Flow of a Self-Similar Non-Newtonian Fluid Using Fractal Dimensions

Abstract

In this paper, the study of the fully developed flow of a self-similar (fractal) power-law fluid is presented. The rheological way of behaving of the fluid is modeled utilizing the Ostwald–de Waele relationship (covering shear-thinning, Newtonian and shear-thickening fluids). A self-similar (fractal) fluid is depicted as a continuum in a noninteger dimensional space. Involving vector calculus for the instance of a noninteger dimensional space, we determine an analytical solution of the Cauchy equation for the instance of a non-Newtonian self-similar fluid flow in a cylindrical pipe. The plot of the velocity profile obtained shows that the rheological behavior of a non-Newtonian power-law fluid is essentially impacted by its self-similar structure. A self-similar shear thinning fluid and a self-similar Newtonian fluid take on a shear-thickening way of behaving, and a self-similar shear-thickening fluid becomes more shear thickening. This approach has many useful applications in industry, for the investigation of blood flow and fractal fluid hydrology.

1. Introduction

Non-Newtonian fluids appear in a variety of practical applications [1,2,3]. Their unusual flow patterns are frequently more complicated than those of Newtonian fluids. Numerous studies and observations have shown that for non-Newtonian fluids such as blood, paints, whipped cream and polymeric solutions, the relationship between viscous shear stress and velocity gradient is nonlinear [4,5,6]. There are a number of empirical or semiempirical formulas that have been developed to accurately measure non-Newtonian viscosity behaviors seen in a variety of disciplines [4,5,6,7,8,9,10], namely the well-known power-law model, the Bingham model and the Casson model. In general, both Newtonian and non-Newtonian fluids are thought of as integer-dimensional homogeneous continuums. However, the interior structures of many complex fluids have scale-invariant characteristics that are described by noninteger fractal dimensions [11,12,13,14]. An object that possesses scale-invariant features is called a self-similar or fractal object (including fluids), whose fundamental characteristic is that they appear similar under different levels of magnification [11,15]. The mass of a self-similar fluid obeys the power law $M\propto R^D$, where M is the mass of a spherical region of the fluid with radius R, and D is the fractal mass dimension. Fractal fluids include solutions containing a fractal distribution of a solute dissolved in a nonfractal solvent [16], emulsions in which one phase is fractally dispersed in the other [17], suspensions in which solid particles are fractally distributed in a liquid [18,19] and classical fluids confined in a fractal configuration space [20]. To explain fractal fluids, there exist four basic ways [21,22]: (a) using the methods of “Analysis on fractals” [23,24,25]; (b) using fractional-differential continuum models [26,27,28,29,30] and also [31,32,33]; (c) applying fractional-integral continuum models [34,35]; and (d) using the theory of integration and differentiation for a noninteger dimensional space [36,37,38].

In this paper, we consider the noninteger dimensional space (NIDS) model developed in [39], where the integrations and differentiations are defined for the spaces with noninteger dimensions. Newtonian fractal fluids have been studied by several scientists, including Tarasov [21,40] and Balankin et al. [41]. The study of a fractal non-Newtonian fluid using the noninteger dimensional space model has not yet been considered. In this regard, this study presents a theoretical investigation of the effects of the self-similarity property on the flow of a non-Newtonian fluid through a cylindrical tube. For simplification, the surface of the tube is considered to be smooth. However, for the case of a rough surface, the fractal approach can be used to quantify the irregularities of the tube’s inner wall. For instance, a recently published study [42] deals with the influence of the surface roughness on the behavior of a non-Newtonian fluid. However, this is not within the scope of our study. This paper focuses mainly on how the self-similarity property of the fluid affects the flow of a non-Newtonian fluid. The behavior of the non-Newtonian fluid is described using the Ostwald–de Waele model, better known as the power-law model [43]. This approach has a wide range of possible applications in any field that involves a non-Newtonian fluid flow that presents a self-similar structure. For instance, blood is a non-Newtonian fluid that is considered to be self-similar because it contains particles such as proteins, hormones, glucose, etc. Thus, this approach can be used to study blood flow [44,45].

The rest of this paper is organized as follows. In Section 2, we introduce the well-known Ostwald–de Waele model that describes the flow of a non-Newtonian fluid. In Section 3, the noninteger dimensional space operators (gradient and divergence) are presented, and we derive an analytical solution using these operators to describe a self-similar non-Newtonian fluid. A discussion of the results obtained is given in Section 4, and an overall conclusion is presented in Section 5.

2. Laminar Flow of an Incompressible Non-Newtonian Fluid

2.1. The Rheological Behavior of Fluids

The classical Newton’s relationship between shear stress and velocity gradient can be expressed as:

$$\stackrel{=}{\tau }=\mu \overline{\overline{\dot{\gamma }}},$$

(1)

where $\stackrel{=}{\tau }$ is the viscous shear stress tensor, $\mu$ is the fluid dynamic viscosity and $\overline{\overline{\dot{\gamma }}}$ is the strain rate tensor. The classical Newton’s expression of the shear stress can be applied to a particular type of fluids called Newtonian fluids. However, in reality, most fluids do not follow this classical relationship. In fact, different models have been developed to describe the complex behavior of fluids (such as blood, rubber and slurry). Fluids that do not follow the classical Newton’s relationship are called non-Newtonian fluids. Several general variations of Equation (1) for various non-Newtonian fluids have been proposed in previous studies. The following power-law model, often known as the Ostwald–de Waele model, is one of the most commonly used formulas:

$$\stackrel{=}{\tau }=K_0{\left(\overline{\overline{\dot{\gamma }}}\right)}^n,$$

(2)

where n and $K_0$ are the power-law index and the consistency index, respectively. For a Newtonian fluid, $n=1$ and $K_0=\mu$.

2.2. Momentum Conservation Equation

Let us consider the flow of a non-Newtonian fluid through a cylindrical tube. To model the hydrodynamics of the fluid, we need two equations, conservation of mass and conservation of momentum or better known as the Cauchy equations, which are written in the following form:

$$\frac{\partial \rho }{\partial t}+\nabla .\left(\rho \mathbf{v}\right)=0,$$

(3)

$$\rho \frac{D\mathbf{v}}{Dt}=\rho \mathbf{f}+\nabla .\stackrel{=}{\sigma },$$

(4)

where $\mathbf{v}$ is the velocity field, $\rho$ is the density of the fluid, $\mathbf{f}$ is a vector representing body forces and $\stackrel{=}{\sigma }$ is the stress tensor characterizing the rheological behavior of the fluid.

The stress tensor can be expressed as follows:

$$\stackrel{=}{\sigma }=-p\stackrel{=}{I}+\stackrel{=}{\tau },$$

(5)

where p and $\stackrel{=}{I}$ are the pressure tensor and unit tensor, respectively. $\stackrel{=}{\tau }$ refers to the shear stress tensor.

Considering the case of the fully developed flow of an incompressible non-Newtonian fluid, the mass and momentum conservation Equations (3) and (4) take the following form:

$$\nabla .\left(\rho \mathbf{v}\right)=0$$

(6)

$$-\nabla p+\nabla .\stackrel{=}{\tau }=0.$$

(7)

For a unidirectional flow (along the x direction in Figure 1), the projection of Equation (7) according to the radial coordinate r gives:

$${\nabla }_r\left(-{\tau }_{xr}{\mathbf{e}}_{\mathbf{x}}\right)=\frac{dp}{dx}{\mathbf{e}}_{\mathbf{x}},$$

(8)

where ${\nabla }_r$ denotes the divergence operator in the radial direction.

The Poiseuille flow differs from flows with inertia in that the pressure field is independent of the velocity field. Therefore, Equation (8) is written as follows:

$$\frac{1}{r}\frac{d}{dr}\left(r{\tau }_{xr}\right)=\frac{\Delta p}{L},$$

(9)

where $\Delta p>0$ is the pressure difference along the pipe, and L is the pipe length.

The integration of Equation (9) between $r=0$ and $r=R$, taking into account the cylindrical symmetry of the pipe at $r=0$, and using expression (2) for the shear stress ${\tau }_{xr}$, yields:

$$\frac{d{v}_x}{dr}=-{\left(\frac{\Delta p}{2K_{0}L}\right)}^{\frac{1}{n}}{r}^{\frac{1}{n}}.$$

(10)

The solution of Equation (10), considering the no-slip condition at the wall, makes it possible to obtain an analytical expression of the velocity field.

$$v_x=v_{max}\left(n\right)\left[1-{\left(\frac{r}{R}\right)}^{\frac{1+n}{n}}\right],$$

(11)

where:

$$v_{max}\left(n\right)=\frac{n}{1+n}{\left(\frac{\Delta p}{2K_{0}L}\right)}^{\frac{1}{n}}{R}^{\frac{n+1}{n}}.$$

(12)

The rheological behavior of non-Newtonian fluids is described by Equation (11), a well-known velocity distribution. The constant n is the power-law index, and the fluid is shear thinning (pseudo-plastic) for $0<n<1$, Newtonian for $n=1$ and shear thickening (dilatant) for $n>1$. Figure 2 illustrates the behavior of these types of fluids.

The non-Newtonian behavior of shear thinning fluids $(0<n<1)$ is characterized by a decrease in the viscosity of the fluid under an applied shear strain. The terms “pseudo-plastic behavior” and “shear-thinning behavior” are occasionally used interchangeably [46,47]. It is the most common type of non-Newtonian fluid behavior, and it can be found in a variety of industrial and everyday applications [48]. Although the precise mechanism of shear thinning is unknown, it is widely assumed to be the result of small changes within the fluid, such as the rearrangement of microscale geometries to facilitate shearing. For shear-thickening fluids $(n>1)$, when shear stress is applied, the viscosity increases. This observable behavior is due to the system crystallizing under stress and acting more like a solid than a solution [49].

3. Laminar Flow of an Incompressible Fractal Non-Newtonian Fluid

Viscous fluids are regarded as being integer-dimensional homogeneous continuums. The internal structures of many complex fluids, on the other hand, have scale-invariant properties, which are characterized by noninteger fractal dimensions. Consider a ball region $V_D$ of radius R of a self-similar fluid; the mass of this ball region is related to its radius by $M\propto R^D$. D is a fractal mass dimension that measures how well the fluid particles fill the region $V_D$. Generally, for a self-similar fluid, the dimension D takes values less than 3 and for a nonfractal viscous fluid $D=3$ [40,41]. The boundary $S_d$ of the ball region $V_D$ is characterized by the dimension d, and the relation $d=D-1$ does not hold in general [21]. Consequently, we define ${\alpha }_r=D-d$, which is a dimension in the radial direction r. For the case of a noninteger dimensional space, Tarasov proposed a generalization of vector calculus [39]. The first- and second-order operators such as gradient, divergence, scalar and vector Laplace operators in a noninteger dimensional space (NIDS) are defined. For the sake of simplicity, the scalar and vector fields are assumed to be merely dependent on r (see Refs. [21,39,40] for more details). The divergence and gradient operators in NIDS are written as follows:

$$\begin{array}{ccc}& & Di{v}_r^{D,d}\mathbf{u}\left(r\right)=\frac{{\pi }^{(1-{\alpha }_r)/2}\Gamma ((d+{\alpha }_r)/2)}{\Gamma \left(\right(d+1)/2)}\left(\frac{1}{r^{{\alpha }_r-1}}\frac{\partial u_r\left(r\right)}{\partial r}+\frac{d}{r^{{\alpha }_r}}{u}_r\left(r\right)\right),\hfill \end{array}$$

(13)

$$\begin{array}{ccc}& & Gra{d}_r^{D,d}\varphi =\frac{\Gamma ({\alpha }_r/2)}{{\pi }^{{\alpha }_r/2}{r}^{{\alpha }_r-1}}\frac{\partial \varphi \left(r\right)}{\partial r}{\mathbf{e}}_r,\hfill \end{array}$$

(14)

where $\mathbf{u}\left(r\right)$ is a vector field and $\varphi \left(r\right)$ is a scalar field. The NIDS operators (13) and (14) enable us to obtain an equation of motion that describes the flow of a fractal fluid.

Considering the flow of a non-Newtonian fluid with the property of self-similarity, substituting Equations (13) and (14) into Equation (8) we get:

$$Di{v}_r^{D,d}(-{\tau }_{xr}{\mathbf{e}}_{\mathbf{x}})=\frac{dp}{dx}{\mathbf{e}}_{\mathbf{x}},$$

(15)

where the shear stress ${\tau }_{xr}$ for a fractal fluid is:

$${\tau }_{xr}=K_0{\left(-\frac{\Gamma ({\alpha }_r/2)}{{\pi }^{{\alpha }_r/2}{r}^{{\alpha }_r-1}}\frac{\partial v_x\left(r\right)}{\partial r}\right)}^n.$$

(16)

Applying the divergence and gradient operators for Equation (15), we have the following differential equation:

$$A_n(d_x,{\alpha }_r)\left[-\frac{n}{r^{(n+1)({\alpha }_r-1)}}\frac{d^2{v}_x}{d{r}^2}{\left(-\frac{d{v}_x}{dr}\right)}^{n-1}+\frac{d_x+n(1-{\alpha }_r)}{r^{(n+1){\alpha }_r-n}}{\left(-\frac{d{v}_x}{dr}\right)}^n\right]=\frac{1}{K_0}\frac{\Delta p}{L}.$$

(17)

where:

$$A_n=\frac{\Gamma ((d_x+{\alpha }_r)/2){(\Gamma ({\alpha }_r/2))}^n}{{\pi }^{({\alpha }_r(n+1)-1)/2}\Gamma ((d_x+1)/2)}.$$

(18)

The boundary conditions for Equation (17) are:

$$v_x\left(0\right)=V_{max},$$

$$v_x\left(R\right)=0.$$

Considering the new function:

$$Y\left(r\right)={\left(-\frac{d{v}_x\left(r\right)}{dr}\right)}^n,$$

(19)

whose derivative is:

$$Y^{\prime }\left(r\right)=-n\frac{d^2{v}_x}{d{r}^2}{\left(-\frac{d{v}_x}{dr}\right)}^{n-1},$$

(20)

we can see that $Y\left(r\right)$ satisfies the first-order differential equation:

$$r{Y}^{\prime }\left(r\right)+[d_x+n(1-{\alpha }_r)]Y\left(r\right)=W{r}^{(n+1){\alpha }_r-n},$$

(21)

where:

$$W=\frac{1}{A_n(d_x,{\alpha }_r)K_0}\frac{\Delta p}{L}.$$

(22)

The solution of Equation (21) is:

$$Y\left(r\right)=\frac{W}{{\alpha }_r+d_x}{r}^{(n+1){\alpha }_r-n}+k_1{r}^{n({\alpha }_r-1)-d_x},$$

(23)

where $k_1$ is a constant. We know that $v_x\left(0\right)=V_{max}$, meaning that $v_x\left(r\right)$ reaches its maximum value at $r=0$. Consequently, the derivative of $v_x\left(r\right)$ is zero at this value:

$$\frac{d{v}_x}{dr}(r=0)=0.$$

(24)

Considering Equations (19) and (24), then $Y(r=0)=0$. Notice that the power $n({\alpha }_r-1)-d_x$ in Equation (23) is negative, which leads to an indefined velocity gradient at $r=0$, implying that $k_1=0$. Equation (23) then becomes:

$$Y\left(r\right)=\frac{W}{{\alpha }_r+d_x}{r}^{(n+1){\alpha }_r-n}.$$

(25)

Notice that for $Y(r=0)=0$, the power $(n+1){\alpha }_r-n$ in expression (25) must be a positive number. Thus, this gives limits to the values permitted for ${\alpha }_r$ and n. Consequently, we get $\frac{n}{n+1}<{\alpha }_r\le 1$.

Considering Equations (19) and (25), we get the following expression for the velocity gradient:

$$\frac{d{v}_x\left(r\right)}{dr}=-{\left(Y\left(r\right)\right)}^{\frac{1}{n}}=-{\left(\frac{W}{{\alpha }_r+d_x}\right)}^{\frac{1}{n}}{r}^{{\alpha }_r(1+\frac{1}{n})-1}.$$

(26)

After integration, we obtain the following expression for $v_x\left(r\right)$:

$$v_x\left(r\right)=-{\left(\frac{W}{{\alpha }_r+d_x}\right)}^{\frac{1}{n}}\frac{r^{{\alpha }_r(1+\frac{1}{n})}}{{\alpha }_r(1+\frac{1}{n})}+k_2,$$

(27)

where $k_2$ is a constant of integration obtained by the means of the second boundary condition $v_x\left(R\right)=0$:

$$k_2={\left(\frac{W}{{\alpha }_r+d_x}\right)}^{\frac{1}{n}}\frac{R^{{\alpha }_r(1+\frac{1}{n})}}{{\alpha }_r(1+\frac{1}{n})}.$$

(28)

The velocity profile $v_x\left(r\right)$ is then given by:

$$v_x\left(r\right)={\left(\frac{W}{{\alpha }_r+d_x}\right)}^{\frac{1}{n}}\frac{R^{{\alpha }_r(1+\frac{1}{n})}}{{\alpha }_r(1+\frac{1}{n})}\left[1-{\left(\frac{r}{R}\right)}^{{\alpha }_r(1+\frac{1}{n})}\right].$$

(29)

Using the effective consistency index $K_{eff}$:

$$K_{eff}=\frac{1}{2}({\alpha }_r+d_x){\alpha }_r^n{A}_n(d_x,{\alpha }_r)K_0,$$

(30)

Equation (29) then becomes:

$$v_x\left(r\right)=V_{max}\left[1-{\left(\frac{r}{R}\right)}^{{\alpha }_r\left(\frac{n+1}{n}\right)}\right],$$

(31)

where:

$$V_{max}=\frac{n}{n+1}{\left(\frac{1}{2K_{eff}}\frac{\Delta p}{L}\right)}^{\frac{1}{n}}{R}^{{\alpha }_r\left(\frac{n+1}{n}\right)}.$$

(32)

Notice that for $d_x={\alpha }_r=1$, we get the usual velocity distribution of a power-law fluid with $K_{eff}=K_0$ and Equation (31) becomes Equation (11).

4. Discussion

The velocity distribution (31) is proportional to the power ${\alpha }_r\left(\frac{n+1}{n}\right)$, meaning that the rheological behavior of a non-Newtonian fluid is heavily dependent on the fractal aspect of the fluid. The fractal dimensions ${\alpha }_r$ and $d_x$ are a measure of the self-similarity present in the fluid, and the fluid is fractal when ${\alpha }_r<1$ and $d_x<1$. Moreover, Equation (31) shows that, in order to have zero viscous shear effects in the center of the tube, the velocity gradient should equal zero at $r=0$, which gives a limit to the values permitted for the radial dimension ${\alpha }_r$, thus, we get $\frac{n}{n+1}<{\alpha }_r\le 1$.

The link between the rheological behavior of a non-Newtonian fluid and its fractal nature can be seen in Figure 3, which plots the effective consistency index $K_{eff}$ defined by Equation (30) normalized by $K_0$ for different values of n. From Figure 3a, we can see that for a fractal fluid $({\alpha }_r<1)$, $K_{eff}$ increases, the rate of increase, however, being higher for shear-thickening fluids $(n>1)$. Figure 3b illustrates that for a fractal fluid with $d_x<1$, $K_{eff}$ decreases in a linear fashion, and the rate of decrease is the same for shear-thinning, Newtonian and shear-thickening fluids. Another interesting result is that the fractal nature of the fluid seems to play a major role in the dynamics of the flow, as it changes completely the rheological behavior of the fluid. Large differences in velocity distributions can be found when different values of fractal dimensions are used. These effects are depicted in Figure 4, Figure 5 and Figure 6. The behavior of fluids with the property of self-similarity is the same as that of shear-thickening fluids. For a particular value of ${\alpha }_r$, shear-thinning fluids $(n<1)$ and Newtonian fluids $(n=1)$ behave like shear-thickening fluids (see Figure 4).

In other words, changing the fractal dimension of a fractal non-Newtonian fluid results in a non-Newtonian fluid with different properties. This is because the higher the degree of similarity of the fluid, the thicker this fluid becomes near the walls, which results in a narrower velocity distribution. Not only does the fractal aspect of the fluid affect the shape of the velocity profile, it also has a significant effect on its amplitude (see Figure 5 and Figure 6). However, it is important to note that shear thinning fluids are the most affected by the self-similarity present in the fluid. For values of ${\alpha }_r<1$ and $d_x<1$, the amplitude of the velocity profile increases significantly, especially for shear-thinning fluids.

It is noteworthy that the radial dimension ${\alpha }_r$ has the most significant effect on the amplitude of the velocity profile. For ${\alpha }_r<1$, the fluid becomes thick near the walls, and because of the conservation of momentum, we see a significant increase in the amplitude of the velocity profile in the center of the tube. By contrast, the dimension $d_x$ only affects the amplitude and not the shape of the velocity profile.

5. Conclusions

We conducted a theoretical investigation of the fractal nature of a power-law fluid. The fluid was modeled using gradient and divergence operators in NIDS. The equations of motion for an incompressible Stokes flow of a non-Newtonian fractal fluid were derived. Then, a radial velocity distribution describing the behavior of fractal non-Newtonian fluids was obtained. The main parameters affecting the behavior of the flow were the fractal dimensions (${\alpha }_r$,$d_x$) and the power-law index $\left(n\right)$, as the fluid was considered to be a nonfractal Newtonian fluid when ${\alpha }_r=d_x=n=1$. We found that for a particular value of ${\alpha }_r$, the behavior of self-similar shear-thinning fluids and self-similar Newtonian fluids was similar to that of shear-thickening fluids. The degree of self-similarity affected shear-thinning fluids more significantly than Newtonian and shear-thickening fluids. The approach presented in this paper is applicable to the flow of a power-law fluid exhibiting a fractal structure, for instance, blood flow in the cardiovascular system. As a result, we anticipate that our findings will spur further experimental and theoretical research on the fractal and hydrodynamic aspects of fractal non-Newtonian fluids.