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In this work, an analytical analysis of the dynamics of a van der Waals gas flow passing through a direct shock wave was performed. For this purpose, modified Rankine-Hugoniot conditions were used. The influence of parameters α and β of the van der Waals model and the pressure jump in the shock adiabat was analyzed. Relations for the velocity jump in flow were obtained, and the influence of parameters α and β on the velocity jump was revealed. Calculations made it possible to estimate the limits of applicability of the van der Waals model, within which it adequately describes the physics of the process under consideration.

The main attention is paid to the analytical analysis of an oblique shock wave in a turbulent adiabatic gas flow. For this purpose, a modified Rankine–Hugoniot model was obtained. On its basis, a solution was derived for the Rankine–Hugoniot conditions for a gas flow with various degrees of turbulence, as well as the equation of the modified Hugoniot adiabat. The behavior of the velocity of an adiabatic turbulent gas flow during its passage through an oblique shock wave at different levels of turbulence is demonstrated. A modification of Prandtl’s law for the velocity coefficients was obtained. The shock polar was also analyzed. The relationship between the angular gas flow and the angle of the shock wave was derived. Finally, the condition for the appearance of an outgoing bow shock wave was obtained.

The paper focuses on a theoretically study of fluid flow and heat transfer of Eyring-Powell fluid in a boundary layer over a flat surface. For the first time, groups of symmetry transformations were obtained for the full rheological law of Eyring-Powell fluid. This enabled obtaining self-similar forms of nonlinear differential equations different from the classical Blasius forms, which describe the limiting case for the large values of the parameter A. At the same time, the self-similar Blasius forms correctly describe the transition from the velocity profiles for a Newtonian fluid to linear profiles for Eyring-Powell fluid. The velocity and temperature profiles become less full due to the increasing parameter A, so that finally these profiles exhibit a linear shape. The friction coefficients and the Nusselt numbers decrease with the increasing parameter А.

The present study outlines a methodology for modeling instantaneous unsteady film condensation of vapor on a vertical flat wall. Both analytical and numerical solutions of the problem were obtained...

The results of the study of heat transfer under forced convection in a flat porous microchannel taking into account the boundary conditions of slippage of the first and second order are considered. The simulation showed that with decreasing porosity the flow velocity in the central part of the microchannel and the slipping velocity on the wall decrease due to the increase in hydrodynamic resistance. Taking into account the influence of the boundary conditions of the second order shows that the magnitude of the velocity jump on the wall varies depending on the value of the parameter A2. The jump decreases with a positive value of A2, with a negative value - increases in comparison with the case A2 = 0 (first order boundary conditions). Qualitatively similar effects of porosity and second-order boundary conditions were also observed with respect to temperature profiles. The results of the calculation of the relative Nusselt number showed that the decrease in porosity contributes to the intensification of heat transfer. The dynamics of the change in the heat transfer coefficient with an increase in the Knudsen number indicates that an increase in the Prandtl number also leads to an improvement in the thermal interaction of the flow with the channel wall. The analysis of taking into account the boundary conditions of the second order showed that at small values of the Prandtl number (Pr ≤ 1) the influence of the parameter A2 was not observed. At A2 < 0 the effects of the boundary conditions of the second order lead to an increase in the relative Nusselt number, whereas at A2> 0 the value of the normalized Nusselt number decreases in comparison with the case A2 = 0 (boundary conditions of the first order).

The paper presents results of the modelling of heat transfer at film boiling of a liquid in a porous medium on a vertical heated wall bordering with the porous medium. Such processes are observed at cooling of high-temperature surfaces of heat pipes, microstructural radiators etc. Heating conditions at the wall were the constant wall temperature or heat flux. The outer boundary of the vapor film was in contact with moving or stationary liquid inside the porous medium. An analytical solution was obtained for the problem of fluid flow and heat transfer using the porous medium model in the Darcy–Brinkman and Darcy–Brinkman–Forchheimer approximation. It was shown that heat transfer at film boiling in a porous medium was less intensive than in the absence of a porous medium (free fluid flow) and further decreased with the decreasing permeability of the porous medium. Significant differences were observed in frames of both models: 20% for small Darcy numbers at Da

A steady heat transfer process of mixed convection in a flat vertical porous microchannel is considered. The results of simulation showed that Knudsen number effects are more significant in the neighborhood of the wall where growth of Knudsen numbers is accompanied with the velocity and temperature jumps on wall. With increasing parameter of porosity M (decreasing permeability), the flow velocity decreases and the velocity jump decrease as well. For all combinations of the criteria Ra, Kn and M increasing Knudsen number reduces heat transfer intensity. This can be attributed to increasing temperature jump on wall which causes deterioration of thermal interaction between the fluid and the wall. For low Rayleigh numbers increasing parameter M leads to increasing heat transfer since the temperature jump decrease on walls. For large Rayleigh numbers the trend becomes reversed, since for larger parameters M, the near-wall velocity decreases. For low Rayleigh numbers increasing the Knudsen number leads to decreasing hydraulic resistance coefficient, but with increasing parameter M leads to increasing this coefficient. At high Ra numbers increasing Knudsen number leads to growth of hydraulic resistance, which is due to increasing velocity gradient on the wall.

Heat transfer with a steady flow of coolant in a vertical microchannel filled with a porous medium is considered. The influence of porosity, the slippage effect on the wall, and the Rayleigh number on heat transfer are analyzed. The simulation results showed that with an increasing of the porosity parameter M (decrease permeability), the flow velocity decreases, the velocity jump on the wall also decreases, and the velocity profile becomes more filled. With an increase in the Rayleigh number, the relative flow velocity decreases, the shape of the velocity profile changes, it becomes M-shaped. At high Rayleigh numbers, the effect of free convection becomes predominant, and the shift of the maximum velocity to the channel wall is associated with a decrease in the density of the medium near the wall. With an increase in the Rayleigh number and the parameter M, the temperature jump on the wall decreases, local temperature values tend to the wall temperature values, the shape of the temperature profile aligns. An increase in the Knudsen number decreases the heat transfer rate. This is due to an increase in the temperature jump on the wall, which causes degradation in the conditions of thermal interaction between the liquid and the wall.The dynamics of change of the relative Nusselt number with increasing Rayleigh number shows that there is an inversion of the influence of the porosity parameter M on the heat transfer coefficient. With small values of Ra, with an increase in the parameter M, the heat transfer coefficient increases, since the temperature jump on the wall decreases. At Ra = 400, the effect of porosity is not observed. At high values of Ra, the intensity of heat transfer decreases but not so sharply as at low of Ra. That effect is caused from decreasing rate flow near the wall.

The authors have analyzed the results of investigations of nucleate and fi lm regimes of boiling of nanofluids. In the nucleate regime of boiling of nanofluids, the increase in the critical heat flux compared to a pure fluid is observed; here, there can be both an increase and a decrease in the heat-transfer coefficient. The increase in the critical heat flux during the boiling of a nanofluid is linked to the formation of a deposited nanoparticle layer on a heated surface, which contributes to the change in the microcharacteristics of the process of boiling and to the decrease in the wetting angle of the surface. Results of theoretical investigation of the fi lm boiling of a nanofluid on a vertical heated surface have been presented. It has been shown that the greatest infl uence on the processes of heat and mass transfer in a vapor film is exerted by the superheating of the wall, the relation of the temperature and Brownian diffusions, and the concentration of nanoparticles in the fluid.

The paper presents results of a study of forced convection in a vertical flat and circular microchannels incorporating porous medium under slip boundary conditions of the first and second orders. The problem was solved analytically and compared with numerical simulations based on the lattice Boltzmann method. Effects of porosity and slip velocity on velocity and temperature profiles were investigated. Behavior of the normalized Nusselt number as a function of the Knudsen and Prandtl numbers, as well as the parameter M characterizing porosity of the medium in the microchannel, was also elucidated. Computations indicated that a decrease in porosity (an increase in the parameter M) causes a decrease in the velocity and temperature jumps on the wall, which contributes to the increase in the Nusselt number. Considering the effect of the second-order slip boundary conditions leads to the decrease in the velocity jump on the wall, when the coefficient A2 (this coefficient considers the second-order slip boundary conditions) changes from negative to positive values. The heat transfer rate at high Prandtl numbers increases with the increasing Knudsen number, because of the improved thermal interaction of the flow with the channel wall. Given the second-order boundary conditions, the effect of parameter A2 was not observed at small Prandtl numbers (Pr ≤ 1). For A2 0 the normalized Nusselt number decreases in comparison with the case of A2 = 0 (the first-order boundary conditions). The results by the analytical solution and the lattice Boltzmann method are in a good agreement with each other for A2 ≥ 0, with the differences ≤ 1%. For A2

Purpose The paper aims to consider heat transfer in incompressible flow in a rotating flat microchannel with allowance for boundary slip conditions of the first and second order. The novelty of the paper encompasses analytical and numerical solutions of the problem, with the latter based on the lattice Boltzmann method (LBM). The analytical solution of the problem includes relations for the velocity and temperature profiles and for the Nusselt number depending on the rotation rate of the microchannel and slip velocity. It was demonstrated that the velocity profiles at high rotation rates transform from parabolic to M-shaped with a minimum at the channel axis. The temperature profiles tend to become uniform (i.e. almost constant). An increase in the channel rotation rate contributes to the increase in the Nusselt number. An increase in the Prandtl number causes a similar effect. The trend caused by the effect of the second-order slip boundary conditions depends on the closure hypothesis. It is shown that heat transfer in a flat microchannel can be successfully modeled using the LBM methodology, which takes into account the second-order boundary conditions. Design/methodology/approach The paper is based on the comparisons of an analytical solution and a numerical solution, which employs the lattice Boltzmann method. Both mathematical approaches used the first-order and second-order slip boundary conditions. The results obtained using both methods agree well with each other. Findings The analytical solution of the problem includes relations for the velocity and temperature profiles and for the Nusselt number depending on the rotation rate of the microchannel and slip velocity. It was demonstrated that the velocity profiles at high rotation rates transform from parabolic to M-shaped with a minimum at the channel axis. The temperature profiles tend to become uniform (i.e. almost constant). The increase in the channel rotation rate contributes to the increase in the Nusselt number. An increase in the Prandtl number causes the similar effect. The trend caused by the effect of the second-order slip boundary conditions depends on the closure hypothesis. It is shown that heat transfer in a flat microchannel can be successfully modeled using the LBM methodology, which considers the second-order boundary conditions. Originality/value The novelty of the paper encompasses analytical and numerical solutions of the problem, whereas the latter are based on the LBM.

This paper focuses on a study of natural convection in a van der Waals gas over a vertical heated plate. In this paper, for the first time, an approximate analytical solution of the problem was obtained using an integral method for momentum and energy equations. A novel simplified form of the van der Waals equation for real gases enabled estimating the effects of the dimensionless van der Waals parameters on the normalized heat transfer coefficients and Nusselt numbers in an analytical form. Trends in the variation of the Nusselt number depending on the nature of the interaction between gas molecules and the wall were analyzed. The results of computations for a van der Waals gas were compared with the results for an ideal gas.

The paper represents an analysis of convective instability in a vertical cylindrical porous microchannel performed using the Galerkin method. The dependence of the critical Rayleigh number on the Darcy, Knudsen, and Prandtl numbers, as well as on the ratio of the thermal conductivities of the fluid and the wall, was obtained. It was shown that a decrease in permeability of the porous medium (in other words, increase in its porosity) causes an increase in flow stability. This effect is substantially nonlinear. Under the condition Da > 0.1, the effect of the porosity on the critical Rayleigh number practically vanishes. Strengthening of the slippage effects leads to an increase in the instability of the entire system. The slippage effect on the critical Rayleigh number is nonlinear. The level of nonlinearity depends on the Prandtl number. With an increase in the Prandtl number, the effect of slippage on the onset of convection weakens. With an increase in the ratio of the thermal conductivities of the fluid and the wall, the influence of the Prandtl number decreases. At high values of the Prandtl numbers (Pr > 10), its influence practically vanishes.

A correction to this paper has been published: https://doi.org/10.1007/s11242-021-01631-0

The paper focuses on a study of turbulence decay in flow with streamwise gradient. For the first time, an analytical solution of this problem was obtained based on the k‐e model of turbulence in one‐dimensional (1D) approximation, as well as on the symmetry properties of the system of differential equations. Lie group technique enabled reducing the problem to a linear differential equation. The analytical solution enabled parametric studies, which are computationally cheap in comparison to CFD based simulations. The lattice Boltzmann method (LBM) in two‐dimensional approximation (2D) was used to validate the analytical results. Large eddy simulation (LES) Smagorinsky approach was used to close the LBM model. Computations revealed that the rate of turbulence decay is significantly different for the cases of positive and negative streamwise pressure gradient. The further comparisons showed that the analytical solution underpredicts the predictions by the numerical methodology, which can be attributed to the simplified problem statement used to derive the closed‐form analytical solution. Comparisons of calculations with experiments revealed that the theoretical models used in the study underpredict the measurements for flows with a positive pressure gradient. Hence it can be concluded that the LBM technique combined with the LES Smagorinsky model requires the further modification.

The paper focused on an analytical analysis of the main features of heat transfer in incompressible steady-state flow in a microconfusor with account for the second-order slip boundary conditions. The second-order boundary conditions serve as a closure of a system of the continuity, transport, and energy differential equations. As a result, novel solutions were obtained for the velocity and temperature profiles, as well as for the friction coefficient and the Nusselt number. These solutions demonstrated that an increase in the Knudsen number leads to a decrease in the Nusselt number. It was shown that the account for the second-order terms in the boundary conditions noticeably affects the fluid flow characteristics and does not influence on the heat transfer characteristics. It was also revealed that flow slippage effects on heat transfer weaken with an increase in the Prandtl number.

The focus of the work is on analytical modeling of normal shock wave propagation in a turbulent adiabatic gas flow. For this, a modified Rankine–Hugoniot model was developed. A solution is obtained for the Rankine–Hugoniot conditions in a turbulent gas flow with different turbulence intensity. Variation of the velocity of an adiabatic turbulent gas flow during its passage through a normal shock wave is elucidated depending on the turbulence intensity. The equation of the modified Hugoniot adiabat is also obtained.

Results of a theoretical investigation of an acceleration liquid flow in a microcylinder beginning sudden rotation are presented. The problem on this flow was solved analytically with the use of the Laplace transform and the method of groups of symmetries as well as numerically using the method of Boltzmann lattices. The nonstationary flow-velocity profiles obtained analytically were compared with the results of a numerical solution. It is shown that the flow in the microcylinder develops by the asymptotic law and that the time of stabilization of this flow increases with increase in the Knudsen number. An analytical expression for the friction coefficient of the indicated flow has been obtained.

The paper focuses on modeling of turbulent flow in a rotating flat microchannel. Based on a linearized model of turbulence, an analytical solution of the two-dimensional problem for fully developed flow was obtained. With the help of a complex function, relations for two velocity components were derived in the form of the Kelvin functions. Validation of the analytical solution in comparison with the lattice Boltzmann model (LBM) technique demonstrated their good agreement except for the region close to the channel wall. Both analytical and numerical solutions were obtained for the first time for the given geometry. Effects of the dimensionless rotation parameter and the Knudsen number on flow structure were analyzed. It was shown how the velocity profiles deform with varying dimensionless rotation parameter and the Knudsen number. It was also demonstrated that higher rotation rates lead to an increase in hydraulic resistance, whereas an increase rate of slippage (Knudsen number) leads to a decrease in this resistance. For the first time, an analogy was revealed with radial outflow between parallel co-rotating disks including existence of the source region, Ekman-type layers on the channel wall and reverse flow near the centerline depending on the magnitude of the rotation parameter. For the limiting case of dominating rotation effects, expressions for the flow swirl parameter and tangent of the flow swirl angle on the wall were derived.

Based on the linear perturbation technique, an equation was derived for the prediction of hydrodynamic stability criteria in vapor layers formed at boiling of nanofluids in the inverse annular-dispersed regime. The equations were obtained in two-dimensional and three-dimensional approximations. A solution of the eigenvalue problem enabled determining criteria of hydrodynamic stability. For the first time, a complete analysis of the thermal-hydrodynamic instability of a vapor layer near a vertical surface has been performed based on the Orr-Sommerfeld approach, where not only the perturbed momentum equation, but also the perturbed energy equation and the perturbed convective-diffusion equation for the nanoparticle concentration were analyzed. This approach accounts for the influence of the unperturbed velocity, temperature, and nanoparticle concentration profiles. The analytical solution for the vapor film stability criteria was validated against experimental studies of the effects of nanoparticles on formation and destruction of a vapor film emerged in nanofluid boiling on the surface of a metal probe during unsteady cooling. In experiments, nanoparticles caused destabilizing influence on the vapor film during boiling.

The paper outlines results of a study of mixed free and forced convection in vertical flat and circular microchannels occupied with porous medium subject to slip boundary conditions. The problem was solved using analytical and numerical methodology including the lattice Boltzmann method. Effects of the Knudsen and Rayleigh numbers and porosity on velocity and temperature profiles, as well as the normalized Nusselt number and friction factor were elucidated. It was revealed that Knudsen number effects dominate in the vicinity of the wall, whereas the Rayleigh number effects are more significant near the channel axis. For high Rayleigh numbers, velocity profiles exhibit M-shapes having a point of minimum at the channel axis. In the same time, temperature profiles become uniform, so that the fluid temperature across the channel is almost equal to the wall temperature. An increase in porosity causes a decrease in the temperature and velocity jumps on the wall. It has been shown that for low Rayleigh numbers a decrease in the Darcy number causes an increase in heat transfer and hydraulic resistance. However, for high Rayleigh numbers, the trend becomes opposite. Comparisons of the quantitative results for the circular and flat channels revealed that the shape of channel cross section makes a significant influence on heat transfer and fluid flow. Results of the analytical studies were validated via comparisons with numerical simulations using the lattice Boltzmann method, which proved to be an accurate and powerful tool able to simulate mixed convection in microchannels, with the deviation from the analytical solution not exceeding 1%.

The article presents an analysis of the convective instability of a nanofluid in a vertical porous cylindrical microchannel with slip boundary conditions on the wall. The microchannel is surrounded by a continuous medium. For the calculation, the Galerkin method with a different order of approximation was used. The influence of the Darcy, Knudsen and Prandtl numbers, the ratio of the thermal conductivities of the fluid and the wall, the intensity of sliding along the wall, as well as the properties of nanoparticles on the Rayleigh number at which instability occurs, is analyzed. It was shown that an increase in the porosity of the medium promotes stabilization. It has been determined that the conditions for the onset of convection are decisively influenced by the thermal conductivity of nanoparticles (Lewis number), whereas the slippage effects (Knudsen number) are of lesser importance.

The paper considers prospects of multi-purpose use of high-temperature gas (helium) nuclear reactors. The processes of hydrodynamics and heat exchange in the modelled core of the hightemperature nuclear reactor with spherical fuel elements were studied. The influence of geometrical and mode parameters on the temperature distribution was analyzed. The paper presents results of calculating unsteady regime with reduction of in consumption of coolant flow in the core.

The paper focuses on a study of mixed convection in a vertically oriented circular microchannel with slip boundary conditions. Novelty of the work consists in analytical and numerical solutions of the problem using the Lattice Boltzmann method (LBM). The novel analytical solution enabled obtaining relations for velocity and temperature profiles and Nusselt numbers depending on the Knudsen, Rayleigh and Prandtl numbers. Knudsen number effects dominate in the vicinity of the channel wall, whereas close to the channel axis the Rayleigh number effects prevail. For high Rayleigh numbers, velocity profiles transform themselves to M-shapes with a minimum point at the channel axis, and temperature profiles flatten out and tend to the wall temperature. The magnitude of the temperature jump on the channel wall is a function of the Prandtl number and diminishes for high Prandtl numbers. For almost all computed cases, larger Knudsen numbers yield diminished heat transfer with an exception for the case of Pr = 10 and Ra = 200. Via comparing the quantitative results for circular and flat channels, it was demonstrated that the shape of channel cross-section makes significant influence on the average Nusselt numbers, so that the results for one channel cannot be extrapolated over the other one. It was also proved that mixed convection in circular vertical microchannels can be successfully modelled using the LBM methodology, whose deviation from the analytical solution is less than 1%.

The algorithm of modeling heat transfer in steady-state flow of a nanofluid in a plane channel on the basis of the Monte Carlo method is described. With the use of this algorithm, the profiles of temperature and of the volume fraction of nanoparticles in the channel are obtained. The dependence of the relative Nusselt number of flow on the volume fraction of nanoparticles in it and on the thermal conductivity of the fluid is shown.

Heat and mass transfer in a model of the core of a nuclear reactor with spherical fuel elements and a helium coolant was studied. The effect of permeability of the pebble bed zone and geometric parameters on the temperature distribution of the coolant in the reactor core is analyzed.

The paper presents results of an investigation into mixed convection in a vertically oriented microchannel with slip boundary conditions. Solutions of the problem were obtained analytically and using a numerical approach based on the Lattice Boltzmann method (LBM). The solution yields relations, which enable estimating velocity and temperature profiles and the Nusselt number as functions of the Knudsen, Rayleigh and Prandtl numbers. It was shown that Knudsen number effects are prevailing in the vicinity of the wall, whereas near the centerline of the channel effects of the Rayleigh number are stronger. If the Rayleigh numbers are high, velocity profiles demonstrate M-shapes with a point of minimum at the channel centerline, whereas temperature profiles flatten so that the fluid temperature in the channel cross-section tends to the wall temperature. The temperature jump magnitude on the wall is dependent on the Prandtl number value and decreases with the increasing Prandtl numbers. For almost all combinations of the parameters considered in this paper, higher Knudsen numbers entail heat transfer deterioration except for the case of Pr = 10 and Ra = 100. Increasing the Knudsen number diminishes hydraulic resistance for low Rayleigh numbers, but for high values of the Rayleigh numbers the trend is reversed. It was shown that mixed convection in microchannels can be successfully simulated using the LBM methodology, whose deviation from the analytical solution and is less than 1%.

In the present paper, renormalization group methods are used to develop a macroscopic turbulence model for thermal diffusivity in turbulent fluid flow under conditions of endothermic and exothermic chemical reactions in flow. The temperature field is divided into slow (large-scale) and fast (small-scale) modes. With the help of the renormalization procedure, energy equations for the large-scale modes and relations for effective turbulent thermal diffusivity were obtained. It was shown how the type of the chemical reaction affects turbulent thermal diffusivity. In addition, the conditions were identified where effective thermal diffusivity undergoes a sharp growth.

A correction to this paper has been published: https://doi.org/10.1007/s10973-021-10906-z

Using the RNG k-e model of turbulence in the approximation for porous media, a numerical study of fluid flow and heat transfer in a model of the core zone of a pebble-bed nuclear reactor was performed. Validations of the results of simulations against experimental data (for the friction factor) and analytical solutions (for the velocity profiles) confirmed applicability of the RNG k-e model of turbulence for the stated problem. An unsteady mode of operation with a decrease in the coolant mass flowrate at the inlet to the reactor core zone model, as well as a steady-state mode were studied. The effect of the pebble bed permeability on the dynamics of flow temperature variation was also investigated.

Different methods of nanofluid preparation and further preserving of their properties have been developed so far. One of them is a so called centrifugal method. During the nanofluid preparation in centrifugal devices, centrifugal instability of different nature may arise, which must be taken into account. Therefore, this paper focuses on a theoretical study of instability of Taylor–Couette flows of nanofluids in a curved channel formed by two concentric cylindrical surfaces. The flow is driven by rotation of the inner cylindrical surface. The collocation method is used to find numerically critical values of the Taylor number, which plays the role of the instability criteria. Effects of the following parameters on the critical values of the Taylor number were studied: the ratio of the radii of the concave and convex walls, the dimensionless parameters that describe the temperature gradient, the relative density of the nanoparticles, the ratio of Brownian diffusion and thermophoretic diffusion, as well as the Prandtl and the Schmidt numbers. It was demonstrated that an increase in the relative density, the ratio of Brownian diffusion and thermophoretic diffusion and Schmidt numbers incurs loss of the flow stability under both positive and negative temperature gradients. In the same time, increasing Prandtl numbers stabilize flow under the negative temperature gradient and destabilize it in the conditions of the positive temperature gradient. The theoretical results obtained in the paper help understand mechanisms of centrifugal instability in nanofluids and thereby optimize the functionality of centrifugal devices used to prepare nanofluids.

The concept of renormalization groups for modeling the parameters of flow in porous media is considered. An algorithm for the renormalization of the equations from the k–e model of turbulence is given. An expression is obtained for the coefficient of renormalized viscosity. Based on the model developed, conditions of the turbulent flow instability in a porous medium have been analyzed.

The paper outlines a novel mathematical model of convective instability in a vertical cylindrical microchannel. The model yields a transcendental equation, whose solution represents a dependence of the critical Rayleigh number on the harmonic number of the perturbation function, the ratio of thermal conductivities of the fluid and the solid wall, as well as the Prandtl and Knudsen numbers. The most dangerous are harmonics with n = 1, which corresponds to diametrically antisymmetric motions. An increase in the Knudsen number causes weakening of the system stability and a decrease in the critical Rayleigh number. This tendency is observed in the entire range of the ratio of thermal conductivities (0 … ∞) and Prandtl numbers (0.01 … 100) studied in the paper. A decrease in the thermal conductivity of the solid wall and the Prandtl numbers causes loss of the system stability, which can be explained by an increase in the amount of heat spent to arouse instability.

The paper presents results of an investigation of the slip flow in a curved microchannel with a rectangular cross-section. Solutions of the problem were obtained analytically using the Fourier method and the method of the eigenfunction decomposition demonstrating thus the applicability of two different analytical approaches to the solution of the fluid flow problems in microchannels. In addition, a numerical approach based on the Lattice Boltzmann method (LBM) was employed. The velocity profiles in the microchannel were obtained with the help of the infinite series solutions validated against the numerical simulations. The solution yields the relations, which enable estimating the hydraulic resistance coefficient and the initial length of the developing flow in the microchannel as the functions of the Knudsen number and the flow curvature parameter.

The paper outlines results of the theoretical study of an incompressible fluid flow in a rectangular microchannel subject to a sudden time-dependent pressure drop. The momentum equation together with the independent and dependent variables was reduced to a self-similar form by means of the symmetry analysis. The problem was solved using two analytical approaches, the Fourier method and the method of eigenfunction decomposition, as well as numerically by means of the lattice Boltzmann method. The unsteady two-dimensional velocity profiles in the microchannel were predicted using the infinite series and validated against the numerical solution. As expected, the flow pattern asymptotically attains the fully developed state, which is reached more rapidly for smaller Knudsen numbers. The analytical solution yielded expressions for the calculation of the hydraulic resistance coefficient.

The paper focuses on modeling of heat, momentum and concentration transport in stable film boiling of a nanofluid over a vertical surface. An approximate analytical model of the transport processes in the vapor film was employed in simulations. The model takes into account effects of the Brownian and thermophoretic diffusion. The novelty of the model consists in pinpointing six major non-dimensional parameters, which describe effects of the nanoparticles on heat transfer and fluid flow in the vapor film. They include: (i) parameter A that accounts for the relation between the thermophoretic and Brownian diffusion; (ii) the nanoparticle concentration φ∞ in the vapor; (iii and iv) the normalized densities of the nanoparticles Rpυ and Rpf; (v) the relative thermal conductivity of the nanoparticles K; and (vi) parameter m that characterizes the viscosity of nanofluids. Novel analytical solutions resulting from this model characterize velocity profiles, the mass flow rate, the thickness of the vapor film and the Nusselt number as the functions of the aforementioned parameters. It was demonstrated that an increase in the nanoparticles concentration fosters the processes of momentum, mass and heat transfer.