Stream Function Research Articles

Utilization of entire new dynamical system approach for peristalsis of Power law fluid with temperature-dependent viscosity

Peristalsis of non-Newtonian fluid with variable viscosity and viscous dissipation is considered in this paper. Modified Brinkman number highlighting both the variable viscosity and viscous dissipation is introduced. By utilizing the long wavelength and small Reynold number approximations, an analytical solution is obtained for stream function. The qualitative analysis is presented using the theory of dynamical system. The phase portraits representing topological structures like trapping, augmented flow and the backward flow are drawn. The change in qualitative flow patterns is discussed from the bifurcation diagrams. The exact size of the bolus is numerically determined. HIGHLIGHTS Analytical solution for Power law fluid with temperature-dependent viscosity. The theory of dynamical system is utilized for stability and bifurcation analysis. The topological structure of the flow is presented using bifurcation theory and phase plane analysis. The size of the bolus is calculated numerically.

Thermal analysis of nanofluid flow within porous enclosure with curved hot wall utilizing numerical approach

To evaluate the efficiency of a system involving wavy shapes of wall, numerical technique has been utilized in this article. The region between two wavy complex surfaces was fabricated from porous material and the working fluid is nanofluid (mixture of water with various shapes of alumina nanoparticles). The new terms related to porous media have been added according to Darcy law. The pressure terms were removed with defining stream function. The last form of equations contains two variables (θ and Ψ) which were solved via CVFEM. The results from the present code have been compared with previous article and good agreement was reported. With augmenting Ha, the speed of nanofluid decreases and Nu drops about 50.48% when Ra = 700. Considering greater buoyancy force makes Nu to increase about 61.36% when Ha = 0. The shape factor has a direct relation with the amount of conductivity, thus, Nu enhances about 11.97% with an increase of this factor when Ha = 15, Ra = 150.

An Analysis of Buoyancy-Driven Heat Behaviour and Flow Pattern of a Water-Copper Nanofluid in a Cylindrical Conduit

The flow of fluids and heat characteristics through free convection within an enclosed space has gained substantial study due to the various applications in manufacturing industries. This work examined the influence of buoyancy factors on normal convection in a heated tube filled with Copper (Cu) nanofluid. The method of finite difference was employed to characterize the regulating fluid formulae, and C++ programming language was employed to evaluate the Navier Stoke and continuity fields. This study examined Cu nanoparticles with particle sizes ranging from 1% to 10% and buoyancy values between 2.6 x 103 and 2.8 x 103 N. Cu nanofluid was used as the working fluid and the findings are presented as temperature gradient, Nusselt number, stream function, and vorticity curves. The findings revealed that an increase in the weight proportions of nanoparticles to 0.04 amplifies the buoyancy parameters to the highest value of 2.75 x 103 N; it yields a substantial enhancement in the heat transport rate by convection. Also, as the buoyancy factor increases, the temperature gradient, vorticity, and stream function of the nanofluid improve, while the local drag coefficient decreases. This study advances the understanding of buoyancy-driven convective flow and heat behavior in the technical design of floating vessels for safety and effectiveness.

Mathematical model for numerical simulations of thermal energy of nano-fluid in a complex peristaltic transport within a curved passage:Pharmacological and engineering biomedical application

The proposed scientific model is based on peristaltic movement of nanofluid within a curved channel of an electrically conducting material. To obtain a numerical solution for the electric potential function, the Debye-Hückel and Lubrication theory approximations are used. Stream function, electroosmotic, axial velocity, and curvature are all estimated using low zeta potentials. The behaviour of electroosmotic flow governs two dimensionless parameters: the Debye-Hückel parameter and the Helmholtz-Smoluchowski velocity. It is also investigated how curvature influences blood flow caused by electroosmosis. It was also investigated how the curvature parameter affects the nano-fluid flow induced by electroosmosis. Nano-biofluid evaluation with varied particle shapes such as cylinder, platelet, brick, and blade flowing in complex travelling wave. It can be stated that the velocity component of nanofluid is decreasing due to magnetic force, particle size, and wall compliance. It is also stated that nanoparticles considerably improve the thermal conductivity of the fluid.

Data-Driven Dynamical System Models of Roughness-Induced Secondary Flows in Thermally Stratified Turbulent Boundary Layers

AbstractThe goal of this work is to investigate the feasibility of constructing data-driven dynamical system models of roughness-induced secondary flows in thermally stratified turbulent boundary layers. Considering the case of a surface roughness distribution which is homogeneous and heterogeneous in the streamwise and spanwise directions, respectively, we describe the streamwise averaged in-plane motions via a stream function formulation, thereby reducing the number of variables to the streamwise velocity component, an appropriately introduced stream function, and the temperature. Then, from the results of large eddy simulations, we perform a modal decomposition of each variable with the proper orthogonal decomposition and further utilize the temporal dynamics of the modal coefficients to construct a data-driven dynamical system model by applying the sparse identification of nonlinear dynamics (SINDy). We also present a novel approach for enforcing spanwise reflection symmetry within the SINDy framework to incorporate a physical bias.

Individual Effect of Spatially Periodic Vertical Surface Temperatures and Nanoparticles on Natural Convection in Water

Abstract This paper considers the thermo-convective boundary-layer flow (BLF) of a water–copper mono-nanofluid over a flat vertical surface which is subjected to three types of periodic temperature variations described by the sinusoidal, sawtooth, and triangular waveforms. The temperature of the fluid at the flat surface is greater than the surrounding ambient temperature. The governing equations describing the BLF have been reduced to a non-similar form using an appropriate stream function formulation. The Keller-Box method is used to obtain numerical solution of the boundary-value problem. The effect of the pertinent parameters on the nature of the flow and the heat transfer has been discussed using actual thermophysical data. The results about the shear–stress and heat transfer rate at the surface are presented as well. To study the nature of BLF, the velocity and thermal boundary-layers, the streamline and isotherm plots have been considered, which reveal that the nanoparticle volume-fraction, amplitude of surface temperature variations, and the Grashof number play a pivotal role in enhancing/diminishing heat transfer. The final outcome reveals that the heat transfer is highest for the sinusoidal waveform, followed by that of the triangular and then, the sawtooth. An important inference is that a symmetric periodic temperature distribution at the surface enhances heat transfer more than that of a constant surface-temperature.

Core and conditionally rare taxa as indicators of agricultural drainage ditch and stream health and function

BackgroundThe freshwater microbiome regulates aquatic ecological functionality, nutrient cycling, pathogenicity, and has the capacity to dissipate and regulate pollutants. Agricultural drainage ditches are ubiquitous in regions where field drainage is necessary for crop productivity, and as such, are first-line receptors of agricultural drainage and runoff. How bacterial communities in these systems respond to environmental and anthropogenic stressors are not well understood. In this study, we carried out a three year study in an agriculturally dominated river basin in eastern Ontario, Canada to explore the spatial and temporal dynamics of the core and conditionally rare taxa (CRT) of the instream bacterial communities using a 16S rRNA gene amplicon sequencing approach. Water samples were collected from nine stream and drainage ditch sites that represented the influence of a range of upstream land uses.ResultsThe cross-site core and CRT accounted for 5.6% of the total number of amplicon sequence variants (ASVs), yet represented, on average, over 60% of the heterogeneity of the overall bacterial community; hence, well reflected the spatial and temporal microbial dynamics in the water courses. The contribution of core microbiome to the overall community heterogeneity represented the community stability across all sampling sites. CRT was primarily composed of functional taxa involved in nitrogen (N) cycling and was linked to nutrient loading, water levels, and flow, particularly in the smaller agricultural drainage ditches. Both the core and the CRT were sensitive responders to changes in hydrological conditions.ConclusionsWe demonstrate that core and CRT can be considered as holistic tools to explore the temporal and spatial variations of the aquatic microbial community and can be used as sensitive indicators of the health and function of agriculturally dominated water courses. This approach also reduces computational complexity in relation to analyzing the entire microbial community for such purposes.

Underlying geology and climate interactively shape climate change refugia in mountain streams

AbstractIdentifying climate‐change refugia is a key adaptation strategy for reducing global warming impacts. Knowledge of the effects of underlying geology on thermal regime along climate gradients and the ecological responses to the geology‐controlled thermal regime is essential to plan appropriate climate adaptation strategies. In the present study, the dominance of volcanic rocks in the watershed is used as a landscape‐scale surrogate for cold groundwater inputs to clarify the importance of underlying geology in stream ecosystems along climate gradients. First, using hundreds of monitoring stations distributed across multiple catchments, we explored the relationship between watershed geology and the mean summer water temperature of mountain streams along climate gradients in the Japanese archipelago. Mean summer water temperature was explained by the interaction between the watershed geology and climate in addition to independent effects. The cooling effect supported by volcanic rocks reached up to 3.3°C among study regions, which was more pronounced in streams with less summer precipitation or lower air temperatures. Next, we examined the function of volcanic streams as cold refugia under contemporary and future climatic conditions. Community composition analyses revealed that volcanic streams hosted distinct stream communities composed of more cold‐water species compared with nonvolcanic streams. Scenario analyses based on multiple global climate models and Representative Concentration Pathways (RCPs) revealed a geology‐related pattern of thermal habitat loss for cold‐water species. Nonvolcanic streams rapidly declined in thermally suitable habitats for lotic sculpins even under the lowest emission scenario (RCP 2.6). In contrast, most volcanic streams will be sustained below the thermal threshold, especially for low‐ and mid‐level emission scenarios (RCP 2.6, 4.5). However, the distinct stream community in volcanic streams and geology‐dependent habitat loss for lotic sculpins was not uniform and were more pronounced in streams with less summer precipitation or lower air temperatures. These findings highlight that underlying geology, climate variability, and their interaction should be considered simultaneously for the effective management of climate‐change refugia in mountain streams.

Asymptotic analysis of boundary layer flow of a fluid with temperature-dependent viscosity

In this work, we study boundary layer flows for thermo-dependent Newtonian fluids. We first show that the governing equations admit a self-similar solution. Furthermore, an asymptotic model is used to obtain correlations on heat transfer and the skin-friction coefficient. In addition, the numerical study of the complete problem enables us to specify the domain of validity of these correlations. Finally, the obtained numerical results are used to analyze the influence of the Prandtl number on the profiles of the stream function and the temperature field, and also on parameters such as the local Nusselt number and the thickness of the boundary layer.

Mathematical modeling of fluid flow and pollutant transport in a homogeneous porous medium in the presence of plate stacks

This study aims at investigating the numerical analysis of pollutant transport in homogeneous porous media with solid plate stacks. The investigation was performed for solid/impervious objects of the same size placed in homogeneous porous media. The pollutant transport equation (i.e., steady-state and time dependent advection-dispersion) chosen in mathematical modeling. Three cases arise on the basis of dispersion coefficients: (a) when dispersion is uniformly constant, (b) when dispersion depends upon magnitude of the velocity, and (c) when dispersion depends upon magnitude of the velocity and directional dispersivities, all these are discussed in detail. Generally, analytical solution of such problems doesn't exist, so all the work is done numerically. The governing partial differential equation of pollutant concentration is approximated by using finite difference technique. Central, one-sided, backward and forward finite difference formulae of the same order are used to discretize the domain.Simulations of velocity potential and stream function are approximated by Matlab software. Then equipotential lines and streamlines are visualized in the form of contours. Both, velocity potential and stream function are harmonic and satisfy Laplace's equation. Fluid flow lines and pollutant concentration are represented graphically for several parameters involved in the study.It is found that entrance/exit length, shape, hydraulic conductivity, the number and position of impervious objects affect the fluid flow and pollutant transport. However, there is no significant affect of heated objects on pollutant transport. Moreover, advection and dispersion depend upon permeability of porous media and properties of solid matrix.To authenticate the Matlab scheme of finite difference, it is verified that fluid as well as pollutant fluxes (in and out) are equal. Moreover, time-dependent problem converges to steady-state form after very long time.For monitoring or forecasting the build up of contamination concentration, the pollutant transport model is considerable. As this model is affected by different parameters which are discussed above, can helps to overcome the pollutant accumulation. The solid object is main key to lessen the contamination in the underground. If the entrance or leakage point of the domain is blocked by impermeable object or filled the vertical column with material of low hydraulic conductivity it ultimately slows down or even refrains the pollutant particles to pass through. The pollutant concentration is also minimized by injecting the bioremedial agents with the help of treatment columns.

Modified pressure and vorticity variables using Helmholtz decomposition for solution of the incompressible flow equations

Incompressible Navier–Stokes equations are reformulated using the Helmholtz decomposition of a velocity vector into rotational and potential components. By substituting the decomposed velocity in the time derivative term in a momentum equation, the potential component representing a gradient of a pressure-like term is combined with the gradient of the pressure modifying the physical pressure field. The rotational component representing a curl of a vortex-like vector is combined with the vorticity vector, making a non-physical vorticity vector that modifies the fluid viscosity. Thus, the unsteady Navier–Stokes equation is transformed into an explicitly steady-state form in terms of non-primitive variables. The stream function vector is governed by a parabolic equation in time, while the vorticity vector is governed by the Poisson function with a source term function of the convection stretching and time dependency of the physical flow vorticity. Therefore, the resulting system of equations is numerically independent of the cell Reynolds number stability condition that hunted the convection–diffusion Navier–Stokes equation. Numerical results are obtained for the two-dimensional driven cavity problem for Reynolds number of 1000 with 21 × 21, 41 × 41, 81 × 81, and 161 × 161 grid points. The computational grids correspond to cell Reynolds numbers 25, 12.5, 6.25, and 3.125, respectively. The computed results are smooth in all cases and validate the method.

Direct Simulation on the Dynamics of Liquid Films Flowing Down a Fiber

Direct simulations on the dynamics of a liquid film coating on the outer surface of a vertical fiber are performed in the present paper. A domain mapping technique has been used on solving the Navier-Stokes problem in stream function form and predicts the dynamics of the various flow regimes with remarkable accuracy. We investigated the morphologies that develop in the coating flow down fiber and analyze the effects of Bond number and fiber diameter on the maximum liquid film height and drop speed. The results showed that the Bond number and fiber diameter have a significant effect on the instability. The transition between the Plateau-Rayleigh and convective regimes occurs at the conditions when the flow transitions from absolutely to convectively unstable.

Influence of MHD and Porous Media on Peristaltic Transport for Nanofluids in An Asymmetric Channel

The aim of this paper is to discuss the influence of nanoparticles and porous media, and magnetic field on the peristaltic flow transport of a couple stress fluid in an asymmetric channel with different wave forms of non-Newtonian fluid. Initially, mathematical modeling of the two dimensions and two directional flows of a couple stress fluid with a nanofluid is first given and then simplified beneath hypothesis of the long wave length and the low Reynolds number approximation. After making these approximations, we will obtain associated nonlinear differential equations. Then, the exact solutions of the temperature distribution, nanoparticle concentration, velocity, stream function, and pressure gradient will be calculated. Finally, the results of drawings for many physical parameters that are of importance for examining the behavior of fluid flow quantities are discussed.

Mixed convection heat transfer and entropy generation of MHD hybrid nanofluid in a cubic porous cavity with wavy wall and rotating cylinders

The current simulations are for the three-dimensional (3D) Fe3O4/MWCNT-water hybrid nanofluid flow within a 3D cubic enclosure filled with a porous medium. It was assumed that the flow area contains two rotating cylinders and has an upper wavy wall. Also the magnetic field is also applied to the problem due to magnetic nanoparticles in the base water. The Galerkin finite element method (GFEM) with a triangular element shape was applied to solve the governing equations. The findings were shown for a range of flow parameters such as angular speed of the cylinder (Ω=-500–1000), Hartmann number (Ha = 0–10), Darcy number (Da = 10−5–10−2), and direction of cylinders rotations and their positions in the cavity. The influence of the various parameters on flow, thermal transport, and entropy production is illustrated by the stream function, isotherms, and isentropic contours. Higher values of Ω Da and lower values of Ha enhanced the heat transfer and Nusselt number. Entropy production is mostly due to heat transfer; however, fluid-friction and magneto effects also contribute.

Peristaltic Transport of a Couple Stress Fluid through a Horizontal Tapered Asymmetric Channel

Abstract: The peristaltic transport of a Couple stress fluid in the horizontal tapered asymmetric channel is examined, concealed by long wavelength and low Reynolds number assumptions. The tapered asymmetric channel is produced due to the peristaltic motion on the walls of the non-uniform channel having alternative amplitudes and phases. Excellent velocity, pressure gradient, and stream function results have been determined. Numerical evaluation has also been found for investigating the pumping characteristics with various flow values of parameters. It has been shown that the peristaltic transport pumping decreases with an increase in non-uniform rheological parameters. Further, our conclusion is created to be very relevant to [19] when the absence of the tapered asymmetric nature of the channel and couple stress fluid parameter.

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.

Application of Keller-Box Method to the Heat and Mass Transfer Analysis of Magnetohydrodynamic Flow of Micropolar Fluid between Porous Parallel Walls of Different Permeability

The present work seeks to investigate the problem of stable laminar micropolar fluid flow through porous walls of varying permeability. Furthermore, the effect of an external magnetic field is investigated. The resulting solution is extended to analyse the nonlinear differential equation of heat and mass transfer in the flow situation. The velocity and microrotation at the channel’s entrance are assumed to be those of Poiseuille flow. Using appropriate transformations, the governing nonlinear equations are transformed to a system of ordinary differential equations, which are then solved using a novel numerical technique based on a finite difference scheme termed the Keller-box method. The results show that the smooth and straightforward Keller-box method is an appropriate method for estimating solutions of complex governing equations since it is simple to implement. Graphs and tables are used to explain the effect of significant parameters such as the suction Reynolds number, micro rotation/angular velocity parameters, and Peclet number on the stream function, temperature distribution, and concentration properties of the fluid. The effect of wall permeability on longitudinal velocity and microrotation has been investigated. The couple stress and skin friction on the walls have also been calculated. The effect of variation in Reynolds number is showing the importance of inertial forces in comparison with viscous forces. The velocity boundary layer is observed to decrease when there is an increase in Reynolds number. Furthermore, the accuracy and convergence of the obtained solutions show that the Keller-box method is applicable to various nonlinear physical problems, even those with significant non-linearity.

A NOVEL CONCEPT FOR FINDING THE FUNDAMENTAL RELATIONS BETWEEN STREAM FUNCTION AND VELOCITY POTENTIAL IN REAL NUMBERS IN TWO-DIMENSIONAL FLUID MOTIONS

In this paper, the author has presented the fundamental relations between stream function or current function,  and velocity potential or velocity function, φ which are ∂φ/∂x= ∂/∂y and ∂φ/∂y= – ∂/∂x where x,y,φ(x_(, ) y),  (x_(, ) y) are all real in two-dimensional fluid motions using real variables only whereas these relations had been established by using complex variables by Cauchy – Riemann which are known as Cauchy – Riemann equations in fluid dynamics.

A study on the bifurcation of the peristaltic driven flow of Ellis fluid

In this work, a bifurcation analysis is presented for the transport of Ellis fluid model due to peristaltic activity. The position, qualitative nature and bifurcations of stagnation-points are deliberated through the dynamical system approach. An exact expression for stream function is found by using the lubrication approach. Based on the stream function, a system of autonomous differential equations is established to analyze the equilibrium-points. Three kinds of flows are observed, namely backward flow, trapping and augmented flow. The transition of one flow into other represents a bifurcation. In the first bifurcation, a saddle point bifurcates into two centers in the normal direction while the second bifurcation emerges where nearby saddle nodes join under the wave trough and then split into two branches in the transverse direction. Ramifications of different embedded parameters on the bifurcations and nature of stagnation-points are explored through graphical representations. It is found that bifurcations transpire earlier for viscous fluids as compared to shear-thinning fluids. Further, bifurcations appear at low flow rate for small values of the material parameter . Moreover, in Newtonian fluids, as opposed to shear-thinning fluids, the trapping starts earlier and the backward region expands for larger values of the material parameter .

Influences of rotation and mass and heat transfer on MHD peristaltic transport of Casson fluid through inclined plane

In the present analysis, we discussed the influence of heat and mass transfer on the hydro-magnetic peristaltic flow of a Casson fluid in a rotating inclined system through an asymmetric channel. The model governing equations are obtained, simplified by applying long wavelength and assumptions of low Reynolds number, and then addressed analytically. DSolve command of Mathematica software was used to solve these equations. The numerical results regarding the impacts of embedded parameters on axial velocity, secondary velocity, stream function, temperature, concentration, and streamline patterns are graphed and explained. It has been discovered that velocity slip reduces the size of the trapped bolus while the permeability parameter increases it. The velocity and temperature profiles decrease as the Casson fluid parameter and magnetic field strength increase. In the fields of biomedical engineering and crude oil refinement, the current problem is quite significant. The trapping phenomenon for different parameters is presented. The visual discussion of numerical results for various values of relevant physical parameters. The current work, which focuses on electromagnetic peristaltic pumps, offers several applications in biomedical engineering.