Abstract
Micropolar fluid model consists of Navier-Stokes equations and microrotational velocity equations describing the dynamics of flows in which microstructure of fluid is important. In this paper, we propose and analyze a decoupled time-stepping algorithm for the evolutionary micropolar flow. The proposed method requires solving only one uncoupled Navier-Stokes and one microrotation subphysics problem per time step. We derive optimal order error estimates in suitable norms without assuming any stability condition or time step size restriction.
1. Introduction
Micropolar fluid model theory considers the interaction between the fluid motion and rotational motion of microparticles suspended in a viscous medium when the deformation of microparticles is ignored. Numerous experimental and numerical studies have indicated that the micropolar fluid theory better describes micro- and nanoflows than the classical Naiver-Stokes theory. Understanding microscale fluid flow phenomena is important in order to effectively design and fabricate microchannels and chambers for microfluidic systems [1]. Growing interest in microscale flow phenomena is also due to the miniaturization of fluid devices for controlling flows in micromachines. Numerical predictions reported in [2] and experimental studies reported in [3–7] show that micropolar fluid models better represent the behavior of flows in microfluidic systems compared to the Navier-Stokes equations. In the experimental work reported in [7], fluids containing minute polymeric additives indicate considerable reduction of the skin friction which can be related to the presence of antisymmetric and coupled stresses in micropolar fluids leading to an increase in the energy dissipation.
There are numerous papers devoted to the mathematical analysis of micropolar fluid flows such as the existence and uniqueness of solutions to micropolar flow equations; see [8–13]. In [14–16], optimal control problems associated with micropolar fluids are studied from the theoretical point of view. Stability problems for micropolar fluids are investigated in [17, 18]. It has also been the subject of many computational simulation based investigations [2, 19–22]. These works mainly focus on the numerical solution of micropolar fluid equations modeling various applied problems such as Hagen-Poiseuille flow and nano/microfluid system [23, 24]. Micropolar fluid models for real and nontrivial flow problems would involve a system of nineteen partial differential equations in nineteen unknowns, therefore computationally very challenging. Despite these challenges in computing micropolar fluid flow, there are very few studies in the literature on numerical analysis and algorithms for efficient computation of micropolar fluid flows. In [25], a numerical scheme based on projection method in time and finite-difference in space is incorporated to solve unsteady incompressible micropolar fluid flow problems. In [26], convergence rate of Galerkin spectral spatial approximation for the micropolar fluid model is studied.
In the present work, we propose and study a decoupled time-stepping scheme for the evolutionary micropolar fluid flow model. It uses a semi-implicit Crank-Nicolson scheme that combines an implicit treatment of the second derivative terms, a semi-implicit second-order extrapolation of the nonlinear convective terms, and explicit treatment of the coupling terms. The proposed scheme solves the Navier-Stokes equations and the microrotational velocity equations separately in each time step without iteration. We derive optimal order error estimates of the scheme without any stability condition or time step size restriction.
An outline of the paper is as follows. In Section 2, we present the governing equations and some preliminary materials. In Section 3, we propose a decoupled Crank-Nicolson time-stepping scheme using extrapolation in time and prove that the proposed decoupled scheme yields the second-order convergence in the temporal direction. Numerical tests are reported in Section 4.
2. Micropolar Fluid System
2.1. Formulation of the Problem
Incompressible flow of micropolar fluids is modeled by the system; see, for example, [27–29]. Given , , , and and time , find , , and such thatwhere is the fluid velocity, the microrotation field interpreted as the angular velocity field of rotation of particles, and the fluid kinematic pressure. Notice that the microrotation vector is equal neither to the flow vorticity nor to average flow angular velocity . The fields and are the external body force and moment (torque), respectively. The positive constants , , , , and represent viscosity coefficients, is the usual Newtonian viscosity, and is the microrotation viscosity. Moreover, the constants , , and satisfy the inequality . The system is supplemented by the Dirichlet boundary conditions,and the initial conditions,Here is a bounded, Lipschitz domain in () and Notice that is a vector variable and the equations satisfied by its components are coupled via the second-order terms which may pose difficulty.
2.2. Weak Formulation of the Evolutionary Micropolar Fluid Model
For a Banach space , we denote by the time-space function space endowed with the norm and We will often use the abbreviated notation for convenience. The symbol denotes the set of continuous functions endowed with the norm For any integer , let be the Sobolev space of functions in with derivatives up to the th order endowed with the norm , where . We denote by the space , when , and drop the subscripts (=2) in referring to the norm in . Moreover, we will use the following simplified norm notations: We introduce the time discrete space associated with ; is the space of -valued sequences with norm defined by
We define the spaces for satisfying , for and
We often use the Sobolev inequality for and the Poincaré inequality We recall also the Gagliardo-Nirenberg interpolation inequalityand Agmon’s inequalityand see, for example, [30].
If we define the bilinear forms , , , and in the following way, for and : and define the trilinear form as for all with on , then the weak formulation of the micropolar fluid model is as follows: seek and such that
We now give the following proposition on the existence and uniqueness result for (1). They were obtained by [8, 9] in a slightly different setting.
Proposition 1. Assume that the given functions , , , , , and satisfy , , , , , and . Then, problem (15) has at least one solution such that , , and . In two spatial dimensions , these solutions are unique.
In order to derive the decoupled time-stepping algorithm, we assume is a convex polyhedral domain, for simplicity, and partition into a mesh with so that and any two closed elements and either are disjoint or share exactly one face, side, or vertex. Suppose further that is a shape regular and quasiuniform triangulation. On the other hand, we divide the time interval into subintervals , satisfying Let be the time step. We introduce the finite element spaces and which are div-stable: there exists a constant , independent of , such thatLet be another finite element space and let and be approximations of and , respectively, such that there exist and satisfying and . We then define , , and .
We make the following assumptions on the finite dimensional subspaces , and .
Assumption A1. We have the approximation properties: there exist an integer and a constant , independent of , , , and , such that
Assumption A2. For any integers and and any real numbers and it holds that
There are many conforming finite element spaces satisfying Assumptions A1 and A2. One may choose, for example, the Taylor-Hood element pair for the velocity and pressure (i.e., piecewise quadratic polynomial for velocity and piecewise linear polynomial for pressure) and piecewise quadratic polynomials for the microrotation vector. Then, Hypotheses A1 and A2 hold with .
We also cite a discrete Grönwall lemma which is useful in our analysis as follows.
Lemma 2 (discrete Grönwall lemma [31]). Let , , , , , and be nonnegative numbers such that Then one has
3. Error Analysis of the Decoupled Time-Stepping Scheme
In this section, we present the decoupled time-stepping algorithm for the micropolar fluid model and derive error estimates. System (15) is discretized by Crank-Nicholson scheme in time and Galerkin finite element in space. The time discretization combines an implicit treatment of the second derivative terms, a semi-implicit second-order extrapolation for the nonlinear convective terms and explicit treatment of the microrotation vector coupling term in the Navier-Stokes equations.
Let denote a typical time subinterval and let be a given algorithmic approximation to . Let denote the difference operator and let denote the extrapolation operator .
Algorithm 3. Given , find such thatfor , where , , and are the intermediate variables defined by , , and , respectively.
3.1. Error Analysis of Decoupled Scheme
In this section, we will derive error estimates of the decoupled Crank-Nicolson scheme proposed above. For simplicity, we will assume the boundary data is independent of time in the subsequent analysis.
Let us define two projections, namely, Stokes and generalized Ritz projections, as follows: given and , we define the Stokes projection as the solution of the problemand the generalized Ritz projection as the solution of the problemUsing the -regularity property of the Stokes and Ritz operators in smooth domains and a duality argument, we can show the following approximation properties hold:Moreover, these approximation properties together with (11)-(12) yield
Moreover, under certain smoothness assumptions on , we have by Taylor expansion with integral remainder
Under the above-mentioned assumptions, we can obtain the following error estimate for the velocity.
Theorem 4. Suppose that Assumptions A1 and A2 hold with a positive number and a positive integer that the solution of (15) satisfies and that the initial conditions satisfy Then, for any the approximate solutions of (23) satisfy the following error estimates: for some constant independent of the mesh size and time step .
Proof. Let us denote the Stokes projection and generalized Ritz projection by and , respectively, for convenience. Moreover, let be the errors defined by Then by the approximation properties (26)-(27), we need to only estimate and in order to furnish the desired error estimates. To this end, we first subtract (15) from (23) and obtain at each time step , where and are defined by Using the definition of Stokes and generalized Ritz projections, we obtain the basic error equations of the methodWe next split the nonlinear terms and on the right-hand side of (37) into several terms as follows: Notice due to skew-symmetry of trilinear form . Therefore, setting into (37) we can write it asWe proceed to bound each term on the right-hand side of (39) and absorb like-terms into the left-hand side. We begin with the first terms on the right-hand side of (39)1 and (39)2. Notice that by triangle inequality It is easy to verify, by Cauchy-Schwarz inequality, that Combining this with Stokes projection approximation property (26) and estimate (30), we obtainIn the same way, we can showBy Cauchy-Schwarz inequality and (42) and (43), we haveUsing Hölder’s inequality, Sobolev inequality, and (26) and (31), we obtain We next estimate and . To this end, first notice that for . Therefore, using this identity, Hölder’s inequality and (27) and (31) we find that Arguing similarly we obtainThus combining these estimates and using Young’s inequality, we haveWe can estimate similarly using Hölder’s inequality, Sobolev inequality, and approximation properties. Therefore, we obtainIn order to estimate terms and , we proceed as follows. We estimate the first term asand the second asApplying estimates (44), (48), (49), (50), and (51) into (39) giveswhere