Correction for non-stationary of hydrodynamic fields in the rheological relationship of liquid

Эксплуатация различного рода технических устройств, в которых реализуются течения жидкости в каналах и трубах, всегда сопровождается нестационарными гидродинамическими процессами. Однако решение задач нестационарных течений жидкостей и газов зачастую приводит к существенных погрешностям, что дает основание исследователям основание сомневаться в справедливости реологических соотношений, учитывающих только неоднородность гидродинамических полей, но не учитывающих их нестационарность. Для устранения этих погрешностей при решении нестационарных задач течения жидкости в каналах и трубах в работах под руководством профессора С.К.Матвеева в выражение для касательного напряжения введена поправка, содержащая производную по времени скорости жидкости. Однако обобщение этой поправки на общий случай течения в тензорном виде оказывается невозможным. Поэтому в данной работе предлагается запись выражения для всего тензора напряжений в жидкости с поправкой на нестационарность, содержащей производную скорости, которая пригодна для описания пространственных течений жидкости. Рассмотрен частный случай нестационарного течения жидкости в плоском канале в одномерной постановке при использовании этой поправки. Показано, что такая модификация реологического соотношения приводит к решениям, согласующимися с решениями С.К.Матвеева. Также эта модификация может привести к уточнениям результатов решения для некоторых задач нестационарных течений. The operation of various technical devices in which fluid flows in channels and pipes are realized is always accompanied by non-stationary hydrodynamic processes. However, the solution of problems of unsteady flows of liquids and gases often leads to significant errors, which gives reason to researchers to doubt the validity of rheological relations, taking into account only the heterogeneity of the hydrodynamic fields, but not taking into account their unsteadiness. To eliminate these errors in solving unsteady problems of fluid flow in channels and pipes, in the work under the guidance of Professor S.K. Matveev, a correction containing the time derivative of the fluid velocity is introduced into the expression for shear stress. However, this correction is generalized to the general case of flow in tensor form turns out to be impossible. Therefore, in this paper, we propose writing an expression for the entire stress tensor in a fluid, adjusted for non-stationarity, containing the derivative of velocity, which is suitable for describing spatial fluid flows. A special case of unsteady fluid flow in a flat channel in a one-dimensional formulation using this correction is considered. It is shown that such a modification of the rheological relation leads to solutions matching the decisions of S.K. Matveev. Also, this modification can lead to more precise results of the solution for some problems of unsteady flows.

Generalization of the Lighthill problem for the viscous fluid filled tubes with complicated wall rheology

A generalization of the Lighthill model of the plane waves propagation along fluid-filled viscoelastic tubes is proposed. The rheological relation of the wall has two relaxation times for strains and stresses. The equations of the generalized model for the averaged pressure, velocity and the cross-sectional area of the tube are obtained. The solution of the equations in the form of the running waves and the dispersion relation are obtained and compared to those for the Lighthill and Shapiro problems, and the viscoelastic Kelvin-Voigt model for the wall material. Numerical calculations for the model parameters corresponded to human circulation system have been carried out. It is shown, the complicated properties of the material allow accounting for both Young and Lame wave modes, and stabilization the modes that were unstable in the case of simpler rheology. The developed model is helpful in performing the numerical calculations on complex models of arterial vasculatures at lower computation time and resources.

Nonlinear effects in the rheology of dilute suspensions

An analysis is presented of the deformation of a solid-like, viscoelastic sphere suspended in the infinite Stokesian flow field of a Newtonian fluid undergoing an arbitrary time-dependent homogeneous deformation far from the particle. The results of the analysis are then used to deduce the macroscopic rheological behaviour of a dilute monodisperse suspension of slightly deformable spheres.Even though inertial effects and second-order terms in the particle deformation are neglected, it is found that non-linear rheological effects can arise, because of the interaction between the deformed particle and the flow. As a consequence, the rheological relation obtained here differs from those presented earlier by Fröhlich & Sack (1946) and by Oldroyd (1955) through the appearance of certain terms which are non-linear in the deformation rate.When the suspended particles are purely elastic in their behaviour the rheological equation presented here reduces for certain flows to a special case of Oldroyd's (1958) phenomenological model, with material constants which can be directly related to suspension properties.