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The study investigates the dynamics of an upper-convected Maxwell fluid flow within the framework of magnetohydrodynamics (MHD) over an stretching surface. The analysis incorporates the Cattaneo–Christov heat flux model to explore the system’s thermal behavior. The inclusion of thermal radiation, heat generation-absorption, and suction-injection effects significantly augments the thermal characteristics. Despite its linear nature, the system demonstrates a pronounced level of nonlinearity in its temporal behavior. Through the application of a two-parameter Lie method, nonlinear partial differential equations (PDEs) are transformed into ordinary differential equations (ODEs). This method amalgamates three independent variables into a single independent similarity variable, ξ . Graphical representations are employed for momentum and thermal analyses, and the ODEs are numerically solved using the bvp4c function in MATLAB. The model’s validity is confirmed through tables and graphs with a consistent assessment. An increase in Hartmann number M s boosts the system’s internal energy but slows down fluid velocity. The momentum Deborah number β 1 escalates velocity amplitude but decreases fluid temperature over time, whereas the heat Deborah number β 2 results in a decrease in fluid temperature.

Giesekus viscoelastic fluid is solved analytically for purely tangential flow in a concentric annulus at laminar and steady state conditions. Flow is created by a relative rotational motion between the cylinders. The analytical expressions for yield dimensionless velocity profile, pressure distribution, (f and Re are Fanning friction factor and Reynolds number) and material functions (viscosity, first and second normal stress difference coefficients) are obtained in cylindrical coordinates. Results show that difference between the values of lower as well as upper critical limits of the velocity ratio (where the minimum velocity happens) with their corresponding Newtonian values increase when mobility factor and Deborah number increase. The results also show that viscometric functions decrease by increasing elasticity because the viscoelastic fluid shows the shear thinning behavior which is strengthened by increasing elasticity. It is found that, for all values, profiles are symmetrical around (  and k are velocity ratio and radius ratio) because no relative motion exists.

We analyse the pressure-driven flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, narrow channels and present a theoretical framework for calculating the relationship between the flow rate $q$ and pressure drop $\\Delta p$. We first identify the characteristic scales and dimensionless parameters governing the flow in the lubrication limit. Employing a perturbation expansion in powers of the Deborah number ($De$), we provide analytical expressions for the velocity, stress and the $q$–$\\Delta p$ relation in the weakly viscoelastic limit up to $O(De^2)$. Furthermore, we exploit the reciprocal theorem derived by Boyko $\\&$ Stone (Phys. Rev. Fluids, vol. 6, 2021, L081301) to obtain the $q$–$\\Delta p$ relation at the next order, $O(De^3)$, using only the velocity and stress fields at the previous orders. We validate our analytical results with two-dimensional numerical simulations in the case of a hyperbolic, symmetric contracting channel and find excellent agreement. While the velocity remains approximately Newtonian in the weakly viscoelastic limit (i.e. the theorem of Tanner and Pipkin), we reveal that the pressure drop strongly depends on the viscoelastic effects and decreases with $De$. We elucidate the relative importance of different terms in the momentum equation contributing to the pressure drop along the symmetry line and identify that a pressure drop reduction for narrow contracting geometries is primarily due to gradients in the viscoelastic shear stresses. We further show that, although for narrow geometries the viscoelastic axial stresses are negligible along the symmetry line, they are comparable or larger than shear stresses in the rest of the domain.

Elastic turbulence is the chaotic fluid motion resulting from elastic instabilities due to the addition of polymers in small concentrations at very small Reynolds ( Re ) numbers. Our direct numerical simulations show that elastic turbulence, though a low Re phenomenon, has more in common with classical, Newtonian turbulence than previously thought. In particular, we find power-law spectra for kinetic energy E ( k ) ~  k −4 and polymeric energy E p ( k ) ~  k −3/2 , independent of the Deborah (De) number. This is further supported by calculation of scale-by-scale energy budget which shows a balance between the viscous term and the polymeric term in the momentum equation. In real space, as expected, the velocity field is smooth, i.e., the velocity difference across a length scale r ,  δ u  ~  r but, crucially, with a non-trivial sub-leading contribution r 3/2 which we extract by using the second difference of velocity. The structure functions of second difference of velocity up to order 6 show clear evidence of intermittency/multifractality. We provide additional evidence in support of this intermittent nature by calculating moments of rate of dissipation of kinetic energy averaged over a ball of radius r ,  ε r , from which we compute the multifractal spectrum. Intermittency is the behavior of extreme fluctuations observed in the flow of a fluid that is often associated with high Reynolds numbers. Here, the authors report intermittency in elastic turbulence at the low Reynolds number and high Deborah number limit.

Pressure-driven flows of viscoelastic fluids in narrow non-uniform geometries are common in physiological flows and various industrial applications. For such flows, one of the main interests is understanding the relationship between the flow rate $q$ and the pressure drop $\\Delta p$, which, to date, is studied primarily using numerical simulations. We analyse the flow of the Oldroyd-B fluid in slowly varying arbitrarily shaped, contracting channels and present a theoretical framework for calculating the $q-\\Delta p$ relation. We apply lubrication theory and consider the ultra-dilute limit, in which the velocity profile remains parabolic and Newtonian, resulting in a one-way coupling between the velocity and polymer conformation tensor. This one-way coupling enables us to derive closed-form expressions for the conformation tensor and the flow rate–pressure drop relation for arbitrary values of the Deborah number ($De$). Furthermore, we provide analytical expressions for the conformation tensor and the $q-\\Delta p$ relation in the high-Deborah-number limit, complementing our previous low-Deborah-number lubrication analysis. We reveal that the pressure drop in the contraction monotonically decreases with $De$, having linear scaling at high Deborah numbers, and identify the physical mechanisms governing the pressure drop reduction. We further elucidate the spatial relaxation of elastic stresses and pressure gradient in the exit channel following the contraction and show that the downstream distance required for such relaxation scales linearly with $De$.

In this article, a Riga plate is exhibited with an electric magnetization actuator consisting of permanent magnets and electrodes assembled alternatively. This exhibition produces electromagnetic hydrodynamic phenomena over a fluid flow. A new study model is formed with the Sutterby nanofluid flow through the Riga plate, which is crucial to the structure of several industrial and entering advancements, including thermal nuclear reactors, flow metres and nuclear reactor design. This article addresses the entropy analysis of Sutterby nanofluid flow over the Riga plate. The Cattaneo–Christov heat and mass flux were used to examine the behaviour of heat and mass relaxation time. The bioconvective motile microorganisms and nanoparticles are taken into consideration. The system of equations for the current flow problems is converted from a highly non-linear partial system to an ordinary system through an appropriate transformation. The effect of the obtained variables on velocity, temperature, concentration and motile microorganism distributions are elaborated through the plots in detail. Further, the velocity distribution is enhanced for a greater Deborah number value and it is reduced for a higher Reynolds number for the two cases of pseudoplastic and dilatant flows. Microorganism distribution decreases with the increased magnitude of Peclet number, Bioconvection Lewis number and microorganism concentration difference number. Two types of graphical outputs are presented for the Sutterby fluid parameter (β = −2.5, β = 2.5). Finally, the validation of the present model is achieved with the previously available literature.

Long and slender liquid filaments are produced during inkjet printing, which can subsequently either retract to form a single droplet, or break up to form a primary droplet and one or more satellite droplets. These satellite droplets are undesirable since they degrade the quality and reproducibility of the print, and lead to contamination within the enclosure of the print device. Existing strategies for the suppression of satellite droplet formation include, among others, adding viscoelasticity to the ink. In the present work, we aim to improve the understanding of the role of viscoelasticity in suppressing satellite droplets in inkjet printing. We demonstrate that very dilute viscoelastic aqueous solutions ($\\text {concentrations} \\sim 0.003\\,\\%$ wt. polyethylene oxide, corresponding to nozzle Deborah number $De_{n}\\sim 3$) can suppress satellite droplet formation. Furthermore, we show that, for a given driving condition, upper and lower bounds of polymer concentration exist, within which satellite droplets are suppressed. Satellite droplets are formed at concentrations below the lower bound, while jetting ceases for concentrations above the upper bound (for fixed driving conditions). Moreover, we observe that, with concentrations in between the two bounds, the filaments retract at velocities larger than the corresponding Taylor–Culick velocity for the Newtonian case. We show that this enhanced retraction velocity can be attributed to the elastic tension due to polymer stretching, which builds up during the initial jetting phase. These results shed some light on the complex interplay between inertia, capillarity and viscoelasticity for retracting liquid filaments, which is important for the stability and quality of inkjet printing of polymer solutions.

A mathematical model of 2D-double diffusive layer flow model of boundary in MHD Maxwell fluid created by a sloping slope surface is constructed in this paper. The numerical findings of non-Newtonian fluid are important to the chemical processing industry, mining industry, plastics processing industry, as well as lubrication and biomedical flows. The diversity of regulatory parameters like buoyancy rate, magnetic field, mixed convection, absorption, Brownian motion, thermophoretic diffusion, Deborah number, Lewis number, Prandtl number, Soret number, as well as Dufour number contributes significant impact on the current model. The steps of research methodology are as followed: a) conversion from a separate matrix (PDE) to standard divisive calculations (ODEs), b) Final ODEs are solved in bvp4c program, which developed in MATLAB software, c) The stability analysis part also being developed in bvp4c program, to select the most effective solution in the real liquid state. Lastly, the numerical findings are built on a system of tables and diagrams. As a result, the profiles of velocity, temperature, and concentration are depicted due to the regulatory parameters, as mentioned above. In addition, the characteristics of the local Nusselt, coefficient of skin-friction as well as Sherwood numbers on the Maxwell fluid are described in detail.

Lubrication theory is adapted to incorporate the large normal stresses that occur for order-one Deborah numbers, $De$, the ratio of the relaxation time to the residence time. Comparing with the pressure drop for a Newtonian viscous fluid with a viscosity equal to that of an Oldroyd-B fluid in steady simple shear, we find numerically a reduced pressure drop through a contraction and an increased pressure drop through an expansion, both changing linearly with $De$ at high $De$. For a constriction, there is a smaller pressure drop that plateaus at high $De$. For a contraction, much of the change in pressure drop occurs in the stress relaxation in a long exit channel. An asymptotic analysis for high $De$, based on the idea that normal stresses are stretched by an accelerating flow in proportion to the square of the velocity, reveals that the large linear changes in pressure drop are due to higher normal stresses pulling the fluid through the narrowest gap. A secondary cause of the reduction is that the elastic shear stresses do not have time to build up to their steady-state equilibrium value while they accelerate through a contraction. We find for a contraction or expansion that the high $De$ analysis works well for $De>0.4$.

A gas bubble sitting at a liquid–gas interface can burst following the rupture of the thin liquid film separating it from the ambient, owing to the large surface energy of the resultant cavity. This bursting bubble forms capillary waves, a Worthington jet and subsequent droplets for a Newtonian liquid medium. However, rheological properties of the liquid medium like elastoviscoplasticity can greatly affect these dynamics. Using direct numerical simulations, this study exemplifies how the complex interplay between elasticity (in terms of elastic stress relaxation) and yield stress influences the transient interfacial phenomenon of bursting bubbles. We investigate how bursting dynamics depends on capillary, elastic and yield stresses by exploring the parameter space of the Deborah number ${{\\textit {De}}}$ (dimensionless relaxation time of elastic stresses) and the plastocapillary number $\\mathcal {J}$ (dimensionless yield-stress of the medium), delineating four distinct characteristic behaviours. Overall, we observe a non-monotonic effect of elastic stress relaxation on the jet development while plasticity of the elastoviscoplastic (EVP) medium is shown to affect primarily the jet evolution only at faster relaxation times (low ${{\\textit {De}}}$). The role of elastic stresses on jet development is elucidated with the support of energy budgets identifying different modes of energy transfer within the EVP medium. The effects of elasticity on the initial progression of capillary waves and droplet formation are also studied. In passing, we study the effects of solvent–polymer viscosity ratio on bursting dynamics and show that polymer viscosity can increase the jet thickness apart from reducing the maximum height of the jet.