Abstract
In the Stokes approximation, the problem of viscous fluid flow through two-dimensional and three-dimensional periodic structures is solved. A system of thin plates of a finite width is considered as a two-dimensional structure, and a system of thin rods of finite length is considered as a three-dimensional structure. Plates and rods are periodically located in space with certain translation steps along mutually perpendicular axes. On the basis of the procedure developed earlier, the authors constructed an approximate solution of the equations for fluid flow with an arbitrary orientation of structures relative to a given vector of pressure gradient. The solution is sought in a finite region (cells) around inclusions in the class of piecewise smooth functions that are infinitely differentiable in the cell, and at the cell boundaries they satisfy the continuity conditions for velocity, normal and tangential stresses. Since the boundary value problem for the Laplace equation is solved, it is assumed that the solution found is unique. The type of functions allows us to separate the variables and to reduce the problem's solution to the solution of ordinary differential equations. It is found that the change in the flow rate of a fluid through a characteristic cross section is determined mainly by the geometric dimensions of the cells of the free liquid in such structures and is practically independent of the size of the plates or rods.
Highlights
A system of thin plates of a finite width is considered as a two-dimensional structure, and a system of thin rods of finite length is considered as a three-dimensional structure
Plates and rods are periodically located in space with certain translation steps along mutually perpendicular axes
On the basis of the procedure developed earlier, the authors constructed an approximate solution of the equations for fluid flow with an arbitrary orientation of structures relative to a given vector of pressure gradient
Summary
В результате взаимодействия дисперсные частицы при определенных условиях образуют периодические структуры в жидкости [1,2,3,4]. Что при обтекании бесконечной периодической решетки сферических частиц при постоянном градиенте давления решении в виде квадратичной функции координат реализуется в определенной конечной области вокруг каждой частицы в решетке. Как одно из подтверждений сказанного, в работе [15] показано, что при учете гидродинамических взаимодействий всех частиц возмущения потока вида R−k не распространяются за границы ячейки Вигнера-Зейтца периодической решетки. Если задан градиент давления, то сначала необходимо найти периодическое распределение скорости в потоке вязкой жидкости в безграничной области, разбитой на ячейки, а затем использовать его для решения задачи о течении вязкой жидкости через периодическую решетку частиц. Ниже приводится метод построения приближенного периодического решения для течения вязкой жидкости с заданным градиентом давления через такие структуры, основанный на результатах работы [18]
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