Stream Function Distributions Research Articles

Mathematical Modeling of Velocity Field Induced by the Vortex

In new technological applications, it is important to use vortex distributions in the area for obtaining large velocity fields. Analogous to electromagnetic and fluid mechanics inductions, vortices are described by the same relationship: the Biot-Savart law. When vortex threads form a vortex track due to fluid mechanics, they induce, analogous to electrical coils on an iron core, a core flow that can be faster than the wind that generates it. In this publication, the velocity field induced in a cylinder using the axially symmetric system of vortex rings and screw vortices and the hydrodynamic flow function in the ideal incompressible fluid are calculated. Also similar problem for mathematical modelling of the heat generation in fluids with alternating current using vortexes is considered. In this paper, it was calculated the distribution of the velocity field and distribution of stream function for ideal incompressible fluid, induced by a different system of finite number of vortex threads: (1) circular vortex lines in a finite cylinder, positioned on its inner, (2) spiral vortex threads, positioned on the inner surface of the finite cylinder or cone, (3) linear vortex lines in the plane channel, positioned on its boundary. An original method was used to calculate the components of the velocity vectors. Such kind of procedure allows calculating the velocity fields inside the domain depending on the arrangement, the intensity, and the radii of vortex lines. In this paper, we have developed a mathematical model for the process in the element of Hurricane Energy Transformer. This element is a central figure in the so-called RKA (ReaktionsKraftAnlage) used on the cars' roofs.

Numerical study of the porous cavity with different square size vanes with a high focus on Nusselt number

Numerical study of the porous cavity with different square size vanes with a high focus on Nusselt number

Mathematical modeling of velocity field induced by the vortex

In new technological applications, it is important to use vortex distributions in the area for obtaining large velocity fields. This paper, it was calculated the distribution of the velocity field and distribution of stream function for ideal incompressible fluid, induced by a different system of the finite number of vortex threads: 1) circular vortex lines in a finite cylinder, positioned on its inner, 2) spiral vortex threads, positioned on the inner surface of the finite cylinder or cone, and 3) linear vortex lines in the plane channel, positioned on its boundary. An original method was used to calculate the components of the velocity vectors. Such kind of procedure allows calculating the velocity fields inside the domain depending on the arrangement, the intensity, and the radii of vortex lines. In this paper, we have developed a mathematical model for the process in the element of Hurricane Energy Transformer. This element is a central figure in the so-called RKA (ReaktionsKraftAnlage) used on the cars’ roofs.

MHD mixed convection flow for Maxwell Hybrid nanofluid with Soret, Dufour and Morphology effects

We have investigated the two-dimensional mixed convective Maxwell hybrid nanofluid boundary layer mass and heat flows over a linearly stretching porous surface with the applied external magnetic flux. Thermal radiations along with the Dufour and Soret effects are also incorporated. The governing model of partial differential equations (PDE) is altered into ordinary differential equations (ODE) with an appropriate similarity transformation. The finite difference-based numerical method BVP4c is applied to solve the system of nonlinear ODEs. The flow features and the heat transfer characteristics have been illustrated with graphical representations and a numerical table. For varied values of the flow-related variables, organized and graphical data for the Nusselt number and skin friction coefficient are indicated. In most cases, spherical-shaped nanoparticles have a better influence on stream function, velocity and temperature distributions. This behavior is the opposite of the mass concentration profile. It has been observed that stream function decreases as increase the value of the magnetic field but opposite for mass concentration distribution and temperature profile. The temperature gradient is enhanced as a result of stronger convective flow when Soret number Sr values increase, which causes the boundary layer thickness to grow. A comparative study of hybrid nanofluid and nanofluid showed that the hybrid nanofluid has superior shear stress/skin friction and Sherwood number/surface mass flux than nanofluid flow.

Electro-Thermo-Convection of a Dielectric Liquid in the External DC and AC Electric Fields

The electro-thermo-convection of a dielectric liquid in a horizontal capacitor is investigated under the autonomous charge injection from the cathode and heating from above. In the case of a DC electric field, the linear stability analysis is carried out, and the thresholds of monotonic and oscillatory instability are determined. The finite difference method is used for the numerical simulation of the nonlinear behavior of electro-thermo-convective patterns: stationary convection and traveling waves. In the case of AC, electric field transient and permanent oscillations are analyzed. Two types of stable solutions are found. The modulated traveling waves are characterized by the quasiperiodic oscillations of convective characteristics. Another solution is modulated electroconvection (MEC). The patterns of MEC oscillate around some average flow synchronously with the external AC field and do not move laterally. The average intensity of convective mixing in modulated traveling waves is several times less than in modulated electroconvection. The spatiotemporal evolution of the stream function, temperature, and charge distributions for different types of transient and permanent solutions are analyzed.

Numerically investigation of Flow and Heat Transfer in differentially heated enclosures

In this research work natural convection in two dimensional cavity is studied numerically for differently heated air filled square and rectangular cavity. Vertical cavity with two adiabatic and hot and cold wall with different aspect ratio of 1, 2.5, 5, 7.5 and 10 has taken for investigation. the heat transfer was investigated over the wide range of Rayleigh number from 103 to 106 . Solutions are obtained for several Rayleigh numbers with Prandtl number Pr = 0.70 and aspect ratio 1, 2.5, 5, 7.5 and 10. The Pressure-Velocity Coupling Method used with Non-Iterative Time Advancement (NITA) scheme used to solve the problem. Flow patterns and isotherm plots were used to display the results, and for every case, local and mean Nusselt values were also produced. The results show that an increase in Rayleigh numbers greatly increases heat transfer, natural convection intensity, and average Nusselt number at a particular aspect ratio. The distribution of temperature and stream function are taken as a function of thermal and geometrical output parameter for given problem. Two stage investigations have been performed to find out the variation in the result between different approaches. Keyword: ANN (average Nusselt number), AR (aspect ratio), LNN (local Nusselt number), Nu (Nusselt number), Pr (Prandtl number), Ra (Rayleigh number).

The Influences of Slippage and Hall Currents on Peristaltic Transport of a Maxwell Fluid with Heat and Mass Transfer Through a Porous Medium

In this paper, the effects of slip velocity and Hall currents on peristaltic motion of a non-Newtonian fluid with heat and mass transfer through a porous medium inside a symmetric horizontal channel with flexible walls are studied. The fluid obeys Maxwell model, the ohmic and viscous dissipations are taken into account. Some of partial differential equations describe the fluid motion with the appropriate boundary conditions are written in dimensionless form and simplified by using the approximations of long wavelength and low Reynolds number. These equations are solved analytically, and the stream function, pressure rise, temperature, and concentration distributions are obtained as functions of physical parameters of the problem. The effects of the parameters of the problem on these solutions are discussed numerically and illustrated graphically through a set of figures. It is found that the physical parameters played important roles in controling the obtained solutions.

Influence Of Rotation, Variable Viscosity And Temperature On Peristaltic Transport In An Asymmetric Channel

In this paper we investigates the effects of each rotation, variable viscosity and temperature on the peristaltic phenomena in an asymmetric channel. The motion and heat equations are obtained in Cartesian coordinates, the dimensionless form of the governing equation are controlled by many dimensionless number e.g. Reynolds, Hartmann, Grashof , Prandle …These equations are nonlinear and to simplify ,the long wave length and low Reynolds number is used. The resulting dimensionless equation are then solved analytically by using perturbation expansion about Reynold model viscosity number. The effects of different parameter on axial velocity, stream function, pressure rise and heat distribution are analysis graphically by using the mathematica package.

Thermal diffusion and diffusion thermoeffects on the peristaltic motion of a non‐Newtonian micropolar fluid inside a nonuniform vertical channel

AbstractIn this paper, the effects of Dufour and Soret numbers on the peristaltic motion of a non‐Newtonian micropolar fluid are discussed. The motion inside a nonuniform vertical channel under the effect of the uniform magnetic field is considered. The Ohmic and elastic dissipations, as well as heat generation and chemical reaction, are taken into account. The problem is modulated mathematically by using continuity, momentum, angular momentum, and heat and mass transfer equations. The nonlinear partial differential equations describing these equations are written in terms of the physical parameters of the problem. The equations are transformed from the laboratory frame to the wave frame and then written in dimensionless form. The approximations of long wavelength and small Reynolds number are applied, then the equations are solved by using the homotopy perturbation method. The velocities, stream function, temperature, and concentration distributions are obtained as functions of the physical parameters of the problem. The effect of these parameters on the obtained solutions are computed mathematically and illustrated graphically through a set of figures. It is found that the parameters play an important role in controlling the solutions. It is found that the stream function decreases by increasing both non‐Newtonian and micropolar parameters on the left side of the channel and vice versa occurs on the right side.

Effects of radiative heat flux and heat generation on magnetohydodynamics natural convection flow of nanofluid inside a porous triangular cavity with thermal boundary conditions

AbstractWhen the nanoparticles are incorporated into the base fluid, the resultant fluid is known as nanofluid. Nanofluids have higher thermal efficiency as compared to base fluid. Some fluids have poor thermal conductivity like, water, air and ethylene glycol and oil. Thus, the thermal efficiency of the work can be increased by inserting the nanoparticles into base fluid. Furthermore, the nanoparticles can be used to enhanced the cooling rate of the system due to higher thermal conductivity. In this investigation, the magnetohydodynamics convective flow phenomenon under the consideration of different nanofluids inside a triangular porous conduit will investigated. For the better understanding of heat transfer characteristics, the thermal radiation and heat generation (or absorption) will also incorporated. We will impose constant and variable temperature on the left inclined wall to analyze the heat transfer mechanism. Furthermore, the heat transfer rate will also be analyzed by considering different nanoparticles. The robust numerical scheme namely the finite element method has been selected to simulate the nonlinear complex flow equations based on iterative scheme. In this technique, at the first stage the penalty method is employed for the purpose of the elimination of pressure from the equations of motion. After that the developed the system in the absence of pressure term is solved. To simulate the problem the value of penalty parameter is chosen 10−7. The contours of stream function and temperature distribution are displayed for several values of physical parameters. Furthermore, the variations of important quantity known as an average heat transfer rate are displayed through bar charts. For constant heating case the aluminum nano particles are the best choice to enhance the heat transfer rate in the system. Moreover, the magnitude of the stream function rises against the radiation and Rayleigh number. Furthermore, the conduction mode of heat transfer is achieved via thermal radiation parameter. The applications of the given study can be found in various industrial processes like, cooling of electronic devices, cooling of house and commercial buildings, cooling of microelectronics, cancer therapy, vehicle thermal management, and heat exchangers.

MHD double-diffusive mixed convection of binary nanofluids through a vertical porous annulus considering Buongiorno’s two-phase model

Binary nanofluids are prepared using a binary liquid (e.g., salt water) instead of a pure liquid as their host fluid. Double-diffusive convection in this type of nanofluids is a kind of triple-diffusive process, which simultaneously includes diffusions of heat, nanoparticles, and solute. This paper is devoted to double-diffusive mixed convection of binary nanofluids flowing through a vertical porous annulus in the presence of an externally applied radial magnetic field. To simulate the problem, a new mathematical model following the Buongiorno’s two-phase model is proposed. The equations are solved numerically by means of the finite-volume method. Thereafter, effects of the involved dimensionless variables on the distributions of stream function, temperature, solute concentration, and nanoparticles fraction as well as the mean values of the Nusselt number and the solute Sherwood number are presented and discussed. Inspection of the results demonstrates the significant contributions of the Peclet number (Pe), the usual Lewis number (Le), and the Soret-solute Lewis number (Ld) on the simulation results. In spite of that, the effects of the Hartmann number (Ha) and the thermophoresis parameter (Nt) on the simulation results are weak, while the consequences of the nanofluid Lewis number (Ln), the double-diffusive ratio (Nc), the buoyancy ratio (Nr), and the Dufour parameter (Nd) on the Nusselt and Sherwood numbers are insignificant.

Thermal radiation and natural convection in a large-scale enclosure heated from below: Building application

A computational research of radiative-convective energy transport in large-scale enclosure with a heat-generating heater under normal room conditions has been conducted. The heater (underfloor heating system) is located at the bottom of the room. Employing the Boussinesq assumption, the control equations have been solved contemporaneously to receive both the velocity fields and temperature patterns. To generate the systems of linear equations using vorticity and stream function, the finite difference technique has been employed. The developed convective-radiative model has been validated through a comparison with several problems. The influence of heater size and location, internal surfaces emissivity from 0 to 1, Ostrogradsky number for a wide range from 0 to 5 on Nusselt numbers and both stream function and temperature distributions has been investigated. The results demonstrate that the influence of the thermal radiation on total heat transfer increases with surface emissivity of walls and heater surfaces.

“Impress of rotation and an inclined MHD on waveform motion of the non-Newtonian fluid through porous canal”

Waveform flow of non-Newtonian fluid through a porous medium of the non-symmetric sloping canal under the effect of rotation and magnetic force, which has applied by the inclined way, have studied analytically and computed numerically. Slip boundary conditions on velocity distribution and stream function are used. We have taken the influence of heat and mass transfer in the consideration in our study. We carried out the mathematical model by using the presumption of low Reynolds number and small wave number. The resulting equations of motion, which are representing by the velocity profile and stream function distribution, solved by using the method of a domain decomposition analysis and we obtained the exact solutions of velocity, temperature, and concentration. The expressions of velocity, temperature, and concentration of the particles of the fluid have obtained and examined graphically by utilizing the soft wave of the Mathematica program. The efforts of various variables on mathematical modeling of motion and energy are discussed in detail. We found that.

Three-dimensional visualization of columnar vortices in rotating Rayleigh–Bénard convection

To enrich the three-dimensional experimental details of vortex structures in rotating Rayleigh–Benard convection, we established a technique visualizing three-dimensional vortex structures using scanning planar particle image velocimetry. Experiments were performed at fixed Rayleigh number, $$\hbox {Ra} = 1.0 \times 10^7$$ and different Taylor numbers from $$\hbox {Ta} = 6.0 \times 10^6$$ to $$1.0 \times 10^8$$ , corresponding to convective Rossby numbers from $$0.1 \le \hbox {Ro} \le 0.5$$ at which gradual transition between vortical plumes and convective Taylor columns regime is observed. Stream function distributions calculated from horizontal velocity vector fields visualize the vortex structure formed in the regimes. As quantitative information extracted from the visualized structures, distances between vortices recognized on the distributions show a good agreement with that evaluated by a theory. With the accumulated planar stream function distributions and vertical velocity component calculated from the horizontal velocity vectors, the three-dimensional representations of vortices indicate that quasi-two-dimensional columnar vortices straighten in the vertical direction with increasing Ta.

Variable Wall Permeability Effects on Flow and Heat Transfer in a Leaky Channel Containing Water-Based Nanoparticles

This work presents the effects of variable wall permeability on two-dimensional flow and heat transfer in a leaky narrow channel containing water-based nanoparticles. The nanofluid is absorbed through the walls with an exponential rate. This situation arises in reverse osmosis, ultrafiltration, and transpiration cooling in industry. The mathematical model is developed by using the continuity, momentum, and energy equations. Using stream function, the transport equations are reduced and solved by using regular perturbation method. The expressions for stream function and temperature distribution are established, which helps in finding the components of velocity, wall shear stress, and heat transfer rate inside the channel. The results show that velocity components, temperature, wall shear stress, and rate of heat transfer are minimum at the entrance region due to the reabsorption of fluid containing nanoparticles. Additionally, with increasing volume fraction of nanoparticles, the rate of heat transfer enhances at all positions inside the channel. Titanium dioxide (TiO 2 ) nanoparticles show higher wall shear stress compared to copper and alumina. The streamlines confirms that all the fluid is reabsorbed before reaching the exit region of the channel for high reabsorption.

Thermal Energy Storage Through Melting of a Commercial Phase Change Material in an Annulus with Radially Divergent Longitudinal Fins

This study reports on the unsteady two-dimensional numerical investigations of melting of a paraffin wax (phase change material, PCM) which melts over a temperature range of 8.7oC. The PCM is placed inside a circular concentric horizontal-finned annulus for the storage of thermal energy. The inner tube is fitted with three radially diverging longitudinal fins strategically placed near the bottom part of the annulus to accelerate the melting process there. The developed CFD code used in Tabassum et al., 2018 is extended to incorporate the presence of fins. The numerical results show that the average Nusselt number over the inner tube surface, the total melt fraction, the total stored energy all increased at every time instant in the finned annulus compared to the annulus without fins. This is due to the fact that in the finned annulus, the fins at the lower part of the annulus promotes buoyancy-driven convection as opposed to the slow conduction melting that prevails at the bottom part of the plain annulus. Fins with two different heights have been considered. It is found that by extending the height of the fin to 50% of the annular gap about 33.05% more energy could be stored compared to the bare annulus at the melting time of 82.37 min for the identical operating conditions. The effects of fins with different heights on the temperature and streamfunction distributions are found to be different. The present study can provide some useful guidelines for achieving a better thermal energy storage system.

周期壁面电势调制下平行板微管道中的电磁电渗流动

Two-dimensional magnetohydrodynamic (MHD) electroosmotic flow (EOF) in zeta potential patterned micro-parallel channels was studied. The flow was driven by the combination of the Lorentz force and the electric field force produced due to an externally imposed vertical magnetic field and two horizontal electric fields. The analytical solutions of stream function and velocity distribution were obtained under the condition of hydrodynamic slippage. The variations of velocities with related non-dimensional parameters, such as Hartmann number Ha,slip length B and electrokinetic width K were addressed in detail. Results show that, the patterned charged surfaces induce a vertical velocity component leading to the formation of the vortexes. Also, the magnitudes of velocities increase with slip length B and electrokinetic width K.Moreover, it is interesting to note that the magnitudes of velocities become small with the increasing value of Ha, unlike the situation where there exists a critical value of Ha in one-dimensional flow. The present theoretical results can be utilized to design efficient microfluidic devices.

Heat Transfer and Slip Effects on the Mhd Peristaltic Flow of Viscous Fluid in A Tapered Microvessels: Application of Blood Flow Research

This research work is proposed at reporting heat transfer on the peristaltic flow of an electrically conducting fluid in a tapered microvessels under the lubrication theory. The proposed geometry analyzes the blood flow in the heart vessels and maintain the pressure level in the human body. The solutions for the distribution of axial velocity, temperature distribution, pressure gradient and stream function have been obtained analytically. The influences of many evolving parameters on the flow characteristics are revealed and deliberated with the assist of figures. The mathematical outcomes show that the trapped bolus enhances in size with increasing slip parameter but decreases with the increase of Grashof number.

Multicellular convection flow in a cylindrical enclosure partitioned by a plate

This work performed a numerical investigation on the laminar natural convection in a cylindrical enclosure with a thin flat plate that is concentrically placed within the enclosure and is inclined with respect to the gravitational direction. The enclosure and plate are kept at low and high temperatures, respectively. The objective of this work is to explore the effects of the two geometrical parameters, i.e., the inclination angle and plate length, on the thermal and flow characteristics, with emphasis on the partitioning effect brought by the plate in generating the multicellular structures and weakening the intensities of fluid circulation and convection. The simulations were carried out using our in-house fourth-order finite difference code which is well validated in earlier studies. The effects of the two parameters are demonstrated by the variations of several characteristic quantities, including the overall heat transfer rate, temperature and stream-function distributions, spatial evolution of multiple vortices and local heat transfer pattern. Numerical results reveal that the inclined plate weakens the fluid circulation and lowers the heat transfer rate of the enclosure-plate system; the partitioning effect is strong for the long plate at nearly horizontal orientations. Depending on the magnitude of the two parameters, there can be at most three vortices within both the left and right halves of the enclosure, in which the primary vortices can be squeezed below the plate and the secondary or tertiary vortex occasionally emerges above the plate, and the thermal field is consequently determined especially in the top half of the enclosure. The spatial variation of the multiple vortices is greatly dependent on the two parameters, especially the two primary vortices. Finally, we observed that for a longer plate, the conduction in the gap between the end of plate and the enclosure becomes pronounced, as reflected by the magnitude and location of maximum local heat transfer rate on the enclosure surface.

Performance prediction of Roots blower based on shape of the rotor-profile using FVM with adaptive mesh redistribution technique and GA-tuned neural network

Attempt is made to define the shape of 2-lobe rotor-profile using three arcs. Mathematical model is proposed, and the rotor-profile is successfully parameterized in terms of two independent parameters (base and waist circle radius). Unsteady internal hydrodynamics and heat interaction of the Roots blower are captured using transient finite volume method with an adaptive mesh redistribution technique (moving mesh method) by considering different shapes of rotor-profile. At first, the present numerical scheme is validated with the results available in the literature. Spatial distribution of pressure, velocity, stream function, and temperature throughout the flow domain is extracted at different time instants considering different shapes of the rotor-profile. For better understanding, time variation in area-weighted average velocity magnitude, mass flux, and temperature at the exit port is also evaluated corresponding to each of the rotor-profiles. Transient distribution of pressure, velocity, temperature in the flow domain, leakage at the inlet, backflow at the outlet as well as the time variation in velocity, mass flowrate, and temperature at the exit port is found very sensitive to the shape of the rotor-profile. Attempt is made to establish relation between performance of the Roots blower and shape of the rotor-profile using multiple regression analysis and neurobiological computational approaches (neural network guided by parametric study and neural network tuned by genetic algorithm). Only using the indigenously developed hybrid neurobiological computational scheme with evolutionary algorithm (obtained through perfect amalgam of local search tool: backpropagation algorithm and global search tool: genetic algorithm), the said relation is perfectly captured and blower performance (in terms of discharge mass flux) is perfectly predicted (by knowing the shape of the rotor-profile).