Viscosity Ratio Research Articles

Influence of viscosity ratio on enhanced oil recovery performance of anti-hydrolyzed polymer for high-temperature and high-salinity reservoir

The viscosity ratio of polymer and oil is a crucial factor for polymer flooding, which can affect the water–oil mobility ratio and oil recovery. However, for high-temperature and high-salinity reservoirs, the reasonable viscosity ratio limit of polymer flooding under the condition of medium–high permeability and low oil viscosity is not clear. Thus, the heterogeneous sand-pack flooding experiments were carried out to analyze the influence of polymer–oil viscosity ratio on the enhanced oil recovery (EOR) performance of anti-hydrolyzed polymer to establish a reasonable viscosity ratio limit. Then the three-dimensional heterogeneous model flooding experiments were performed to clarify the mechanism. The results showed that when the permeability ratio was the same, as the viscosity ratio increased from 0.15 to 2.0, the incremental oil recovery increased from 3.2% to 27.2%. When the viscosity ratio was the same, the incremental oil recovery decreased with the increase in the permeability ratio. The reasonable viscosity ratio ranges from 1.0 to 1.5. For three-dimensional heterogeneous model flooding experiments, as the polymer–oil viscosity ratio increased from 0.45 to 1.0, the swept area of high and low permeability area was expanded and the oil saturation near the injection well in the mainstream channel was greatly reduced. Moreover, when the polymer–oil viscosity ratio was 1, the difference in the width of the mainstream channels between high and low permeability layers in the saturation field decreased, and the degree of utilization in low permeability layers increased significantly. As the polymer–oil viscosity ratio increased from 0.45 to 1.0, the incremental oil recovery increased from 16.2% to 24%.

Numerical study on the mechanism of drag modulation by dispersed drops in two-phase Taylor–Couette turbulence

The presence of a dispersed phase can significantly modulate the drag in turbulent systems. We derived a conserved quantity that characterizes the radial transport of azimuthal momentum in the fluid–fluid two-phase Taylor–Couette turbulence. This quantity consists of contributions from advection, diffusion and two-phase interface, which are closely related to density, viscosity and interfacial tension, respectively. We found from interface-resolved direct numerical simulations that the presence of the two-phase interface consistently produces a positive contribution to the momentum transport and leads to drag enhancement, while decreasing the density and viscosity ratios of the dispersed phase to the continuous phase reduces the contribution of local advection and diffusion terms to the momentum transport, respectively, resulting in drag reduction. Therefore, we concluded that the decreased density ratio and the decreased viscosity ratio work together to compete with the presence of a two-phase interface for achieving drag modulation in fluid–fluid two-phase turbulence.

Viscoelasticity of suspension of red blood cells under oscillatory shear flow

We present a numerical analysis of the rheology of a suspension of red blood cells (RBCs) for different volume fractions in a wall-bounded, effectively inertialess, small amplitude oscillatory shear (SAOS) flow for a wide range of applied frequencies. The RBCs are modeled as biconcave capsules, whose membrane is an isotropic and hyperelastic material following the Skalak constitutive law. The frequency-dependent viscoelasticity in the bulk suspension is quantified by the complex viscosity, defined by the amplitude of the particle shear stress and the phase difference between the stress and shear. SAOS flow basically impedes the deformation of individual RBCs as well as the magnitude of fluid-membrane interactions, resulting in a lower specific viscosity and first and second normal stress differences than in steady shear flow. Although it is known that the RBC deformation alone is sufficient to give rise to shear-thinning, our results show that the complex viscosity weakly depends on the frequency-modulated deformations or orientations of individual RBCs but rather depends on combinations of the frequency-dependent amplitude and phase difference. The effect of the viscosity ratio between the cytoplasm and plasma and of the capillary number is also assessed.

Experimental observations and modeling of the effect of wettability on immiscible viscous fingers at the Darcy scale

In this work, we present a Darcy-scale study of immiscible unstable displacement in large (15 × 15 × 1 cm3) slabs of Bentheimer sandstone. Oil is displaced by water under two wetting conditions, a weakly water-wet system and a weakly oil-wet system (at two initial water saturations), with a viscosity ratio of μo/μw = 100. The system is directly visualized using x rays and the fluid distribution monitored throughout the displacement. Following a period of water flooding, the injection water is viscosified with glycerol to act as a pseudo tertiary polymer flood. The experiments are then modeled directly using numerical simulation applying a recently developed fractional flow approach, which has been shown to reproduce experimental immiscible fingering, before upscaling the results to the reservoir scale. From the observed results, we can conclude that typical wettability modification using crude oil aging may generate a wettability similar to that found in oil reservoirs, but this may still result in an exaggerated capillary dampening of the viscous fingers. However, as the system is scaled up in size to the field dimensions, viscous forces will dominate, and fingering will re-emerge due to the increase in the viscous/capillary force balance. But if the input relative permeabilities are constructed from an excessively capillary stabilized experiment, the shock front mobility of the fractional flow will still overestimate the stability of the system. Indeed, we derive the same water saturation above Swi for the three floods, but with different shock front mobilities.

Asymptotic analysis of electrohydrodynamic flow through a swarm of porous cylindrical particles

The present article reveals the study of an electrohydrodynamic flow through a membrane composed of a swarm of porous layered cylindrical particles adopting a heat transfer approach. The configuration of the proposed theoretical model is segregated into two regions in which the region proximate to the solid core of the cylindrical particle is a porous region. However, a region surrounded by a porous region is a non-porous (clear fluid) region. The thermal equations are employed under steady-state conditions to establish the temperature distribution when heat conduction prevails over heat convection. The Brinkman and Stokes equations regulate fluid flow through a swarm of porous layered cylindrical particles in porous and non-porous regions, respectively. With the purpose of addressing an electric field in the fluid flow process through a swarm of porous layered cylindrical particles to understand the role of a Hartmann electric number, the momentum equation and the charge density are coupled and nonlinear. The nonlinear second-order differential equation governs the momentum equation and regulates fluid flow through a swarm of porous cylindrical particles. The solutions of the energy equations for both regions are analytically obtained. The asymptotic expansions of velocities for porous and non-porous regions have been derived using the perturbation technique for the small and large values of the nonlinearity parameter α. The effects of various parameters like Hartmann electric number, Grashof number, radiation parameter, viscosity ratio parameter, and porosity of the porous material on the hydrodynamical permeability, Kozeny constant of the membrane, and temperature are analyzed graphically. A noteworthy observation is that a rising Hartmann electric number, the ratio of electric force to the viscous force, enhances the velocity, which is relatively more significant for higher permeability and hence enhances the membrane permeability; however, decay in Kozeny constant is reported with a rising Hartmann electric number. Significant velocity and membrane permeability growth are described with a rising Grashof number, a ratio of thermal buoyancy and viscous forces. The observations from the present study hold promise for advancing our understanding of critical physical and biological applications, including wastewater treatment filtration processes, petroleum reservoir rocks, and blood flow through smooth muscle cells.

Effect of surfactant adsorption on the efficiency of drainage displacement at different viscosity ratios: A lattice Boltzmann study

Surfactant flooding is an effective method for enhanced oil recovery, accompanied by the adsorption of surfactant molecules during their transport in porous media. This paper studies the effects of surfactant adsorption on immiscible two-phase flows in porous media and focuses on two primary objectives. The first is to identify the influence of the water/oil viscosity ratio on the maximum adsorbed amount of surfactant. The second scope involves establishing a relationship between the viscosity ratio and the effect of surfactant adsorption on the displacement efficiency during drainage displacement. The novelty lies in the identification of new patterns characterizing the adsorption effect on the displacement efficiency depending on the viscosity ratio. The main results are based on statistical data collected from numerical simulations performed on various synthetic 2D micromodels. The lattice Boltzmann method was used as a mathematical model. It has been established that the adsorbed amount is influenced by the viscosity ratio as follows: the higher the oil viscosity, the less the adsorbed amount. However, the effect is not strong. An increase in oil viscosity leads to a decrease in the adsorption effect on displacement efficiency. This finding is determined by two factors: the maximum adsorbed amount and the adsorbed amount at the moment of surfactant breakthrough. It has been found that if surfactant adsorption is neglected, the viscosity ratio does not affect the increment in oil recovery compared to the injection of pure water. However, if adsorption is taken into account, an increase in oil viscosity enhances the effect of surfactant flooding on displacement efficiency. The dependence of the viscosity ratio on the adsorption effect on oil recovery obeys linear law. This trend was confirmed in a digital 3D model of natural sandstone obtained using X-ray computed tomography.

Reversal of particle Migration for viscoelastic solution at high solvent viscosity

The imbalance of normal stress around a particle induces its transverse migration in pressure-driven viscoelastic flow, offering possibilities for particle manipulation in microfluidic devices. Theoretical predictions align with experimental evidence of particles migrating towards the center-line of the flow. However, these arguments have been challenged by both experimental and numerical investigations, revealing the potential for a reversal in the direction of migration for viscoelastic shear-thinning fluids. Yet, a significant property of viscoelastic liquids that remains largely unexplored is the ratio of solvent viscosity to the sum of solvent and polymer viscosities, denoted as β. We computed the lift coefficients of a freely flowing cylinder in a bi-dimensional Poiseuille flow with Oldroyd-B constitutive equations. A transition from a negative (center-line migration) to a positive (wall migration) lift coefficient was demonstrated with increasing β values. Analogous to inertial lift, the changes in the sign of the lift coefficient were strongly correlated with abrupt (albeit small) variations in the rotation velocity of the particle. We established a scaling law for the lift coefficient that is proportional, as expected, to the Weissenberg number, but also to the difference in rotation velocity between the viscoelastic and Newtonian cases. If the particle rotates more rapidly than in the Newtonian case, it migrates towards the wall; conversely, if the particle rotates more slowly than in the Newtonian case, it migrates towards the center-line of the channel. Finally, experiments in microfluidic slits confirmed migration towards the wall for viscoelastic fluids with high viscosity ratio.

Dynamic Pore Network Modeling of Imbibition in Real Porous Media with Corner Film Flow.

Wetting films can develop in the corners of pore structures during imbibition in a strongly wetting porous medium, which may significantly influence the two-phase flow dynamics. Due to the large difference in scales between main meniscus and corner film, accurate and efficient modeling of the dynamics of corner film remains elusive. In this work, we develop a novel two-pressure dynamic pore network model incorporating the interacting capillary bundle model to analyze the competition between the main meniscus and corner film flow in real porous media. A pore network with four-point star-shaped pore bodies and throat bonds is extracted from the real porous medium based on the pore shape factor and pore cross-sectional area, which is then decomposed into several layers of sub-pore networks, where the first layer of sub-pore network simulates the main meniscus flow while the upper layers characterize the corner film flow. The two-phase flow conductance of throat bonds for different layers of sub-pore networks are determined by high-resolution two-phase lattice Boltzmann modeling, thus inherently considering the viscous coupling effect. In addition, two artificial neural network models are developed to predict the two phases' flow conductance based on the shape of the throat cross section and the fluid properties. The accuracy of the developed model is validated with a lattice Boltzmann simulation of imbibition in a strongly wetting square tube. Then the model is used to simulate imbibition in a strongly wetting sandstone porous medium, and the competition between the main meniscus and the corner film flow is analyzed. The results show that with decreasing capillary number and viscosity ratio between wetting and nonwetting fluids, the development of the wetting corner film becomes more significant.

Improvement of the properties of poly (L-lactide)/ethylene-ethyl acrylate-glyceryl methacrylate terpolymer (EGA) by the introduction of stereocomplex polylactide crystals

Ternary blends comprising poly(L-lactide) (PLLA), ethylene-ethyl acrylate-glyceryl methacrylate terpolymer (EGA), and poly(D-lactide) (PDLA) were melt blended using low-temperature method to obtain blends with balanced properties. The formation of stereocomplex polylactide (SC-PLA) crystals was verified by torque changes and differential scanning calorimetry (DSC) results. The SC-PLA crystals formed a percolating network in the PLLA matrix at a concentration of 2 wt% PDLA, and blend melt presented solid-like rheological behavior. Attributing to the increased viscosity ratio between PLLA and EGA, the size of dispersed EGA phase increased. The crystallization of PLLA was significantly promoted because of the nucleation effect of SC-PLA crystals. The PLLA/EGA/2 %PDLA ternary blends with elongation at break of 138 % and impact strength of 72.3 kJ/m2 exhibited excellent toughness. These ternary blends demonstrated exceptional crystallization ability and tailored rheological and mechanical properties, opening a new path for developing high-performance PLLA-based materials in packaging industry.

Correlations in a weakly interacting two-dimensional random flow.

We analytically examine fluctuations of vorticity excited by an external random force in two-dimensional fluid. We develop the perturbation theory enabling one to calculate nonlinear corrections to correlation functions of the flow fluctuations found in the linear approximation. We calculate the correction to the pair correlation function and the triple correlation function. It enables us to establish the criterion of validity of the perturbation theory for different ratios of viscosity and bottom friction. We find that the corrections to the second moment are anomalously weak in the cases of small bottom friction and small viscosity and relate the weakness to the energy and enstrophy balances. We demonstrate that at small bottom friction the triple correlation function is characterized by universal scaling behavior in some region of lengths. The developed perturbation method was verified and confirmed by direct numerical simulations.

Direct numerical simulation of a single contaminated droplet microextraction in a yield-stress fluid

The present work investigates the extraction of solute from the contaminated micro-droplet suspended in non-Newtonain viscoplastic fluids (Herschel Bulkley). The fluid flow dynamics and interfacial mass transfer phenomena near a contaminated micro-droplet have been investigated prior to the estimation of extraction efficiency. The stagnant-cap model is employed to depict the droplet that has been contaminated. Dispersed droplet systems are commonly observed in diverse facets of our everyday encounters. These systems are found in various industries, encompassing flotation, energy storage, biofuels, printing, airlift reactors, and foam manufacture, among others. The study has focused on a linear thermodynamic equilibrium relationship that exists at the interface between two fluid phases. The momentum and species transport have been solved for a range of dimensionless parameters. These parameters include θcap, which varies between 120∘ and 150∘, the Reynolds number (1≤Re≤100), the Bingham number (0≤Bn≤5), the dispersed to continuous fluid viscosity ratio (0.5≤μratio≤4), and the shear-thinning index (0.4≤n≤1). The results have been conveyed in the form of streamlines, contours, and by delineating the regions of deformation and non-deformation. The size of the zone that undergoes yielding tends to diminish as a result of the impact of the yield stress, while the presence of inertia promotes the enlargement of the zone that remains unyielded. The extraction efficiency and Sherwood number were estimated to be a function of θcap, Bingham number, Reynolds number, and shear-thinning index.

Impact of wall-slip on the soluto-marangoni instability in a two-fluid system in a channel–Creeping flow scenario

The role of wall-slip on the soluto-Marangoni instability that emerges in a two-fluid creeping two-dimensional Poiseuille flow is assessed within the framework of Orr-Sommerfeld analysis and species-transport system. Different wall-slip scenarios, namely S1 (slip at the upper wall), S2 (slip at the lower wall), S1S2 (symmetric slip at the upper and lower walls) are considered. The system instability is characterized by identifying unstable and stable zones in the m-n (viscosity ratio - depth ratio) plane for each of the slip conditions at the wall using asymptotic analysis for long-wave disturbances. Numerical simulations based on Chebyshev spectral collocation method reveals the emergence of two modes (M1 and M2) and corresponding stability features are displayed for typical values of Marangoni numbers, n, m and slip lengths. The occurrence of short-wave modes and the features of exchanging dominance between the short-wave modes in the presence of wall-slip for different slip condition are elucidated and the physical mechanism responsible for triggering the instability in the system are revealed through energy budget analysis. The channel system hosting less viscous fluid adjacent to the upper wall demonstrates interesting characteristics such as (among other things): stabilizing/destabilizing the emerging M1/M2 modes in system with no-slip by imposing S1/S2/S1S2 wall conditions and increasing the slip length; exchange of dominance of unstable modes beyond a threshold slip length for S1 slip type. The study thus presents possible means of stabilizing/destabilizing a creeping channel flow system by designing the walls of the channel with appropriate slip type relevant to the application.

Measurements of the Viscosity of Hydrogen and a (Hydrogen + Methane) Mixture with a Two-Capillary Viscometer

AbstractMeasurements of the viscosity of pure hydrogen and a binary (hydrogen + methane) mixture with a nominal composition 90 mol % hydrogen are presented. The measurements were conducted with a two-capillary viscometer relative to helium along three isotherms of (298.15, 323.15, and 348.15) K and at pressures of up to 18 MPa. Expanded relative combined uncertainties in viscosity range from (0.65 to 2.7) % (k = 2) for the hydrogen data, and from (0.91 to 3.2) % (k = 2) for the (hydrogen + methane) data. The viscosity data are compared to experimental literature data and viscosity correlations implemented in the NIST REFPROP v10.0 database. Good agreement between this work’s data, literature data, and the viscosity correlation was achieved for pure hydrogen. The (hydrogen + methane) mixture was compared to the Extended Corresponding States (ECS) model implemented in REFPROP v10.0. Relative deviations between the experimental data and the ECS model exceed the experimental uncertainty and were found to exhibit a positive trend with increasing density and a weakly pronounced negative trend with increasing temperature. No experimental literature data are available at overlapping state regions. Nonetheless, deviations to the ECS model imply reasonable consistency of this work’s data and literature data. In addition to experimental viscosities, experimental zero-density viscosity ratios of the fluids under investigation and helium are reported. Fairly good agreement within the experimental uncertainty of this work with a highly accurate literature value and a value obtained from accurate ab initio calculated data was achieved for hydrogen.

Nonisothermal analysis of two uniform boundary layer shear flows

AbstractThe heat transfer phenomenon on the multiphase flows plays an imperative role in several biological and industrial processes, like the oil production industries, catalytic reactors, energy losses in several thermal systems, energy storage, paper manufacture, and heat exchanger systems. Therefore, this study scrutinized the heat transfer characteristic between two uniform boundary layer shear flows with different strengths flowing in the same direction. This problem involves the simulation of the convergence of boundary layers with varying shear strengths. The nonlinear differential system is scrutinized and converted into dimensionless ordinary differential equations by employing appropriate dimensionless variables. The resulting set of highly nonlinear ordinary differential equations is resolved numerically utilizing the Matlab solver “bvp4c.” The resulting numerical solutions are used to construct graphs that illustrate and elucidate the impact of numerous physical parameters on flow patterns. For both fluids, results for the Nusselt number and shear stress are also presented graphically for various parameters. The acquired results demonstrate that the velocity profile upsurges for both upper and lower fluids by enhancing the shear parameter. Furthermore, the number of streamlines is enhanced by augmenting the values of the viscosities ratio parameter. The present problem applies to the trailing edge flow over a thin airfoil with camber.

Stable production of fluid jets with vanishing diameters via tip streaming

We study numerically the microjetting mode obtained when a fluid is injected through a tube submerged in a uniaxial extensional flow. The steady solution to the full nonlinear Navier–Stokes equations is calculated. We obtain the linear global modes determining the linear stability of the steady solution. For sufficiently large outer viscosity, the flow remains stable for infinitely small values of the injected flow rate. This implies that jets with vanishing diameters can be produced regardless of the jet viscosity and outer flow strength. For a sufficiently small inner-to-outer viscosity ratio, the microjetting instability is associated only with the flow near the entrance of the jet. The tapering meniscus stretches and adopts a slender quasiconical shape. Consequently, the cone tip is exposed to an intense outer flow, which stabilizes the flow in the cone–jet transition region. This work presents the first evidence that fluid jets with arbitrarily small diameters can be stably produced via tip streaming. The results are related to those of a droplet in a uniaxial extensional flow with its equator pinned to an infinitely thin ring. The pinning of the equator drastically affects the droplet stability and breakup.

Effect of bulk viscosity on the hypersonic compressible turbulent boundary layer

The impact of bulk viscosity is unclear with considering the increased dilatational dissipation and compressibility effects in hypersonic turbulence flows. In this study, we employ direct numerical simulations to conduct comprehensive analysis of the effect of bulk viscosity on hypersonic turbulent boundary layer flow over a flat plate. The results demonstrate that the scaling relations remain valid even when accounting for large bulk viscosity. However, the wall-normal velocity fluctuations $v_{rms}^{\prime \prime }$ decrease significantly in the viscous sublayer due to the enhanced bulk dilatational dissipation. The intensity of travelling-wave-like alternating positive and negative structures of instantaneous pressure fluctuations $p_{rms}^{\prime }$ in the near-wall region decreases distinctly after considering the bulk viscosity, which is attributed mainly to the reduction of compressible pressure fluctuations $p_{c,rms}^{+}$ . Furthermore, the velocity divergence $\partial u_{i} / \partial x_{i}$ undergoes a significant decrease by bulk viscosity. In short, our results indicate that bulk viscosity can weaken the compressibility of the hypersonic turbulent boundary layer and becomes more significant as the Mach number increases and the wall temperature decreases. Notably, when the bulk-to-shear viscosity ratio of the gas reaches a few hundred levels ( $\mu _b/\mu =O(10^2)$ ), and mechanical behaviour of the near-wall region ( $\kern 0.06em y^+\le 30$ ) is of greater interest, the impact of bulk viscosity on the hypersonic cold-wall turbulent boundary layer may not be negligible.

Numerical simulation investigation of the mechanism of lacquer curtain formation in curtain coating

To study the key factors on integrated curtain formation and breakup in curtain coating production process, we used the numerical simulation method to analyze the various stages of the curtains from full curtain to full thread state by changing the inlet flow rate, which is, changing the Weber number (We). The numerical model was based on an actual production machine, and the physical properties of the lacquer were accurately measured for the simulation. It is found that with the gradual breakup of the curtain, the oscillation of the curtain gradually increased, there was obvious distortion and increase in Turbulent Viscosity Ratio (TVR) at the breakup, and there was a decrease in Dynamic Pressure (DP) at the top of the breakup. Different curtain state reflects a shift from inertia force dominance to a balance of forces and finally to surface tension dominance. The finding from the simulation, also the comparison between the results of simulation and experiments provided the basis to understand the mechanism of curtain coating, also inspired the engineers to optimize the geometry of curtain coater and the coating production parameters for easier integrated curtain forming.

Behavior of Compound Materials on Superhydrophobic Cylinders: Effects of Droplet’s Size and Interface Angle

Compound droplets can consist of two or more immiscible substances sharing an interface. Among such droplets, the low-viscosity component of Janus droplets can exhibit peculiar bouncing behavior on nonwettable surfaces. There have been recent advances in droplet control technologies, however the impact dynamics of droplets on complex surfaces, and strategies to control their behavior, have not been extensively studied. This study employs the volume of fluid method to analyze the effects of Janus droplet size and the initial interface angle on the dynamics of the two fluidic components in droplets on superhydrophobic cylinders. Janus droplets are composed of low-viscosity (W-) and high-viscosity liquid (G-component). The dynamic characteristics of Janus droplets are investigated as a function of Weber number (<i>We</i>), initial interface angle, the ratio of the droplet’s diameter to the cylinder’s diameter, and viscosity ratio (α). Numerical models provide a regime map of the separation ratio of Janus droplets based on We and α, and the influence of droplet size on asymmetric bouncing is discussed. This study also examines the threshold We at which separation begins after impact, varying with droplet size and α. In addition, the shape evolutions of the droplets are discussed for various initial interface angles to understand the bouncing behavior and separation efficiency. This study is expected to provide valuable strategies for controlling droplet behavior and separation in applications such as liquid purification, rheology, and solidification.

An Analytical Solution for Variable Viscosity Flow in Fractured Media: Development and Comparative Analysis With Numerical Simulations

AbstractExplicit fracture models often use analytical solutions for predicting flow in fractured media, usually assuming uniform fluid viscosity for simplicity. This assumption, however, can be inaccurate as fluid viscosity varies due to factors like composition, temperature, and dissolved substances. Our study, recognizing these discrepancies, abandons this uniform viscosity assumption for a more realistic model of variable viscosity flow, focusing on viscous displacement scenarios. This includes instances like injecting viscous surfactants for hydrocarbon recovery in fractured reservoirs or soil decontamination. This presents a significant challenge, enhancing our understanding of transport within fractures, mainly governed by advection. Our study centers on a low‐permeability rock matrix intersected by two fractures with variable apertures. We employ two methods: an analytical approach with a new solution and numerical simulations with two distinct in‐house codes, discretizing both the rock matrix and fractures with two‐dimensional triangular elements. The first code uses a Discontinuous Galerkin finite element method, while the second utilizes a finite‐volume method, allowing a comprehensive comparison of solutions. Additionally, we investigate parameter identifiability, like fracture apertures and viscosity ratios, using breakthrough curves from our analytical solution, applying the Markov Chain Monte Carlo technique.

Deformation and breakup of a ferrofluid droplet in shear flow under magnetic field

Effects of magnetic field applied perpendicular to a shear plane in shear flow on the deformation of a ferrofluid droplet are numerically investigated. The boundary integral method is employed to solve the two-phase Stokes flow under a uniform magnetic field. When the magnetic field is applied perpendicular to the shear plane, the deformation of the droplet in the shear plane decreases. The magnetic field causes the droplet to elongate in the y-direction, and its cross-sectional radius in shear plane decreases. Consequently, the apparent capillary number in the shear plane decreases, thereby suppressing the droplet deformation. Droplet breakup is also suppressed by imposing a magnetic field perpendicular to the shear plane, thereby increasing the critical capillary numbers. The critical capillary numbers for the magnetic Bond numbers Bo = 2.0 and 4.0 increase to approximately 110% and 130%, respectively, than those without magnetic field. Furthermore, an equation for the theoretical prediction of the droplet deformation under a magnetic field in shear flow is presented, which is based on the small deformation theory, the decrease in the cross-sectional radius, and the boundary conditions at the droplet interface. The theoretical prediction agrees well with the numerical results for the variation in the magnetic susceptibility of the droplet as well as the viscosity ratio between the external fluid and the ferrofluid droplet under a small deformation. The critical capillary numbers under a magnetic field can also be predicted by using the numerical results without a magnetic field.