Viscous incompressible fluids

A refinement of previous hypotheses concerning the local structure of turbulence in viscous incompressible fluid for high Reynolds numbers. Journal of Fluid Mechanics 13, 82-85. 

We consider a single polymer molecule in solution. The solvent is treated as a viscous incompressible fluid. The polymer chain contains N monomer units or beads, and each individual bead is regarded as a point source of friction in the solvent. This is the standard model. 

Because Fj is the frictional force exerted on the bead by the medium, then - Fj is the force that the bead exerts on the medium. Now we recall that when a force (regarded as precisely localized at the position R ) is exerted on a viscous incompressible fluid, in steady motion at low Reynolds number, it produces an extra velocity field AV(R) everywhere in the medium. This extra velocity field... 

The most general equations for the laminar flow of a viscous incompressible fluid of constant physical properties are the Navier-Stokes equations. In terms of the rectangular coordinates x, y, z, these may be written ... 

Let us write the model of nonstationary flow distribution as applied to the problem of search for the maximum pressure rise at a given node of the hydraulic circuit at a fast cut off of the flow in one of its branches (or the largest drop at pipe break) provided that there is an isothermal motion of viscous incompressible fluid subjected to the action of the pressure, friction, and inertia forces (Gorban et al., 2006). find... 

Let us consider the problem, similar to Stokes first problem, in which a stagnant viscous incompressible fluid occupying the half-space Y > 0 is set in motion for t > 0 by a constant tangential stress to acting on the fluid surface (the simplest model of flow in the near-surface layer of water under the action of wind) [140]. In this case, the initial and boundary conditions for Eq. (1.9.1) are written as follows ... 

For steady-state flows of viscous incompressible fluid, by neglecting the inertia terms in (1.1.4) and by including all conservative mass forces in the pressure P, we arrive at the Stokes equations [464]... 

Low Reynolds numbers. In [216, 382] the problem on a circular cylinder of radius a in translational flow of viscous incompressible fluid with velocity Ul at low Reynolds numbers was solved by the method of matched asymptotic expansions. The study was carried out on the basis of the Navier-Stokes equations (1.1.4) in the polar coordinates 1Z, 6. Thus, the following expression for the stream function was obtained for IZ/a 1 ... 

Fixed cylinder. Let us consider a fixed circular cylinder in an arbitrary steady-state linear shear flow of viscous incompressible fluid in the plane normal to the cylinder axis. The velocity field of such a flow remote from the cylinder in the Cartesian coordinates X, X% can be represented in the general case as follows ... 

Statement of the problem. Thermal boundary layer. Let us consider heat transfer to a flat plate in a longitudinal translational flow of a viscous incompressible fluid with velocity U at high Reynolds numbers. We assume that the temperature on the plate surface and remote from it is equal to the constants Ts and 7], respectively. The origin of the rectangular coordinates X, Y is at the front edge of the plate, the X-axis is tangent, and the Y-axis is normal to the plate. 

Here we use the model of a viscous incompressible fluid, which is described by Navier-Stokes equations. 

Slezkin, N. A., Dynamics of Viscous Incompressible Fluid, Gostekhizdat, Moscow, 1955 [in Russian]. 

Ladyszenskaya O. A., Mathematical problems in viscous incompressible fluid dynamics, Nauka, Moscow, 1970 (in Russian). 

The method used to research stability of small perturbations of a cylindrical jet of non-viscous incompressible fluid is similar to the previously discussed method of handling the small perturbation problem. The main difference from the case of a plane surface is the axial symmetry of the problem, which consequently features the characteristic linear size a (radius of the jet). In a coordinate system moving with the jet s velocity, the jet itself is motionless. Let us neglect gravity and take into account only the force of surface tension. Then the pressure along the jet is equal to p = pa+ 2/a (since l i = oo, i 2 = a). Now proceed to linearize the equations and the boundary conditions. Assume the perturbations of the flow to be small and consider the equation of motion. After linearization, i.e. after rejecting the second order terms, one obtains Eq. (17.36) for velocity perturbations and full pressure. Since the flow is a potential one, any perturbation of the velocity potential satisfies the Laplace equation = 0, which in a cylindrical coordinate system (r, 0, z) is written as... 

We consider the steady flow of a viscous incompressible fluid in an infinitely long stationary channel of breadth h with no body forces present. The flow is everywhere parallel to the x-axis and the y-axis is placed at the bottom of the channel. 

Atabek H. B., and Lew, H. S., Wave propagation through a viscous incompressible fluid contained in a initially stressed elastic tube, Biophys. /., 6 481-503, 1966. 

Consider a spherical bubble of radius a which is moving through a viscous incompressible fluid which contains soluble surfactants. This motion can be due to buoyancy or thermocapillary migration in microgravity applications. It is easier to consider the equivalent problem which has a coordinate system fixed at the center of the bubble, with a uniform stream, of velocity C/qo say, far away. The fluid has density p and kinematic viscosity u and we assume that the surfactant concentration is uniform far from the bubble and has value Coo-... 

These steps are illustrated by applying the B.E.M. to the problem described by low Reynolds number fluid flow. By assuming the flow of a viscous, incompressible fluid at low Reynolds number the Navier-Stokes equations reduce to, in non-dimensional form,... 

Verma, P.D., Bansal, J.L., 1966. Flow of a viscous incompressible fluid between two parallel plates, one in uniform motion and the other at rest with uniform suction at the stationary plate. Proc. Indian Acad. 

We propose to describe the blood flow in the flexible large blood vessels and the artificial heart valve as a three dimensional nonstationary flow of viscous incompressible fluid with variable viscosity and density (see [13], [14], [15], [16]).Thus,the... 

Peskin, C.S., McQueen, D.M. A three-dimensional computational method for blood flow in the heart (I) immersed elastic fibers in a viscous incompressible fluid. J. Comput. Phys. 81, 372-405 (1989)... 

The main output stages of the two-component viscous incompressible fluid model were considered and numerical algorithm for solving the resulting model was chosen as well in this work. Calcrflations for two-dimensional and three-dimensional problems of the wave emergence and propagation on the free surface were carried out. 

Balaganckii, M.Y., Zakharov, Y.N., Shokin, Y.I. Comparison of two- and three-dimensional steady flows of a homogeneous viscous incompressible fluid. Russian Journal of Numerical Analysis and Mathematical Modelling 24(1), 1-14 (2009)... 

Moin, P. and Kim, J. (1980) On the numerical solution of time-dependent viscous incompressible fluid flows involving solid botmdaries. J, Comput, Phys, 35, 381-392. 

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