Equations For non-Newtonian Fluids Research Articles

Topology Optimization and Efficiency Evaluation of Short-Fiber-Reinforced Composite Structures Considering Anisotropy

The current study aims to develop a methodology for obtaining topology-optimal structures made of short fiber-reinforced polymers. Each iteration of topology optimization involves two consecutive steps: the first is a simulation of the injection molding process for obtaining the fiber orientation tensor, and the second is a structural analysis with anisotropic material properties. Accounting for the molding process during the internal iterations of topology optimization makes it possible to enhance the weight efficiency of structures—a crucial aspect, especially in aerospace. Anisotropy is considered through the fiber orientation tensor, which is modeled by solving the plastic molding equations for non-Newtonian fluids and then introduced as a variable in the stiffness matrix during the structural analysis. Structural analysis using a linear anisotropic material model was employed within the topology optimization. For verification, a non-linear elasto-plastic material model was used based on an exponential-and-linear hardening law. The evaluation of weight efficiency in structures composed of short-reinforced composite materials using a dimensionless criterion is addressed. Experimental verification was performed to confirm the validity of the developed methodology. The evidence illustrates that considering anisotropy leads to stiffer structures, and structural elements should be oriented in the direction of maximal stiffness. The load-carrying factor is expressed in terms of failure criteria. The presented multidisciplinary methodology can be used to improve the quality of the design of structures made of short fiber-reinforced composites (SFRC), where high stiffness, high strength, and minimum mass are the primary required structural characteristics.

Time decay of solutions for compressible isentropic non-Newtonian fluids

In this paper, we consider the Cauchy problem of a compressible Navier–Stokes system of Eills-type non-Newtonian fluids. We investigate the time decay properties of classical solutions for the compressible non-Newtonian fluid equations. More specifically, we construct a new linearized system in terms of a combination of the solutions, and then we investigate the long-time behavior of the Cauchy problem for the three-dimensional isentropic compressible Eills-type non-Newtonian fluids with an initial perturbation.

Well-posedness for heat conducting non-Newtonian micropolar fluid equations

<abstract><p>In this paper, we consider the first boundary value problem for a class of steady non-Newtonian micropolar fluid equations with heat convection in the three-dimensional smooth and bounded domain $ \Omega $. By using the fixed-point theorem and introducing a family of penalized problems, under the condition that the external force term and the vortex viscosity coefficient are appropriately small, the existence and uniqueness of strong solutions of the problem are obtained.</p></abstract>

The stability of a class of 2D non-newtonian fluid equations with unbounded delays

ABSTRACTWe address the stability of stationary solutions to a class of 2D non-newtonian fluid equations, when the external force contains hereditary characteristics involving unbounded delays. Firstly, when the unbounded variable delay is driven by a continuously differential function, we establish the stability of nontrivial weak stationary solutions and the asymptotic stability of trivial stationary solution. Then when the general unbounded delay is continuous with respect to time, the stability of nontrivial strong stationary solutions is also obtained. Eventually, when the proportional delay is considered, the polynomial stability of trivial stationary solution is verified.

Numerical evaluation of heat-triggered drug release via thermo-sensitive liposomes: A comparison between image-based vascularized tumor and volume-averaged porous media models

In this study, two numerical models for controlled drug delivery through thermo-sensitive liposomes under mild microwave hyperthermia are compared. The first model simulates blood flow directly in a realistic tumor vascular network. The second one, at the volume-averaged macro scale, employs the porous media model. Using the vascular-scale approach is helpful for these kind of applications since the porous media assumption does not allow to account for local vascular blood flow, which is a crucial variable to quantify outputs like local drug accumulations. In the present study, the vascularized tumor model is obtained from a photo-acoustic scan and used as computational domain. The vascular-scale model considers heat and drug transfer in vascular and intracellular spaces, employing energy equations for heat transfer and a three-compartment convection-diffusion model for drug diffusion. Blood flow is simulated using the momentum equation for non-Newtonian fluids. The volume-averaged model shows good agreement with the vascular-scale model even if provides less precise information about temperature field distribution and drug localization within the tumor tissue. Also, the volume-averaged model underestimates Fraction of Killed Cells by 3 % post-treatment, presenting a more uniform temperature field and drug distribution compared to the vascular-scale one. Despite these small limitations, the porous media model is reliable for obtaining average results, while the vascular-scale model is suitable when more accurate results are needed. This study's findings aid those who aim to predict hyperthermia-driven drug delivery, suggesting the vascular-scale approach for precision and accuracy and the volume-averaged approach for average outcomes.

Relative energy inequality and weak-strong uniqueness for an isothermal non-Newtonian compressible fluid

Our paper deals with three-dimensional nonsteady Navier-Stokes equations for non-Newtonian compressible fluids. It contains a de­ri­va­tion of the relative energy inequality for the weak solutions to these equations. We show that the standard energy inequality implies the relative energy inequality. Consequently, the relative energy inequality allows us to achieve a weak-strong uniqueness result. In other words, we present that the weak solution of the Navier-Stokes system coincides with the strong solution emanated from the same initial conditions as long as the strong solution exists. For this purpose, a new assumption on the coercivity of the viscous stress tensor was introduced along with two natural examples satisfying it.

Fiber trajectory tracking of centrifugal spinning and coupled-forces field effects on jet motion stability

High-speed centrifugal spinning holds excellent potential as a promising approach for nanofiber manufacturing due to its various raw materials, low environmental requirements, and high preparation efficiency. However, continuous jet movement and trajectory tracking on the action of gravity, centrifugal force, Coriolis force, viscous force, and air resistance remain significant challenges. Therefore, the present article has developed a polymer jet motion model based on the momentum equation, continuity equation, and constitutive equation of non-Newtonian fluid to explore the influence of spinning parameters on jet motion. Moreover, curved fiber trajectory in the centrifugal field, jet velocity distribution, and variation of air velocity around the nozzle have been simulated using the discrete phase model in terms of different angular velocities and solution concentrations. It is found that the angular velocity affects the formation of the jet trajectory. Furthermore, the solution concentration determines the attenuation of the jet velocity. Therefore, centrifugal spinning experiments have been carried out to verify the correctness of the results and theoretical analysis gained from the research. Scanning electron microscope observations show nanofibers with 6 wt% polyethylene oxide solution at 5500 rpm have better morphology and quality.

Weak solutions to the generalized Navier–Stokes equations with mixed boundary conditions and implicit obstacle constraints

This paper is concerned with the investigation of a generalized Navier–Stokes equation for non-Newtonian fluids of Bingham-type (GNSE, for short) involving a multivalued and nonmonotone slip boundary condition formulated by the generalized Clarke subdifferential of a locally Lipschitz superpotential, a no leak boundary condition, and an implicit obstacle inequality. We obtain the weak formulation of (GNSE) which is a generalized quasi-variational–hemivariational inequality. By introducing an Oseen model as an auxiliary (intermediated) problem and employing Kakutani-Ky Fan theorem for multivalued operators as well as the theory of nonsmooth analysis, an existence theorem to (GNSE) is established.

ANALYTICAL STUDY OF NON-NEWTONIAN FLUID AND NAVIER-STOKES EQUATIONS

This article gives a brief discussion of study on Non-Newtonian fluid and Navier-Stokes equation. A non-Newtonian fluid is a fluid that does not follow Newton's law of viscosity, that is, it has variable viscosity dependent on stress. Navier-Stokes equation in fluid mechanics is a partial differential equation that describes the flow of incompressible fluids. The aim of the present work is to study the Non-Newtonian fluid and Navier-Stokes equation. In this article, the non-Newtonian fluid can be characterized by a viscosity that varies with motion. For non-Newtonian fluids, the viscosity varies with the shear rate are studied. The behaviour of a fluid flow can be explained by the Navier Stokes equation. Key words: Mathematical methods, Non-Newtonian Fluids, Navier-Stokes equation

A note on the slip effects of an Oldroyd 6-constant fluid: Optimal homotopy asymptotic method

We study here the effectiveness of the optimal homotopy asymptotic method (OHAM) in solving non-linear differential equations of non-Newtonian fluids. To this consequence, we consider the Oldroyd 6-constant fluid when it demonstrates slippage between the plate and fluid generating non-linear boundary value problems. The problems of plane Couette flow, generalized Couette flow, and plane Poiseuille flow are considered. Graphs of the results are plotted to show the performance of the method in terms of velocity profile. It is observed that the method is quite easy to implement, having latent potential to handle such kinds of non-linear problems and yield accurate results at minimum to low computational work.

Determination of pressure from nonlinear equations of nonhomogeneous non-newtonian fluids

In this paper, the initial-boundary problem for a system of nonlinear equations modified by p-laplacian diffusion and nonlinear damping term, in which describe the motion of a non-homogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids is considered. Usually, a pressure does not include in the definition of weak generalized solution to equations of hydrodynamics for incompressible fluids, because weak solutions consider in the solenoidal space due to the incompressibility (continuity) equation. In the classical hydrodynamic equations, after determining a velocity and a density, a pressure can be determined by using the theorem of a representation of the space L2(Ω) into the direct sum of two orthogonal subspaces. The recovering of a pressure from the nonlinear equations of hydrodynamics modified by p-laplacian and nonlinear damping term requires a new approches. Here, under a suitable conditions on the data of the problem, a pressure has been uniquelly determined from the initial-boundary value problem for the system of nonlinear modified equations describing the motion of nonhomogeneous (with unknown and non constant density ) incompressible viscoelastic non-Newtonian fluids. The main functional spaces and necessary axuillary conclusions are introduced. A space of generalized weak solutions is identified. The pressure uniquelly recovered by using the known de-Ram lemma.

On a non-Newtonian fluid type equation with variable diffusion coefficient

<abstract><p>Since the non-Newtonian fluid type equations arise from a broad and in-depth background, many research achievements have been gained from 1980s. Different from the usual non-Newtonian fluid equation, there is a nonnegative variable diffusion in the equations considered in this paper. Such a variable diffusion reflects the characteristic of the medium which may not be homogenous. By giving a generalization of the Gronwall inequality, the stability and the uniqueness of weak solutions to the non-Newtonian fluid equation with variable diffusion are studied. Since the variable diffusion may be degenerate on the boundary $ \partial \Omega $, it is found that a partial boundary value condition imposed on a submanifold of $ \partial\Omega\times (0, T) $ is enough to ensure the well-posedness of weak solutions. The novelty is that the concept of the trace of $ u(x, t) $ is generalized by a special way.</p></abstract>

Transport pattern of Non-Newtonian mass and thermal energy under two diverse flow conditions by using modified models for thermodynamics properties

Background and purposeThe viscosity and thermal conductivity play an important role in transport of mass and thermal energy. The effect of temperature on these properties could not be ignored during fluid flow because flow pattern determines the performance and economic situation of the overall process. In view of these facts, the models of these properties are modified to measure the temperature effects and used in the transport equations of non-Newtonian fluid. Design of modelThe mathematical model is based on continuity, momentum and energy equations. The model is simplified by applying the boundary layer and similarity transformations and then solved by numerical method. The influence of affected properties is examined on the velocity and temperature profiles as well as on the Skin friction coefficient and Nusselt number under two flow conditions: a) when mainstream fluid moves faster than surface motion and b) vice versa. FindingsIn the results, the viscosity and thermal conductivity are reduced with increasing of temperature which became the cause of enhancement in velocity profile and reduction in temperature profile. Moreover, Nusselt number is increased with average rate 2.8% and 0.45% respectively under both conditions when viscosity is reduced by temperature.

Optimal partial boundary condition for degenerate parabolic equations

For the stability of the non-Newtonian fluid equation∂u∂t−div(a(x)|∇u|p−2∇u)−∑i=1Nbi(x)Diu+c(x,t)u=f(x,t), where a(x)|x∈Ω>0, a(x)|x∈∂Ω=0 and bi(x)∈C1(Ω‾), we know that the degeneracy of a(x) may make the usual Dirichlet boundary value condition overdetermined and only a partial boundary value condition is expected. How to depict the geometric characteristic of the partial boundary value condition has been a long-time standing open problem. In this study, an optimal partial boundary value condition has been proposed, and the stability of weak solutions based on this partial boundary value condition is established. When the rate of the diffusion coefficient decays to zero, we explore how it affects the stability of weak solutions.

Stability of non-Newtonian fluid and electrorheological fluid mixed-type equation

In this paper, we consider a non-Newtonian fluid and electrorheological fluid mixed-type equation where p>1, and . The existence of weak solution is investigated by means of the parabolically regularized method. Due to , the stability of weak solution is established by choosing a suitable test function and the local stability is also discussed without any boundary value condition.

Upper and lower convergence rates for weak solutions of the 3D non-Newtonian flows

This paper focuses on the 3D shear thickening non-Newtonian fluid equation with the nonlinear constitutive relations τijv=2(μ0+μ1|e(u)|r−2)eij(u)−2μ2Δeij(u) for r≥3. We consider the difference between a weak solution u of the aforementioned equation with the initial data u0 and the weak solution u˜ of the same equation with perturbed initial data u0+w0. The goal is to find the exact large-time behavior of the difference u˜(t)−u(t). By invoking some new observations on the nonlinear parts of the aforementioned non-Newtonian fluid equation and using an iterative argument together with a generalized Fourier splitting method, we are able to show thatC1(1+t)−5−ϵ4≤‖u˜(t)−u(t)‖L2(R3)≤C2(1+t)−5−ϵ4,for t>1large, where ϵ>0 is taken to be sufficiently small. The initial perturbation w0 is not required to be small.

On Liouville Type Theorem for Stationary Non-Newtonian Fluid Equations

In this paper, we prove a Liouville type theorem for non-Newtonian fluid equations in $$\mathbb R^3$$, having the diffusion term $${\varvec{A}}_p( u)=\nabla \cdot ( |{\varvec{D}}(u)|^{p-2} {\varvec{D}}(u))$$ with $$ {\varvec{D}}(u) = \frac{1}{2} (\nabla u + (\nabla u)^{ \top })$$, $$3/2<p< 3$$. In the case $$3/2< p\le 9/5$$, we show that a suitable weak solution $$u\in W^{1, p}(\mathbb R^3)$$ satisfying $$ \liminf _{R \rightarrow \infty } |u_{ B(R)}| =0$$ is trivial, i.e., $$u\equiv 0$$. On the other hand, for $$9/5<p<3$$ we prove the following Liouville type theorem: if there exists a matrix valued function $${\varvec{V}}= \{V_{ ij}\}$$ such that $$ \partial _jV_{ ij} =u_i$$(summation convention), whose $$L^{\frac{3p}{2p-3}} $$ mean oscillation has the following growth condition at infinity, $$\begin{aligned} {\int \!\!\!\!\!\!-}_{B(r)} |{\varvec{V}}- {\varvec{V}}_{ B(r)} |^{\frac{3p}{2p-3}} \mathrm{d}x \le C r^{\frac{9-4p}{2p-3}}\quad \forall 1< r< +\infty , \end{aligned}$$then $$u\equiv 0$$.

Analysis of Transient Rivlin-Ericksen Fluid and Irreversibility of Exothermic Reactive Hydromagnetic Variable Viscosity

This study analyzes the unsteady Rivlin-Ericksen fluid and irreversibility of exponentially temperature dependent variable viscosity of hydromagnetic two-step exothermic chemical reactive flow along the channel axis with walls convective cooling. The non-Newtonian Hele-Shaw flow of Rivlin-Erickson fluid is driven by bimolecular chemical kinetic and unvarying pressure gradient. The reactive fluid is induced by periodic changes in magnetic field and time. The Newtons law of cooling is satisfied by the constant heat coolant convection exchange at the wall surfaces with the neighboring regime. The dimensionless non-Newtonian reactive fluid equations are numerically solved using a convergent and consistence semi-implicit finite difference technique which are confirmed stable. The response of the reactive fluid flow to variational increase in the values of some entrenched fluid parameters in the momentum and energy balance equations are obtained. A satisfying equations for the ratio of irreversibility, entropy generation and Bejan number are solved with the results presented graphically and discussed quantitatively. From the study, it was obtained that the thermal criticality conditions with the right combination of thermo-fluid parameters, the thermal runaway can be prevented. Also, the entropy generation can minimize at low dissipation rate and viscosity.

Strong solutions for the steady incompressible MHD equations of non-Newtonian fluids

In this paper we deal with a system of partial differential equations describing a steady motion of an incompressible magnetohydrodynamic fluid, where the extra stress tensor is induced by a potential with p-structure (p = 2 corresponds to the Newtonian case). By using a fixed point argument in an appropriate functional setting, we proved the existence and uniqueness of strong solutions for the problem in a smooth domain Ω ⊂ Rn (n = 2, 3) under the conditions that the external force is small in a suitable norm.

Soft lubrication of cornstarch-based shear-thickening fluids

The behavior of complex fluids in thin films under the isoviscous elastohydrodynamic lubrication regime is a topic of current interest. A vast majority of the materials used to lubricate compliant contacts are shear-thinning fluids and, in comparison, very little is known about the lubrication properties of shear-thickening (ST) fluids. In this work we carried out both rheological and tribological experiments on ST fluids based on cornstarch suspensions. We investigated the influence of the polarity and viscosity of the carrier fluid and the cornstarch particles’ concentration. From a rheological perspective, the ST response is diminished (as the shear-thinning region enlarges) when non-polar carriers are used (e.g. silicone oil) while a marked thickening is observed for polar carriers (e.g. glycerol-water mixtures). From a tribological perspective, the friction coefficient at slow sliding speeds is strongly affected by the nature of the carrier fluid, and in particular its wettability in contact with hydrophobic PDMS tribopairs. We demonstrate that experimental data, plotted in terms of Stribeck curves, provide useful information on the entrainment of particles/fluid through the contact. Finally, experimental results are also compared with numerical solutions of the Reynolds equation for inelastic non-Newtonian fluids in the isoviscous elastohydrodynamic regime. A Carreau–Yasuda constitutive equation for the viscosity-shear rate relationship is used.