**2. Mathematical Modeling**

The peristaltic (or "sinusoidal") motion of Synovial fluid described by generalized incompressible fluid possesses the Navier-Stokes equations with a viscosity depending on a shear rate and concentration. We must couple this system with one extra convection-diffusion equation for a concentration of hyaluronic acid. The fundamental equations of governing flow with synovial fluid model are described in reference [8] as follows:

$$\text{div}V = 0\tag{1}$$

$$\frac{\partial V}{\partial t} + V \cdot \nabla V + \frac{\nabla p}{\rho} = \frac{2}{\rho} \text{div}(\Theta) \tag{2}$$

$$\frac{\partial \mathcal{C}}{\partial t} = \text{div}(F(\mathcal{C})) - \mathbf{V} \cdot \nabla \mathcal{C} \tag{3}$$

in above equation:

$$F(\mathbb{C}) = D\_{\mathbb{C}} \nabla \mathbb{C}, \; \Theta = \mu(\mathbb{C}, D) \; D \tag{4}$$

in which Δ*V*(*U*,*V*) is velocity, μ is viscosity, *D* is symmetric part of velocity gradient, *P* is pressure, ρ is density, *F* is concentration flux, *C* is concentration of hyaluronan/hyaluronic, and *D*<sup>C</sup> is constant diffusivity.

Let us focus on two-dimensional peristaltic flows in an asymmetric channel containing width d1 + d2 due to wave traveling in direction of flow with constant velocity *c*. The flow is discussed in Cartesian coordinates. The mass concentrations upon the upper wall are *C*0, whereas on the bottom wall they are *C*1. Peristaltic motion on the upper and lower internal surfaces is recognized as

$$H\_1(X, t) = \,^\vee Y = \,^l d\_1 + b\_1 \cos 2\pi (X - ct) \frac{1}{\lambda} \tag{5}$$

*Coatings* **2018**, *8*, 407

$$H\_2(X,t) = \,^\prime Y = \,^\prime - d\_2 - b\_2 \cos[(X - ct)2\pi + \lambda\phi]\frac{1}{\lambda} \tag{6}$$

To translate the coordinates, we use the same procedure that was used in [13]. The consequent relations of the boundaries of channel are described as

$$h\_1(\mathbf{x}) = y - 1 = \text{ acos}2\pi\mathbf{x} \tag{7}$$

$$h\_2(\mathbf{x}) = y = -b\cos(\phi + 2\pi\mathbf{x}) - d\tag{8}$$

#### *Synovial Fluid Model*

The peristaltic motion of viscous synovial fluid (see [33,34]) with thin film coating at the walls is considered in a two-dimensional channel. The flow patterns corresponding to Models 1 and 2 are markedly different. We shall ignore the detailed discussion here. However, fewer essential points associated with the model are presented. The models under consideration present exciting features. Model 1 is a simple generalized form of a power-law mathematical model for a shear-dependent viscosity that is helpful to define various non-Newtonian fluids in biological and polymer fluid mechanics, food rheology, and geology, to consider the basis of viscosity that affects the concentration of a reactant. Model 2 describes that exponent is a function of concentration.

Model 1: The generalized power-law model and the viscosity are exponentially dependent on concentration, then the Model 1 is written as:

$$\mu(\mathbb{C}, D) = \mu\_0 \mathbf{e}^{\alpha \mathbb{C}} \left( 1 + \gamma^2 \Big| D^2 \Big| \right)^n \tag{9}$$

Model 2: In this model, a shear-thinning index depends upon the concentration (i.e., zero concentration):

$$
\mu(\mathcal{C}, D) = \mu\_0 \left( 1 + \gamma^2 \left| D^2 \right| \right)^{n(\mathcal{C})} \tag{10}
$$

in which,

$$|D| = \sqrt{2\left(\frac{\partial u}{\partial x}\right)^2 + 2\left(\frac{\partial u}{\partial y}\right)^2 + \left(\frac{\partial v}{\partial x} + \frac{\partial u}{\partial y}\right)^2} \tag{11}$$

and

$$m(\mathbb{C}) = -\frac{\mathbf{e}^{\alpha \mathbb{C}} - 1}{2\mathbf{e}^{\alpha \mathbb{C}}} \tag{12}$$

in which *n* is index of shear-thinning comprising values between −0.5 and 0. It is worth mentioning that results of Newtonian fluid are obtained as a particular case of current fluid when *n* = 0.

The governing equations are too arduous to be acquiescent to stability analysis. Therefore, it is necessary to simplify the modeled equations. Make sure that the simplification process is congruous for such problems. Henceforth, we shall assume the long wavelength constraint, i.e., δ 1 and less Reynolds number Re ≈ *O*(1). Now, it is suitable to make the observing equations dimensionless by defining the following ratios:

$$\begin{array}{l} \mathsf{3} = \frac{\mathsf{y}}{\mathsf{d}\_{1}}, a = a^\*(\mathsf{C}\_{1} - \mathsf{C}\_{0}), \ \sigma = \frac{\mathsf{C} - \mathsf{C}\_{0}}{\mathsf{d}\_{1} - \mathsf{C}\_{0}}, \mathsf{7} = \frac{\mathsf{a}}{\|\mathsf{a}\_{0}\|}, \mathsf{7} = \frac{\mathsf{a}}{\|\mathsf{a}\_{1}\|}, \mathsf{7} = \frac{\mathsf{a}\_{1}}{\|\mathsf{a}\|}, \mathsf{8} = \frac{\mathsf{a}\_{1}}{\|\mathsf{a}\|}, \mathsf{a} = \frac{\mathsf{b}\_{1}}{\|\mathsf{a}\|},\\\ \mathsf{b} = \frac{\mathsf{b}\_{2}}{\mathsf{d}\_{1}}, d = \frac{\mathsf{a}\_{2}}{\mathsf{d}\_{1}}, \mathsf{We} = \frac{\mathsf{a}\_{1}}{\mathsf{d}\_{1}}, \ |\mathsf{D}| = \frac{\mathsf{d}\_{1}}{\mathsf{d}\_{1}} |D|, \mathsf{S}\_{\mathsf{c}} = \frac{\mathsf{a}\_{1}}{\|\mathsf{D}\_{\mathsf{c}}}, \ \mathsf{\mathsf{D}} = \frac{\mathsf{a}\_{1}}{\mathsf{d}\_{\mathsf{c}}}, \ \mathsf{Re} = \frac{\mathsf{g} \mathsf{d}\_{\mathsf{c}}}{\mathsf{g}}, \ \mathsf{\mathsf{T}} = \frac{\mathsf{g}}{\mathsf{g}}, \mathsf{T} = \frac{\mathsf{g}}{\mathsf{k}} \end{array} \tag{13}$$

In above expression, *S*<sup>c</sup> denotes Schmidt number, Re stands for Reynolds number, α represents concentration production, and γ is a material parameter.

The resulting non-dimensional governing equations along with Models 1 and 2 after exempting bar symbols in a wave frame will observe the following form:

$$\frac{\partial v}{\partial y} = -\frac{\partial u}{\partial x} \tag{14}$$

$$\frac{\partial p}{\partial \mathbf{x}} = \begin{cases} \frac{1}{2} \frac{\partial}{\partial y} \left[ (1 + n\sigma) \left\{ 1 + n\mathcal{W} \mathbf{e}^2 \left( \frac{\partial \mathbf{u}}{\partial y} \right)^2 \right\} \frac{\partial \mathbf{u}}{\partial y} \right] & \text{(Model 1)}\\ \frac{\partial^2 \boldsymbol{u}}{\partial y^2} - \frac{n\mathcal{W}\mathbf{e}^2}{2} \frac{\partial}{\partial y} \left[ \sigma \left( \frac{\partial \mathbf{u}}{\partial y} \right)^3 \right] & \text{(Model 2)} \end{cases} \tag{15}$$

Concentration equation for Models 1 and 2 is simplified to the following form:

$$\frac{1}{S\_{\mathbb{C}}} \frac{\partial^2 \sigma}{\partial y^2} = 0 \tag{16}$$

The no slip boundary conditions become:

$$
\mu(h\_1) = -1 \,\upmu(h\_2) = -1, \,\text{\textbullet}(h\_1) = 0, \,\text{\textbullet}(h\_2) = 1 \tag{17}
$$
