Flow Differential Equations Research Articles

A Two-Phase Numerical Model of Well Test Analysis to Characterize Formation Damage in Near-Well Regions of Injection Wells

Formation damage usually occurs in near-well regions for injection wells completed in offshore oilfields under the development of line drive patterns. However, current works on characterizing the damage by well test analysis were basically focused on using single-phase analogy to solve two-phase flow issues, resulting in errors on the diagnosis and interpretation of transient pressure data. In this paper, we developed a two-phase model to simulate the pressure transient behavior of a water injection well in a multiwell system. To solve the model more efficiently, we used the finite volume method to discretize partially differential flow equations in a hybrid grid system, including both Cartesian and radial meshes. The fully implicit Newton-Raphson method was also employed to solve the equations in our model. With this methodology, we compared the resulting solutions with a commercial simulator. Our results keep a good agreement with the solutions from the simulator. We then graphed the solutions on a log-log plot and concluded that the effects of transitional zone and interwell interference can be individually identified by analyzing specific flow regimes on the plot. Further, seven scenarios were raised to understand the parameters which dominate the pressure transient behavior of these flow regimes. Finally, we showed a workflow and verified the applicability of our model by demonstrating a case study in a Chinese offshore oilfield. Our model provides a useful tool to reduce errors in the interpretation of pressure transient data derived from injection wells located in a line drive pattern.

Unsteady Coupled Heat Transfer Simulation of Unsteady numerical simulation of slot flow in anti-heating tile

An unsteady coupled heat transfer solver is developed in the study. The coupled solver, solving the partial differential equations for fluid flow and thermal conduction in the solid, is employed to predict the unsteady heat and transfer in the anti-heating tile. The influence of gap depth on the fluid flow and solid thermal conduction is studied. The results show that the flow field distribution between the different gap depths is similar, otherwise reducing the gap depth is conducive to reducing the thermal load in the bottom plate.

A Semi-Analytical Analysis of Gas Slippage Effect in Pressure Transient Behavior of Non-Condensate Gas Reservoirs with the Different Boundary Condition

The conventional models utilized to study flow behavior in a Non-Condensate gas reservoir did not consider the effects of gas slippage and different outer boundary conditions. For gas wells with a constant production flow rate in a bounded oil or gas reservoir, commonly, the outer boundary conditions are infinite boundary conditions or zero flux. In this study, the dimensionless pseudo pressure and dimensionless derivative of pseudo pressure in the presence of 4 different outer boundary conditions (infinite reservoir, constant pressure, exponential, and power-law) besides effects of gas slippage, wellbore storage, and skin factor are analyzed. To do this, the dimensionless pseudo pressure partial differential equation of radial flow was derived from the combination of continuity equation with Darcy’s law, the equation of state, compressibility equation, and dimensionless parameters. Then the derived partial differential equation is solved analytically in the Laplace domain. The obtained results of this work have important significance to understand the effects of different conditions on the transient pressure behavior of Non-Condensate gas reservoirs.

A computationally-efficient finite element method for the hydraulic analysis and design of subsurface drip irrigation subunits

A finite element model was developed for the hydraulic design of subsurface drip irrigation subunits. The governing differential equation for steady-state water flow in a multi-outlet pipeline was derived according to the energy and mass conservation equations and was discretized using the Galerkin method. An equation system containing unknown hydraulic heads was constructed and solved using direct stiffness algorithm and Gaussian elimination. The local head loss from the emitter was calculated as a fraction of the kinetic head. The emitter flow rate and its variation were evaluated using an analytical expression of positive soil water pressure and a log transformation of saturated hydraulic conductivity. The model adopted a virtual emitter concept, which allowed for rapid convergence on an ordinary computer. The finite element formulation made the model capable of analyzing both branched and looped pipe network systems. A MATLAB software script was developed, and the hydraulic performance of subsurface drip irrigation subunits was analyzed. The results indicate that a higher number of elements discretized in a pipe led to a higher accuracy of the model, with 20 elements being sufficient. The subsurface drip irrigation subunit can be recognized as an integrated water energy dissipation system. The average value and uniformity of emitter flow rate were determined by the pipe system and soil hydraulic properties. The designed maximum lateral length of a drip and a subsurface drip irrigation subunit was different. In the presented design example, that difference was determined by the emitter nominal flow rate and variation of the soil saturated hydraulic conductivity.

Numerical investigation of a boundary layer water-based nanofluid flow with induced magnetic field

The investigation concerns the induced magnetic field influence on the hydrodynamics and thermal build-up of water-based nanofluids containing nanoparticles in the non-isothermal and constant heat flux limits. Partial differential equations for the Boundary Layer Flow (BLF) are formulated and non-dimensionalized. Appropriate similarity variables are further introduced to arrive at a system of coupled nonlinear differential problems. Symbolic solutions are presented graphically and explained for various governing parameters. From the result, the thermal boundary layer is seen to be on the increase with increasing values of the fluid Prandtl number for both non-isothermal and constant heat flow conditions.

Finding a weak solution of the heat diffusion differential equation for turbulent flow by Galerkin's variation method using p-version finite elements

The stochastic turbulence model developed by Professor Czibere provides a means of clarifying the flow conditions in pipes and of describing the heat evolution caused by shear stresses in the fluid. An important part of the theory is a consideration of the heat transfer-diffusion caused by heat generation. Most of the heat is generated around the pipe wall. One part of the heat enters its environment through the wall of the tube (heat transfer), the other part spreads in the form of diffusion in the liquid, increasing its temperature. The heat conduction differential equation related to the model contains the characteristics describing the turbulent flow, which decisively influence the resulting temperature field, appear. A weak solution of the boundary value problem is provided by Bubnov-Galerkin’s variational principle. The axially symmetric domain analyzed is discretized by a geometrically graded mesh of a high degree of p-version finite elements, this method is capable of describing substantial changes in the temperature gradient in the boundary layer. The novelty of this paper is the application of the p-version finite element method to the heat diffusion problem using Czibere’s turbulence model. Since the material properties depend on temperature, the problem is nonlinear, therefore its solution can be obtained by iteration. The temperature states of the pipes are analyzed for a variety of technical parameters, and useful suggestions are proposed for engineering designs.

Analytical Solution of the MHD Forced Flow and Heat Transfer of a non-Newtonian Visco-Inelastic Fluid between Two Infinite Rotating Disks

In the present article, the model of forced flow and heat transfer in a non-Newtonian Reiner-Rivlin fluid between two infinite rotating disks in the presence of a magnetic field has been studied analytically. We converted the higher-order, partial differential equations (PDEs) into a system of nonlinear ordinary differential equations (ODEs) of flow and heat transfer by using the Von Karman similarity transformation. Homotopy Perturbation Method (HPM) is adopted to solve the system of nonlinear ODEs associated with boundary conditions. The viscous dissipation function is included in the energy equation. Therefore, the differential equation of the energy equation involves the Eckert number. The influence of physical parameters like Reynolds number, visco-inelastic parameter, forced parameter, Prandtl number, magnetic field on radial, transverse, axial velocity components, and temperature profile are discussed in detail. A comparison of the results obtained by HPM is made with numerical results, which proves the accuracy of HPM. The Nusselt number on the lower and upper disks are computed and presented graphically. The shear and couple stress on the lower disk is also discussed. The Maple software package is used for numeric and symbolic calculation of velocity components and temperature profiles.

Thermal radiation and Joule heating effects on a magnetohydrodynamic Casson nanofluid flow in the presence of chemical reaction through a non-linear inclined porous stretching sheet

The present study explores the thermal and Joule heating effect of Casson nanofluid flow with chemical reaction over an inclined porous stretching surface. The results of heat source/sink, viscous dissipation, and suction are regarded. The new physical governing equations of partial differential flow equations are converted into nonlinear ordinary differential equations and are numerically resolved employing the implicit finite difference technique. The influence on velocity, temperature, and concentration fields of many flow variables are addressed. The numerical and graphical findings are defined for the numerous related attentiveness flow parameters. The empirical data reported are compared with the published outcomes.

Significance of Coriolis force on the dynamics of water conveying copper and copper oxide nanoparticles

The worth of the study is to explore how in the presence of Coriolis force, heat and mass transfer rates are optimized through water carrying nanoparticles of more than one type and how this force affects various factors of the flow to make them easy come easy go. The aim of the current study is to analyze the vital characteristic of flow in particular velocity, temperature, concentration, heat flux at the surface, and tangential stress for both nanofluid CuO-water and hybrid nanofluid Cu/Cuo-water over a stretching surface in presence of Coriolis force. Partial differential equations for flow, energy, and concentration are constructed and transformed into ordinary differential equations by using suitable transformations. A highly reliable numerical scheme namely midpoint integration accompanied by Richardson’s extrapolation is launched to solve these equations. A computer software MAPLE is taken in to account to tackle the problem. Results are expressed through graphs and tables. The major conclusions are that the Coriolis force declines the velocity boundary and heat transmission rate but increases the thermal boundary of nanofluid. The heat sink/source factor uplifts the thermal boundary while Schmidt number enhances the mass concentration.

Analysis of Ethylene Glycol (EG)-based ((Cu-Al2O3), (Cu-TiO2), (TiO2-Al2O3)) Hybrid Nanofluids for Optimal Car Radiator Coolant

Coolants are vital in any automotive since they manage the heat in the internal combustion of the engines by preventing corrosion in the cooling system as well as assist in eradicating the engine’s waste heat. This paper examines three different types of ethylene glycol-based hybrid nanofluids ((Cu-Al2O3), (Cu-TiO2), (TiO2-Al2O3)) to establish their cooling capabilities for industrial cooling applications. The vertical flow of these hybrid nanofluids combination through a semi-infinite convectively heated flat plate mimicking the flow of coolant in car radiator is modeled. The governing non-linear partial differential equations of fluid flow are transformed into a system of coupled non-linear ordinary differential equations using a suitable similarity transformation variables and the numerical solution executed using the shooting technique together with the fourth-order Runge-Kutta-Fehlberg integration scheme. The numerical simulation is executed using MATLAB and results are displayed graphically. The effects of pertinent parameters on velocity, temperature, skin friction, and local Nusselt number are investigated. From the study (Cu-Al2O3 hybrid nanocoolant leads to a rapid decrease in temperature at the boundary layer.

Effect of Thermal Radiation and Chemical Reaction on MHD Flow of Blood in Stretching Permeable Vessel

This paper focuses on the theoretical analysis of blood flow in the presence of thermal radiation and chemical reaction under the influence of time dependent magnetic field intensity. Unsteady non linear partial differential equations of blood flow consider time dependent stretching velocity, the energy equation also accounts time dependent temperature of vessel wall and the concentration equation includes the time dependent blood concentration. The governing non linear partial differential equations of motion, energy and concentration are converted into ordinary differential equations using similarity transformations solved numerically by applying ode45. The effect of physical parameters, viz., the permeability parameter, unsteadiness parameter, Prandtl number, Hartmann number, thermal radiation parameter, chemical reaction parameter and Schmidt number on flow variables, viz., velocity of blood flow in vessel, temperature and concentration of blood, has been analyzed and discussed graphically. From the simulation study the following important results are obtained: velocity of blood flow increases with the increment of both permeability and unsteadiness parameter. The temperature of blood increases at the vessel wall as the Prandtl number and Hartmann number increase. Concentration of blood decreases as time dependent chemical reaction parameter and Schmidt number increases.

THE ROLE OF FOX-H FUNCTION IN ANALYTIC AND FRACTIONAL MODELING OF HELICITY OF CYLINDER: FRACTIONAL GENERALIZED BURGER FLUID

In this paper, the analytic and fractional solutions of governing differential equations for helical flow of cylindrical nature have been presented. The series expansions and Laplace and Hankel transforms are applied to the governing equation of generalized Burger fluid flow for generating gamma functions. The analytical solutions of velocity fields and shear stresses are obtained through Caputo fractional approach. In order to justify the initial and boundary conditions, infinite series are invoked for expressing the analytical results of velocity fields and shear stresses in terms of [Formula: see text] Fox-H function. At the end, few rheological parameters have been analyzed on four different types of models as shown in graphs. Finally, a comparative analysis of ordinary and fractional models has been focussed for angular and oscillating velocities of helical flow generated by circular cylinder.

Stability Analysis of MHD Fluid Flow over a Moving Plate with Pressure Gradient Using the Chebyshev Spectral Method

The purpose of this paper is to investigate the magnetohydrodynamic stability external flow of a viscous, incompressible and electrically conducting fluid over a moving flat plate using temporal linear stability analysis. Using a similarity variables based on the Skan-Falkner transformation, the governing differential equations of mean flow are transformed into a nonlinear ordinary differential equation, which is then solved numerically by the Runge-Kutta method. The MHD stability equation is solved numerically by using the Chebyshev spectral collocation method, which is based on the eigenfunction expansion in terms of Chebyshev polynomials, collocation points, and the subsequent solution of the resulting generalized eigenvalue problem with the QZ algorithm. The influence of the pressure gradient, magnetic field and wall velocity on the dimensionless mean velocity are presented graphically and discussed. For the disturbance flow in the presence of magnetic field and moving wall, the critical Reynolds number increases with increasing the magnetic field parameter and wall velocity, indicating that these parameters stabilize the flow.

Group theoretical analysis for magnetohydrodynamic generalized Stokes’ flow and radiative heat transfer model of a non-Newtonian nanofluid with heat generation/absorption

The fully developed boundary layer flow and heat transfer model of an unsteady, incompressible, electrically conducting, optically thick and thermodynamically compatible third-grade nanofluid are investigated from the viewpoint of Lie symmetry approach. We have studied the Stokes’ model in which flow is generated due to motion of the bounding surface with an impulsive time-dependent velocity. The mathematical model also includes the effect of thermal radiation and internal heat source or sink in the flow regime. Lie symmetry approach is employed to obtain the symmetry algebra of the governing nonlinear partial differential equations for both flow and heat transfer models. The governing nonlinear partial differential equations describing the flow and heat transfer model are reduced to different classes of nonlinear ordinary differential equations under the implication of symmetry generators. The reduced ordinary differential equations are investigated using the compatibility and generalized group theoretical method. The exact closed-form solutions for nanofluid motion and temperature distribution within the boundary layer are obtained. The effect of various thermophysical parameters on the characteristics of flow and heat transfer for Cu-water nanofluid model is explored from a physical point of view.

Effect of buoyancy and suction on Sisko nanofluid over a vertical stretching sheet in a porous medium with mass flux condition

The present article investigates the flow and heat transfer of Sisko nanofluid over a permeable vertical stretching surface in a porous medium. The effect of buoyancy, suction, and viscous dissipation has been taken into account. Buongiorno’s model of nanofluid consisting of thermophoresis and Brownian diffusion has been considered. Moreover, zero nanoparticle mass flux condition is employed at the boundary which leads to a more realistic physical problem. Using a suitable transformation governing partial differential equations of fluid flow are transformed into a set of nonlinear ordinary differential equations (ODEs). The numerical solution of nonlinear ODEs are obtained using the finite difference technique in MATLAB. The influence of physical parameters viz. buoyancy parameter (λ*), porosity parameter (β*), thermophoresis parameter (Nt*), suction parameter (f), Sisko material parameter (A*), Brinkman number (Br*), Brownian diffusion parameter (Nb*) and Lewis number (Le*) on velocity, temperature and nanoparticle volume fraction are shown graphically. Moreover, to understand the physical phenomenon in the boundary layer region, the numerical values of skin friction and Nusselt number are calculated and presented through table values. It has been found that the Brownian diffusion has a negligible impact on Nusselt number relative to the results obtained in previous studies, where nanoparticle volume fraction on the boundary was actively controlled. The obtained results disclose that the buoyancy parameter increases the velocity of fluid while it reduces the temperature. Suction parameter reduces both velocity and temperature, whereas the porosity parameter reduces velocity and enhances the temperature and nanoparticle volume fraction.

Mechanism of Soret‐Dufour, magnetohydrodynamics, heat and mass transfer flow with buoyancy force, and viscous dissipation effects

AbstractThis paper examined the mechanism of both positive and negative effects of Soret‐Dufour with heat and mass transfer processes over an accelerating permeable surface. The partial differential flow equations were simplified using similarity variables, and the resulting equations were solved numerically using the spectral homotopy analysis method (SHAM). The SHAM is used in separating nonlinear equations into linear and nonlinear. The physics of each pertinent flow parameters was used to examine their influence on velocity, temperature, and concentration fields. The effect of Soret‐Dufour was examined separately, and its negative effect was used to determine its influence on velocity, temperature, and concentration fields. The result revealed that positive Soret‐Dufour enhances the boundary layer, whereas negative Soret‐Dufour parameter decreases the boundary layer. The result presented in this paper is in good agreement with existing works in literature.

Temperature distribution and entropy generation during Darcy–Forchheimer–Brinkman electrokinetic flow in a microfluidic tube subject to a prescribed heat flux

Electrokinetically modulated flow through a hydrophobic microtube embedded in a Darcy–Forchheimer porous medium is investigated in this paper. The steric effect has been taken into account in the electrical double layer (EDL) region. Heat transfer is analysed in the case of Darcy–Forchheimer–Brinkman flow subject to Joule heating. The flow is supposed to take place under the combined influence of electroosmosis and imposed pressure gradient. The governing nonlinear partial differential equations for electric potential, fluid flow and heat transfer are solved numerically by developing an iterative finite difference method that has second order accuracy. The thermal efficiency is discussed under the purview of the second law of thermodynamics. Influences/impact of different physical parameters on velocity, temperature and entropy are investigated and demonstrated graphically. The paper shows that with an increase in the steric effect, the electrokinetic velocity diminishes and that thermal irreversibility is very high in the electrical double layer region, but it reduces drastically in the neighbourhood of the central region of the microtube. Results of the study are likely to be of profuse interest in the design and development of microfluidic devices that deal with critical types of fluid transport mechanism in non-Darcian porous media.

Calculation of Interphase Mass Transfer in a Spray Flow Produced by a Nozzle with Account of Crisis

This paper describes the numerical model and the results of calculating the interphase mass transfer in a two-phase flow generated by atomizing a liquid in a gas using a nozzle. The proposed mathematical model is based on differential equations of unsteady flow in a compressible medium, complemented by the equation of mass transfer from gas to droplets. The popular explicit Lax–Wendroff scheme is used for transition to difference analogs of continuity and phase motion equations. Axial profiles of droplet and gas velocities and gas impurity concentrations in a free spray cone are calculated and presented together with radial profiles of impurity concentrations in a two-phase flow through a cylindrical apparatus, taking into account early drag crisis of droplets and crisis of interphase mass transfer, as well as the features of turbulent friction in gas, discovered in previous experiments. The calculation established the dependencies of the gas flow rate, the concentration of gas impurities at the outlet of the apparatus, and the amount of impurities absorbed by the liquid from the height and cross-sectional area of the apparatus.

Computational analysis of bioconvective flow of nanofluid containing gyrotactic microorganisms over a nonlinear stretching sheet with variable viscosity using HAM

Abstract This paper is concerned with the investigation of variable viscosity bioconvection flow of nanofluid containing motile gyrotactic microorganisms over a nonlinear stretching sheet in the presence of nonlinear thermal radiation, chemical reaction, internal heat source, and suction/injection effects. The homotopy analysis method has been developed for solving the governing nonlinear differential equations of the boundary layer flow of nanofluid over a stretching sheet. The scaling group transformation (a special form of Lie group transformation) has been applied to find the similarity variable $\eta $. Figures are drawn by using Mathematica software to analyze the results that correspond to some important physical parameters and bioconvection parameters on velocity, temperature, nanoparticle concentration, and density of gyrotactic microorganisms. It is found that the influence of variable viscosity on velocity profiles showed that there is an increase in the velocity profiles of nanofluid and the reverse effect is observed on its temperature distribution. It is seen that the thermal radiation parameter increases the temperature distribution, whereas it decreases the nanoparticle concentration distribution. It is also found that the inverse Darcy number reduces the velocity profile, whereas it enhances the temperature distribution. This work may find applications in advanced nanomechanical bioconvection energy conversion devices, bio-nanocoolant systems, etc.

The influence of magnetized couple stress heat, and mass transfer flow in a stretching cylinder with convective boundary condition, cross-diffusion, and chemical reaction

This paper addresses an analysis for axisymmetric boundary layer couple stress fluid flow by stretching cylinder under the influence of the magnetic field, chemical reaction, Soret, and Dufour. The leading partial differential equations of boundary layer flow are transformed into nonlinear differential equations by applying similarity transformations then solved by the Bvph2.0 based on Homotopy analysis method (HAM). The effect of pertinent parameters shown graphically and observed that velocity decreases when Reynold number and magnetic parameters are increased. The velocity, temperature, and concentrations are enhanced with curvature parameter rise. The dimensionless skin shear stress (skin friction coefficient), heat and mass transfer rates are computed and its convergence analysis also studied. It is assumed that the axial velocity decelerates for the couple stress parameter while the thermal and solutal profile reverse situation is observed.