Contraction Ratio Research Articles

Research on the Criteria for Determining the Starting Performance of an Inward-Turning Inlet by Integrating the Concept of the Equivalent Contraction Ratio

The prediction of hypersonic inlet starting performance is crucial for the successful ignition of the combustion chamber, directly impacting the overall performance of the propulsion system. This challenge arises especially when freestream conditions vary. Therefore, this paper proposes the concept of the equivalent contraction ratio, and establishes and analyzes the intrinsic correlation between the geometric contraction ratio and angle of attack on the starting performance of three-dimensional inward-turning inlet. The results indicate the following: (1) The startability index can be applied to determine the start boundary of the three-dimensional inward-turning inlet under conditions of the freestream Mach number of 6.0 and an altitude of 27 km, with a deviation of no more than 6.6% from the optimal SI = 0.087 criterion; (2) The start boundary after applying the equivalent contraction ratio shows deviations not exceeding 4.0% under positive angle-of-attack conditions compared to the startability index, while the deviation is larger under negative angle-of-attack conditions, reaching a maximum of 13.3%. After applying a correction formula, the deviations can be reduced to within 2.0%; (3) For the same equivalent contraction ratio, the differences in starting performance between different positive and negative angle-of-attack conditions may fundamentally arise from the degree of compression of the inlet. Finally, the equivalent contraction ratio theory is proven to be able to quickly and easily predict the accurate starting performance of the inward-turning inlet at different angles of attack, improving the breadth and efficiency of engineering predictions.

Study on the influence of expansion waves on the reflection type of shock wave in double-wedge configurations

Abstract This paper investigates how variations in the relative positioning between the expansion fan emitted from the triple point in the shock/shock interference on double-wedge geometries in confined spaces and the incidence point of oblique shocks influence the transition of oblique shock reflection types, and understands the roles of expansion waves in inhibiting MR. Predictive analysis through the shock polar line provides critical values for the transition of oblique shock reflection types under the influence of expansion fan effects. The position where the shock/expansion fan undergoes post-oblique reflection on the opposite sidewall can be controlled by adjusting the model’s contraction ratio (H 1/H 2). This study uses the density-based solver blastFOAM in openFOAM to numerically solve the Euler equations, conducting inviscid calculations of shock/shock interactions induced by double wedges in confined spaces with ideal gases. This study compares pressure values in different regions obtained from numerical simulations and examines the reflection types of oblique shocks on opposite sidewalls, confirming the feasibility of shock polar analysis and the accuracy of theoretical predictions.

The investigation of ultrasound to assess lateral abdominal wall activation with different types of core exercises

BackgroundCore training is the foundation of physical exercise. The activation of the lateral abdominal wall (LAW) muscles in the core muscles, particularly the transversus abdominal (TrA) muscles, has a stabilizing effect on the chest and abdomen. Therefore, we need to focus on the training effect of the TrA. There are many ways to measure the LAW. Ultrasound can assess the effect of training in real time and intuitively. Therefore, we intend to evaluate the activation of the LAW in different types of core training using ultrasound, to determine the best movements that can activate the TrA and train the core muscles.Methods22 healthy subjects (male 10, female 12, age 22.82 ± 0.98, BMI 20.78 ± 2.27) were included. The subjects were given the following instructions to perform breathing exercises at different positions: calm breathing and deep breathing at 0° hip flexion and 0° knee flexion; calm breathing, deep breathing, abdominal crunches and ball crunches at 45° hip flexion and 90° knee flexion; and calm breathing, deep breathing, abdominal crunches and ball crunches at 90° hip flexion and 90° knee flexion. The muscle thicknesses of the bilateral transversus abdominis (TrA), internal oblique (IO), external oblique (EO), and LAW muscles were measured using ultrasonography at the end of expiration during the above movements.Results(1) The action with the greatest contraction ratio of the TrA was deep exhalation, which was significantly greater than crunch and ball crunch; (2) During deep exhalation, the TrA had the greatest contraction ratio, significantly greater than the IO and EO. (3) The TrA was thinnest during deep exhalation at 90°, followed by 45° and 0°.ConclusionIn healthy young people, deep expiration with 90° hip flexion and 90° knee flexion was the optimal action for activating the LAW, especially the TrA.

Movement behavior and morphological change of droplets on vertically crossed fibers

Droplet capture and coalescence are critical processes for enhancing emulsion separation efficiency in oily wastewater treatment. In this paper, four kinds of motion behaviors and secondary droplet formation behaviors of oil droplets on vertically crossed fibers in a water environment were investigated using high-speed camera technology. The research shows that during the droplet capture process, droplets on large-diameter horizontal fiber exhibit greater lateral spreading, and smaller vertical elongation. Increasing the diameter of horizontal fiber significantly shortens droplet capture time. The capture time on lipophilic fiber intersection of 190–420 μm is 14 ms less than that of 190–190 μm. During the droplet coalescence process, the liquid bridge on large-diameter horizontal fiber is wider, and the droplet contraction ratio is smaller. The droplet contraction ratio on fiber intersection of 190–420 μm is 0.3 smaller than that of 190–190 μm. Increasing the diameter of horizontal fiber can promote rapid fusion between droplets, but lipophobic fibers hinder the degree of fusion and deformation. Furthermore, the diameter and wettability of horizontal fibers have significant effects on the volume of secondary droplets. The volume of secondary droplets increases with the diameter of lipophilic horizontal fiber in the capture process. The volume of secondary droplets on lipophilic horizontal fiber is approximately three times larger than that on lipophobic horizontal fiber in the coalescence process. These findings contribute to understanding the details of coalescence separation and optimizing the design of fiber fillers.

Ultrasound Evaluation of Onset Core Muscle Activity in Subjects with Non-Specific Lower Back Pain and Without Lower Back Pain: An Observational Case-Control Study.

Lower back pain (LBP) has been the leading cause of disability since 1990. Objectives: The main objective of this observational case-control study was to evaluate, using ultrasound, whether there were differences in the onset and ratio of core muscle contraction between subjects with non-specific chronic lower back pain and healthy subjects. Methods: A total of 60 participants (52% women), split between those with non-specific chronic lower back pain (n = 26) and healthy (n = 34) subjects, were recruited. Initial muscle contraction of the lateral abdominal wall, pelvic floor, lumbar multifidus, and respiratory diaphragm was measured using ultrasound. The abdominal drawing-in maneuver, contralateral arm elevation, the Valsalva maneuver, and voluntary contraction of the pelvic floor in seated and standing positions were performed. The muscle thickness of the lateral abdominal wall and lumbar multifidus and excursion of the pelvic floor and diaphragm at rest and during testing were also analyzed. Results: No differences were found between the groups in the initial contraction. Statistically significant differences were found in the following variables: diaphragm excursion (p = 0.032, r = 0.277) and lumbar multifidus ratio (p = 0.010, r = 0.333) in the standing-abdominal retraction maneuver; pelvic floor excursion (p = 0.012, r = 0.325) in the standing-contralateral arm raise; and transverse abdominis ratio (p = 0.033, r = 0.275) in the sitting-contralateral arm raise. A statistically significant interaction between the groups and body mass index was observed in resting diaphragm excursion (p = 0.018, partial eta squared = 0.096) during sitting-voluntary pelvic floor contraction. Conclusions: It cannot be concluded that there is a specific pattern of core activation in any of the groups. However, statistically significant differences were found in the contraction indexes of the lumbopelvic musculature.

Optimal box contraction for solving linear systems via simulated and quantum annealing

Solving linear systems of equations is an important problem in engineering. Many quantum algorithms, such as the Harrow–Hassidim–Lloyd algorithm and the box algorithm, have been proposed for solving such systems. The focus of this article is on improving the efficiency of the box algorithm. The basic principle behind this algorithm is to transform the linear system into a series of quadratic unconstrained binary optimization (QUBO) problems, which are then solved on annealing machines. The computational efficiency of the box algorithm is entirely determined by the number of iterations, which, in turn, depends on the box contraction ratio, typically set to 0.5. Here, it is shown through theoretical analysis that a contraction ratio of 0.5 is sub-optimal and that a computational speed-up can be achieved with a contraction ratio of 0.2. This is confirmed through numerical experiments where a computational speed-up between 20 % to 60 % is observed when the optimal contraction ratio is used.

An exact solution of the lubrication equations for the Oldroyd-B model in a hyperbolic pipe

An exact analytical solution of the lubrication equations for the steady, isothermal, incompressible flow of a viscoelastic Oldroyd-B fluid in a hyperbolic cylindrical contracting pipe is derived. The solution is valid for values of the Deborah number, De, up to order unity (De is defined as the ratio of the longest relaxation time of the polymer to the characteristic residence time of the fluid in the pipe), all values of the ratio of the polymer viscosity to the total viscosity of the fluid, η, and typical values of the contraction ratio, Λ, encountered in experiments and practical applications. It is provided in terms of the streamfunction only and is used in the momentum balance to derive a strongly non-linear ordinary differential equation of second order with unknown a function which corresponds to a modified fluid velocity along the main flow direction. The final equation is solved semi-numerically using a fully spectral (Legendre)-Galerkin approach to resolve the unknown function almost down to machine accuracy. The exact solution for the polymer extra-stresses, which is emphasized is not the full solution of the complete lubrication equations, allows for the derivation of a variety of theoretical expressions for the average pressure-drop along the pipe. In all cases, a decrease in the pressure drop compared to the Newtonian value with increasing De, η and/or Λ is predicted. The differences between the corresponding analytical solution for the planar geometrical configuration are also identified and discussed.

Experimental investigation of slit weir discharge

ABSTRACT A rectangular slit weir is used to measure small flows, namely, those flows less than 5 L/s and contraction ratios (b/B) less than 0.25. In the present study, flow tests were conducted in a laboratory setup with various contraction ratios (b/B) under models A (b/B = 1/12), B (b/B = 1/8), C (b/B = 1/6), D (b/B = 5/24), and E (b/B = 1/4). The dimensional analysis revealed that b/B, the height of the flow on the crest of the weir to the height of the weir crest (h/P), the Reynolds number (Re), and the Weber number (We) are effective parameters for predicting the discharge coefficient of the slit weir. The experimental results indicated that the discharge coefficient is variable for different contraction ratios; the maximum average discharge coefficient was achieved at the lowest b/B (Model A), with a value of 0.72. The minimum average discharge coefficient obtained in Model B equals 0.639. In addition, former prediction equations were compared with the obtained experimental data, and the minimum, average, and maximum errors were calculated. Furthermore, a relationship was established for deriving the discharge coefficient in slit weirs with R2 = 0.9212, RMSE = 0.02244, and MAE = 0.0134 based on the obtained experimental data with four effective parameters, b/B, h/P, Re, and We.

The Effect of Adding the Mishraq Sulfur Ore on the Physical and Chemical Properties of Al-Angor and Al-Nahrawan Clays to Producing Lightweight Bricks

This paper aims to examine the possibility of producing lightweight laboratory clay Bricks by adding sulfur ore powder in different weight ratios ranging from 0, 10, 20, 30, 40, to 50% as supplementary materials to two types of clays taken from two different sites, which is the Al-Angor clay region near the city of Ramadi and the Al-Nahrawan clay region. The bricks burned at temperatures of 950 and 1050 C°٬ and the longitudinal Shrinkage tests, bulk density, water absorption, compression strength, and efflorescence were measured. The X-ray diffraction analysis was executed to determine the formed metal phases. The results of the tests showed a direct proportion between longitudinal contraction and water absorption ratio with the addition of sulfur ore and high burning temperature. While bulk density and compression strength are inversely proportional to the sulfur ratio and high burning temperature. Efflorescence tests showed a decrease in the efflorescence rate with an increase in the proportion of sulfur ore.

2-dimensional impact-damping electrostatic actuators with elastomer-enhanced auxetic structure

Biomimetic robots yearn for compliant actuators that are comparable to biological muscle in both functions and structural properties. For that, electrostatic actuators have been developed to imitate bio-muscle in features of fast response, high power, energy-efficiency, etc. However, those actuators typically lack impact damping performance, making them vulnerable and unstable in real applications. Here, we present auxetic electrostatic actuators that address this issue and demonstrate muscle-like performance by using elastomer-enhanced auxetics and electrostatic zipping mechanism. The proposed actuators contract linearly on applied voltage, producing large actuation strength (15 N) and contraction ratio (59%). Fabricated from readily available materials, our prototypes can quickly attenuate vibrations caused by impacts and absorb shock energy in 0.3 s. Furthermore, leveraging their 2-dimensional working mode and self-locking mechanism, a stiffness-changing muscle for a robotic arm and an active tensegrity device exemplify the potential applications of auxetic electrostatic actuators to a wide range of bionic robots.

Simultaneous effects of capillary number, viscosity ratio, and contraction ratio on droplet dynamics in contraction microchannel

Abstract In droplet-based microfluidic systems, fluid properties, flow conditions, and channel geometry play crucial roles in the movement and manipulation of droplets. This study employs a three-dimensional numerical simulation method to investigate droplet dynamics in a contraction microchannel under the simultaneous influence of different factors, such as capillary number (Ca), viscosity ratio (λ), and contraction ratio (C). Specifically, a theoretical model is introduced for predicting the transition of droplets from trap to squeeze, and it is observed to be 〖Ca〗_I = f(C,λ). Meanwhile, droplet dynamics are investigated for the transition from squeeze to breakup for finding critical capillary number value (CaII). In addition, the critical viscosity ratio for the process from squeeze to breakup has been figured out by numerical results for a wide range of contraction ratios. Eventually, droplet dynamics including deformation, velocity ratio, and breakup during contraction microchannel are also quantitatively studied under the factors above.

Open-jet facility for bio-inspired micro-air-vehicle flight experiment at low speed and high turbulence intensity

Bio-inspired micro-air-vehicles (MAVs) usually operate in the atmospheric boundary layer at a low Reynolds number and complex wind conditions including large-scale turbulence, strong shear, and gusts. We develop an open jet facility (OJF) to meet the requirements of MAV flight experiments at very low speed and high turbulence intensity. Powered by a stage-driven fan, the OJF is capable of generating wind speeds covering 0.1 – 16.8 m/s, with a velocity ratio of 100:1. The contraction section of the OJF is designed using an adjoint-driven optimization method, resulting in a contraction ratio of 3:1 and a length-to-diameter ratio of 0.75. A modularized design of the jet nozzle can produce laminar or high-turbulence wind conditions. Flow field calibration results demonstrate that the OJF is capable of producing a high-quality baseline flow with steady airspeed as low as 0.1 m/s, uniform region around 80% of the cross-sectional test area, and turbulence intensity around 0.5%. Equipped with an optimized active grid (AG), the OJF can reproduce controllable, fully-developed turbulent wind conditions with the turbulence intensity up to 24%, energy spectrum satisfying the five-thirds power law, and the uniform region close to 70% of the cross-sectional area of the test section. The turbulence intensity, integral length scale, Kolmogorov length scale, and mean energy dissipation rate of the generated flow can be adjusted by varying the area of the triangular through-hole in the wings of the AG.

Newtonian flow with slip and pressure-drop predictions in hyperbolic confined geometries

We study theoretically the steady Newtonian flow in confined and hyperbolic long tubes (symmetric channels and axisymmetric pipes) considering slip along the walls. Using a stream function formulation, and the extended (or high-order) lubrication method in terms of the square of the aspect ratio of the tube, ε, the solution for the stream function is found analytically up to twentieth order in ε. At the classic lubrication limit, i.e. i.e. for a vanishing small aspect ratio, and for perfect slip conditions, the analysis predicts a plug-like velocity profile and a constant strain-rate on the midplane/axis of symmetry of the tube. A constant strain-rate is also predicted for the non-slip case. Furthermore, the high order asymptotic results for the stream function and fluid velocity are post-processed with an acceleration technique to investigate the convergence and accuracy of the solution. The results reveal the existence of a boundary layer at the inlet of the tube, the influence of which diminishes in a very short distance from the entrance. We discuss the effect of the contraction ratio of the tube and the dimensionless slip coefficient on the midplane/centerline and wall (slip) velocities, as well as on the average pressure-drop, required to maintain a constant flow-rate. The acceleration of converge technique on the solution for the pressure-drop revealed a remarkable convergence at a value slightly larger (∼1 %) than the value predicted by the classic lubrication theory. Finally, we comment on the common practice in the literature for approaching the velocity profile with the velocity profile at the classic lubrication limit, and we compare the high-order results for the strain rate at the midplane/centerline with the effective strain rate previously derived in the literature by Housiadas & Beris, J. Rheology, 68(3), 327–339, 2024.

Usage of coke dry quenching dust to improve coke mean size

ABSTRACT Improving the mean size of surface coke is an important subject in coke-making process. Focusing on usage of crushed inertinite, we investigated on addition of coke dry quenching (CDQ) dust in coal blend in different compositions. The effect of addition of CDQ dust on coke quality including coke mean size, changes in size fractions and coke CSR was studied. CDQ dust acts as crushed inertinite and helps in minimizing the decrease in coke strength occurred because of small cracks generated during carbonization. These cracks are generated because of different contraction ratio of vitrinite and inertinite. After taking plant trials with varied CDQ dust composition from 1% to 3% in coal blend, 1% CDQ dust usage in coal blend revealed as the optimized percentage for achieving higher mean size.

Numerical Calculations of Liquid-Metal Magnetohydrodynamic Flows in Rectangular Ducts with Sudden Contractions

For the design of the liquid-metal blanket in a fusion reactor, numerical calculations have been carried out on liquid-metal magnetohydrodynamic flows in rectangular ducts with sudden contractions. Conservation equations of fluid mass and fluid momentum and the Poisson equation for electrical potential have been solved numerically. The numerical calculations have been conducted for a Hartmann number of ~10 000; a Reynolds number of ~10 000; and contraction ratios (CRs) of 2, 3, and 4. The pressure loss through the contraction has been estimated by the loss coefficient ζ divided by the interaction parameter N, i.e. ζ/N. The loss coefficient ζ/N through the contraction parallel to the magnetic field is much larger than that through the corresponding contraction perpendicular to the magnetic field. The loss coefficient ζ/N increases consistently with the CR and does not change very much with N. While ζ/N also does not change very much with the wall conductance ratio for the contraction parallel to the magnetic field, ζ/N increases gradually with the wall conductance ratio for the contraction perpendicular to the magnetic field.

Unsteady flow characteristics in an over-under TBCC inlet during mode transition under unthrottled and throttled conditions

The study presents an experimental exploration into the mode transition of an over-under TBCC (Turbine-Based Combined Cycle) inlet, with a specific emphasis on the flow characteristics at off-design transition Mach number. A systematic investigation was undertaken into the mode transition characteristics in both unthrottled and throttled conditions within a high-speed duct, employing high speed Schlieren and dynamic pressure acquisition systems. The results show that the high-speed duct faced flow oscillations primarily dictated by the separation bubble near the duct entrance during the downward rotation of splitter, leading to the duct’s unstart under the unthrottled condition. During the splitter’s reverse rotation, a notable hysteresis of unstart/restart of the high-speed duct was observed. Conversely, hysteresis vanishes when the initial flowfield nears the critical state owing to downstream throttling. Moreover, the oscillatory diversity, a distinctive characteristic of the high-speed duct, was firstly observed during the mode transition induced by throttling. The flow evolution was divided into four stages: an initial instability stage characterized by low-frequency oscillations below 255 Hz induced by shock train self-excitation oscillation and high-frequency oscillations around 1367 Hz caused by the movement of separation bubble. This stage is succeeded by the “big buzz” phase, comprised of pressure accumulation/release within the overflow-free duct and shock motion outside the duct to retain dynamic flow balance. The dominant frequency escalated with the increase of the internal contraction ratio in the range of 280 Hz to 400 Hz. This was followed by a high-frequency oscillation stage around 453 Hz dominated by a large internal contraction ratio with low pulsating energy, accompanied by a continuous supersonic overflow. Lastly, as the splitter gradually intersected the boundary layer of the first-stage compression surface, the capture area and the turbulence intensity of the incoming flow underwent a sudden shift, leading to a more diverse flow oscillation within the duct, manifested as various forms of mixed buzz.

Design and aerodynamic characteristics of variable-geometry hypersonic inlet based on shock wave elimination

To solve the problem that the control of shock wave elimination is weakened under off-design conditions, the design concept of a variable-geometry inlet scheme that combines the variable-geometry cowl (translating and diagonalizing) with regulating shock wave elimination is introduced in this paper. The variable-geometry inlet is designed by the theories of oblique shock wave and isentropic wave as well as the Oswatitsch theory. Regulatory law of the variable-geometry cowl based on shock wave elimination is obtained by the geometric relationships between cowl compression angle, cowl shock wave angle, and optimal control point or range. Numerical simulations are conducted to investigate flow field characteristics, control mechanism, and working performance of the inlet. Results reveal that expansion waves have a significant impact on the cowl shock wave and boundary layer interaction, and flow separation. Furthermore, variable-geometry inlet with translating and diagonalizing cowl based on the regulation of shock wave elimination effectively controls and even completely eliminates the flow separation. In terms of inlet performance, the total pressure loss of the variable-geometry inlet decreases such that the total pressure recovery coefficients of the translating cowl and diagonalizing cowl inlets are increased by maximum values of 3.39 % and 9.97 %, respectively. However, the mass flow coefficient of translating cowl inlet decreases, whereas that of the diagonalizing cowl inlet is equivalent to that of the fixed-geometry inlet. The working range can be widened by changing the internal contract ratio of the inlet through translating or diagonalizing the cowl. The results confirm that the scheme of variable geometry inlet with diagonalizing cowl is practicable and reliable and has important guiding significance and value for inlet design.

Effects of the outlet contraction ratio on the performance of liquid kerosene/air two-phase rotating detonation combustors

Experiments and numerical simulations were conducted across a range of contraction ratio to investigate the detonation combustion characteristics of liquid aviation kerosene/air rotating detonation combustor. The research indicates that extremely low and high contraction ratio adversely affect the stable operation of the two-phase rotating detonation combustors. At a contraction ratio that is too low (19.03 % in the RDC discussed in this paper), the atomization and mixing properties of the two-phase fuel are poor, with a lower reaction pressure in the fresh mixture, hindering the initiation of detonation waves. On the other hand, a contraction ratio (85.7 % in the RDC discussed in this paper) that is too high enhances the energy of the detonation wave; it triggers oblique shockwave feedback, preventing the fresh mixture from entering the combustor and thus leading to the quenching of detonation waves. With an increase in the contraction ratio of outlet configurations, there is a noticeable rise in both the peak pressure and combustion efficiency of detonation waves. In contrast, the wave height gradually decreases, and the wave velocity shows a pattern of initially increasing and then decreasing. Along the axial position inside the two-phase rotating detonation combustor, the upstream detonation wave is caused by the reflected shockwaves increasing the pressure of the upstream fresh mixture. In contrast, the downstream detonation wave results from increased fresh mixture pressure and enhanced fuel atomization and mixing due to the increased contraction ratio of outlet configurations and wave feedback. So, the detonation wave exhibits a strong-weak-strong distribution characteristic.

Liquid Crystal Elastomer Hollow Fibers as Artificial Muscles with Large and Rapid Actuation Enabled by Thermal‐Pneumatic Enhanced Effect

AbstractLiquid crystal elastomer (LCE) has large and reversible deformation under stimulation, making it an ideal material for artificial muscles. Currently, multi‐stimulus responsive LCE actually responds to each stimulus independently rather than responding to several stimuli simultaneously. Achieving an enhanced effect from concurrent stimuli is still a challenge, which requires a new stimulus‐response mechanism. This work develops a novel and facile solvent evaporation‐assisted template method to prepare LCE hollow fiber (LCEHF) with axial alignment. Taking advantage of the enhanced effect of heat‐induced phase transition and mechanical force‐induced orientation transition of mesogens, the LCEHF under both heat and pressure can produce a large contraction ratio of ≈50%, which is higher than 42% of thermal stimulation or 27% of pneumatic stimulation. It can also enhance the respective response and recovery speed of LCEHF to 300 and 3700 times of pneumatic actuation. Additionally, the enhanced effect lowers the actuation temperature much below the phase transition temperature, imparting LCEHF with high mechanical performance during actuation. Furthermore, the dual thermal‐pneumatic responsive LCEHF can mimic human biceps and cause large and rapid bending of an artificial arm reversibly. This new actuation methodology sheds new light on improving LCE actuation performance and broadens its application scenarios.

Multiple solution of MHD Casson fluid flow in porous channel with chemical reactions and thermal radiation effects

This study focused on evaluating the synergistic effects of magnetohydrodynamics, Casson fluid properties, thermal radiation, chemical reactions, and heat transfer mechanisms within a porous channel. Through the application of similarity transformation, nonlinear governing equations are converted into nonlinear ordinary differential equations. These modified equations were then numerically solved in MATLAB using the Runge–Kutta method alongside the shooting technique to ensure precise results. The investigation yielded various numerical solutions for the dimensionless velocity, temperature, concentration, skin friction coefficient, heat transfer coefficient, and Sherwood number, all of which are presented in both tabular and graphical formats. Three distinct solutions emerged from the analysis, differentiated by variations in the Reynolds number, Casson number, expansion or contraction ratio, and magnetic parameters. These solutions demonstrate differences in the parameters under investigation. Solutions within the ranges of γ[5,∞] and R[31.07,∞] are influenced by the specific values of the magnetic number M[0,2.0]. For each M value, there exists a unique set of solutions for γ and R that satisfy the established parameters. As the magnetic number varies, so does the range of feasible solutions, highlighting the sensitivity of the system to magnetic influence. The values of γ and R are adjusted according to the chosen M value.