Governing Equations Of Fluid Flow Research Articles

Numerical Investigation of Effect Partially Corrugated Heated Wall on Behaviour Natural Convection Heat Transfer in A Square Enclosure

For fluid flow and heat transfer with engineering industrial applications important methods called natural convection are used. The current numerical study focuses on the effect of changing the heating positions of the corrugated left wall on heat transfer by natural convection with laminar flow in a closed square two-dimensional cavity filled with air as a working fluid. The code of a commercial program (COMSOL Multiphysics 6.0) was used to solve the mathematical equations governing fluid flow represented by continuity, momentum, and conservation of thermal energy. The right wall is exposed to a cold temperature, the other walls are thermally insulated. The effect of partial heating of the left corrugated wall and the Rayleigh number on the process of heat transfer by free convection was studied. At different Rayleigh numbers, flow field patterns and temperature distribution lines appear. The average Nusselt number, the average air temperature, and the flow field are slightly affected by low Rayleigh numbers, while with high Rayleigh numbers, these parameters are significantly and noticeably affected and behave differently by changing the heating sites. Numerical results show negligible effects of wall heating sites at low Rayleigh numbers and significant effects at high Rayleigh numbers. Also, the Nusselt number increases progressively with increasing Rayleigh number, so the heat transfer process gets improved.

Numerical Examination of a Squeezing Casson Hybrid Nanofluid Flow Considering Thermophoretic and Internal Heating Mechanisms

One of the main areas of study in the field is increasingly the flow of non-Newtonian fluids. These liquids find extensive use in nuclear reactors, food processing, paint and adhesives, drilling rigs, and cooling systems, among other industrial and engineering domains. However, hybrid nanofluids are crucial to the process of heat transfer. Considering this, this study investigates the motion of a Casson hybrid nanofluid squeezing flow between two parallel plates under the influence of a heat source and thermophoretic particle deposition. The Runge–Kutta–Fehlberg fourth–fifth-order approach is utilized to numerically solve the ordinary differential equations derived from the partial differential equations governing fluid flow, by utilizing suitable similarity variables. The diagrams show how several important parameters affect fluid profiles both with and without the Casson parameter. These figures demonstrate how fluid velocity increases as the local porosity parameter increases. When the heat source/sink parameter is increased, thermal dispersal increases, and when the thermophoretic parameter is increased, the concentration profile increases.

Impact of chemical reaction on hybrid nanofluid (GO + MoS2) flow over an exponentially stretching sheet with Soret and Dufour effects

AbstractThe Soret and Dufour effects play a crucial role in various fields such as geosciences, groundwater pollutant migration, chemical reactor operations, binary alloy solidification, and isotope separation. This study focuses on examining the impact of mixed convective flow on hybrid nanofluid through an exponentially stretching sheet with Soret and Dufour effects. The flow is affected by factors like variable viscosity, radiation, viscous dissipation, and activation energy. Instead of the no‐slip condition at the boundary, velocity slip, thermal slip, and concentration slip are considered. The physical problem is modeled using boundary layer theory, and flow patterns are expressed using partial differential equations (PDEs). These governing fluid flow equations are transformed into non‐linearly coupled ordinary differential equations (ODEs) using exponential similarity transformations. These simplified ODEs are resolved using the MATLAB bvp4c package. The effects of physical parameters on velocity, temperature, and concentration are illustrated through figures. Additionally, the drag force coefficient and heat and mass transfer rates are calculated for various parameters and presented graphically and in tabular form. It is observed that compared to nanofluids, the drag force coefficient of hybrid nanofluids increases by up to 21.05% with various solute buoyancy parameters (δ). Also, the mass transfer rate of hybrid nanofluids can be increased by .96% by the chemical reaction rate (σm). A comparison of this work with previously published research has been reported.

Study of pulse wave phenomena associated with blood flow model in human viscoelastic artery

Cardiac output monitoring has proven to be a promising hemodynamic management tool especially for critically ill patients. Pulse wave analysis is a noninvasive method used to quantify cardiac output continuously with respect to time. In this article, we have proposed a novel methodology to quantify the contribution of pulse waves to further study the role of arterial wall relaxation with respect to time. The relaxation time can further help in the diagnosis of disease. The pulse wave velocity component is derived by transforming governing fluid flow equations into hyperbolic equations for laminar incompressible blood flow in an artery of viscoelastic walls. The viscoelastic behavior of the wall is analogously modeled by the modified Zener model that has the capability to measure creep, stress relaxation, and hysteresis. The derived model equations are solved numerically by the multiderivative Runge–Kutta implicit–explicit time integration method with a weighted, essentially non-oscillatory, discretization scheme. The results are well validated with clinical trial-based Riemann problems for the case of elastic walls. It is observed that the proposed modified Zener model is well suitable to identify the location of arterial stiffness where the pulse wave manifests various types of phenomena like discontinuity and reflection.

Magnetic dipole interactions with tangent hyperbolic ferromagnetic fluid over a moving plate: A numerical study

The present study aims to numerically analyze a non-Newtonian-based ferrofluid for its heat transfer characteristics and flow properties. The non-Newtonian nature of the ferrofluid, coupled with the application of a magnetic field, induces heat dissipation, which is further modified by incorporating viscous dissipation effects into the mathematical model. The mathematical model utilized in this study is based on the classical dynamic equations governing fluid flow. Similarity transformations were applied to derive a system of coupled nonlinear ordinary differential equations (ODEs). The resulting system was efficiently solved using the numerical technique known as the Keller Box method (KBM). Graphical results illustrate that the strengthening of an external magnetic field disrupts the flow in non-Newtonian ferrofluid, primarily due to the Lorentz forces induced by thte applied magnetic field. Moreover, the magnetic field enhances the thermal conductivity of the TH-fluid, leading to an increased heat transfer rate at the surface of the sheet as well as within the boundary layer.

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

On the Existence of Steady-State Solutions to the Equations Governing Fluid Flow in Networks

Heat and mass transfer in peristaltic flow of MHD non-miscible micropolar and Newtonian fluid through a porous saturated asymmetric channel

The current flow problem primarily focuses on the production of entropy as a result of radiative heat and mass transfer during peristaltic transport of non-miscible Newtonian and micropolar fluid through a porous saturated channel. The movement of immiscible fluid has been simulated using a two-phase flow model with Newtonian fluid at the channel's periphery and micropolar fluid in the core area. Here, under the long-wavelength approximation and low Reynolds number assumption, the linearized governing fluid flow equations were solved by classical technique and obtained closed-form solutions for pressure rise, volume flow rate, velocity, temperature distribution, and concentration. In the present peristaltic flow problem, authors evaluated the entropy analysis in non-miscible type of Newtonian and micropolar fluid through the porous saturated channel under the influence of heat radiation, mass transfer, and an oriented magnetic field. The production of entropy and Bejan number due to concentration distribution in peristaltic motion of non-miscible type of Newtonian and micropolar fluid in a porous medium is the novelty of the work. From this study, we obtained that on raising the Soret and Schmidt number, the concentration of the immiscible micropolar and Newtonian fluid drops, and entropy generation number rises. Also, the temperature and flow distribution of the immiscible micropolar and Newtonian fluid get decreased on raising the permeability parameter and Hartmann number. The authors come to the significant conclusion that, during the peristaltic motion of immiscible fluid in a porous saturated asymmetric channel, naturally occurring porous material achieved higher values of flow properties like axial pressure gradient, temperature profile, concentration, and entropy generation number than man-made porous material. The newly obtained findings of the present peristaltic flow problem are validated with the past published research work. The findings of this study may be used in biomedical engineering, applications, including the thermal therapy procedure.

Analysis of Bejan number and Entropy generation of Non-Newtonian nanofluid through an asymmetric microchannel

This study examines the electro-kinetic peristaltic movement of a Sutterby nanofluid under the influence of entropy generation and Bejan number. The fluid flow was introduced into the undulating asymmetrical microchannel. The Sutterby nanofluid flow was considered in the static electric field in the horizontal direction and the applied external magnetic field in the transverse direction. The mathematical formulation of the implications of Bejan number and entropy generation are also explored. The governing fluid flow equations, such as the continuity, momentum, and temperature equations, are reduced considering the tiny wave number and the small Reynolds number approximation. The resulting nonlinear equations were resolved numerically with the help of the built-in NDSolve coding through the computational mathematical software MATHEMATICA. Graphical illustrations of the fluid velocity profile, temperature, entropy generation, Bejan number, and trapping phenomenon (streamlines) are presented in detail. The results from the Sutterby nanofluid model might have a broad range of applications, such as cancer tissue destruction, disease diagnosis, etc. It was concluded that there is a decrease in the fluid velocity and an increase in the Hartmann number.

Numerical Investigation of Fluid Flow and Heat Transfer Characteristics over Double Backward-Facing Step with Obstacles

Many engineering device, including nuclear reactors, heat exchangers, and electronic circuit boards, exhibit flow separation and reattachment. To reduce thermal stress impacts, the components of these devices, which resemble stepped channel designs and are susceptible to high heat flux, must be properly cooled. The present work numerically investigates the fluid flow and heat transfer characteristics through a double backward-facing step with elliptic obstacles located after each step. For this purpose, equations governing fluid flow and heat transfer are solved in a Cartesian framework using an in-house code based on streamline upwind/Petrov-Galerkin finite element method. The effect of various parameters (axis ratios (0.25, 0.5, 0.75, 1), vertical location of obstacles (0.38, 0.577, 0.769), Reynolds number (300, 500, 800, 1000)) on fluid flow and heat transfer characteristics in the channel with and without obstacles are compared and quantified. The recirculation region length of the double backward-facing stepped channel with obstacles decreases after the first and second steps when compared with no obstacles. The local Nusselt number distribution along the corner regions of the stepped wall is enhanced due to obstacles. The axis ratio of obstacles has less influence on the convective heat transfer enhancement than the vertical location of the obstacles.

Observing a dynamical skeleton of turbulence in Taylor-Couette flow experiments.

Recent work shows that recurrent solutions of the equations governing fluid flow play an important role in structuring the dynamics of turbulence. Here, an improved version of an earlier method (Krygier et al. 2021 J. Fluid. Mech. 923, A7 and Crowley et al. 2022 Proc. Natl Acad. Sci. USA 119, e2120665119) is used for detecting and analyzing intervals of time when turbulence 'shadows' (spatially and temporally mimics) recurrent solutions in both numerical simulations and laboratory experiments. We find that all the recurrent solutions shadowed in numerics are also shadowed in experiment, and the corresponding statistics of shadowing agree. Our results set the stage for experimentally grounded dynamical descriptions of turbulence in a variety of wall-bounded shear flows, enabling applications to forecasting and control. This article is part of the theme issue 'Taylor-Couette and related flows on the centennial of Taylor's seminal Philosophical Transactions paper (part 1)'.

Quadratic convection in a power-law fluid with activation energy and suction/injection effects

Current research demonstrates the effects of activation energy and quadratic convection on the non-Newtonian power-law fluid flow through an inclined permeable plate. In addition, the fluid’s heat transfer is carefully investigated while taking convective boundary conditions and thermal radiation into account. Also, mass transfer between the flow and the inclined plate was investigated for binary chemical reactions. By appropriate transformations, governing fluid flow equations are changed into the dimensionless form. Numerical solution of this system of equations is obtained by both the local non-similarity method and the successive linearisation method. The effect of emerging parameters on the flow and physical quantities are examined with Biot number ( B i ), activation energy coefficient ( E ), the chemical reaction parameter ( M ), quadratic convection parameters ( α 1 , α 2 ), inclination angle ( Ω ), the suction/injection parameter ( f w ) and the radiation parameter ( R d ). Further, the rate of mass and heat transfer are compared for all the instances of linear and quadratic convective flows. Power-law fluids concentration is increased with activation energy, while it diminishes with chemical reaction. The thickness of thermal boundary layer is considerably enhanced by both Biot number and radiation parameters. The thermal and solutal rates are increasing functions of quadratic convection parameters.

High-order compact difference schemes on wide computational stencils with a spectral-like accuracy

In this paper, we propose a generalized framework for the derivation of compact difference schemes for the approximation of an arbitrary order derivative. We formulate a methodology for calculating the approximation coefficients involving a large number of mesh points leading to very high-order formulas, e.g., 20th or 30th order. The main emphasis is on the first and second order derivative occurring in the Navier-Stokes equations governing fluid flow problems. Contrary to most studies on compact schemes in which the computational stencils employed in the discretization are narrow and consist of only a few mesh points K, in the presented paper we consider wide stencils that in practice are limited only by the overall number of the nodes in a computational grid. We analyze the features of such wide stencil based schemes in physical and spectral space. It is shown that the solutions obtained applying the schemes of the same order but involving different number of the mesh points characterize significantly different accuracy. This is presented by direct comparisons of the analytically derived formula for the leading terms of the truncation errors as well as in practical simulations. The test cases include a second order ordinary differential equation with variable coefficients, 1D Burgers equation and three typical benchmark problems used in Computational Fluid Dynamics (CFD), i.e., inviscid vortex advection, convected Taylor-Green vortices and a double jet flow. The obtained results are compared with available analytical solutions and the solutions obtained with the help of a pseudo-spectral method. We found that very high-order compact schemes reach the spectral accuracy characterized by the error decrease proportional to O(hK). In some cases, they show even higher accuracy, and moreover, reaching the limit of the machine precision they are substantially more resistant to the round-off errors.

A well test analysis model of generalized tube flow and seepage coupling

“Generalized mobility” is used to realize the unification of tube flow and seepage in form and the unification of commonly used linear and nonlinear flow laws in form, which makes it possible to use the same form of motion equations to construct unified governing equations for reservoirs of different scales in different regions. Firstly, by defining the generalized mobility under different flow conditions, the basic equation governing fluid flow in reservoir coupling generalized tube flow and seepage is established. Secondly, two typical well test analysis models for coupling tube flow and seepage flow are given, namely, pipe-shaped composite reservoir model and partially open cylindrical reservoir model. The log-log pressure draw-down type-curve of composite pipe-shaped reservoir model can show characteristics of two sets of linear flow. The log-log pressure drawdown plot of partially opened cylindrical reservoir model can show the characteristics of spherical flow and linear flow, as well as spherical flow and radial flow. The pressure build-up derivative curves of the two models basically coincide with their respective pressure drawdown derivative curves in the early stage, pulling down features in the late stage, and the shorter the production time is, the earlier the pulling down feature appears. Finally, the practicability and reliability of the models presented in this paper are verified by three application examples.

An efficient parallel scheme based on the nodal discontinuous Galerkin method for fluid flow simulations

An efficient parallel scheme based on the nodal discontinuous Galerkin finite element method (nodal-DGFEM) for the numerical solution of the partial differential equations governing fluid flow phenomena is discussed. The flow solver is demonstrated to perform numerical simulation of two-dimensional flow regimes on unstructured triangular grids. The parallel implementation serves to fulfill the requisition of the numerical method regarding high-performance computing resources. The distributed memory programming model with the domain decomposition approach is adopted. The message passing interface library is used for communication among the parallel processes, which are assigned domain-decomposed subproblems. The presented parallelization strategy accurately and efficiently tackles the communication of multi-node data on the element edges between the neighboring parallel processes. The efficacy and efficiency of the parallel solver are demonstrated through solving the well-known problem of non-viscous isentropic convecting vortex flow on parallel systems. The parallelization would extend the scope of the DGFEM by producing solutions in reasonable time frames.

Time-normalized conductivity concept for analytical characterization of dynamic-conductivity hydraulic fractures through pressure-transient analysis in tight gas reservoirs

This paper presents a time-normalized conductivity concept to allow an analytical model for characterization of dynamically-closed fractures through pressure-transient analysis in tight gas reservoirs. Specifically, the fracture conductivity in our flow model is considered to be a function of time. With the definition of the time-dependent fracture conductivity in the model, calculation of variable-rate pressure-transient behavior (e.g., pressure buildup tests) can be still based on the solution of linear differential equations governing fluid flow in a gas reservoir. Therefore, the superposition principle can be legitimately utilized in our solution to model pressure buildup tests. We then introduce a modified exponential function to represent the decline behavior of the time-normalized fracture conductivity during the production stage, and the dynamic fracture conductivity is essentially handled in a stair-step manner to calculate bottom-hole pressures. The accuracy of the calculated transient pressures was validated by comparing with a commercial simulator. The pressure drawdowns at early-times, because of dynamically-closed fractures, clearly increase more than in the invariant-fracture-conductivity case. We examine the effects of fracture conductivity behavior and production history on the pressure-transient response extensively, and use them to help identify the presence of dynamically-closed fractures. Finally, we demonstrate the influence of dynamically-closed fractures in synthetic and field cases to show the utility of the methodology. Results indicate that integrating multiple pressure build-up data obtained in different production periods enable us to diagnose and then to understand the changes in fracture conductivity during reservoir depletion. Using the results from the multiple pressure-transient analyses, this method can further be used to forecast the trend of fracture conductivity changes.

Simulation of coupled multiphase flow and geomechanics in porous media with embedded discrete fractures

AbstractIn fractured natural formations, the equations governing fluid flow and geomechanics are strongly coupled. Hydrodynamical properties depend on the mechanical configuration, and they are therefore difficult to accurately resolve using uncoupled methods. In recent years, significant research has focused on discretization strategies for these coupled systems, particularly in the presence of complicated fracture network geometries. In this work, we explore a finite‐volume discretization for the multiphase flow equations coupled with a finite‐element scheme for the mechanical equations. Fractures are treated as lower dimensional surfaces embedded in a background grid. Interactions are captured using the embedded discrete fracture model (EDFM) and the embedded finite element method (EFEM) for the flow and the mechanics, respectively. This nonconforming approach significantly alleviates meshing challenges. EDFM considers fractures as lower dimension finite volumes that exchange fluxes with the rock matrix cells. The EFEM method provides, instead, a local enrichment of the finite‐element space inside each matrix cell cut by a fracture element. Both the use of piecewise constant and piecewise linear enrichments are investigated. They are also compared to an extended finite element approach. One key advantage of EFEM is the element‐based nature of the enrichment, which reduces the geometric complexity of the implementation and leads to linear systems with advantageous properties. Synthetic numerical tests are presented to study the convergence and accuracy of the proposed method. It is also applied to a realistic scenario, involving a heterogeneous reservoir with a complex fracture distribution, to demonstrate its relevance for field applications.

Free convective heat transfer of a non-Newtonian fluid in a cavity containing a thin flexible heater plate: an Eulerian–Lagrangian approach

In the present study, the free convection heat transfer of a power-law non-Newtonian fluid is considered in a cavity containing a flexible hot thin heater. The sidewalls of the square cavity are maintained at cold temperatures, while the hot heater is placed inside the cavity. The top and bottom walls of the cavity are kept insulated. The thin heater plate can undergo large deformations due to the interaction between the fluid flow and the heater. The arbitrary Lagrangian–Eulerian moving mesh method is employed to track the displacement of the heater in the fluid domain. Appropriate non-dimensional parameters are utilized to transform the governing equations into a general non-dimensional form. The equations governing fluid flow and heat transfer are solved using the finite element method with an automatic time-stepping scheme. The effect of control parameters such as the non-Newtonian power index (0.6 1) behavior reduces the fluid circulation and heat transfer rate in the cavity, but it increases the magnitude of the exerted tensions on the element. Moreover, raising Ra from 104 to 106 enhances the average heat transfer the value of Nuav by up to 3.5 times in pseudoplastic fluids and by 1.5 times in dilatant ones. In addition, it is found that shifting the heater upward deteriorates the heat transfer rate by suppressing the convection flow intensity. A 35% rise in the average heater can be obtained when the height of the plate was divided by 4 in the cse of dilatant fluid, and an increase by up to 100% is found for pseudoplastic fluids. Increasing the length of the element is also found to reduce the average Nusselt number and to increase the tensions in the heater. The average Nusselt number can be doubled when the length of the plate is reduced seven times.

Heat transfer and fluid flow of Biodiesel at a backward-Facing step

You should Three-dimensional simulation of a biodiesel fluid flow within a rectangular duct over a backward-facing step is investigated in the present paper. The fluid, which obeys to the Newtonian rheological behavior, is obtained by transformation of Algerian waste cooking oil into a biodiesel. Flow through a rectangular channel subjected to a constant wall temperature or constant heat flux as boundary conditions. The partial differential equations governing fluid flow and heat transfer are solved by the Fluent CFD computational code based on the Finite Volume Method. The numerical experiments are carried out to examine the effect of the Reynolds number by fluid inlet velocity variation for the two boundary conditions. The results are analyzed through the distribution of the temperature and the velocity contours. The variation of the Reynolds number and boundary conditions affects greatly the heat transfer and the fluid flow, in particular near the step region.

Mathematical analysis of magnetohydrodynamic (MHD) flow of micropolar nanofluid under buoyancy effects past a vertical shrinking surface: dual solutions

In this paper, we explore dual solutions of MHD flow, heat and mass transfer of micropolar nanofluid over a linear vertical shrinking surface with buoyancy effects, which was not considered in the previous works. The governing fluid flow equations of this problem are transformed into nonlinear boundary value problems (BVPs) of ordinary differential equations (ODEs) by applying similarity variables. The resultant BVPs are converted into initial value problems (IVPs) by using shooting method which then resolved by employing Runge Kutta of order four. The impacts of the governing parameters, such as suction parameter, material parameter, Richardson number, magnetic parameter, Prandtl number, thermophoresis and Brownian motion parameters on velocity, angular velocity, temperature, and concentration are illustrated graphically. The results indicate that the existence of a range of dual solutions and no-solutions. When Richardson number (δ) is increased, the reduction of the velocity of micropolar nanofluid has occurred in the second solution. The stability analysis on dual solutions, however, reveals that only the first solution is stable.

Verification of Solutions of Equations Governing Fluid Flows

We have carried out the verifications of the solutions for the equations governing fluid flows. The equations solved are the similar equation of the boundary layer flow and the two-dimensional Navier-Stokes equations in the formulation of the stream function and the vorticity. The methods of the verification are the interval analysis and the Krawczyk operator based on the fixed point theorem. A modified Gaussian elimination method is introduced to solve the system of linear interval equations. The Krawczyk operator presents the results with a high accuracy. The modified elimination method gives the results with an accuracy of one or two order(s) of magnitude higher than those obtained by the original interval analysis.