Investigation of Relaxation Time on Viscoelastic Two-Dimensional Flow Characteristics Using FreeFem++

Viscoelastic uid ow models have shown promising scope in modeling the behavior of many industrial materials such as polymeric materials, microuidics, biological liquids, gels, plastic melts, and geomaterials. e relaxation time in these models is of great physical signicance. In this article, we study the impact of relaxation time on the viscoelastic ow characteristics in a twodimensional baed cavity. To the best of authors’ knowledge, relaxation time impact on the chosen ow characteristics in the present context has not been studied and presented in the literature before. e constitutive theory of upper convected Maxwell viscoelastic ow incorporating the viscosity ad relaxation time is taken into consideration. To this account, the ow governing PDEs are derived, and an unsteady variational numerical approach based on classical variational setting is presented. A numerical algorithm based on characteristic Galerkin nite elements method is designed and implemented using the programming language FreeFem++. Computations are carried out and drag and lift forces along with other parameters of interests are calculated. Impact of relaxation time on these ow characteristics are studied and analyzed. e relaxation time Rf is assumed to be in the range of 0≤Rf ≤ 1 × 10 − 3.e ow simulations are carried out for large Reynolds number in the range of 200≤Re≤ 5000. In addition to the application of FreeFem++, some new and interesting features of the ow characteristics are presented and discussed.


Introduction
In nature complex uids such as viscoelastic or micro uidic mixtures surround many important phenomena, and therefore, investigation of such uids can play an important role in the development of many industrial processes. ese industrial processes range from polymeric melting processes [1,2], pulp bers [3,4], mineral processing [5], food processing [6]; and references therein, cement pastes and cosmetics processing [7]; and reference therein, and biophysical processes [8][9][10]. Also, viscoelastic ows are seen to be generated in simple liquids by the vibration of nanostructures, see for instance [8], and references present therein.
Viscoelastic behavior of ows has been studied in the literature so far in varying contexts. For instance, Boyang in his PhD dissertation [11] investigated the ow of polymeric viscoelastic uid in three di erent geometries where it was observed that uid relaxation in uences the onset of turbulence in shear ows. For a review and state of the art on the e ect of viscoelastic uid ows and their applications, a/ the reader is referred to the work of Yuan et al. [12]. For a review and state-of-the-art research on viscoelastic uids in particle focusing and related particle manipulation applications, the reader is referred to the work of Chen [9]. Swimming of ciliated cells in a viscoelastic Giesekus uid is investigated in the work of Zhu et al. [13]. ey observe a decay in the ow velocity of the uid in the presence of polymeric stresses. e main characteristic of their investigation was the behavior of the Weissenberg number on the polymeric swimmers using numerical simulation through the nite element method. Having applications in health and diseases, see for instance the review article by Sebastian and Dittrich [14], the microfluidics which are viscoelastic in nature are used to mimic the blood flow in the/a human body. Also, viscoelastic transport of the female sperm is evidenced in the work of Suarez and Pacey [15]. To understand the practical application of viscoelastic fluids in porous media, the reader is referred to the work of Haward et al. [16]. e kinematics of DNA flows is quantified with the application of viscoelastic microflows in the work of Gulati et al. [17]. On the practical problems involving viscoelastic behaviour in biological dynamics, the review article by Goldstein [18] is referred to the reader. In connection to the clinical experimentation and microfluidics to study the behaviour of neutrophil genomics and proteomics, the reader is referred to the work of Kotz et al. [19]. Liu et al. [20] used viscoelastic fluids to study the separation of Bacillus population in microfluidics. e flow of these complex fluids is resolved by using a number of numerical algorithms in the literature; in this regard, the reader is referred to a comprehensive review article of Alves et al. [21] on the numerical methods for viscoelastic fluid flows.
Although many important applications [1][2][3][4][5][6][7][8][9][10] are realizing the investigation of viscoelastic flows, the effect of fluid relaxation time in these models has been rarely investigated. Among the existing literature only few studies are dealing with the study of fluid relaxation time but in different contexts. For instance, Mahapatra and Bandopadhyaya [22] observed the effect of fluid relaxation time and retardation time over a nonuniformly charged surface in the electroosmosis of a viscoelastic fluid. In an annular domain, Dey [23] observed that the fluid relaxation time affects the hydromagnetic viscoelastic flow maximum at the center portion of the geometry. Eldesoky [24] analyzed various effects of relaxation time in a compressible Maxwellian peristaltic slip flow, but in one-dimensional settings. e influence of relaxation time on the locomotion of the microorganisms is analyzed by Josef [25] where it was observed that the fluid relaxation time can increase the speed and thus the efficiency of a swimmer swimming freely in a viscoelastic fluid medium. Looking towards the importance and applications of viscoelastic fluids, it is therefore believed that to understand the behavior of complex fluids and microfluidic mixtures having viscoelastic like structure, the study of relaxation time needs further attention. e aim of this article is to present an investigation of fluid relaxation time on the evolution of viscoelastic flow dynamics where different parameters of study are taken into consideration. e governing flow equations are modeled, and the variational framework based on the characteristic Galerkin method is developed and implemented using FreeFem++ [26]. Numerical simulations are performed, and some important features of the flow dynamics are observed and discussed. e rest of the paper is organized as follows: e flow governing equations of viscoelastic fluid are introduced in Section 2, and a time relaxation mathematical model is presented. In Section 3, classical variational formulation of the flow governing equations is constructed. In Section 4, numerical simulations are shown, and results are presented and discussed. Finally, conclusions are drawn in section 5.

Viscoelastic Relaxation-Time Mathematical Model
One of the widely appreciated models to describe the viscoelastic fluids is the upper convected Maxwell (UCM) model that incorporates viscosity and relaxation time with the following constitutive equation: where λ and η are the fluid characteristic relaxation time and the fluid viscosity, respectively. e deformation strain tensor d(v) in (1) is defined as ∇ τ being the upper convected derivative is defined as Here, λ and η are assumed to be constant, whereas in general, they may depend on the local shear rate, pressure, and temperature, see for instance [21]. After following [27], the governing equation of motion for the unsteady incompressible flow is stated as together with the mass conservation within the medium Consider a two-dimensional unsteady incompressible viscoelastic flow confined in a lid-driven baffled cavity as depicted in Figure 1. In the absence of body forces b, the momentum balance and continuity equations in (4) and (5) read in the component form as zv zt Let us introduce the following transformations to obtain the dimensionless forms of the set of PDEs in (7) and (8) where L is the width of the cavity. e dimensionless forms of the x-momentum in equation (7) and y momentum in (8) along with the continuity equation thus read as In (11) and (12), the nondimensional parameters R e represent Reynolds number, and R f is the fluid relaxation time, respectively, defined by 183 Figure 1: Computational meshes (from left to right): coarser, refined, and finer.

Numerical Strategy and Classical Weak Formulation
In order to solve the above strong form of the PDEs in (10)-(12), we use the finite element method where the weak formulation of the time discretized model is presented in this section. Let us discretize the time domain by a time step length Δt > 0 in n subintervals with t n � nΔt. To obtain a time discretized variational formulation, we use the method of characteristics whereby defining the characteristic trajectory at any time t through x as where X| τ�t � x.
We define the total derivative by assuming a function Now, since x � X(x, t n+1 , t n+1 ) x, it is therefore the total derivative of ϕ becomes where X n (x) approximates the X(x, t n+1 , t n+1 ). For detail description of this method, the reader is referred to the work by Pironneau and Tezduyar [28]. Define the functional space e time incremental variational formulation thus reads.
Given U n , V n { } and P n at time step t n find U n+1 , V n+1 ∈ H 1 0 (Ω) and P n+1 ∈ L 2 (Ω) at time step t n+1 , such that subjected to the following boundary and initial conditions

Numerical Results and Discussion
e set of governing (17) and (18) along with the associated boundary and initial conditions in (19), (20), and (21) are implemented through FreeFem++ [26]. e numerical results presented and tests are carried out with Reynolds number Re ranging between 200 ≤ Re ≤ 5000. e relaxation time R f is assumed to be in the range of 0 ≤ R f ≤ 1 × 10 − 3 .
ese numerical values for the relaxation times are taken phenomenological. However, exact values can be calculated by using the novel device [29] which is capable of measuring relaxation times of complex fluids. In Figure 5, the time evolution of drag forces for different Reynolds numbers are shown with varying values of the relaxation time. It can be seen that the drag force fluctuates in time once the flow enters in the transitional regime, whereas in the laminar regime, the drag force fluctuations are not observed. Moreover, higher the Reynolds numbers in the transitional regime, larger are the fluctuations in the drag at the bottom surface of the cavity. e drag force at the bottom surface decreases as the relaxation time is increased in both the laminar and transitional regimes. e drag force is influenced by the particle velocities at the bottom surface as the moving particle near the bottom surface observes resistance in motion with the increasing value of the fluid relaxation time; therefore, the drag force is decreased as the fluid relaxation time is increased. In Figure 6, the drag forces are computed at the lid surface and are graphically presented over time for varying values of the Reynolds numbers Re and relaxation time R f ; it is observed that the drag force gets fluctuations in the transitional regime, whereas in the laminar regime of the flow, these fluctuations are not present. Also, these fluctuations in the drag force disappear and become negligibly small once the time is evolved, and the flow gets developed. e drag force has a nonmonotone behavior in the regime of small Reynolds numbers at the lid surface and decrease by increasing the relaxation time when the flow gets near the transitional regime. Once it enters the transitional regime, then the drag force get increased by increasing the relaxation time.
is behavior of the drag force is due to the eddies produced in the regime of high Reynolds numbers which affect the evolution of the drag force at the lid of the cavity. In Tables 1 and 2, the horizontal  and vertical velocities along the width and height of the  cavity are studied with varying values of the relaxation time. e horizontal velocities near the top lid and near the bottom of the cavity increase in both the test cases by increasing the fluid relaxation time. It is observed that by increasing the relaxation time, the horizontal velocities are increasing over a larger depth of the cavity from the top lid in case of small Reynolds number as compared to the case of large Reynolds number whereas these flow velocities are increasing over a larger depth upward from the bottom surface of the cavity in case of large Reynolds number as compared to small Reynolds number by increasing the fluid relaxation time. However, in case of the small Reynolds number, the vertical velocities decrease by increasing the relaxation time, and this decrease in velocities is observed over a larger depth from the right end boundary of the cavity as compared to the case of the large Reynolds number.
Nevertheless, in case of large Reynolds number, vertical velocities attain increasing trend over a larger depth from the left end boundary of the cavity as compared to the case of small Reynolds number.
In Table 3, the kinetic energies are calculated in different presented test cases for varying values of the relaxation time R f . It is observed that for increasing value of relaxation time, the kinetic energies of the system of particles are increasing.
is shows that the relaxation time influences the magnitude of the fluid velocities of the system particles in the flow, and the translational motions of the particles get increased by increasing the fluid relaxation time. In Table 4, different parameters of interests are computed and presented for two different test cases, i.e., for Re � 200 and Re � 1000, with varying values of the relaxation time R f . It can be observed that the minimum stream value ψ min decreases with an increasing value of the relaxation time. e maximum stream value ψ max increases with increasing relaxation time monotonically in the case of Re � 200, whereas this relation is nonmonotone in the case of Re � 1000. e maximum value of ψ max for chosen relaxation times in the case of Re � 1000 is corresponding to R f � 2.0x10 − 3 . e mass flow rate is increased by increasing the relaxation time. e drag force at the lid is increased by increasing relaxation time in the case of Re � 200, whereas in the case of R � 1000, a nonmonotone behavior is observed.
is behavior of nonmonotonicity in the case of large Reynolds numbers is due to vortex nature of the flow field. In conjunction to this, the drag force at the bottom of the cavity surface is increasing for both the test cases by increasing the relaxation times. Based on the minimum values of the flow field velocities u and v, it can be observed that these velocities are increasing and decreasing, respectively, by increasing the relaxation time. However, the maximum values of the vertical velocity are increasing for both the test cases by increasing the value of relaxation time.   Journal of Mathematics

Conclusions
is study aims to understand the rheology of complex liquids important to model different physical problems arising in a wide range of industrial applications. In this respect, a fluid relaxation time nonlinear PDE model is presented and numerically investigated. In particular, the effect of fluid relaxation time on the characteristics of viscoelastic flow dynamics is examined. As a model test problem, a two-dimensional lid-driven baffled cavity is chosen for the analysis. Computations are performed using characteristic Galerkin finite elements, and a developed code is implemented through FreeFem++. Different features of the flow characteristics are computed and presented through graphs and tables. Some of the main findings are summarized in the following points: (i) In the laminar regime of flow, the relaxation time raises the mass flow rate, whereas in the transitional flow regime, the relaxation time drops the mass flow rate. (ii) e kinetic energy of the system of particles in flow gets increased by increasing the fluid relaxation time. is result is in accordance with the observations of Josef et al. [17] where an increase in the swimmer speed was observed by increasing the fluid relaxation time.

Data Availability
No data were used to support this study.