolds numbers of steady to unsteady transition by observing the behaviors of the unsteady solutions. The present results of the critical Reynolds number nearly agreed with those by the 2D linear stability analysis (Yamamoto et al. [9]) except for the lowest critical Reynolds number region, where the present study gave the critical Reynolds number much less than that obtained by the 2D linear stability analysis. We found the vortical structures in the form of a standing wave slightly above the marginal instability state, which is a trigger of explosive 3D instability.

SIMULATION OF WASTE WATER TREATMENT BASED ON CFD MODEL: EXPRESS CALCULATION

Purpose. Development of CFD model to evaluate the efficiency of wastewater treatment in a horizontal settler. The CFD model can be used to calculate flow hydrodynamics and mass transfer in settlers with complex geometric shape in the area of wastewater flow. Methodology. For numerical simulation of the process of wastewater movement in a horizontal settler, two mathematical models are used. The first model is based on the motion equations of a viscous incompressible fluid – the Navier-Stokes equations. The Navier-Stokes equations are written in the variables «vorticity - flow function». A two-dimensional mass transfer equation is used to calculate the concentration of a pollutant in a horizontal settler. To numerically integrate the two-dimensional mass transfer equation, a finite-difference splitting scheme is used. The splitting of the modeling equation of mass transfer is carried out so that at each fractional step to determine the unknown value of the pollutant concentration by an explicit formula. For numerical integration of the vortex transfer equation and the equation for the flow function (the Navier-Stokes system of equations), finite-difference splitting schemes are used. Findings. Based on the developed CFD model, a complex of computer programs has been developed, which makes it possible to determine the efficiency of water treatment in a horizontal settler with additional elements. The results of a computational experiment to assess the efficiency of water treatment in a horizontal settler with additional elements in the form of plates are presented. Originality. An efficient CFD model has been created, which allows to quickly evaluate the efficiency of wastewater treatment in a horizontal settler with additional elements. The developed CFD model takes into account the geometric shape of the facility and the most significant physical factors, that influence the efficiency of the waste water treatment in horizontal settler: non-uniform flow velocity, diffusion, different position of inlet and outlet openings. Practical value. The developed CFD model belongs to the class of «diagnostic models» and can be used to assess the efficiency of treatment facilities at the stage of their preliminary design.

Multiphase CFD simulation of the nearshore spilled oil behaviors

Oil spills are a serious environmental problem. To better support risk assessment and pollution control for oil spills, a good understanding of oil transport in the environment is required. This study focused on the numerical simulation of the nearshore oil behaviors based on computational fluid dynamics. Based on the Navier-Stokes momentum equations for an incompressible viscous fluid and volume of fluid (VOF) method, a 3D numerical model of three-phase transient flow was developed. The wave number, averaged flow velocity and oil properties would affect the oil spread extent and the oil volume fraction. The higher the averaged flow velocity and wave number, the lower the oil concentration, and the faster the horizontal movement of the oil. The spilled oil may move to contact the seafloor by increasing the averaged flow velocity at the inlet boundary. Through increasing the wave number, the oil would stay near the water surface. In the nearshore, where the wave is the main seawater motion, the oil containment boom should be set preferentially to the direction of wave transmission for oil cleaning. This study shows that by doubling the wave number and increasing the averaged flow velocity (ten times) at the same time, the maximum oil volume fraction would be reduced by around 32%. Finally, the water temperature had no significant impact on the oil migration, and the impact of evaporation should be considered in the simulation.

On the $$C_0$$ semigroup generated by the Oseen operator around a steady flow exterior to a rotating obstacle

We consider the motion of an incompressible viscous fluid filling the whole space exterior to a moving with rotation and translation obstacle. We show that the Stokes operator around the steady flow in the exterior of this obstacle generates a $$C_0$$ -semigroup in $$L^p$$ space and then develop a series of $$L^p$$ – $$L^q$$ estimates of such semigroup. As an application, we give out the stability of such steady flow when the initial disturbance in $$L^3$$ and the steady flow are sufficiently small.

A Hopf Bifurcation in the Planar Navier\u2013Stokes Equations

We consider the Navier–Stokes equation for an incompressible viscous fluid on a square, satisfying Navier boundary conditions and being subjected to a time-independent force. As the kinematic viscosity is varied, a branch of stationary solutions is shown to undergo a Hopf bifurcation, where a periodic cycle branches from the stationary solution. Our proof is constructive and uses computer-assisted estimates.

STABILITY OF A VISCOUS INCOMPRESSIBLE CONDUCTING LIQUID LAYER OF A CYLINDRICAL SHAPE IN AN INHOMOGENEOUS TEMPERATURE FIELD AND A MAGNETIC FIELD OF A VACUUM ARC CURRENT THROUGH IT

Convective mass transfer in a cylindrical viscous incompressible conductive fluid layer in an inhomogeneous temperature field and in the external magnetic field of the vacuum arc current through it is theoretically investigated in this work. For a horizontal layer of a viscous, incompressible, conducting liquid of a cylindrical shape, located in a temperature field inhomogeneous in height and in an external magnetic field of a vacuum arc current flowing through it, the original equations are written. These equations consist of linearized equations for small velocity perturbations, small deviations from the equilibrium values of temperature, pressure, and magnetic field strength. The considered boundary value problem is solved for the case of free boundaries. Comparison of the experimental data with theoretical calculations made it possible to determine the rotation velocity of the steel melt during vacuum arc melting.

Stability analysis of an anisotropic rotor partially filled with viscous incompressible fluid based on Andronov-Hopf bifurcation

Instability of a rotor partially filled with viscous incompressible fluid is a common issue in rotary machines, which leads the amplitude of perturbations to increase exponentially. Currently, different models of an isotropic rotor partially filled with fluid are established to investigate its stability, but the research on the stability of an anisotropic rotor is rarely reported. To explore the instability of an anisotropic rotor partially filled with fluid, a continuous model is established to represent an isotropic rotor system partially filled with fluid and dimensionless hydrodynamic forces are calculated, and then D-decomposition method is introduced to obtain stable and unstable regions of this isotropic rotor. The transitions of different regions are called as Andronov-Hopf bifurcation. Then, a novel predicting model, where the elliptical motion of a rotor partially with fluid is assumed, is established to obtain the same unstable regions, comparing with the results from D-decomposition method. The correctness of this novel predicting method for an isotropic rotor partially filled with fluid is verified by this comparison. The influence of dimensionless damping coefficient on the stability is also analyzed. Last, this novel predicting model is applied to an anisotropic rotor partially filled with fluid, the unstable regions of rotational frequencies are analyzed. The results show that dimensionless damping coefficient and dimensionless stiffness coefficient have significant influences on the stability of a rotor partially filled with fluid, there especially exist two unstable regions for a lower dimensionless damping coefficient. In addition, the asymmetric degrees of an anisotropic rotor have great influences on the unstable low boundary.

Behavior of fluid flow and heat transfer induced by a stretching surface in the presence of polymers

This article discusses the effects of polymers on flow of a viscous incompressible fluid induced by a heated, permeable, stretching surface. Our main interest is to investigate the response of skin friction and heat transport in the presence of polymers. In consequence of polymer stretching, an additional stress arises in the equation of motion for the velocity field. The simplest Oldroyd-B model of polymers is used to study polymer presence. A set of equations is derived accordingly and solved numerically by using an iterative scheme. We find that the zero-shear viscosity reduces near the surface because of the polymers’ interaction with the velocity gradients in the boundary layer, but then goes back to zero-shear viscosity on moving away from the plate. Reduction in skin friction and heat flux is also noticed as a result of polymer addition in the base fluid. Dependence of this reduction on different parameters is also investigated.

Hydromagnetic Flow Over a Heated Stretching Surface

In the present paper on steady flow of a viscous incompressible fluid past a heated stretching sheet with thermal radiation is studied. Magnetic field is applied normal to the flow. With suitable similarity transformations, the momentum and energy equations are reduced to ordinary differential equations. The governing differential equations with corresponding boundary conditions are solved numerically using MATLAB inbuilt solver bvp4c,. Graphical results for velocity and temperature fields, tabular values of Skin-friction and Nusselt numbers are presented and discussed. It is found the temperature of the fluid increases for the increasing values of radiation parameter.

Analytical study of the Ekman angle for the Benard–Marangoni convective flow of viscous incompressible fluid

The paper considers the convective flow of a viscous incompressible fluid over a rotating surface. It studies the angle between the fluid velocity vector in the upper layer and the temperature gradient vector on the free surface. For the study, an analytical solution to the Oberbeck–Boussinesq equations is constructed, which describes the stratified Ekman flow taking into account two components of the Coriolis force. The temperature gradient and the conditions of Marangoni thermocapillary convection are set at the upper (free) boundary, and the condition of fluid adhesion is set on the lower (solid) boundary. The representation of velocities in the form of linear functions of horizontal coordinates is used. It is shown that, when the flow depth tends to infinity, the angle between the upper layer fluid velocity vector and the temperature gradient vector tends to π/2 .

On the Non-Smooth Solutions of 3D Navier-Stokes Equations for the Incompressible Fluid Flows

In the paper non-stationary 3D incompressible viscous fluid flow over the point, the infinite line, the plane, the rectangular prism and the octahedron are studied. The corresponding Navier-Stokes equations (NSE) with the appropriate initial-boundary conditions are considered. NSE is a very important equation and has various applications in Plasma Physics, Astrophysics, magma physics, geophysical fluids, biophysics, nanofluids, etc. NSE describes significant characteristics of different fluids. The exact solutions are obtained in a very few cases and especially in 2D. In the paper the novel exact non-smooth solutions blow-up in time are obtained for the specific pressure and initial conditions by means of the methods of mathematical physics (the main result). Besides, the solutions for the turbulent flows are given. Those solutions are new and are applied to solving of the problem of some substance transportation in the space by means of the turbulent flow. The profiles of the velocity and substance distribution are constructed by means of “Maple” for the different parameters. The results have applications to the description of atmospheric and ocean currents, nanosciences.

Benchmark solution for the stability of plane Couette flow with net throughflow

This paper investigates the stability of an incompressible viscous fluid flow between relatively moving horizontal parallel plates in the presence of a uniform vertical throughflow. A linear stability analysis has been performed by employing the method of normal modes and the resulting stability equation is solved numerically using the Chebyshev collocation method. Contrary to the stability of plane Couette flow (PCF) to small disturbances for all values of the Reynolds number in the absence of vertical throughflow, it is found that PCF becomes unstable owing to the change in the sign of growth rate depending on the magnitude of throughflow. The critical Reynolds number triggering the instability is computed for different values of throughflow dependent Reynolds number and it is shown that throughflow instills both stabilizing and destabilizing effect on the base flow. It is seen that the direction of throughflow has no influence on the stability of fluid flow. A comparative study between plane Poiseuille flow and PCF has also been carried out and the similarities and differences are highlighted.

A sharp interface Lagrangian-Eulerian method for rigid-body fluid-structure interaction

This paper introduces a sharp interface method to simulate fluid-structure interaction (FSI) involving rigid bodies immersed in viscous incompressible fluids. The capabilities of this methodology are benchmarked using a range of test cases and demonstrated using large-scale models of biomedical FSI. The numerical approach developed herein, which we refer to as an immersed Lagrangian-Eulerian (ILE) method, integrates aspects of partitioned and immersed FSI formulations by solving separate momentum equations for the fluid and solid subdomains, as in a partitioned formulation, while also using non-conforming discretizations of the dynamic fluid and structure regions, as in an immersed formulation. A simple Dirichlet-Neumann coupling scheme is used, in which the motion of the immersed solid is driven by fluid traction forces evaluated along the fluid-structure interface, and the motion of the fluid along that interface is constrained to match the solid velocity and thereby satisfy the no-slip condition. To develop a practical numerical method, we adopt a penalty approach that approximately imposes the no-slip condition along the fluid-structure interface. In the coupling strategy, a separate discretization of the fluid-structure interface is tethered to the volumetric solid mesh via stiff spring-like penalty forces. Our fluid-structure coupling scheme relies on an immersed interface method (IIM) for discrete geometries, which enables the accurate determination of both velocities and stresses along complex internal interfaces. Numerical methods for FSI can suffer from instabilities related to the added mass effect, but computational tests indicate that the methodology introduced here remains stable for selected test cases across a broad range of solid-fluid mass density ratios, including extremely small, nearly equal, equal, and large density ratios. Biomedical FSI demonstration cases include results obtained using this method to simulate the dynamics of a bileaflet mechanical heart valve in a pulse duplicator, and to model transport of blood clots in a patient-averaged anatomical model of the inferior vena cava.

First-order necessary conditions of optimality for the optimal control of two-dimensional convective Brinkman–Forchheimer equations with state constraints

This work is concerned about some distributed optimal control problems governed by two-dimensional convective Brinkman–Forchheimer (CBF) equations, which describe the motion of incompressible viscous fluid through a rigid, homogeneous, isotropic, porous medium. We address a distributed optimal control problem of minimizing a suitable cost functional subjected to 2D CBF equations with state constraints. We establish the existence of an optimal solution and obtain the first-order necessary conditions of optimality for such optimal control problems.