Immiscible Newtonian Fluids Research Articles

Flow and heat transfer of non-miscible micropolar and Newtonian fluid in porous channel sandwiched between parallel plates

An analytical approach has been considered to inspect the flow and heat transfer of immiscible Newtonian and micropolar fluid in a porous medium. This study aims to elucidate the Darcy effect on the model in which isotropic porous regions with different permeability are used. The model of the current problem explains that the flow region is divided into two regions: Newtonian fluid flows in the upper layer, and micropolar fluid flows in the lower layer. Different constant temperatures imposed at boundary walls, heat transfer does not affect the pressure gradient. The governing equations are solved analytically by applying a linear differential equation (LDE). The effect of associated physical parameters such as material parameter, Darcy number, Eckert number, Prandtl number, and viscosity ratio on velocity, micro-rotation, heat transfer, flow rate, heat transfer rate, wall shear stress, and Nusselt number have been inspected. The most significant finding of this research work is that increasing the Darcy number signifies enhanced permeability, resulting in a higher flow rate. The heat transfer rate at the top occurs maximum when the material parameter’s range is minimum while raising the viscosity ratio leads to an increasing heat transfer rate at the bottom. Enhancement in material parameter influences Nusselt number and decreases in nature. The findings of our study are verified with the previously established results. The present work has a setup that is useful in petroleum extraction, transport problems in reservoir rock of an oil field, improving nutrient transport and thermal regulation in tissue engineering, and designing more efficient drug delivery systems and biomedical devices.

Rheological impact of viscous fluid in the core region of heated curved pipe surrounded by Casson rheological fluid

The immiscible fluids can help design more efficient systems for drug delivery and understanding the blood flow through diseased arteries. It can also have implications for the design of flow control devices or medical implants that involve curved geometries. Therefore, the current article scrutinizes the characteristics of heat transfer on the flow of two immiscible Casson and Newtonian fluids through a curved pipe induced due to a pressure gradient in an axial direction. The pipe is divided into two distinct regions: namely, (i) Region 1 (core) is filled with Newtonian fluid while (ii) Region 2 (peripheral) is filled with non-Newtonian Casson fluid. The complex curvilinear coordinates called toroidal coordinates are used to form a mathematical model for the present problem. Equations governing the flow are considered without ignoring any curvature ratio terms and are solved analytically by considering the perturbation series. The continuity of velocities and shear stresses at the fluid-fluid interface is taken into account along with no-slip, symmetric, and regularity conditions while solving the two-fluid flow problem under consideration. The effect of various fluid parameters such as curvature ratio, Casson fluid parameter, Reynolds number, viscosity ratio, density ratio, Eckert number, and conductivity ratio on the axial velocity and temperature is studied through 2-dimensional graphs and contour plots. In addition, the volumetric flow rate and shear stress are also presented to evaluate the effects of various key parameters. The results show that the axial velocity of immiscible fluids flow increases with an increase in the values of the viscosity ratio parameter and the fluid temperature is decreased by increasing the conductivity ratio parameter.

Energy identity for the incompressible Cahn–Hilliard/Navier–Stokes system with non–degenerate mobility

We consider the Cahn–Hilliard/Navier–Stokes system with non–degenerate mobility in the space–periodic case, describing the flow of two viscous immiscible and incompressible Newtonian fluids with matched densities. We identify sufficient conditions on the velocity field for weak solutions to satisfy an energy identity, improving previous results on the literature.

Radial displacement patterns of shear-thinning fluids considering the effect of deformation

Radial injection of shear-thinning fluids into rock fractures is ubiquitous in subsurface engineering practices, including drilling, hydraulic fracturing, and rock grouting. Yet, the effect of injection-induced fracture deformation on radial displacement behavior of shear-thinning fluids remains unclear. Through radial injection experiments of shear-thinning fluids displacing an immiscible Newtonian fluid in a Hele–Shaw cell, we investigate the fracture deformation behavior during injection and the fluid–fluid displacement patterns under this impact. A mixed displacement pattern is observed where the invasion front gradually evolves from unstable (viscous fingering) to stable (compact displacement) as the injection proceeds. We demonstrate that the combined effect of shear-thinning property and radial flow geometry plays a controlling role in the evolution of the patterns. At high flow rates, the fracture dilation induced by high injection pressure tends to reduce the displacement efficiency in stages. Based on linear stability analysis, we propose a theoretical criterion for the transition of interfacial stability considering the viscosity of injected fluids and fracture deformation, which agrees well with the experimental observations. This research underscores the importance of rock deformation on two-phase flow dynamics in fractured media.

An efficient unconditional energy stable scheme for the simulation of droplet formation

We have developed an efficient and unconditionally energy-stable method for simulating droplet formation dynamics. Our approach involves a novel time-marching scheme based on the scalar auxiliary variable technique, specifically designed for solving the Cahn-Hilliard-Navier-Stokes phase field model with variable density and viscosity. We have successfully applied this method to simulate droplet formation in scenarios where a Newtonian fluid is injected through a vertical tube into another immiscible Newtonian fluid. To tackle the challenges posed by nonhomogeneous Dirichlet boundary conditions at the tube entrance, we have introduced additional nonlocal auxiliary variables and associated ordinary differential equations. These additions effectively eliminate the influence of boundary terms. Moreover, we have incorporated stabilization terms into the scheme to enhance its numerical effectiveness. Notably, our resulting scheme is fully decoupled, requiring the solution of only linear systems at each time step. We have also demonstrated the energy decaying property of the scheme, with suitable modifications. To assess the accuracy and stability of our algorithm, we have conducted extensive numerical simulations. Additionally, we have examined the dynamics of droplet formation and explored the impact of dimensionless parameters on the process. Overall, our work presents a refined method for simulating droplet formation dynamics, offering improved efficiency, energy stability, and accuracy.

Fluid–structure-interactive elasto- and thermo-hydrodynamics of electrokinetic binary fluid flows in compliant micro-confinements

We explore the intricate two-way fluid structure interaction arising due to the flow of a binary system of immiscible Newtonian fluids, composed of upper electrically conducting and lower electrically insulating fluids, flowing within a compliant microchannel, whose walls behave as linear elastic solids. The transport of the fluids along the domain occurs due to the collective impact of pressure gradient and applied electric field. We solve the closed-form system of equations and obtain semi-analytical expressions for the velocity fields and channel wall deformation from the coupled elasto-hydrodynamic problem. We then delineate the effect of four pivotal parameters: (a) Debye–Hückel parameter κ¯, (b) upper wall slip length, ls¯, (c) viscosity ratio, μr, and (d) elasticity ratio, Nr, on the morphological evolution of the wall deformation characteristics and the spatial distribution of the velocity profile of the fluids. Observations establish a positive co-relationship of wall deformation with fluid pressure, showcasing an increased collapsibility with augmented pressure gradients. Consequently, the channel walls show enhanced deformation with a decrease in κ¯, ls¯, μr and with an increase in Nr. We also demonstrate from our model that by properly tuning the applied pressure gradient and electric field, it is possible to achieve counterflow of the two fluids. We also consider the effect of heat generation in the fluids due to viscous dissipation and Joule heating, which dissipates to the surrounding by the mechanism of conjugate heat transfer. Accordingly, we provide semi-analytical expressions for the temperature profile distribution within the channel, and discuss their variation with three thermo-physical parameters: (a) Biot number of the top wall (Bi1), (b) Peclet number of the top fluid (Pe1), and (c) ratio of the thermal conductivities of the upper conducting fluid to that of the upper solid wall (kr2). We establish from our investigation that with the increase in Pe1 and with the decrease in Bi1 and kr2, the overall system temperature enhances. Finally, in order to design a thermally efficient system, we compute the global entropy generation rate in the system and evaluate optimum values of, Pe1, Bi1, and kr2 for which the system exhibits highest second law efficiency. We expect our findings to contribute toward the development of optimized microfluidic devices fabricated from deformable elastic materials.

Interplay of interface and rheological properties in the interfacial flow of two uniformly rotating immiscible Jeffrey and Newtonian fluids

Interplay of interface and rheological properties in the interfacial flow of two uniformly rotating immiscible Jeffrey and Newtonian fluids

Physics of generalized couette flow of immiscible fluids in anisotropic porous medium

This study presents an investigation on the generalized Couette flow of two immiscible Newtonian fluids in an anisotropic porous medium. The flow occurs between two parallel plates, driven by the combined effects of an axial pressure gradient and the movement of the upper plate. The region between the plates is filled with an anisotropic porous medium characterized by directional permeability. An exact solution is derived for the proposed problem by considering appropriate boundary and interface conditions. The slip condition is employed at the lower plate of the channel which has a significant role in the flow mechanics of fluids flowing through porous medium. The effects of directional permeability on the flow of immiscible fluids are explored, and specific cases are presented. This study reveals a significant influence of directional permeability on the flow velocity of both fluids. Furthermore, quantitative conclusions are drawn regarding the shear stress at the lower and upper wall of the channel, with implications for the anisotropic nature of blood arteries. The findings contribute to a deeper understanding of fluid flow in anisotropic porous media and have potential applications in various engineering domains.

Thermal and multilayer analysis of magnetised dusty fluids under electroosmosis and pressure-driven effects in a microchannel

The study of an immiscible fluid flow models is complex in nature and it has crucial applications in engineering and industry, such as powder technology, dust assortment, dust in gas cooling systems, retrieval of crude oil, waste water treatment, sedimentation process and nuclear reactors. Therefore, the current study deals with the steady flow of two immiscible dusty Newtonian and Casson fluids in a horizontal porous microchannel. The effect of magnetic field, electroosmotic forces, thermal radiation, viscous dissipation, Joule heating and chemical reactions have been considered into account. Initially, the flow model is considered with the well-known set of partial differential equations. Under the assumptions and non-dimensional quantities, the coupled governing differential equations have been transmuted to the non-dimensional ordinary differential equations. The exact solutions for the velocity, temperature and concentration of the fluid and dusty phases have been obtained in both the regions. The behaviour of non-dimensional emerging constraints on the flow, thermal and mass characteristics have been expressed through graphical representations. It is noticed that the dust particles volume fraction enhances the dusty Newtonian fluid as well as the dusty Casson fluid temperature. Escalating the values of Schmidt number declines the concentration profile of Newtonian and non-Newtonian dusty fluids.

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 Entropy Production of Immiscible Micropolar and Newtonian Fluids Flow through a Channel: Effect of Thermal Radiation and Magnetic Field

Analysis of Entropy Production of Immiscible Micropolar and Newtonian Fluids Flow through a Channel: Effect of Thermal Radiation and Magnetic Field

Magnetohydrodynamics of immiscible Newtonian fluids in porous regions of different variable permeability functions

This work is an analysis of the flow model often occurs in crude oil extraction, blood flow in the arteries, filtration of underground different fluids flowing together. In this analysis, a model of three-layered porous horizontal channel is considered. This problem is significant because of the different permeability functions used for each porous layer of the channel and flow of different Newtonian fluids takes place in these porous regions. The problem is solved for the general case which can be reduced into several particular cases. The flows inside the channel and in every porous layer is governed by the Brinkman's momentum equation for the porous medium. The momentum equations in each region of the present model are the Airy's inhomogeneous differential equations respectively. An analytical closed form solution of the flow in each region have been obtained. Authors have discussed various results of velocity profile, flow rate and wall shear stresses graphically. Our results agree with the previous published results.

Non-Newtonian Rheology in a Capillary Tube with Varying Radius

Fluid blobs in an immiscible Newtonian fluid flowing in a capillary tube with varying radius show highly nonlinear behavior. We consider here a generalization of previously obtained results to blobs of non-Newtonian fluids. We compute here the yield pressure drop and the mean flow rate in two cases: (i) When a single blob is injected, (ii) When many blobs are randomly injected into the tube. We find that the capillary effects emerge from the non-uniformity of the tube radius and contribute to the threshold pressure for flow to occur. Furthermore, in the presence of many blobs the threshold value depends on the number of blobs and their relative distances which are randomly distributed. For a capillary fiber bundle of identical parallel tubes, we calculate the probability distribution of the threshold pressure and the mean flow rate. We consider two geometries: tubes of sinusoidal shape, for which we derive explicit expressions, and triangular-shaped tubes, for which we find that essential singularities are developed. We perform numerical simulations confirming our analytical results.

Exploration of Thermophoresis and Brownian motion effect on the bio-convective flow of Newtonian fluid conveying tiny particles: Aspects of multi-layer model

This research deals with the analysis of bioconvection caused by the movement of gyrotactic microorganisms. The multi-layer immiscible Newtonian fluid flowing through the vertical channel conveying tiny particles is accounted. The immiscible fluids are arranged in the form of a sandwich where the middle layer has a different base fluid that does not mix with the base fluid of the adjacent fluid layer. This separation of the fluid layers gives rise to the interface boundary conditions. Such flows have found applications in electronic cooling and solar reactors processes. Buongiorno’s model has been incorporated to design the mathematical model that describes the three-layer flows of Newtonian fluid conveying tiny (metal/oxide) particles under thermophoretic force and Brownian motion. The model thus formed is in the form of the ordinary differential system of equations that are solved using the DTM-Pade approximant after non-dimensionalization. The limited results have an excellent comparison with the existing literature results. The results are discussed through graphs and tables. It is seen that thermophoresis enhances the temperature and particle concentration of the fluid whereas, the Brownian motion is found to enhance the temperature and decrease the concentration. The presence of bioconvection helps in achieving enhanced energy and mass transportation. Moreover, the heat transfer occurring between the different base fluids helps to maintain the optimum temperature in the systems.

Diffusiophoretic propulsion of an isotropic active colloidal particle near a finite-sized disk embedded in a planar fluid–fluid interface

Breaking spatial symmetry is an essential requirement for phoretic active particles to swim at low Reynolds number. This fundamental prerequisite for swimming at the micro scale is fulfilled either by chemical patterning of the surface of active particles or alternatively by exploiting geometrical asymmetries to induce chemical gradients and achieve self-propulsion. In the present paper, a far-field analytical model is employed to quantify the leading-order contribution to the induced phoretic velocity of a chemically homogeneous isotropic active colloid near a finite-sized disk of circular shape resting on an interface separating two immiscible viscous incompressible Newtonian fluids. To this aim, the solution of the phoretic problem is formulated as a mixed-boundary-value problem that is subsequently transformed into a system of dual integral equations on the inner and outer domains. Depending on the ratio of different involved viscosities and solute solubilities, the sign of phoretic mobility and chemical activity, as well as the ratio of particle–interface distance to the radius of the disk, the isotropic active particle is found to be repelled from the interface, be attracted to it, or reach a stable hovering state and remain immobile near the interface. Our results may prove useful in controlling and guiding the motion of self-propelled phoretic active particles near aqueous interfaces.

Numerical Study of Interface Tracking for the Unsteady Flow of Two Immiscible Micropolar and Newtonian Fluids Through a Horizontal Channel with an Unstable Interface

The dynamics of the interaction between immiscible fluids is relevant to numerous complex flows in nature and industry, including lubrication and coating processes, oil extraction, physicochemical separation techniques, etc. One of the most essential components of immiscible flow is the fluid interface, which must be consistently monitored. In this article, the unsteady flow of two immiscible fluids i.e., an Eringen micropolar and Newtonian liquid is considered in a horizontal channel. Despite the no-slip and hyper-stick shear stress condition at the channel edge, it is accepted that the liquid interface is dynamic, migrating from one position to the next and possibly get absolute change; as a result, The CS (continuum surface) model is integrated with the single moment equation based on the VOF (volume of fluid) approach to trace the interface. The immiscible fluids are considered to flow under three applied pressure gradients (constant, decaying, and periodic) and flow is analyzed under seamless shear stress over the entire interface. The modified cubic b-spline differential quadrature method (MCB-DQM) is used to solve the modeled coupled partial differential equations for the fluid interface evolution. The advection and tracking of the interface with time, wave number, and amplitude are illustrated through graphs. It is observed that the presence of micropolar parameters affects the interface with time. The novelty of the current study is that previous studies (which considered the smooth and unstable movement of the micropolar fluid, the steady stream of two immiscible fluids, and interface monitoring through different modes) are extended and generalized to consider the time-dependent flow of two immiscible fluids namely Eringen micropolar and Newtonian with a moving interface in a horizontal channel. For the decaying pressure gradient case, which requires more time to achieve the steady-state, the peak of the waves resembles those for the constant pressure gradient case. The interface becomes steady for a more extensive time when a constant pressure gradient is applied. The interface becomes stable quickly with time as the micropolar parameter is decreased for the constant pressure gradient case i.e., weaker micropolar fluids encourage faster stabilization of the interface. With periodic pressure gradient, the interface takes more time to stabilize, and the crest of the waves is significantly higher in amplitude compared to the constant and decaying pressure cases. The simulations demonstrate the excellent ability of MCB-DQM to analyze complex interfacial immiscible flows.

Analysis of two immiscible Newtonian and micropolar fluid flow through an inclined porous channel

In this work, the flow behaviour of two immiscible fluids is analysed through an inclined channel which is made of two rigid plates. The flow model consists of two porous regions of different permeability. Newtonian and micropolar fluids are allowed to flow in the region‐I and region‐II, respectively. The flow of Newtonian and micropolar fluid in respective porous region is governed by the Brinkman's equation. An exact solution of the proposed mathematical model is obtained by using well‐known and appropriate boundary conditions. The expressions for linear velocity, microrotational velocity, flow rate and stresses are evaluated. The effect of various emerging non‐dimensional parameters like viscosity ratio, couple stress parameter, gravitational parameter, Reynolds number, and so on, on linear velocity, microrotational velocity, flow rate and stresses is presented graphically. The results are validated with the help of the previous established results.

Numerical Solution of the Time-Depending Flow of Immiscible Fluids with Fuzzy Boundary Conditions

Fluid flow modeling using fuzzy boundary conditions is one of the viable areas in biofluid mechanics, drug suspension in pharmacology, as well as in the cytology and electrohydrodynamic analysis of cerebrospinal fluid data. In this article, a fuzzy solution for the two immiscible fluid flow problems is developed, which is motivated by biomechanical flow engineering. Two immiscible fluids, namely micropolar and Newtonian fluid, are considered with fuzzy boundary conditions in the horizontal channel. The flow is considered unsteady and carried out by applying a constant pressure gradient in the X-direction of the channel. The coupled partial differential equations are modeled for fuzzy profiles of velocity and micro-rotation vectors then the numerical results are obtained by the modified cubic B - spline differential quadrature method. The evolution of membership grades for velocity and microrotation profiles has been depicted with the fuzzy boundaries at the channel wall. It is observed that Micropolar fluid has a higher velocity change than Newtonian fluid, and both profiles indicate a declining nature toward the interface.

Slip effect on the unsteady electroosmotic and pressure-driven flows of two-layer fluids in a rectangular microchannel

Abstract of the paper: Electroosmotic flows of two-layer immiscible Newtonian fluids under the influence of time-dependent pressure gradient in the flow direction and different zeta potentials on the walls have been investigated. The slippage on channel walls is, also, considered in the mathematical model. Solutions to fluid velocities in the transformed domain are determined by using the Laplace transform with respect to the time variable and the classical method of the ordinary differential equations. The inverse Laplace transforms are obtained numerically by using Talbot’s algorithm and the improved Talbot’s algorithm. Numerical results corresponding to a time-exponential pressure gradient and translational motion with the oscillating velocity of the channel walls have been presented in graphical illustrations in order to study the fluid behaviour. It has been found that the ratio of the dielectric constant of fluid layers and the interface zeta potential difference have a significant influence on the fluid velocities.

A magnetic field coupling lattice Boltzmann model and its application on the merging process of multiple-ferrofluid-droplet system

The present study concerns the dynamics of ferrofluid droplets in a Newtonian fluid. The incompressible Navier-Stokes (NS) equations are applied for the dynamics of the immiscible magnetic and Newtonian fluids while the Cahn-Hilliard (C-H) equation is adopted to describe the behavior of their interface. A magnetic field coupling lattice Boltzmann (LB) model is developed to solve both the NS equations and the C-H equation. Specially, a mass-correcting term is introduced into the C-H equation, which strongly enforces the mass conservation. The magnetic field is evaluated by a Poisson equation solver with a self-correcting procedure for the static Maxwell equations. The magnetic dipole force is transformed into the magnetic surface force by a rigorous mathematical procedure, which can physically describe the magnetic effect on the interface. Moreover, the magnetic force after this treatment becomes easy-to-implement, which can be directly incorporated into the external force term of the LB model. The capability of the present combination method to simulate the magnetic multiphase flows is demonstrated by three typical numerical examples, i.e., Laplace law for a stationary droplet, a stationary cylinder under an external uniform magnetic field, the deformation of a single ferrofluid droplet. Further, the merging process of multiple-ferrofluid-droplet system in organic oil is investigated. It is found that the elongation of ferrofluid droplet in the direction of magnetic field shows positive impact on the merging process when the orientation of ferrofluid droplets is parallel to the external magnetic field, while negative impact on the merging process when their orientation is perpendicular to the external magnetic field.