Introduction. We proposed a mathematical description of the flow of a Bingham fluid with transverse circulation in a rectangular channel of a worm machine. Materials and methods. As a material, we chose Bingham fluids with two constant parameters-viscosity and fluidity thresholds. We overviewed the influence of the transverse circulation on such characteristics of the visco-plastic flow as the size of the solid core, velocity of the core and the flow rate. The flow in a rectangular channel is formed as a result of superposition of flows in two plane channels which are crossed at right angles. Results. During the simulation of the three-dimensional flow of Bingham fluid in the channel of a rectangular cross-section with transverse circulation, two basic elements are applied. The first one consists of dividing the rectangle into a solid core and four rectangular sections of the viscous flow. The second element is that the viscous flow in each of the plots is two-dimensional, that is, longitudinal and transverse, but depends on the one coordinate. This means that such flows are equivalent to the flows with transverse circulation in the flat channel, the core of the flow of the Bingham fluid has a rectangle at the intersection. This approach allows to calculate all the basic characteristics of a complex three-dimensional flow in the explicit analytical form and analyze its dependence on the boundary conditions, taking into account the influence of all eight longitudinal and transverse conditions with any possible distribution on the boundaries of the channel. We proposed the calculation formulas for determining the velocity of the core of the current, flow rates and the values of dissipation energy in the symmetric form with respect to the coordinates. Conclusion. The mathematical description of the longitudinal flow of a Bingham fluid with transverse circulation in a rectangular channel of a worm machine allows to carry out the simulation of various flows of visco-plastic liquids and to determine the macro-kinetic characteristics at each point of the channel.

The problem of heat transfer of non-Newtonian fluids in the channels of chemical-technological equipment is considered. A mathematical model is proposed for determining heat transfer characteristics during the flow of Bingham fluids, generalized displaced fluids and power fluids in channels of different geometries. During the Bingham fluid flow, for the calculation of heat transfer coefficients, the convective temperature transfer equation is given in the approximation of the thermal boundary layer so that only the transverse derivative with respect to y is stored on the right side, and the x coordinate is assumed to be aligned along the tangent component of the fluid flow velocity. Nusselt numbers are determined by the derivatives of the tangent velocity on the walls of the channels and at the boundaries of the solid core. If the tangent of the fluid velocity on the wall has two components, then the velocity, the derivative of the Nusselt number, is determined through these components in accordance with the Pythagorean theorem. When a generalized shear fluid is used to calculate the Nusselt numbers, it must be taken into account that in a flat channel with longitudinal and longitudinal-transverse flows there are two heat transfer coefficients, and in a rectangular channel there are four heat transfer coefficients. The determination of the heat transfer coefficients of a power-law fluid is considered only for longitudinal flow in a flat channel and is carried out similarly to the calculation procedure for Bingham and generalized-shear fluids. The obtained expressions, when carrying out engineering calculations, allow us to calculate the corresponding heat transfer and heat transfer coefficients during the flow of non-Newtonian fluids in the channels and with the environment.