A Molecular-Based Q-Tensor Hydrodynamic Theory of Smectic Liquid Crystals

Abstract

The Doi–Onsager molecular theory is capable of providing a rather accurate description of the local behavior of molecules; however, its computation is extremely time-consuming, since some higher-dimensional variables are typically involved. Therefore, establishing a computable reduced model that can capture essential physical properties is an important issue. In this work, we derived a reduced Q-tensor hydrodynamic theory that described smectic phases with density variations from the Doi–Onsager molecular theory using the Bingham closure approximation. The coefficients in the tensor model were derived from those in the molecular model. The energy dissipation law was inherited from the tensor model. Some special cases for the model were also discussed.

1. Introduction

Liquid crystals, which are composed of anisotropic non-spherical molecules, are featured in the local orientational order. There are several types of liquid crystal phases that can be classified by their orientational and positional orders, such as the nematic phase, the cholesteric phase and the smectic phase. Smectic liquid crystals are characterized by orientational and partial translation orders of the anisotropic molecules. The simplest of the smectics is the smectic-A phase, which exhibits a one-dimensional lamellar order in the direction of the preferred molecular orientational axis; it can be regarded as layers of two-dimensional fluids that are stacked upon each other. Details on smectic liquid crystals can be found in [1].

There are three main theories that are used to model smectic liquid crystals: the molecular theory, the tensor theory and the vector theory. The first is a microscopic kinetic theory derived from viewpoints of statistical mechanics, and the latter two are macroscopic theories based on continuum mechanics. The primary purpose of this work was to derive a new Q-tensor dynamical theory for smectic liquid crystals from the molecular theory of the Bingham closure. Before introducing the above three theories, some notations and conventions are provided.

Notations and conventions. Throughout this paper, we adopt the Einstein summation convention on repeated indices. We denote by $\mathcal{Q}$ the space of symmetric traceless tensors,

$$\begin{array}{c}\hfill \mathcal{Q}\stackrel{\mathrm{def}}{=}\{Q\in {\mathbb{R}}^{3\times 3}:Q_{ij}=Q_{ji},Q_{ii}=0\},\end{array}$$

which is endowed with the inner product $Q_1:Q_2=Q_{1ij}{Q}_{2ij}$. The set $\mathcal{Q}$ is a five-dimensional subspace of ${\mathbb{R}}^{3\times 3}$. The matrix norm on $\mathcal{Q}$ is denoted by $\left|Q\right|\stackrel{\mathrm{def}}{=}\sqrt{tr\left(Q^2\right)}=\sqrt{Q_{ij}{Q}_{ij}}$. For any two tensors $Q_1,Q_2\in \mathcal{Q}$, we define ${(Q_1·Q_2)}_{ij}=Q_{1ik}{Q}_{2kj}$. We use $\mathbf{I}$ to denote the $3\times 3$ identity tensor.

We use a small bold-faced letter $\mathbf{m}$ to represent a vector. We denote by ${\mathbf{m}}_1\otimes {\mathbf{m}}_2\otimes \cdots \otimes {\mathbf{m}}_k$ the tensor product of vectors ${\mathbf{m}}_1,{\mathbf{m}}_2,\cdots ,{\mathbf{m}}_k$, and we usually omit the symbol ⊗ for simplicity. Again, the notation $\nabla ·\sigma$ is defined by ${(\nabla ·\sigma )}_i={\partial }_j{\sigma }_{ij}$, and ${\mathbb{S}}^2$ denotes the unit sphere of ${\mathbb{R}}^3$. We also employ $f_{,i}$ to denote ${\partial }_{i}f$, such as $Q_{ij,j}={\partial }_j{Q}_{ij}$.

1.1. Molecular Theory

Based on the Virial expansion, Onsager [2] first put forward the molecular theory to study the isotropic-nematic phase transition. We use $\mathbf{x}\in \Omega \subseteq {\mathbb{R}}^3$ to denote the material point and $f(\mathbf{x},\mathbf{m})$ to represent the number density for the number of molecules whose orientation is parallel to $\mathbf{m}\in {\mathbb{S}}^2$ at point $\mathbf{x}$. Onsager’s energy functional is given by (also see [3,4])

$$\begin{array}{cc}\hfill & A_{\epsilon }\left[f\right]={\int }_{\Omega }{\int }_{{\mathbb{S}}^2}\left(f(\mathbf{x},\mathbf{m})(lnf(\mathbf{x},\mathbf{m})-1)+\frac{1}{2}{U}_{\epsilon }(\mathbf{x},\mathbf{m})f(\mathbf{x},\mathbf{m})\right)\mathrm{d}\mathbf{m}\mathrm{d}\mathbf{x},\hfill \\ \hfill & U_{\epsilon }(\mathbf{x},\mathbf{m})={\int }_{\Omega }{\int }_{{\mathbb{S}}^2}{G}_{\epsilon }(\mathbf{x},\mathbf{m};{\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime })f({\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime })\mathrm{d}{\mathbf{x}}^{\prime }\mathrm{d}{\mathbf{m}}^{\prime }.\hfill \end{array}$$

(1)

In (1), the first term is the entropy, and the second term represents the interaction energy between each pair of two molecules. $G_{\epsilon }(\mathbf{x},\mathbf{m};{\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime })$ is the interaction kernel between two molecules in the configurations $(\mathbf{x},\mathbf{m})$ and $({\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime })$, and it takes one of the following forms:

(i) The hard-core interaction potential:

$$\begin{array}{c}\hfill G_{\epsilon }(\mathbf{x},\mathbf{m};{\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime })=\left\{\begin{array}{c}1,\mathrm{molecule}(\mathbf{x},\mathbf{m})\mathrm{is}\mathrm{disjoint}\mathrm{with}\mathrm{molecule}({\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime }),\hfill \\ 0,\mathrm{joint}\mathrm{with}\mathrm{each}\mathrm{other}.\hfill \end{array}\right.\end{array}$$

(2)

(ii) The long-range Maier–Saupe interaction potential:

$$\begin{array}{c}\hfill G_{\epsilon }(\mathbf{x},\mathbf{m};{\mathbf{x}}^{\prime },{\mathbf{m}}^{\prime })=\frac{1}{{\epsilon }^{3/2}}g\left(\frac{\mathbf{x}-{\mathbf{x}}^{\prime }}{\sqrt{\epsilon }}\right)\alpha {|\mathbf{m}\times {\mathbf{m}}^{\prime }|}^2,\end{array}$$

(3)

where the small parameter $\sqrt{\epsilon }$ represents the typical interaction distance, and $g\left(\mathbf{x}\right)\in C^{\infty }\left({\mathbb{R}}^3\right)$ is the radial function with ${\int }_{{\mathbb{R}}^3}g\left(\mathbf{x}\right)\mathrm{d}\mathbf{x}=1$. Again, the parameter $\alpha$ measures the intensity of the potential. It should be noted that (3) can be regarded as a smooth approximation of (2).

The kinetic theory of liquid crystal polymers, starting from the pioneering work of Doi [5], is based on Onsager’s molecular theory. This is the so-called Doi–Onsager theory, which had been greatly successful in predicting some dynamical properties of liquid crystal polymers in a solvent, such as tumbling, wagging, flow-aligning, log-rolling [6,7,8], etc; however, it is only valid for spatially homogeneous flows. The molecular theory for nonhomogeneous flows [3,9,10,11] was subsequently developed by introducing a nonlocal intermolecular potential.

The chemical potential ${\mu }_{\epsilon }$ is defined as

$$\begin{array}{c}\hfill {\mu }_{\epsilon }=\frac{\delta A_{\epsilon }\left[f\right]}{\delta f}=lnf(\mathbf{x},\mathbf{m},t)+U_{\epsilon }(\mathbf{x},\mathbf{m},t).\end{array}$$

The non-dimensional Doi–Onsager equation takes the following form [3]:

$$\begin{array}{cc}\hfill \frac{\partial f}{\partial t}+\mathbf{v}·\nabla f=& \frac{\epsilon }{De}\nabla ·\left\{\left({\gamma }_{\parallel }\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{I}-\mathbf{m}\mathbf{m})\right)·(f\nabla {\mu }_{\epsilon })\right\}\hfill \\ \hfill & +\frac{1}{De}\mathcal{R}·\left(f\mathcal{R}{\mu }_{\epsilon }\right)-\mathcal{R}·(\mathbf{m}\times \kappa ·\mathbf{m}f),\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}=& -\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v}+\frac{1-\gamma }{2Re}\nabla ·(\mathbf{D}:{\langle \mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}\rangle }_f)\hfill \\ \hfill & +\frac{1-\gamma }{DeRe}(\nabla ·{\tau }^e+{\mathbf{F}}^e),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}=& 0,\hfill \end{array}$$

(6)

where $De$ and $Re$ are called the Deborah number and the Reynolds number, respectively, $\mathcal{R}=\mathbf{m}\times {\nabla }_{\mathbf{m}}$ is the rotational gradient operator and $\gamma \in (0,1)$ is a constant. Here p is the pressure to maintain the incompressibility (6), $\kappa ={(\nabla \mathbf{v})}^T$ is the gradient tenor of the velocity field $\mathbf{v}$ and $\mathbf{D}=\frac{1}{2}(\kappa +{\kappa }^T)$, while ${\tau }^e$ and ${\mathbf{F}}^e$ represent the elastic stress and the body force, which are respectively defined as

$$\begin{array}{c}\hfill {\tau }^e=-{\langle \mathbf{m}\mathbf{m}\times \mathcal{R}{\mu }_{\epsilon }\rangle }_f,{\mathbf{F}}^e=-{\langle \nabla {\mu }_{\epsilon }\rangle }_f.\end{array}$$

In (4), the constants ${\gamma }_{\parallel }=\frac{L_{0}De}{V_0{L}^2}{D}_{\parallel }$ and ${\gamma }_{\perp }=\frac{L_{0}De}{V_0{L}^2}{D}_{\perp }$ denote the translational diffusion coefficients parallel to and normal to the orientation of the liquid crystal molecule, respectively.

The system (4)–(6) has the following energy dissipation law ([3,12]):

$$\begin{array}{cc}\hfill & -\frac{\mathrm{d}}{\mathrm{d}t}\left({\int }_{\Omega }\frac{1}{2}{\left|\mathbf{v}\right|}^2\mathrm{d}\mathbf{x}+\frac{1-\gamma }{DeRe}{A}_{\epsilon }\left[f\right]\right)\hfill \\ \hfill & ={\int }_{\Omega }(\frac{\gamma }{Re}\mathbf{D}:\mathbf{D}+\frac{1-\gamma }{2Re}{\langle {(\mathbf{m}\mathbf{m}:\mathbf{D})}^2\rangle }_f+\frac{1-\gamma }{D{e}^{2}Re}{\langle \mathcal{R}{\mu }_{\epsilon }·\mathcal{R}{\mu }_{\epsilon }\rangle }_f\hfill \\ \hfill & +\frac{\epsilon (1-\gamma )}{D{e}^{2}Re}{\langle \nabla {\mu }_{\epsilon }·({\gamma }_{\parallel }\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{I}-\mathbf{m}\mathbf{m}))·\nabla {\mu }_{\epsilon }\rangle }_f)\mathrm{d}\mathbf{x}.\hfill \end{array}$$

(7)

1.2. Vector Theory

In this theory, the configuration at the point $\mathbf{x}\in \Omega$ is described by the director $\left(\mathbf{x}\right)\in {\mathbb{S}}^2$, a unit vector. The corresponding Oseen–Frank free energy takes the form [13,14],

$$\begin{array}{cc}{F}_{\mathrm{OF}}={\int }_{\Omega }\hfill & (\frac{K_1}{2}{(\nabla ·\mathbf{n})}^2+\frac{K_2}{2}{(\mathbf{n}·(\nabla \times \mathbf{n}))}^2+\frac{K_3}{2}{|\mathbf{n}\times (\nabla \times \mathbf{n})|}^2\hfill \\ \hfill & +\frac{K_2+K_4}{2}\left(\mathrm{tr}{(\nabla \mathbf{n})}^2-{(\nabla ·\mathbf{n})}^2\right))\mathrm{d}\mathbf{x},\hfill \end{array}$$

(8)

where $K_1,\cdots ,K_4$ are the Frank elastic constants. The free energy (8) is only suitable for characterizing uniaxial nematic phases.

In order to characterize the smectic phase, besides the director $\mathbf{n}\left(\mathbf{x}\right)\in {\mathbb{S}}^2$, de Gennes [15] introduced a complex-valued order parameter,

$$\begin{array}{c}\hfill \mathsf{\Psi }\left(\mathbf{x}\right)=\psi \left(\mathbf{x}\right)e^{iq\phi \left(\mathbf{x}\right)}.\end{array}$$

(9)

The molecular mass density is defined by

$$\begin{array}{c}\hfill \delta \left(\mathbf{x}\right)={\rho }_0\left(\mathbf{x}\right)+\frac{1}{2}\left(\mathsf{\Psi }\left(\mathbf{x}\right)+{\mathsf{\Psi }}^{*}\left(\mathbf{x}\right)\right)={\rho }_0\left(\mathbf{x}\right)+\rho \left(\mathbf{x}\right)cosq\phi \left(\mathbf{x}\right),\end{array}$$

where ${\rho }_0$ is a locally uniform mass density, $\rho \left(\mathbf{x}\right)$ is the mass density of the smectic layer and $\phi$ parameterizes the layers so that $\nabla \phi$ is the direction of a layer’s normal. In addition, q is the wave number and $2\pi /q$ is the layer thickness. If $\mathsf{\Psi }\equiv 0$ then the smectics can become the nematics.

In (8), if we increase the terms with density variations, then it becomes the famous Chen–Lubensky model [16]:

$$\begin{array}{c}\hfill F_{\mathrm{CL}}=F_{\mathsf{\Psi }}+F_{\mathrm{OF}},\end{array}$$

where $F_{\mathsf{\Psi }}$ contains terms up to the second order derivatives of the density $\mathsf{\Psi }\left(\mathbf{x}\right)$,

$$\begin{array}{c}\hfill \begin{array}{cc}{F}_{\mathsf{\Psi }}=\frac{1}{2}{\int }_{\Omega }& (a{\mathsf{\Psi }}^2+D_1{\left[{(\mathbf{n}·\nabla )}^2\mathsf{\Psi }\right]}^2-C_1{(\mathbf{n}·\nabla \mathsf{\Psi })}^2+\frac{C_1^2}{4D_1}{\mathsf{\Psi }}^2\hfill \\ & +C_2({\delta }_{ij}-n_i{n}_j){\partial }_i\mathsf{\Psi }{\partial }_j\mathsf{\Psi }+D_2{\left[({\delta }_{ij}-n_i{n}_j){\partial }_i{\partial }_j\mathsf{\Psi }\right]}^2)\mathrm{d}\mathbf{x},\hfill \end{array}\end{array}$$

(10)

where $a=a_0[(T-T_{NA})/T_{NA}]$, and $T_{NA}$ is the transition temperature from the nematic state to the smectic-A state. In (10), the coefficients $C_1$ and $D_1$ determine the horizontal period, $C_2$ and $D_2$ determine the perpendicular period.

The hydrodynamic theory of nematic phases was established by Ericksen [17] and Leslie [18] in the 1960s. Based on the Ericksen–Leslie theory, the works [1,19,20] established the linear dynamical continuum theory of smectic-A liquid crystals. E [21] first set up a general nonlinear continuum theory for smectic-A liquid crystals that was applicable to situations with large deformations and non-trivial flows. Motivated by the approach of E, Stewart [22] also presented a nonlinear continuum theory for the dynamics of smectic-A liquid crystals, which included the usual Oseen constraint and the permeation.

1.3. Tensor Theory

The state of the nematic liquid crystals is described by a macroscopic tensor value order parameter $Q\left(\mathbf{x}\right)$, which is a symmetric and traceless $3\times 3$ matrix. Physically, it can be interpreted as the second-order traceless moment of the orientational distribution function f, that is,

$$\begin{array}{c}\hfill Q(\mathbf{x},\mathbf{m})={\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})f(\mathbf{x},\mathbf{m})\mathrm{d}\mathbf{m}.\end{array}$$

(11)

Under this interpretation, the so-called physical constraint is that the eigenvalues of Q should satisfy

$$\begin{array}{c}\hfill {\lambda }_i\left(Q\right)\in (-\frac{1}{3},\frac{2}{3}),\mathrm{for}1\le i\le 3.\end{array}$$

The classic Landau–de Gennes energy functional takes the following general form [1],

$$\begin{array}{c}\hfill \begin{array}{cc}{F}_{LG}(Q,\nabla Q)=\hfill & {\int }_{\Omega }\{\underset{\mathrm{bulk}\mathrm{energy}}{\underbrace{-\frac{a}{2}tr\left(Q^2\right)-\frac{b}{3}tr\left(Q^3\right)+\frac{c}{4}(tr{\left(Q^2\right)}^2}}\hfill \\ \hfill & +\underset{\mathrm{elastic}\mathrm{energy}}{\underbrace{\frac{1}{2}\left(L_1{|\nabla Q|}^2+L_2{Q}_{ij,j}{Q}_{ik,k}+L_3{Q}_{ij,k}{Q}_{ik,j}+L_4{Q}_{ij}{Q}_{kl,i}{Q}_{kl,j}\right)}}\}\mathrm{d}\mathbf{x},\hfill \end{array}\end{array}$$

(12)

where $a,b,c$ are material-dependent and temperature-dependent non-negative constants and $L_i(i=1,2,3,4)$ are material-dependent elastic constants. We refer to [1,23] for more details.

There are two types of Q-tensor hydrodynamics for nematic liquid crystal. The first type is based on variational methods, such as the Beris–Edwards model [24] and the Qian–Sheng model [25]. The second type is derived from the molecular kinetic theory of closure approximations, which is called the molecular-based Q-tensor dynamical theory [4]. In such models, the evolution of Q is derived from the evolution of the probability density function $f(\mathbf{x},\mathbf{m})$ by the relation (11); however, the equations of lower-order moments of $f(\mathbf{x},\mathbf{m})$ obtained from kinetic theory are coupled with higher-order moments, which causes a problem. So, we need to express the higher-order moment as a function of the lower-order moment using the closure approximation.

Since the above three theories are derived from different physical considerations, exploring the connection between the different theories is not only of significance in mathematics but it is also directly related to many physical properties. In this respect, Wang, Zhang and Zhang [12] presented a rigorous derivation from the Doi–Onsager molecular theory to the Ericksen–Leslie theory. Based on the same idea, Li, Wang and Zhang [26] rigorously derived the Ericksen–Leslie theory starting from the molecular-based Q-tensor theory for nematic phases. By the Hilbert expansion, Li and Wang [27] rigorously justified the connection between the inertial Qian–Sheng theory and the full Ericksen–Leslie theory. Based on the moment method, Degond, Frouvelle and Liu [28] presented the convergence of the Doi–Onsager theory to the Ericksen–Leslie theory. These works were mainly concerned with the hydrodynamic theory of nematic phases. For a detailed survey on modeling, analysis and computation of liquid crystals, the work of [29] can be refered to.

To our knowledge, Q-tensor hydrodynamics had not be applied to smectic phases prior to this work. In this paper, by using the Bingham closure, we aimed to derive new Q-tensor hydrodynamics that could describe the essential physical properties of smectic liquid crystal flows. Meanwhile, the new reduced Q-tensor model would be computable for future numerical works.

2. Derivation of $\mathbf{Q}$-Tensor Hydrodynamics for Smectic Liquid Crystals

Before deriving the new Q-tensor hydrodynamics for smectic phases, we introduce the Bingham closure and the previous hydrodynamics for nematic phases established in [4].

For a given density function $f(\mathbf{x},\mathbf{m})$ satisfying

$$\begin{array}{c}\hfill {\int }_{{\mathbb{S}}^2}f(\mathbf{x},\mathbf{m})\mathrm{d}\mathbf{m}=c\left(\mathbf{x}\right),{\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})f(\mathbf{x},\mathbf{m})\mathrm{d}\mathbf{m}=c\left(\mathbf{x}\right)Q\left(\mathbf{x}\right),\end{array}$$

the Bingham closure uses the quasi-equilibrium distribution [4],

$$\begin{array}{c}\hfill f_Q=\frac{c\left(\mathbf{x}\right)}{Z_Q}exp(B_Q\left(\mathbf{x}\right):\mathbf{m}\mathbf{m}),Z_Q={\int }_{{\mathbb{S}}^2}exp(B_Q\left(\mathbf{x}\right):\mathbf{m}\mathbf{m})\mathrm{d}\mathbf{m},\end{array}$$

to approximate f. Notice that if the density $c\left(\mathbf{x}\right)=c$ is constant, then the state is the nematic phase; if $c\left(\mathbf{x}\right)$ is varied, then the state is the smectic phase. Again, $B_Q$ depends on Q and can be determined by the following relation (see [4,26,30])

$$\begin{array}{c}\hfill {\int }_{S^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})f_Q\mathrm{d}\mathbf{m}=Q.\end{array}$$

It can be seen from [26,30] that $B_Q$ can be uniquely determined for Q and satisfies the physical constraint. Then, the fourth-order moment and the sixth-order moment of f can be viewed as functions of Q.

Now, we introduce the molecular-based Q-tensor dynamical model for nematic liquid crystals presented in [4]. For a given free energy functional $\mathcal{F}(Q,\nabla Q)$, we define

$$\begin{array}{c}\hfill {\mu }_Q=\frac{\delta \mathcal{F}(Q,\nabla Q)}{\delta Q}.\end{array}$$

Here the energy functional $\mathcal{F}(Q,\nabla Q)$, derived from Onsager’s molecular theory, is given by ([4])

$$\begin{array}{c}\hfill \mathcal{F}(Q,\nabla Q)={\mathcal{F}}_b\left(Q\right)+{\mathcal{F}}_e(Q,\nabla Q),\end{array}$$

(13)

where the bulk energy ${\mathcal{F}}_b$ and the elastic distortion energy ${\mathcal{F}}_e$ are respectively expressed as

$$\begin{array}{cc}\hfill {\mathcal{F}}_b\left(Q\right)=& L_0{\int }_{\Omega }\left(-ln{Z}_Q+Q:B_Q-\frac{\alpha }{2}{\left|Q\right|}^2\right)\mathrm{d}\mathbf{x},\hfill \\ \hfill {\mathcal{F}}_e(Q,\nabla Q)=& \frac{\epsilon }{2}{\int }_{\Omega }\{L_1{|\nabla Q|}^2+L_2\left({\partial }_i\left(Q_{ik}\right){\partial }_j\left(Q_{jk}\right)+{\partial }_i\left(Q_{jk}\right){\partial }_j\left(Q_{ik}\right)\right)+L_3{|\nabla Q^{\left(4\right)}|}^2\hfill \\ \hfill & +L_4\left({\partial }_i\left(Q_{iklm}^{\left(4\right)}\right){\partial }_j\left(Q_{jklm}^{\left(4\right)}\right)+{\partial }_i\left(Q_{jklm}^{\left(4\right)}\right){\partial }_j\left(Q_{iklm}^{\left(4\right)}\right)\right)+L_5{\partial }_i\left(Q_{ijkl}^{\left(4\right)}\right){\partial }_j\left(Q_{kl}\right)\}\mathrm{d}\mathbf{x},\hfill \end{array}$$

where the coefficients $L_i(i=0,\cdots ,5)$ are explicitly expressed as functions of molecular parameters, and $Q^{\left(4\right)}=Q^{\left(4\right)}\left(Q\right)$ is the fourth order symmetric traceless moment of $f_Q$. The parameter $\epsilon$ appears in ${\mathcal{F}}_e$ due to the fact that the ratios between the coefficients in ${\mathcal{F}}_e$ and the ones in ${\mathcal{F}}_b$ are at the order of the square of the molecule length.

We introduce two linear operators

$$\begin{array}{cc}\hfill {\mathcal{M}}_Q\left(A\right)=& \frac{1}{3}A+Q·A-A:M_Q^{\left(4\right)},\hfill \\ \hfill {\mathcal{N}}_Q{\left(A\right)}_{\alpha \beta }=& {\partial }_i\left\{\right[{\gamma }_{\perp }(M_{\alpha \beta kl}^{\left(4\right)}{\delta }_{ij}-\frac{1}{3}{\delta }_{\alpha \beta }{Q}_{kl}{\delta }_{ij})\hfill \\ \hfill & +({\gamma }_{\Vert }-{\gamma }_{\perp })(M_{\alpha \beta klij}^{\left(6\right)}-\frac{1}{3}{\delta }_{\alpha \beta }{M}_{klij}^{\left(4\right)})]\partial A_{kl}\},\hfill \end{array}$$

(14)

where the fourth-order moment $M_Q^{\left(4\right)}$ and the sixth-order moment $M_Q^{\left(6\right)}$ are respectively defined as follows:

$$\begin{array}{c}\hfill M_Q^{\left(4\right)}={\int }_{{\mathbb{S}}^2}\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}{f}_Q\mathrm{d}\mathbf{m},M_Q^{\left(6\right)}={\int }_{{\mathbb{S}}^2}\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}{f}_Q\mathrm{d}\mathbf{m}.\end{array}$$

Based on Doi–Onsager’s dynamical theory, the molecular-based Q-tensor model for the nematic phase [4] is given by

$$\begin{array}{cc}\hfill \frac{\partial Q}{\partial t}+\mathbf{v}·\nabla Q& =\frac{\epsilon }{De}{\mathcal{N}}_Q\left({\mu }_Q\right)-\frac{2}{De}\left({\mathcal{M}}_Q\left({\mu }_Q\right)+{\mathcal{M}}_Q^T\left({\mu }_Q\right)\right)+{\mathcal{M}}_Q(\nabla \mathbf{v})+{\mathcal{M}}_Q^T(\nabla \mathbf{v}),\hfill \\ \hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}& =-\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v}+\frac{1-\gamma }{2Re}\nabla ·(\mathbf{D}:M_Q^{\left(4\right)})\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & +\frac{1-\gamma }{DeRe}\left(2\nabla ·{\mathcal{M}}_Q\left({\mu }_Q\right)+{\mu }_Q:\nabla Q\right),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}& =0,\hfill \end{array}$$

(17)

where the term ${\mathcal{N}}_Q\left({\mu }_Q\right)$ represents the translational diffusion. The system (15)–(17) obeys the following basic energy dissipative law (see [4]):

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \frac{d}{dt}\left(\frac{1}{2}{\int }_{\Omega }{\left|\mathbf{v}\right|}^{2}d\mathbf{x}+\frac{1-\gamma }{ReDe}\mathcal{F}(Q,\nabla Q)\right)=-{\int }_{\Omega }(\frac{\gamma }{Re}{|\nabla \mathbf{v}|}^2+\frac{1-\gamma }{2Re}\mathbf{D}:M_Q^{\left(4\right)}:\mathbf{D}\hfill \\ \hfill & -\frac{\epsilon (1-\gamma )}{ReD{e}^2}{\mu }_Q:\mathcal{N}\left({\mu }_Q\right)+\frac{4(1-\gamma )}{ReD{e}^2}{\mu }_Q:{\mathcal{M}}_Q\left({\mu }_Q\right))\mathrm{d}\mathbf{x}.\hfill \end{array}\end{array}$$

2.1. From the Doi–Onsager Model to the New Q-Tensor Dynamical Model

The energy functional of the molecular-based Q-tensor static model in a modified version from the works [4,31] is

$$\begin{array}{c}\hfill \frac{F[c,Q]}{k_{B}T{c}_0}=F_{\mathrm{bulk}}+F_{\mathrm{elasitc},1}+F_{\mathrm{elasitc},2},\end{array}$$

(18)

which is measured by the product of the Boltzmann constant $k_B$ and the absolute temperature T, and $c=c\left(\mathbf{x}\right)$ is the density of molecules. In (18), $F_{\mathrm{bulk}}$ contains the entropy and the quadratic terms of the order parameters, $F_{\mathrm{elasitc},1}$ and $F_{\mathrm{elasitc},2}$ contain the first derivative terms and the second derivative terms of the order parameters. They are respectively given by

$$\begin{array}{cc}\hfill F_{\mathrm{bulk}}=& {\int }_{\Omega }\left[c\left(lnc+B_Q:Q-lnZ\right)+\frac{\pi c_0{L}^3\eta }{2}\left(A_1{c}^2+A_2{\left|cQ\right|}^2\right)\right]\mathrm{d}\mathbf{x},\hfill \\ \hfill F_{\mathrm{elasitc},1}=& -\frac{\pi c_0{L}^5\eta }{2}{\int }_{\Omega }[G_1{|\nabla c|}^2+G_2{|\nabla \left(cQ\right)|}^2+G_3{\partial }_i\left(c{Q}_{ij}\right){\partial }_{j}c\hfill \\ \hfill & +G_4{\partial }_i(c{Q}_{ik}{\partial }_j\left(c{Q}_{jk}\right)]\mathrm{d}\mathbf{x},\hfill \\ \hfill F_{\mathrm{elasitc},2}=& \frac{\pi c_0{L}^7\eta }{2}{\int }_{\Omega }[H_1|{\nabla }^2{c|}^2+H_2{\left|{\nabla }^2\left(cQ\right)\right|}^2+H_3{\partial }_{ij}\left(c{Q}_{ij}\right){\partial }_{kk}\left(c\right)\hfill \\ \hfill & +H_4{\partial }_{ik}\left(c{Q}_{il}\right){\partial }_{jk}\left(c{Q}_{jl}\right)+H_5{\partial }_{ij}\left(c{Q}_{ij}\right){\partial }_{kl}\left(Q_{kl}\right)]\mathrm{d}\mathbf{x},\hfill \end{array}$$

where the constant coefficients $A_i,G_i,H_i\propto 1$ only depend on the dimensionless molecular parameter $\eta =D/L$, where D and L are the molecular radius and molecular length, respectively. The coefficient $c_0$ is the average density of molecules in the region $\Omega$, which is defined by

$$\begin{array}{c}\hfill c_0=\frac{1}{|\Omega |}{\int }_{\Omega }c\left(\mathbf{x}\right)\mathrm{d}\mathbf{x}.\end{array}$$

Let ${\mathbf{x}}^{*}=\mathbf{x}/L_0$, where $L_0$ is the typical size of the flow region, then the nondimensional form of (18) can be expressed by

$$\begin{array}{c}\hfill \begin{array}{cc}F[c,Q]=\hfill & {\epsilon }^{-\frac{3}{2}}{\int }_{\Omega }\{\left[c(lnc+B_Q:Q-ln{Z}_Q)+\frac{\alpha }{2}(A_1{c}^2+A_2{\left|cQ\right|}^2)\right]\hfill \\ \hfill & -\frac{\epsilon }{2}\alpha \left[G_1{|\nabla c|}^2+G_2{|\nabla \left(cQ\right)|}^2+G_3{\partial }_i\left(c{Q}_{ij}\right){\partial }_{j}c+G_4{\partial }_i\left(c{Q}_{ik}\right){\partial }_j\left(c{Q}_{jk}\right)\right]\hfill \\ \hfill & +\frac{{\epsilon }^2}{2}\alpha [H_1|{\nabla }^2{c|}^2+H_2{\left|{\nabla }^2\left(cQ\right)\right|}^2+H_3{\partial }_{ij}\left(c{Q}_{ij}\right){\partial }_{kk}\left(c\right)+H_4{\partial }_{ik}\left(c{Q}_{il}\right){\partial }_{jk}\left(c{Q}_{jl}\right)\hfill \\ \hfill & +H_5{\partial }_{ij}\left(c{Q}_{ij}\right){\partial }_{kl}\left(c{Q}_{kl}\right)\left]\right\}\mathrm{d}\mathbf{x}.\hfill \end{array}\end{array}$$

(19)

A direct calculation of the first variational derivative of (19) yields

$$\begin{array}{cc}\hfill \frac{\delta F}{\delta \left(cQ\right)}=& {\epsilon }^{-\frac{3}{2}}\{B_Q-A_2\alpha cQ-\frac{\epsilon }{2}\alpha [-2G_2\Delta \left(cQ\right)+G_3({\partial }_{ij}c-\frac{1}{3}{\delta }_{ij}\Delta c)\hfill \\ \hfill & +G_4(Q_{jk,ji}+Q_{ji,jk})]+\frac{{\epsilon }^2}{2}\alpha [-2H_2{\Delta }^2\left(cQ\right)+H_3({\partial }_{ij}\Delta c-\frac{1}{3}{\delta }_{ij}{\Delta }^{2}c)\hfill \\ \hfill & +H_4({\partial }_{ji}\Delta \left(c{Q}_{jl}\right)+{\partial }_{jl}\Delta \left(c{Q}_{ji}\right))+H_5\left(Q_{kl,klij}+Q_{kl,klji}\right)\left]\right\},\hfill \end{array}$$

(20)

$$\begin{array}{cc}\frac{\delta F}{\delta c}=& {\epsilon }^{-\frac{3}{2}}\{lnc+1-ln{Z}_Q+A_1\alpha c-\frac{\epsilon }{2}\alpha \left(-2G_1\Delta c+G_3{\left(c{Q}_{ij}\right)}_{,ij}\right)\\ \hfill & +\frac{{\epsilon }^2}{2}\alpha \left(2H_1{\Delta }^{2}c+H_3{\partial }_{ij}\Delta \left(c{Q}_{ij}\right)\right)\}.\hfill \end{array}$$

(21)

In light of the above two variational derivatives, (20) and (21), we can define

$$\begin{array}{c}\hfill \widehat{\mu }={\mu }_{cQ}:\mathbf{m}\mathbf{m}+{\mu }_c,{\mu }_{cQ}=\frac{\delta F}{\delta \left(cQ\right)},{\mu }_c=\frac{\delta F}{\delta c}.\end{array}$$

For convenience, we also define the following linear operators:

$$\begin{array}{cc}\hfill {\mathcal{S}}_Q\left(b\right)=& {\partial }_i\left\{\left[({\gamma }_{\Vert }-{\gamma }_{\perp })c{M}_{ij}^{\left(2\right)}+c{\gamma }_{\perp }{\delta }_{ij}\right]{\partial }_{j}b\right\},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill {\mathcal{B}}_Q\left(A\right)=& {\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })c{M}_{ijkl}^{\left(4\right)}+{\gamma }_{\perp }{\delta }_{ij}c{M}_{kl}^{\left(2\right)}\right]{\partial }_j{A}_{kl}\right\},\hfill \end{array}$$

(23)

$$\begin{array}{cc}\hfill {\mathcal{K}}_Q{\left(b\right)}_{\alpha \beta }=& {\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })\left(c{M}_{\alpha \beta ij}^{\left(4\right)}-\frac{1}{3}c{\delta }_{\alpha \beta }{M}_{ij}^{\left(2\right)}\right)+{\gamma }_{\perp }c{\delta }_{ij}{Q}_{\alpha \beta }\right]{\partial }_{j}b\right\},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill {\mathcal{N}}_Q{\left(A\right)}_{\alpha \beta }=& {\partial }_i\left\{\right[({\gamma }_{\Vert }-{\gamma }_{\perp })\left(c{M}_{\alpha \beta klij}^{\left(6\right)}-\frac{1}{3}{\delta }_{\alpha \beta }c{M}_{klij}^{\left(4\right)}\right)\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & +{\gamma }_{\perp }\left(c{M}_{\alpha \beta kl}^{\left(4\right)}{\delta }_{ij}-\frac{1}{3}{\delta }_{\alpha \beta }c{Q}_{kl}{\delta }_{ij}\right)\left]{\partial }_j{A}_{kl}\right\}.\hfill \end{array}$$

(26)

Lemma 1.

For any vector field $\mathbf{u}$ defined on ${\mathbb{S}}^2$, it holds that

$$\begin{array}{c}\hfill {\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathcal{R}·\left(\mathbf{u}f\right)\mathrm{d}\mathbf{m}={〈\mathbf{m}\times \mathbf{u}\otimes \mathbf{m}+\mathbf{m}\otimes \mathbf{m}\times \mathbf{u}〉}_f.\end{array}$$

The proof of the lemma, which can be found in [3], is easy.

Based on the Doi–Onsager dynamical model (4)–(6), we are now in a position to derive the new Q-tensor dynamical model for smectic liquid crystals. We first derive the time evolution equation for $cQ$. Multiplying Equation (4) by $\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I}$ and integrating over a unit ball ${\mathbb{S}}^2$, and using the Bingham closure, we have

$$\begin{array}{cc}\hfill & \frac{\partial }{\partial t}\left(cQ\right)+\mathbf{v}·\nabla \left(cQ\right)\hfill \\ \hfill & =\underset{I_1}{\underbrace{\frac{\epsilon }{De}{\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\nabla ·\left\{\left({\gamma }_{\parallel }\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{I}-\mathbf{m}\mathbf{m})\right)·(f_Q\nabla \widehat{\mu })\right\}\mathrm{d}\mathbf{m}}}\hfill \\ \hfill & +\underset{I_2}{\underbrace{\frac{1}{De}{\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathcal{R}·\left(f_Q\mathcal{R}\widehat{\mu }\right)\mathrm{d}\mathbf{m}}}-\underset{I_3}{\underbrace{{\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathcal{R}·(\mathbf{m}\times \kappa ·\mathbf{m}{f}_Q)\mathrm{d}\mathbf{m}}}.\hfill \end{array}$$

(27)

By a straightforward calculation, we infer that

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill I_1=& \frac{\epsilon }{De}{\int }_{{\mathbb{S}}^2}(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\nabla ·\left\{\left({\gamma }_{\parallel }\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{I}-\mathbf{m}\mathbf{m})\right)·(f_Q\nabla \widehat{\mu })\right\}\mathrm{d}\mathbf{m}\hfill \\ \hfill =& \frac{\epsilon }{De}\nabla ·\left\{{\int }_{{\mathbb{S}}^2}\left[({\gamma }_{\parallel }-{\gamma }_{\perp })(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathbf{I}\right]·(\nabla {\mu }_c)f_Q\mathrm{d}\mathbf{m}\right\}\hfill \\ \hfill & +\frac{\epsilon }{De}\nabla ·\left\{{\int }_{{\mathbb{S}}^2}\left[({\gamma }_{\parallel }-{\gamma }_{\perp })(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{m}\mathbf{m}-\frac{1}{3}\mathbf{I})\mathbf{I}\right]·\nabla \left({\mu }_{cQ}\right)f_Q\mathrm{d}\mathbf{m}\right\}\hfill \\ \hfill =& \frac{\epsilon }{De}\left({\mathcal{N}}_Q\left({\mu }_{cQ}\right)+{\mathcal{K}}_Q\left({\mu }_c\right)\right).\hfill \end{array}\end{array}$$

(28)

According to Lemma 1, taking $\mathbf{u}={\mu }_{cQ}:\mathcal{R}\left(\mathbf{m}\mathbf{m}\right)$, we have

$$\begin{array}{cc}\hfill I_2& ={〈\mathbf{m}\times \left({\mu }_{cQ}:\mathcal{R}\left(\mathbf{m}\mathbf{m}\right)\right)\otimes \mathbf{m}+\mathbf{m}\otimes \mathbf{m}\times \left({\mu }_{cQ}:\mathcal{R}\left(\mathbf{m}\mathbf{m}\right)\right)〉}_{f_Q}\hfill \\ \hfill & =-\frac{2}{De}\left(\frac{2}{3}c{\mu }_{cQ}+{\mu }_{cQ}·cQ+cQ·{\mu }_{cQ}-2{\mu }_{cQ}:c{M}_Q^{\left(4\right)}\right)\hfill \\ \hfill & =-\frac{2}{De}c\left({\mathcal{M}}_Q\left({\mu }_{cQ}\right)+{\mathcal{M}}_Q^T\left({\mu }_{cQ}\right)\right).\hfill \end{array}$$

(29)

Similarly, by Lemma 1 and taking $\mathbf{u}=\mathbf{m}\times \kappa ·\mathbf{m}$, we obtain

$$\begin{array}{cc}\hfill I_3& ={〈\mathbf{m}\times (\mathbf{m}\times \kappa ·\mathbf{m})\otimes \mathbf{m}+\mathbf{m}\otimes \mathbf{m}\times (\mathbf{m}\times \kappa ·\mathbf{m})〉}_{f_Q}\hfill \\ \hfill & ={〈2(\mathbf{m}\times \kappa ·\mathbf{m})\mathbf{m}\mathbf{m}-(\kappa ·\mathbf{m})\mathbf{m}-\mathbf{m}(\kappa ·\mathbf{m})〉}_{f_Q}\hfill \\ \hfill & =2\mathbf{D}:{〈\mathbf{m}\mathbf{m}\mathbf{m}\mathbf{m}〉}_{f_Q}-\mathbf{D}·{〈\mathbf{m}\mathbf{m}〉}_{f_Q}+\Omega ·{〈\mathbf{m}\mathbf{m}〉}_{f_Q}-{〈\mathbf{m}\mathbf{m}〉}_{f_Q}·(\mathbf{D}+\Omega )\hfill \\ \hfill & =c\left(2\mathbf{D}:M_Q^{\left(4\right)}-\mathbf{D}·(Q+\frac{1}{3}\mathbf{I})+\Omega ·(Q+\frac{1}{3}\mathbf{I})-(Q+\frac{1}{3}\mathbf{I})·(\mathbf{D}+\Omega )\right)\hfill \\ \hfill & =-c\left({\mathcal{M}}_Q(\nabla \mathbf{v})+{\mathcal{M}}_Q^T(\nabla \mathbf{v})\right).\hfill \end{array}$$

(30)

Therefore, substituting (28)–(30) into (27) we obtain the time evolution equation of $cQ$

$$\begin{array}{c}\hfill \begin{array}{cc}\frac{\partial }{\partial t}\left(cQ\right)+\mathbf{v}·\nabla \left(cQ\right)=& \frac{\epsilon }{De}\left({\mathcal{N}}_Q\left({\mu }_{cQ}\right)+{\mathcal{K}}_Q\left({\mu }_c\right)\right)-\frac{2}{De}c\left({\mathcal{M}}_Q\left({\mu }_{cQ}\right)+{\mathcal{M}}_Q^T\left({\mu }_{cQ}\right)\right)\hfill \\ & +c\left({\mathcal{M}}_Q(\nabla \mathbf{v})+{\mathcal{M}}_Q^T(\nabla \mathbf{v})\right).\hfill \end{array}\end{array}$$

(31)

Replacing f with $f_Q$ in the elastic stress ${\tau }^e$ of (5), it follows that

$$\begin{array}{c}\hfill -{\langle \mathbf{m}\mathbf{m}\times \mathcal{R}\widehat{\mu }\rangle }_{f_Q}=2c{\mathcal{M}}_Q\left({\mu }_{cQ}\right).\end{array}$$

In a similar way, the force ${\mathbf{F}}^e$ can be also rewritten as

$$\begin{array}{c}\hfill {\langle \nabla \widehat{\mu }\rangle }_{f_Q}=c\nabla {\mu }_c-cQ:\nabla {\mu }_{cQ}.\end{array}$$

Thus, from Equation (5) we know that the evolution equation of the fluid velocity $\mathbf{v}$ is given by

$$\begin{array}{c}\hfill \begin{array}{cc}\frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}=& -\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v}+\frac{1-\gamma }{2Re}\nabla ·\left(\mathbf{D}:c{M}_Q^{\left(4\right)}\right)s\hfill \\ & +\frac{1-\gamma }{DeRe}\left(2\nabla ·\left[c{\mathcal{M}}_Q\left({\mu }_{cQ}\right)\right]-c\nabla {\mu }_c+cQ:\nabla {\mu }_{cQ}\right).\hfill \end{array}\end{array}$$

(32)

Finally, from Equation (4) we derive the evolution equation of the molecular density $c\left(\mathbf{x}\right)$. Integrating over a unit ball ${\mathbb{S}}^2$ for two sides of (4) gives

$$\begin{array}{c}\hfill \begin{array}{cc}\frac{\partial c}{\partial t}+\mathbf{v}·\nabla c& =\underset{J_1}{\underbrace{\frac{\epsilon }{De}{\int }_{{\mathbb{S}}^2}\nabla ·\left\{\left({\gamma }_{\parallel }\mathbf{m}\mathbf{m}+{\gamma }_{\perp }(\mathbf{I}-\mathbf{m}\mathbf{m})\right)·(f_Q\nabla \widehat{\mu })\right\}\mathrm{d}\mathbf{m}}}\hfill \\ & +\underset{J_2}{\underbrace{\frac{1}{De}{\int }_{{\mathbb{S}}^2}\mathcal{R}·\left(f_Q\mathcal{R}\widehat{\mu }\right)\mathrm{d}\mathbf{m}}}-\underset{J_3}{\underbrace{{\int }_{{\mathbb{S}}^2}\mathcal{R}·(\mathbf{m}\times \kappa ·\mathbf{m}{f}_Q)\mathrm{d}\mathbf{m}}}.\hfill \end{array}\end{array}$$

(33)

A straightforward calculation leads to

$$\begin{array}{c}\hfill \begin{array}{cc}{J}_1=& \frac{\epsilon }{De}{\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })c{M}_{ijkl}^{\left(4\right)}+{\gamma }_{\perp }{\delta }_{ij}c{M}_{kl}^{\left(2\right)}\right]{\partial }_j{\left({\mu }_{cQ}\right)}_{kl}+\left[({\gamma }_{\parallel }-{\gamma }_{\perp })c{M}_{ij}^{\left(2\right)}+{\gamma }_{\perp }c{\delta }_{ij}\right]{\partial }_j{\mu }_c\right\}\hfill \\ =& \frac{\epsilon }{De}\left({\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mathcal{S}}_Q\left({\mu }_c\right)\right).\hfill \end{array}\end{array}$$

(34)

Note that an important property of the rotational operator $\mathcal{R}$ is the formula of integration by parts [5],

$$\begin{array}{c}\hfill {\int }_{{\mathbb{S}}^2}A\left(\mathbf{m}\right)\mathcal{R}B\left(\mathbf{m}\right)\mathrm{d}\mathbf{m}=-{\int }_{{\mathbb{S}}^2}\left[\mathcal{R}A\left(\mathbf{m}\right)\right]B\left(\mathbf{m}\right)\mathrm{d}\mathbf{m}.\end{array}$$

(35)

Then, it is easy to see from (35) that

$$\begin{array}{c}\hfill J_2=0,J_3=0.\end{array}$$

(36)

Thus, plugging (34) and (36) into (33), the evolution equation of $c\left(\mathbf{x}\right)$ is given by

$$\begin{array}{c}\hfill \frac{\partial c}{\partial t}+\mathbf{v}·\nabla c=\frac{\epsilon }{De}\left({\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mathcal{S}}_Q\left({\mu }_c\right)\right).\end{array}$$

To sum up, we obtain the new molecular-based Q-tensor dynamical system of smectic liquid crystals:

$$\begin{array}{cc}\hfill \frac{\partial }{\partial t}\left(cQ\right)+\mathbf{v}·\nabla \left(cQ\right)=& \frac{\epsilon }{De}\left({\mathcal{N}}_Q\left({\mu }_{cQ}\right)+{\mathcal{K}}_Q\left({\mu }_c\right)\right)-\frac{2}{De}c\left({\mathcal{M}}_Q\left({\mu }_{cQ}\right)+{\mathcal{M}}_Q^T\left({\mu }_{cQ}\right)\right)\hfill \\ \hfill & +c\left({\mathcal{M}}_Q(\nabla \mathbf{v})+{\mathcal{M}}_Q^T(\nabla \mathbf{v})\right),\hfill \end{array}$$

(37)

$$\begin{array}{cc}\hfill \frac{\partial c}{\partial t}+\mathbf{v}·\nabla c=& \frac{\epsilon }{De}\left({\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mathcal{S}}_Q\left({\mu }_c\right)\right),\hfill \end{array}$$

(38)

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}=& -\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v}+\frac{1-\gamma }{2Re}\nabla ·\left(\mathbf{D}:c{M}_Q^{\left(4\right)}\right)\hfill \\ \hfill & +\frac{1-\gamma }{DeRe}\left(2\nabla ·\left[c{\mathcal{M}}_Q\left({\mu }_{cQ}\right)\right]-c\nabla {\mu }_c+cQ:\nabla {\mu }_{cQ}\right),\hfill \end{array}$$

(39)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}=& 0,\hfill \end{array}$$

(40)

where the operator ${\mathcal{M}}_Q$ is expressed by (14), the other operators ${\mathcal{S}}_Q,{\mathcal{B}}_Q,{\mathcal{K}}_Q,{\mathcal{N}}_Q$ are expressed by (22)–(26), respectively.

2.2. Energy Dissipation Law

One of the most important properties of the Bingham closure approximation is that it maintains the energy dissipation of the kinetic model. We derived the energy dissipation relation of the new molecular-based Q-tensor system (37)–(40). Combining the first variational derivative (20)–(21) and the dynamical system (37)–(40), it follows that

$$\begin{array}{cc}\hfill \frac{\mathrm{d}}{\mathrm{d}t}F[c,Q]& ={\int }_{\Omega }\left(\frac{\delta F}{\delta \left(cQ\right)}:\frac{\partial \left(cQ\right)}{\partial t}+\frac{\delta F}{\delta c}\frac{\partial c}{\partial t}\right)\mathrm{d}\mathbf{x}\hfill \\ \hfill & ={\int }_{\Omega }\{\frac{\epsilon }{De}{\mu }_{cQ}:\left({\mathcal{N}}_Q\left({\mu }_{cQ}\right)+{\mathcal{K}}_Q\left({\mu }_c\right)\right)-\frac{2}{De}{\mu }_{cQ}:c\left({\mathcal{M}}_Q\left({\mu }_{cQ}\right)+{\mathcal{M}}_Q^T\left({\mu }_{cQ}\right)\right)\hfill \\ \hfill & +{\mu }_{cQ}:c\left({\mathcal{M}}_Q(\nabla \mathbf{v})+{\mathcal{M}}_Q^T(\nabla \mathbf{v})-\mathbf{v}·\nabla \left(cQ\right)\right)\hfill \\ \hfill & +\frac{\epsilon }{De}{\mu }_c\left({\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mathcal{S}}_Q\left({\mu }_c\right)-\frac{De}{\epsilon }\mathbf{v}·\nabla c\right)\}\mathrm{d}\mathbf{x},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & \frac{\mathrm{d}}{\mathrm{d}t}{\int }_{\Omega }\frac{1}{2}{\left|\mathbf{v}\right|}^2\mathrm{d}\mathbf{x}={\int }_{\Omega }\{-\frac{\gamma }{Re}{\left|\mathbf{D}\right|}^2-\frac{1-\gamma }{2Re}\mathbf{D}:c{M}_Q^{\left(4\right)}:\mathbf{D}\hfill \\ \hfill & +\frac{1-\gamma }{DeRe}\left(-2c{\mathcal{M}}_Q\left({\mu }_{cQ}\right):\nabla \mathbf{v}-c\mathbf{v}·\nabla {\mu }_c+cQ:(\mathbf{v}·\nabla ){\mu }_{cQ}\right)\}\mathrm{d}\mathbf{x}.\hfill \end{array}$$

(42)

It is easy to see that ${\mathcal{M}}_Q\left(A\right):B={\mathcal{M}}_Q\left(B\right):A,$ if A or B is a symmetric tensor. Then, we deduce from (41) and (42) that

$$\begin{array}{c}\hfill \begin{array}{cc}& \frac{\mathrm{d}}{\mathrm{d}t}\left(\frac{1}{DeRe}F[c,Q]+{\int }_{\Omega }\frac{1}{2(1-\gamma )}{\left|\mathbf{v}\right|}^2\mathrm{d}\mathbf{x}\right)\hfill \\ =& {\int }_{\Omega }\{\frac{\epsilon }{ReD{e}^2}{\mu }_{cQ}:\left({\mathcal{N}}_Q\left({\mu }_{cQ}\right)+{\mathcal{K}}_Q\left({\mu }_c\right)\right)-\frac{4}{ReD{e}^2}{\mu }_{cQ}:c{\mathcal{M}}_Q\left({\mu }_{cQ}\right)\hfill \\ & +\frac{1}{ReDe}{\mu }_{cQ}:\left(2c{\mathcal{M}}_Q(\nabla \mathbf{v})-\mathbf{v}·\nabla \left(cQ\right)\right)\hfill \\ & -\frac{\gamma }{Re(1-\gamma )}{\left|\mathbf{D}\right|}^2-\frac{1}{2Re}\mathbf{D}:c{M}_Q^{\left(4\right)}:\mathbf{D}\hfill \\ & +\frac{1}{ReDe}\left(-2c{\mathcal{M}}_Q\left({\mu }_{cQ}\right):\nabla \mathbf{v}-c\mathbf{v}·\nabla {\mu }_c+cQ:(\mathbf{v}·\nabla ){\mu }_{cQ}\right)\hfill \\ & +\frac{\epsilon }{ReD{e}^2}{\mu }_c\left({\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mathcal{S}}_Q\left({\mu }_c\right)-\frac{De}{\epsilon }\mathbf{v}·\nabla c\right)\}\mathrm{d}x\hfill \\ =& {\int }_{\Omega }\{\frac{\epsilon }{ReD{e}^2}{\mu }_{cQ}:{\mathcal{N}}_Q\left({\mu }_{cQ}\right)-\frac{4}{ReD{e}^2}{\mu }_{cQ}:c{\mathcal{M}}_Q\left({\mu }_{cQ}\right)\hfill \\ & -\frac{\gamma }{Re(1-\gamma )}{\left|\mathbf{D}\right|}^2-\frac{1}{2Re}\mathbf{D}:c{M}_Q^{\left(4\right)}:\mathbf{D}\hfill \\ & +\frac{\epsilon }{ReD{e}^2}\left({\mu }_c{\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mu }_c{\mathcal{S}}_Q\left({\mu }_c\right)+{\mu }_{cQ}:{\mathcal{K}}_Q\left({\mu }_c\right)\right)\}\mathrm{d}x.\hfill \end{array}\end{array}$$

(43)

Note that ${\partial }_j\widehat{\mu }={\partial }_j{\mu }_{cQ}:\mathbf{m}\mathbf{m}+{\partial }_j{\mu }_c,$ we have

$$\begin{array}{c}\hfill \begin{array}{cc}0\le & {\int }_{\Omega }{〈\nabla \widehat{\mu }·\left(({\gamma }_{\parallel }-{\gamma }_{\perp })\mathbf{m}\mathbf{m}+{\gamma }_{\perp }\mathbf{I}\right)·\nabla \widehat{\mu }〉}_{f_Q}\mathrm{d}\mathbf{x}\hfill \\ =& {\int }_{\Omega }\{{\int }_{{\mathbb{S}}^2}{\partial }_i{\mu }_c\left[({\gamma }_{\parallel }-{\gamma }_{\perp })m_i{m}_j+{\gamma }_{\perp }{\delta }_{ij}\right]{\partial }_j{\mu }_c{f}_Q\mathrm{d}\mathbf{m}\hfill \\ & +{\int }_{{\mathbb{S}}^2}{\partial }_i{\mu }_c\left[({\gamma }_{\parallel }-{\gamma }_{\perp })m_i{m}_j+{\gamma }_{\perp }{\delta }_{ij}\right](m_k{m}_l-\frac{1}{3}{\delta }_{kl}){\partial }_j{\left({\mu }_{cQ}\right)}_{kl}{f}_Q\mathrm{d}\mathbf{m}\hfill \\ & +{\int }_{{\mathbb{S}}^2}{\partial }_i{\left({\mu }_{cQ}\right)}_{\alpha \beta }(m_{\alpha }{m}_{\beta }-\frac{1}{3}{\delta }_{\alpha \beta })\left[({\gamma }_{\parallel }-{\gamma }_{\perp })m_i{m}_j+{\gamma }_{\perp }{\delta }_{ij}\right]{\partial }_j{\mu }_c{f}_Q\mathrm{d}\mathbf{m}\hfill \\ & +{\int }_{{\mathbb{S}}^2}{\partial }_i{\left({\mu }_{cQ}\right)}_{\alpha \beta }(m_{\alpha }{m}_{\beta }-\frac{1}{3}{\delta }_{\alpha \beta })\left[({\gamma }_{\parallel }-{\gamma }_{\perp })m_i{m}_j+{\gamma }_{\perp }{\delta }_{ij}\right]\hfill \\ & (m_k{m}_l-\frac{1}{3}{\delta }_{kl}){\partial }_j{\left({\mu }_{cQ}\right)}_{kl}{f}_Q\mathrm{d}\mathbf{m}\}\mathrm{d}\mathbf{x}\hfill \\ =& -{\int }_{\Omega }\{{\mu }_c{\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })c{M}_{ij}^{\left(2\right)}+{\gamma }_{\perp }{\delta }_{ij}c\right]{\partial }_j{\mu }_c\right\}\hfill \\ & +{\mu }_c{\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })c{M}_{ijkl}^{\left(4\right)}+{\gamma }_{\perp }{\delta }_{ij}c{M}_{kl}^{\left(2\right)}\right]{\partial }_j{\left({\mu }_{cQ}\right)}_{kl}\right\}\hfill \\ & +{\left({\mu }_{cQ}\right)}_{\alpha \beta }{\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })\left(c{M}_{\alpha \beta ij}^{\left(4\right)}-\frac{1}{3}c{\delta }_{\alpha \beta }{M}_{ij}^{\left(2\right)}\right)+{\gamma }_{\perp }c{\delta }_{ij}{Q}_{\alpha \beta }\right]{\partial }_j{\mu }_c\right\}\hfill \\ & +{\left({\mu }_{cQ}\right)}_{\alpha \beta }{\partial }_i\left\{\right[({\gamma }_{\Vert }-{\gamma }_{\perp })(c{M}_{\alpha \beta klij}^{\left(6\right)}-\frac{1}{3}{\delta }_{\alpha \beta }c{M}_{klij}^{\left(4\right)})\hfill \\ & +{\gamma }_{\perp }(c{M}_{\alpha \beta kl}^{\left(4\right)}{\delta }_{ij}-\frac{1}{3}{\delta }_{\alpha \beta }c{Q}_{kl}{\delta }_{ij})\left]{\partial }_j{\left({\mu }_{cQ}\right)}_{kl}\right\}\}\mathrm{d}\mathbf{x}\hfill \\ =& -{\int }_{\Omega }\left\{{\mu }_c{\mathcal{S}}_Q\left({\mu }_{cQ}\right)+{\mu }_{cQ}{\mathcal{B}}_Q\left({\mu }_{cQ}\right)+{\mu }_{cQ}:{\mathcal{K}}_Q\left({\mu }_c\right)+{\mu }_{cQ}:{\mathcal{N}}_Q\left({\mu }_{cQ}\right)\right\}\mathrm{d}\mathbf{x}.\hfill \end{array}\end{array}$$

(44)

Therefore, from (43) and (44), the system (37)–(40) fulfills the following energy dissipation law:

$$\begin{array}{ccc}& & \frac{\mathrm{d}}{\mathrm{d}t}\left[{\int }_{\Omega }\frac{1}{2}{\left|\mathbf{v}\right|}^2\mathrm{d}\mathbf{x}+\frac{1-\gamma }{DeRe}F(c,Q)\right]\hfill \\ & & =-{\int }_{\Omega }\{\frac{\gamma }{Re}\mathbf{D}:\mathbf{D}+\frac{1-\gamma }{2Re}\mathbf{D}:c{M}_Q^{\left(4\right)}:\mathbf{D}+\frac{4(1-\gamma )}{D{e}^{2}Re}{\mu }_{cQ}:c{\mathcal{M}}_Q\left({\mu }_{cQ}\right)\hfill \\ & & +\frac{\epsilon (1-\gamma )}{D{e}^{2}Re}{〈\nabla \widehat{\mu }·\left(({\gamma }_{\parallel }-{\gamma }_{\perp })\mathbf{m}\mathbf{m}+{\gamma }_{\perp }\mathbf{I}\right)·\nabla \widehat{\mu }〉}_{f_Q}\}\mathrm{d}\mathbf{x}.\hfill \end{array}$$

(45)

Similarly to system (15)–(17), the parameters in the system (37)–(40) have clear physical significance but not are phenomenological. The coefficients $A_i,G_i$ and $H_i$ are also explicitly calculated in terms of physical molecular parameters. Another important feature of the new system (37)–(40) is that the translational and rotational diffusions are still maintained.

3. Discussion on the New $\mathbf{Q}$-Tensor Model

In this section, we give some special cases for the molecular-based Q-tensor dynamical system (37)–(40).

Case I. If the order parameter tensor Q and the molecular density $c\left(\mathbf{x}\right)$ are all equal to zero, i.e., $Q(\mathbf{x},t)=0,c(\mathbf{x},t)=0$, then the liquid crystal system will become the isotropic phase. At the moment, the system (37)–(40) is degenerated into

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}=& -\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v},\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}=& 0.\hfill \end{array}$$

(47)

The system (46) and (47) is just the standard incompressible Navier–Stokes equation for ordinary isotropic liquids. Here $\frac{\gamma }{Re}$ represents the viscous coefficient of isotropic flows.

Case II. If the order parameters Q and c satisfy

$$\begin{array}{c}\hfill Q(\mathbf{x},t)=0,c(\mathbf{x},t)\ne 0,\end{array}$$

then it follows that

$$\begin{array}{cc}\hfill F\left[c\right]=& {\epsilon }^{-\frac{3}{2}}\int \left(cln\frac{c}{4\pi }+\frac{1}{2}\alpha \left(A_1{c}^2-\epsilon G_1{|\nabla c|}^2+{\epsilon }^2{H}_1{\left|{\nabla }^{2}c\right|}^2\right)\right)\mathrm{d}\mathbf{x},\hfill \\ \hfill {\mu }_c=& \frac{\delta F}{\delta c}={\epsilon }^{-\frac{3}{2}}\left(ln\frac{c}{4\pi }+1+\alpha \left[A_{1}c+\epsilon G_1\Delta c+{\epsilon }^2{H}_1{\Delta }^{2}c\right]\right),\hfill \\ \hfill {\mu }_{cQ}=& 0.\hfill \end{array}$$

Consequently, the system (37)–(40) can be reduced to

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}=& -\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v}+\frac{1-\gamma }{2Re}\nabla ·\left(\frac{\epsilon }{2De}{\mathcal{K}}_0\left({\mu }_c\right)+\frac{1}{3}c\mathbf{D}\right)-\frac{1-\gamma }{DeRe}c\nabla {\mu }_c,\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill \frac{\partial c}{\partial t}+\mathbf{v}·\nabla c=& \frac{\epsilon }{De}{\mathcal{S}}_0\left({\mu }_c\right),\hfill \end{array}$$

(49)

$$\begin{array}{cc}\hfill \nabla ·\mathbf{v}=& 0,\hfill \end{array}$$

(50)

where ${\mathcal{K}}_0\left(b\right)={\mathcal{K}}_Q\left(b\right){{|}_{Q=0},{\mathcal{S}}_0\left(b\right)={\mathcal{S}}_Q\left(b\right)|}_{Q=0}$. We further simplify the above system (48). Note that $B_Q=0,Z_Q=4\pi$ if $Q=0$. Let $\mathbf{m}={(sin\theta cos\phi ,sin\theta sin\phi ,cos\theta )}^T,$ then we have

$$\begin{array}{cc}\hfill M_{ij}^{\left(2\right)}{|}_{Q=0}=& \frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}{m}_i{m}_j\mathrm{d}\mathbf{m}\hfill \\ \hfill =& \frac{1}{4\pi }{\int }_0^{2\pi }\mathrm{d}\phi {\int }_0^{\pi }\mathrm{d}\theta sin\theta m_i{m}_j=\frac{1}{3}{\delta }_{ij},\hfill \end{array}$$

(51)

$$\begin{array}{cc}\hfill M_{\alpha \beta ij}^{\left(4\right)}{|}_{Q=0}=& \frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}{m}_{\alpha }{m}_{\beta }{m}_i{m}_j\mathrm{d}\mathbf{m}\hfill \\ \hfill =& \frac{1}{4\pi }{\int }_{{\mathbb{S}}^2}\left[\frac{1}{7}{\left(m_{\alpha }{m}_{\beta }{\delta }_{ij}\right)}_{sym}-\frac{1}{35}{\left({\delta }_{\alpha \beta }{\delta }_{ij}\right)}_{sym}\right]\mathrm{d}\mathbf{m}\hfill \\ \hfill =& \frac{1}{4\pi }\left[\frac{2}{7}·\frac{4\pi }{3}{\left({\delta }_{\alpha \beta }{\delta }_{ij}\right)}_{sym}-\frac{4\pi }{35}{\left({\delta }_{\alpha \beta }{\delta }_{ij}\right)}_{sym}\right]\hfill \\ \hfill =& \frac{1}{15}{\left({\delta }_{\alpha \beta }{\delta }_{ij}\right)}_{sym}.\hfill \end{array}$$

(52)

Thus, using the definition of ${\mathcal{S}}_Q\left(b\right)$ and ${\mathcal{K}}_Q\left(b\right)$, it can be seen from (51) and (52) that

$$\begin{array}{cc}\hfill {\mathcal{S}}_0\left(b\right)=& {\partial }_i\left\{\left[({\gamma }_{\Vert }-{\gamma }_{\perp })c{M}_{ij}^{\left(2\right)}{|}_{Q=0}+c{\gamma }_{\perp }{\delta }_{ij}\right]{\partial }_{j}b\right\}\hfill \\ \hfill =& (\frac{1}{3}{\gamma }_{\Vert }+\frac{2}{3}{\gamma }_{\perp }){\partial }_i\left(c{\partial }_{j}b\right){\delta }_{ij}\hfill \\ \hfill \stackrel{\mathrm{def}}{=}& \tilde{\gamma }\nabla ·(c\nabla b),\hfill \\ \hfill {\mathcal{K}}_0{\left(b\right)}_{\alpha \beta }=& {\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })\left(c{M}_{\alpha \beta ij}^{\left(4\right)}{{|}_{Q=0}-\frac{1}{3}c{\delta }_{\alpha \beta }{M}_{ij}^{\left(2\right)}|}_{Q=0}\right)\right]{\partial }_{j}b\right\}\hfill \\ \hfill =& {\partial }_i\left\{\left[({\gamma }_{\parallel }-{\gamma }_{\perp })\left(\frac{1}{15}{\left({\delta }_{\alpha \beta }{\delta }_{ij}\right)}_{sym}-\frac{1}{9}{\delta }_{\alpha \beta }{\delta }_{ij}\right)c\right]{\partial }_{j}b\right\}\hfill \\ \hfill \stackrel{\mathrm{def}}{=}& \left[{\widehat{\gamma }}_1({\delta }_{\alpha i}{\delta }_{\beta j}+{\delta }_{\alpha j}{\delta }_{\beta i})+{\widehat{\gamma }}_2{\delta }_{\alpha \beta }{\delta }_{ij}\right]{\partial }_i\left(c{\partial }_{j}b\right)\hfill \\ \hfill =& {\widehat{\gamma }}_1\left({\partial }_{\alpha }\left(c{\partial }_{\beta }b\right)+{\partial }_{\beta }\left(c{\partial }_{\alpha }b\right)\right)+{\widehat{\gamma }}_2{\delta }_{\alpha \beta }\nabla ·(c\nabla b),\hfill \end{array}$$

where $\tilde{\gamma }=\frac{1}{3}{\gamma }_{\Vert }+\frac{2}{3}{\gamma }_{\perp },{\widehat{\gamma }}_1=\frac{1}{15}({\gamma }_{\parallel }-{\gamma }_{\perp }),{\widehat{\gamma }}_2=-\frac{2}{45}({\gamma }_{\parallel }-{\gamma }_{\perp })$. Let

$$\begin{array}{c}\hfill {\tau }_{\alpha \beta }^{\epsilon }={\widehat{\gamma }}_1\left({\partial }_{\alpha }\left(c{\partial }_{\beta }{\mu }_c\right)+{\partial }_{\beta }\left(c{\partial }_{\alpha }{\mu }_c\right)\right)+{\widehat{\gamma }}_2{\delta }_{\alpha \beta }\nabla ·(c\nabla {\mu }_c)+\frac{2}{3}\frac{De}{\epsilon }c{\mathbf{D}}_{\alpha \beta },\end{array}$$

then the system (48)–(50) becomes the following more simplified incompressible system for the fluid velocity $\mathbf{v}(\mathbf{x},t)$ and the molecular density $c(\mathbf{x},t)$:

$$\begin{array}{cc}\hfill \frac{\partial \mathbf{v}}{\partial t}+\mathbf{v}·\nabla \mathbf{v}=& -\nabla p+\frac{\gamma }{Re}\Delta \mathbf{v}+\frac{(1-\gamma )\epsilon }{4DeRe}\nabla ·{\tau }^{\epsilon }-\frac{1-\gamma }{DeRe}c\nabla {\mu }_c,\hfill \\ \hfill \frac{\partial c}{\partial t}+\mathbf{v}·\nabla c=& \frac{\tilde{\gamma }\epsilon }{De}\nabla ·(c\nabla {\mu }_c),\hfill \\ \hfill \nabla ·\mathbf{v}=& 0.\hfill \end{array}$$

Case III. Suppose that the molecular density $c(\mathbf{x},t)$ is constant (particularly $c(\mathbf{x},t)=1$) and $Q(\mathbf{x},t)\ne 0$, then the system will become the hydrodynamics of the nematic phases. Consequently, we have

$$\begin{array}{c}\hfill \widehat{\mu }={\mu }_Q:\mathbf{m}\mathbf{m},{\mu }_c=0,{\mu }_Q=\frac{\delta F}{\delta Q}.\end{array}$$

The non-dimensional energy functional (19) can be reduced to

$$\begin{array}{c}\hfill \begin{array}{cc}F\left[Q\right]=& {\int }_{\Omega }\{\left(B_Q:Q-ln{Z}_Q+\frac{\alpha }{2}{A}_2{\left|Q\right|}^2\right)-\frac{\epsilon }{2}\alpha \left(G_2{|\nabla Q|}^2+G_4{\partial }_i{Q}_{ik}{\partial }_j{Q}_{jk}\right)\hfill \\ & +\frac{{\epsilon }^2}{2}\alpha \left(H_2{\left|{\nabla }^{2}Q\right|}^2+H_4{\partial }_{ik}{Q}_{il}{\partial }_{jk}{Q}_{jl}+H_5{\partial }_{ij}{Q}_{ij}{\partial }_{kl}{Q}_{kl}\right)\}\mathrm{d}\mathbf{x}.\hfill \end{array}\end{array}$$

(53)

It should be noted that there some second derivative terms of the order parameter tensor Q appear in (53). For this reason we needed to truncate at the fourth order derivatives of the Taylor expansion for the molecular order parameter $f(\mathbf{x},\mathbf{m})$ with regards to $\mathbf{x}$, when deriving the molecular-based Q-tensor static model for smectic liquid crystals using the Bingham closure. If we only considered describing the nematic liquid crystals, then we did not need such high-order Taylor expansions. Therefore, after removing the second order derivative terms of Q in (53), the corresponding energy functional is

$$\begin{array}{cc}\hfill F\left[Q\right]=& {\int }_{\Omega }\left\{\left(B_Q:Q-ln{Z}_Q+\frac{\alpha }{2}{A}_2{\left|Q\right|}^2\right)-\frac{\epsilon }{2}\alpha \left(G_2{|\nabla Q|}^2+G_4{\partial }_i{Q}_{ik}{\partial }_j{Q}_{jk}\right)\right\}\mathrm{d}\mathbf{x}.\hfill \end{array}$$

(54)

The functional (54) is a special case of the free energy functional (13) with the coefficients $L_0=1,L_3=L_4=L_5=0$. Hence, under the condition of incompressible flows, the system (37)–(40) is degenerated into the molecular-based Q-tensor dynamical model (15)–(17) that describes the nematic liquid crystal flows.

4. Conclusions

In this paper, based on the Doi–Onsager molecular theory, a novel Q-tensor hydrodynamic model for smectic phases was derived using the Bingham closure. The model demonstrated some new features: its coefficients were all expressed as those in the molecular model, and the energy dissipation was maintained. The hydrodynamics could be further reduced to the version of nematic liquid crystals if the density was constant. The new model not only described many interesting physical phenomena of smectic liquid crystals, but it also included complex structures of equations. More importantly, the new model will be a computable model when considering the quasi-entropy proposed by recent work [32], since quasi-entropy is an elementary function that maintains the same asymptotic behavior and underlying physics of the Bingham closure. Therefore, we aim to investigate this model from numerical computations and well-posedness analysis in future works. Furthermore, exploring the connection between the new Q-tensor hydrodynamics and the Ericksen–Leslie model with density variations will be a rather challenging problem.