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This study presents a mathematical model for the coupled growth of two interacting wax particles in a non-isothermal laminar flow. The formulation is based on a diffusion-controlled framework, in which the particle evolution is governed by a Stefan-type moving boundary condition with temperature-dependent interfacial concentration. An asymptotic analysis is developed in the limit where the particles’ size is small compared to their separation distance. This leads to a reduced system of nonlinear ordinary differential equations that captures the combined effects of particle interaction and thermal asymmetry. The analysis reveals that both mechanisms enter at leading order and jointly determine the growth dynamics. Numerical simulations are performed to investigate symmetric and asymmetric configurations. The results demonstrate that temperature differences induce a symmetry-breaking mechanism, leading to distinct growth rates even for initially identical particles. Furthermore, the interaction between particles amplifies this asymmetry through a competitive growth process. A key finding is the monotonic increase in the asymmetry ratio, reflecting progressive divergence driven by thermal effects. The proposed model extends the classical method of reflections for interacting Stefan problems to account for thermally induced asymmetry, incorporating non-identical boundary conditions governed by a prescribed temperature field.

Using a thermally stratified water-based nanofluid and a permeable stretching sheet as a simulation environment, this research examines the impact of nanoparticle aggregation on MHD mixed convective stagnation point flow. Nanoparticle aggregation is studied using two modified models: the Krieger–Dougherty and the Maxwell–Bruggeman. The present problem's governing equations were transformed into a solvable mathematical model utilizing legitimate similarity transformations, and numerical solutions were then achieved using shooting with Runge–Kutta Fehlberg (RKF) technique in Mathematica. Equilibrium point flow toward permeable stretching surface is important for the extrusion process because it produces required heat and mass transfer patterns and identifies and clarifies fragmented flow phenomena using diagrams. Nanoparticle volume fraction was shown to have an impact on the solutions' existence range, as well. Alumina and copper nanofluids have better heat transfer properties than regular fluids. The skin friction coefficients and Nusselt number, velocity, temperature profiles for many values of the different parameters were obtained. In addition, the solutions were shown in graphs and tables, and they were explained in detail. A comparison of the current study's results with previous results for a specific instance is undertaken to verify the findings, and excellent agreement between them is observed.

Ethylene glycol is commonly used as a cooling agent in the engine, therefore the study associated with EG has great importance in engineering and mechanical fields. The hybrid nanofluid has been synthesized by adding copper and graphene nanoparticles into the Ethylene glycol, which obeys the power-law rheological model and exhibits shear rate-dependent viscosity. As a result of these features, the power-law model is utilized in conjunction with thermophysical characteristics and basic rules of heat transport in the fluid to simulate the physical situations under consideration. The Darcy Forchhemier hybrid nanofluid flow has been studied under the influence of heat source and magnetic field over a two-dimensionally stretchable moving permeable surface. The phenomena are characterized as a nonlinear system of PDEs. Using resemblance replacement, the modeled equations are simplified to a nondimensional set of ODEs. The Parametric Continuation Method has been used to simulate the resulting sets of nonlinear differential equations. Figures and tables depict the effects of physical constraints on energy, velocity and concentration profiles. It has been noted that the dispersion of copper and graphene nanoparticulate to the base fluid ethylene glycol significantly improves velocity and heat conduction rate over a stretching surface.

A magneto couple stress nanofluid flow along with double diffusive convection is presented for peristaltic induce flow through symmetric nonuniform channel. A comprehensive mathematical model is scrutinized for couple stress nanofluid magneto nanofluids and corresponding equations of motions are tackled by applying small Reynolds and long wavelength approximation in viewing the scenario of the biological flow. Computational solution is exhibited with the help of graphical illustration for nanoparticle volume fraction, solutal concentration and temperature profiles in MATHEMTICA software. Stream function is also computed numerically by utilizing the analytical expression for nanoparticle volume fraction, solutal concentration and temperature profiles. Whereas pressure gradient profiles are investigated analytically. Impact of various crucial flow parameter on the pressure gradient, pressure rise per wavelength, nanoparticle volume fraction, solutal concentration, temperature and the velocity distribution are exhibited graphically. It has been deduced that temperature profile is significantly rise with Brownian motion, thermophoresis, Dufour effect, also it is revealed that velocity distribution really effected with strong magnetic field and with increasing non-uniformity of the micro channel. The information of current investigation will be instrumental in the development of smart magneto-peristaltic pumps in certain thermal and drug delivery phenomenon.

The analysis of nanofluids under various physical scenarios convinced the researchers and scientists because of their broad range of applications in potential area of the current time like chemical engineering, biomedical engineering and applied thermal engineering etc. To give the final shape of many industrial and engineering processes, enhanced heat transfer desired, therefore, the study of Al2O3-H2O, γAl2O3-H2O, Al2O3-C2H6O2, and γAl2O3- C2H6O2 nanofluids is reported. The model successfully achieved after mathematical operations and by appealing similarity transforms. To examine the behavior of heat transfer, numerical tools utilized and performed the results. It is observed that enhanced heat transfer in Al2O3-H2O, γAl2O3-H2O, Al2O3-C2H6O2, and γAl2O3-C2H6O2 could be attained by setting nanoparticles concentration up to 20%. For Al2O3-H2O, γAl2O3-H2O, optimum heat transfer trends noticed due to their prominent thermophysical values. Also, fewer effects of combined convection on θ ( η ) examined.

After applying a magnetic field, the behavior of the partly ionized liquids is completely different from that of the ordinary fluids. In this study, we concentrated on the Cattaneo–Christov heat flux model-based three-dimensional partly ionized bio-convective flow of a second-order fluid on a bidirectional permeable stretching surface. The development of the thermal and solutal flow models takes into account the impacts of non-uniform sources and sinks, Ohmic viscous dissipation, and chemical reactions. In addition, the surface boundary effects of electron and ion collisions with convective boundary conditions are seen. The mathematical flow model is transformed appropriately to create an ordinary differential equations, which is then numerically solved with MATLAB’s BVP4C approach. To demonstrate the physical relevance of the flow field along various developing parameters, graphical and tabular results are created. It is noteworthy to note that while fluid temperature decreases with stronger values of the second-order fluid parameter, fluid velocity improves in both directions. In addition, it is shown that raising the thermal and concentration relaxation parameters, respectively, causes a drop in the fluid temperature and nanoparticle concentration. Graphical Abstract Graphical Abstract The graphical description of the second-grade fluid parameter along velocity field. It is seen from the figure that stronger values of second-grade fluid parameter improve the fluid velocity consequently, as a result momentum boundary layer thickness boosts up.

The dissolution of solid particles in liquids is commonly modeled as a diffusion-controlled process coupled with a moving boundary. While such models allow for analytical solutions under stationary conditions, they do not account for the effects of fluid motion that commonly arise in practical situations. In the present work, this classical model is extended by incorporating advective transport due to an imposed uniform flow. The dissolution process is formulated as a free boundary problem governed by an advection-diffusion equation in the exterior of the particle, coupled with a Stefan-type condition describing the interface motion. By introducing a non-dimensionalization, the problem is characterized by the Péclet number, which measures the relative importance of advection and diffusion. An asymptotic expansion in the Péclet number is employed to develop semi-analytical asymptotic solutions. The leading-order term recovers the diffusion controlled self-similar behavior, while higher-order corrections reveal how uniform flow modifies the concentration field and increases the dissolution rate. In particular, it is shown that the first non-zero flow-induced correction to the interface motion arises at second order, reflecting the inherently nonlinear nature of advection effects. The analysis provides a mathematically consistent extension of diffusion-only dissolution models and offers new insight into flow-induced dissolution mechanisms.

Many crude oils contain dissolved waxes that can cause significant problems, such as blockage of pipeline, because of the precipitation and deposition of wax particles during the production and transportation of the oil. The waxy oils are transported through very long pipelines from warm natural reservoirs to cooler conditions in the surrounding of the pipe. An important phenomenon occurring during the under-cooling of the pipeline is the formation of solid matter inside the pipe. The wax deposition is one of the most serious problems, potentially restricting flow and plugging the pipe. Wax deposits begin to form when the temperature is below the wax appearance temperature. We model a particle's growth in the oil pipe once the temperature falls below the wax appearance temperature. We determine the temperature distribution in the oil, formulate and solve a self-similar problem of wax particle growth from a single point. A corresponding initial boundary value problem is studied numerically by a time-stepping numerical algorithm. The numerical algorithm is used to compute the non-linear solution of an initial value problem of diffusion and transport of wax towards the particle. The numerical solution is compared with and validated against the self-similar solution derived for specific conditions. Then the boundary value problem is formulated for the general case. A coupled diffusion/flow problem of a single wax particle is formulated. An asymptotic analysis is used to describe the motion and growth of the wax particle. At the leading order, we consider a spherical wax particle and assume the velocity of the particle to be close to the local velocity of the flow. The flow relative to the wax particle is negligible in the leading order problem. In the first-order correction problem, the wax particle is treated as spherical initially, and the correction to the particle shape is caused by a small difference between the wax particle speed and the speed of the local flow. We draw conclusions and recommendation for further work at the end of this thesis.