Search Results (68)

In this manuscript, we study a general incompressible Oldroyd-B system and first establish a new interpretation for the Green’s matrix and then establish the pointwise estimates for the Green’s matrix, especially for the high-frequency part, where the dependence on the viscosity constants is carefully analyzed. Full article

This study theoretically investigates blood rheology in arteries by modeling blood as an Oldroyd-B nanofluid with uniformly suspended $Au$, $Cu$, and $A{l}_2{O}_3$ nanoparticles. A fractional order framework is employed to capture memory and hereditary effects while preserving the invariance of governing equations. The influence of nanoparticle geometry is examined by considering spherical (isotropic), cylindrical (axial), and platelet-like (planar) shapes. Using integral transform techniques, a comparative analysis highlights how particle symmetry and system parameters affect flow behavior and heat transfer. Thermal effects are further analyzed as both a contributor to flow resistance and a source of symmetry breaking in conduction, with implications for optimizing nanofluid-based blood rheology in biomedical applications such as cryosurgery. Full article

The optimization of polymer extrusion processes is crucial for improving product quality and manufacturing efficiency in plastic industries. This study aims to investigate the viscoelastic flow behavior of high-density polyethylene (HDPE) through an extrusion die with an internal mandrel, focusing on the effects of die geometry and flow parameters. A two-dimensional (2D) numerical model is developed in COMSOL Multiphysics using the Oldroyd-B constitutive equation, solved using the Galerkin/least-square finite element method. The simulation results indicate that the Weissenberg number (Wi) and die geometry significantly influence the dimensionless drag coefficient (Cd) and viscoelastic stress distribution along the die wall. Furthermore, filleting sharp edges of the die wall surface effectively reduces stress oscillations, enhancing flow uniformity. These findings provide valuable insights for optimizing die design and improving polymer extrusion efficiency. Full article

In this paper, we are concerned with the Cauchy problem of the compressible Oldroyd-B model without stress diffusion in ${\mathbb{R}}^n(n=2,3)$. The absence of stress diffusion introduces significant challenges in the analysis of this system. By employing tools from harmonic analysis, particularly the Littlewood–Paley decomposition theory, we establish the global well-posedness of solutions with initial data in $L^p$ critical spaces, which accommodates the case of large, highly oscillating initial velocity. Furthermore, we derive the optimal time decay rates of the solutions by a suitable energy argument. Full article

Cardiovascular diseases represent one of the leading causes of mortality worldwide, underscoring the need for accurate simulations of blood flow to improve diagnosis and treatment. This study examines blood flow dynamics in two different vascular structures—the aorta and the right coronary artery (RCA)—using Computational Fluid Dynamics (CFD). Utilizing COMSOL Multiphysics^®, various mathematical models were applied to simulate blood flow under physiological conditions, assuming a steady-flow regime. These models include both Newtonian and non-Newtonian approaches, such as the Carreau and Casson models, as well as viscoelastic frameworks like Oldroyd-B, Giesekus, and FENE-P. Key metrics—such as velocity fields, pressure distributions, and error analysis—were evaluated to determine which model most accurately describes hemodynamic behavior in large vessels like the aorta and in smaller and more complex vessels like the RCA. The results highlight the importance of shear-thinning and viscoelastic properties in small vessels like the RCA, which contrasts with the predominantly Newtonian behavior observed in the aorta. While computational challenges remain, this study contributes to a deeper understanding of blood rheology, enhancing the accuracy of cardiovascular simulations and offering valuable insights for diagnosing and managing vascular diseases. Full article

This paper simulates the blood clot extraction process inside an idealized cylindrical blood vessel model using the aspiration-based thrombectomy technique. A fully Eulerian technique is used within the finite volume method where incompressible Navier–Stokes equations are solved in the fluid region. In contrast, the Cauchy stress equation is solved in the clot region. Blood is assumed to be a Newtonian fluid, while the clot is either hyperelastic or viscoelastic material. In the hyperelastic formulation, the clot deformation is calculated based on the left Cauchy–Green deformation tensor, while the stresses are based on the linear Mooney–Rivlin model. In the viscoelastic formulation, the Oldroyd B model is used within the log conformation approach to calculate the viscoelastic stresses in the clot. The interface between the blood and the clot is tracked with the help of the geometric volume-of-fluid method. We focus on the role of flow variables like the pressure, velocity, and proximity between the clot and the catheter tip to successfully capture the clot under catheter suction. We observe that, once the clot is attracted to the catheter port due to pressure forces, the viscous stresses try to drag it inside the catheter. On the other hand, if the clot is not initially attracted, it is carried downstream by the viscous stresses. If the suction velocity is low (∼0.2 m/s), the clot cannot be sucked inside the catheter, even if it is touching the catheter. At a higher suction velocity of 0.4 m/s, the suction effect is strong enough to capture the clot despite the larger initial distance from the catheter. Hence, the pressure distribution and viscous stresses play essential roles in the suction or escape of the clot during the thrombectomy process. Also, the viscoelastic model predicts the rupture of the clot inside the catheter during suction. Full article

This paper investigates the linear stability of the liquid film of Oldroyd-B fluid on an oscillating plate. The time-dependent Orr–Sommerfeld boundary-value problem is formulated through the assumption of a normal modal solution and the introduction of the stream function, which is further transformed into the Floquet system. A long-wavelength expansion analysis is performed to derive the analytical solution of the Orr–Sommerfeld equation. The results indicate that long-wave instability occurs only within specific bandwidths related to the Ohnesorge number ($Oh$). Fixing the elasticity parameter ($El$) and increasing the relaxation-to-delay time ratio ($\tilde{\lambda }$) from 2 to 4 or fixing ($\tilde{\lambda }$) and increasing ($El$) from 0.001 to 0.01 decreases the number of unstable bandwidths while enhancing the intensity of unstable modes. Increasing the surface-tension-related parameter (${\zeta }^{\ast }$) from 0 to 100 suppresses the wave growth rate, stabilizing the system. Additionally, increasing ($\tilde{\lambda }$) from 2 to 4 reduces the maximum values of the coupling of viscoelastic, gravitational, and surface-tension forces, as well as the maximum value of the Floquet exponent, further stabilizing the system. These findings provide supplements to the theoretical research on the stability of viscoelastic fluids and also offer a scientific basis for engineering applications in multiple fields. Full article

According to Prandtl’s boundary layer theory, the entrance length refers to the axial distance required for a flow to transition from its initial entry condition to a fully developed flow where the velocity profile stabilizes downstream. However, this theory remains applicable only under the assumption of Re ≫ 1, while its validity diminishes under low-Reynolds-number conditions. This study utilizes OpenFOAM based on the finite volume method to numerically examine Newtonian and viscoelastic fluids in a laminar circular pipe flow. The objective is to determine the range of Reynolds numbers for which the differential equations from within the Prandtl boundary layer theory are strictly valid. Additionally, the study explores the effects of Reynolds numbers (Re) ranging from 50 to 100, s solvent viscosity ratio (β) fixed at 0.3 and 0.7, and Weissenberg numbers (Wi) ranging from 0.2 to 5 on the entrance length and friction factor for the Oldroyd-B model. The results indicate the presence of a lower Reynolds number that impedes the attainment of the outcomes predicted by the Prandtl boundary layer theory for the entrance length. The inertia effect, the increase in solvent viscosity contribution, and the elastic effect exhibit a linear relationship with the entrance length and friction factor. Full article

Symmetry transformation methods are widely used in fluid flow problems. One such method is renormalization group analysis. Renormalization group methods are used to develop a macroscopic turbulence model for non-Newtonian fluids (Oldroyd-B type). This model accounts for the large-distance and large-time behavior of velocity correlations generated by the momentum equation for a randomly stirred, incompressible flow and does not account for empirical constants. The aim of this mathematical study was to develop a k-ε RNG turbulence model for non-Newtonian fluids (Oldroyd-B type). For the first time, using the renormalization procedure, the transport equations for the large-scale modes and expressions for effective transport coefficients are obtained. Expressions for the renormalized turbulent viscosity are also derived. This model explains the phenomenon of the abrupt growth of the irregularity of velocity at low values of the Reynolds number. Full article

The Taylor–Couette flow of electrically conducting incompressible Oldroyd-B fluids induced by time-dependent couples in an annulus is analytically investigated when magnetic and porous effects are taken into account. Closed-form expressions are established for the dimensionless shear stress, fluid velocity and Darcy’s resistance by means of the integral transforms. Similar solutions for the MHD Taylor–Couette flow of the same fluids through a porous medium induced by a time-dependent couple in an infinite circular cylinder are obtained as limiting cases of previous results. In both cases, the obtained results can generate exact solutions for any motion of this kind of the respective fluids. Consequently, the two MHD motions of the respective fluids through a porous medium are completely solved. For illustration, two case studies are considered and the fluid behavior is graphically investigated. The convergence of the starting solutions to their permanent components is proved and the required time to touch the permanent state is determined. Full article

In this paper, we investigate the effect of singular viscosity on the stability of a thin film of Oldroyd-B viscoelastic fluid flowing along a porous inclined surface under the influence of a normal electric field. First, we derive the governing equations and boundary conditions for the flow of the film and assume that the film satisfies the Beavers–Joseph sliding boundary condition when it flows on a porous inclined surface. Second, through the long-wave approximation, we derive the nonlinear interfacial evolution equation. Then, linear and nonlinear stability analyses are performed for the interfacial evolution equation. The stability analyses show that the singular viscosity has a stabilizing effect on the flow of the film, while the strain delay time of the Oldroyd-B fluid, the electric field, and the parameters of the porous medium all have an unsteady effect on the flow of the film. Interestingly, in the linear stability analysis, the parameters of the porous medium have an unsteady effect on the flow of the film after a certain value is reached and a stabilizing effect before that value is reached. In order to verify these results, we performed numerical simulations of the nonlinear evolution equations using the Fourier spectral method, and the conclusions obtained are in agreement with the results of the linear stability analysis, i.e., the amplitude of the free surface decreases progressively with time in the stable region, whereas it increases progressively with time in the unstable region Full article

The instability of the double-diffusive convection of an Oldroyd-B fluid in a vertical Brinkman porous layer caused by temperature and solute concentration differences with the Soret effect is studied. Based on perturbation theory, an Orr–Sommerfeld eigenvalue problem is derived and numerically solved using the Chebyshev collocation method. The effects of dimensionless parameters on the neutral stability curves and the growth rate curves are examined. It is found that Lewis number Le, Darcy–Prandtl number Pr_D, and normalized porosity η have critical values: When below these thresholds, the parameters promote instability, whereas exceeding them leads to suppression of instability. In addition, for Le < Le_c2 (a critical value of Le), S_r strengthens the instability of the flow, while for Le > Le_c2, S_r suppresses it. These results highlight the complex coupling of heat and mass transfer in Oldroyd-B fluids within porous media. Full article

In this study, five different time-dependent incompressible non-Newtonian boundary layer models in two dimensions are investigated with the self-similar Ansatz, including external magnetic field effects. The power-law, the Casson fluid, the Oldroyd-B model, the Walter fluid B model, and the Williamson fluid are analyzed. For the first two models, analytical results are given for the velocity and pressure distributions, which can be expressed by different types of hypergeometric functions. Depending on the parameters involved in the analytical solutions of the nonlinear ordinary differential equation obtained by the similarity transformation, a vast range of solution types is presented. It turned out that the last three models lack self-similar symmetry; therefore, no analytic solutions can be derived. Full article

Two classes of magnetohydrodynamic (MHD) motions of the incompressible Oldroyd-B fluids through an infinite cylinder are analytically investigated. General expressions are firstly established for shear stress and velocity fields corresponding to the motion induced by longitudinal shear stress on the boundary. For validation, the expression of the shear stress is determined by two different methods. Using an important remark regarding the governing equations for shear stress and fluid velocity corresponding to the two different motions, this expression is then used to provide the dimensionless velocity field of the MHD motion of the same fluids generated by a cylinder that rotates around its symmetry axis. Obtained results can generate exact solutions for any motion of this kind of Oldroyd-B fluids. Consequently, both types of motions are completely solved. For illustration, some case studies are considered, and adequate velocity fields are provided. The steady-state components of these velocities are presented in different forms whose equivalence is graphically proved. The influence of the magnetic field on the fluid behavior is graphically investigated. It was found that the fluid flows slower, and a steady state is earlier reached in the presence of a magnetic field. The fluid behavior when shear stress is given on the boundary is also investigated. Full article

In this article, we analytically investigate the isothermal magnetohydrodynamic axial symmetric flows of ordinary and fractional incompressible Oldroyd-B fluids through a porous medium in an annular channel. The fluid’s motion is generated by an outer cylinder, which moves along its symmetry axis with an arbitrary time-dependent velocity $Vh(t)$. Closed-form expressions are established for the dimensionless velocity fields of both kinds of fluids, generating exact solutions for any motion of this type. To illustrate the concept, two particular cases are considered, and the velocity fields corresponding to the flow induced by the outer cylinder are presented in simple forms, with the results validated graphically. The motion of fractional and ordinary fluids becomes steady over time, and their corresponding velocities are presented as the sum of their steady and transient components. Moreover, the steady components of these velocities are identical. The influence of magnetic fields and porous media on the flow of fractional fluids is graphically depicted and discussed. It was found that a steady state is reached earlier in the presence of a magnetic field and later in the presence of a porous medium. Moreover, this state is obtained earlier in fractional fluids compared with ordinary fluids. Full article