# Investigation of the fluid flow dynamic parameters for Newtonian and non-Newtonian materials: an approach to understanding the fluid flow-like structures within fault zones

- H. Tanaka
^{1}, - Y. Shiomi
^{1, 3}and - K.-F. Ma
^{2}Email author

**Received: **9 May 2017

**Accepted: **25 October 2017

**Published: **2 November 2017

## Abstract

## Keywords

## Introduction

Understanding the flow speed and direction in non-Newtonian fluids is important in many fields, such as for estimating the energy loss for airplanes and characterizing viscous-fluid flow within pipes. Several groups, particularly in the industrial community, have studied ductile deformation structure from fluid dynamics at the boundary layer near plates (Abdelhafez 1985; Hussaini et al. 1987; Ishak et al. 2007). Field investigations of the fault zones associated with ancient earthquakes (e.g., Additional file 1: Fig. S1 from Ramsay and Huber, 1983, and Additional file 1: Fig. S2 from Hobbs and Ord, 2015) exhibit similar ductile deformation features, but these features remain to be explained dynamically because most previous simulations of earthquake dynamics considered Newtonian materials. Recently, Ishak and Bachok (2009) extended fluid dynamics calculations to include non-Newtonian fluids, and similar studies were also performed by Zheng et al. (2008) and Dabrowski (2009). These studies estimated the dynamics of the fluid flow parameters. However, no thorough study has reconstructed the dynamics of fluid flow parameters based on geometric information, which is particularly important for the understanding of the ductile deformation structure within fault zones. These geological features observed in the field could extend from meters to kilometers in scale. This ductile, often asymmetric, deformation structure mainly made by viscous deformation of rock behaved as fluid flow. Because investigations of the fluid flow within and nearby fault zones are primarily performed via field surveys, observing the geometric information is a key to understanding the fluid flow dynamics. Therefore, in this paper, we propose an inverse-calculation technique based on the geometry of the streak lines formed by fluid flow at the boundary layers of Newtonian and non-Newtonian fluids. Via this analysis, we obtain a better understanding of the time, flow rate and shear stress involved in the deformation of the streak lines at the boundary of infinite and semi-infinite plates. Estimation of the flow rate could be important to understand the evolution of the fault zone material, and the time and shear stress, thus, could help to quantify the process to understand the earthquake energy budget, or stress partition in a fault system. The proposed technique should be useful for estimating the fluid dynamics parameters based on the geometrical information of the ductile deformation structure.

To model nonlinear viscous fluids, we use the dimensionless velocity profile at the laminar boundary layer with a semi-infinite plate, as Ishak and Bachok (2009). Next, we calculate the dimensionless profiles of the deformed streak lines for fluid flows over infinite and semi-infinite plates and then estimate the time required for the streak lines to deform. Although the geological feature is more related to the “infinite plate” model, the “semi-infinite” model was used to reflect the further application from the semi-infinite model basis of Ishak and Bachok (2009), and the comparison made in the experiments in the later section, as that the ductile deformation structure could be correctly estimated from integrating the flow rate distribution. For geological indication, the “infinite” model could be comparable to the geological field observation of a long extension of the boundary (e.g., fault zone and fault line) to the ductile deformation structure in scale. The “semi-infinite” model could be comparable to the injection of the gouge from the main fault plain. From the profile of the deformed streak lines, we obtain the shear stress and shear rate at the boundary layer with infinite and semi-infinite plates. Initially, we derive the dimensionless profile of the deformed streak lines at the boundary layer by integrating the dimensionless velocity profile. Next, we compare the deformed dimensionless structure obtained from the calculation with the results of the flow experiments on Newtonian fluids performed by Nakayama (2011). Although Nakayama (2011) represented the motion of the flat plate to the stationary fluid, which is opposite to our model, the relative motion between the plate and fluid in both models is the same. We, thus, considered the experiment of Nakayama (2011) for our comparison.

## Estimation of the dynamic parameters

### Equations and models

To estimate the flow time, shear rate, and shear stress based on the geometry of the deformed streak lines at the laminar boundary layer of viscous fluids, we introduced the equations following Ishak and Bachok (2009).

*τ*, over an infinite area is generally expressed as (Metzner and Reed 1955)

*n*is the power-law index,

*u*as the flow rate in

*x*-component, and

*K*the viscosity. The fluid is Newtonian when \(n = 1\), and when \(n < 1\), the fluid is non-Newtonian for shear thinning. In this study, we consider the cases of Newtonian material of

*n*= 1 and non-Newtonian of

*n*< 1, as fault zone material is more considered as shear softening. For ductile shear zones as counterflow boundaries in pseudoplastic fluids by Talbot (1999), it considered the power-law fluid with different exponents

*n*, which is exponent to the stress rather than to the strain of our Eq. (1), of 1.9–5. It is equivalent to our power-law index

*n*, as the inverse to the exponent

*n*, of ~ 0.2–0.53. Brodsky et al. (2009) studied a ductile texture in a series of asymmetrical intrusion structure suggested that non-Newtonian material can explain better the intrusion vein and the deformation of the ductile shear zone with

*n*= 1.5 and

*n*= 5, where

*n*is also as the power of shear stress proportional to the strain rate, and is inverse to our power-law index

*n*. It is equivalent to our values of 0.2–0.67. Their result is also consistent with the study of Smith (1977), which suggested that non-Newtonian behavior is necessary and plays an important role in the formation of boudins and mullions observed in the field.

To understand the influence of non-Newtorian material in the fluid flow-like structure, in this model, we neglect the elastic, plastic and thermal properties of fluids and the heat generated by viscous friction first. The boundary layer in the fluid is assumed to be laminar, with a sufficiently small Reynolds number.

*x*-component flow,

*u*, and

*y*-component flow,

*v*, with density,

*ρ*, are

*y*-direction,

*η*, is

*Re*is the Reynolds number and

*L*is the characteristic length. The dimensional stream function,

*ψ*, is

*f*(

*η*) is the non-dimensional stream function, and \(\psi\) satisfies

*x*-component flow rate in

*η*, as shown in Fig. 2. In comparable to the recent study of Deswita et al. (2010), our case is for impermeable and stationary plate with boundary conditions such as \(f(0)\) = 0, \(f^{'} (0)\) = 0, and \(f^{'} (\eta )\) ~ 1 as \(\eta \to \infty\). In the present calculation, the velocity

*u*(

*y*) of the fluid at a distance

*y*from the plate is \({\text{U}}_{\infty } f^{'} (\eta )\).

### Deformed streak lines in laminar flow

#### In the boundary layer of an infinite plate

For the infinite plate model, the flow rate is constant along a line parallel to the *x*-axis. However, with time, the laminar boundary layer grows in the direction perpendicular to the plate. As the boundary layer grows, streak lines are deformed within the boundary layer.

*η*, in the

*y*-direction. By combining these equations, we obtain

*t*

_{2}denotes the time that the displacement structure began to be generated, and with

*x*=

*U*

_{ ∞ }

*t*

_{1}, and

*X*=

*U*

_{ ∞ }

*t*

_{2}, we obtained

To calculate the nonlinear profile of deformed streak lines, for each y, we calculate the corresponding values of \(\eta (x,y)\) and \(f^{'} (\eta )\) for *x* = 0 to \(X\) using an increment of \(\Delta X\). Then, the results of the calculations are summed for each \(y\). The results are divided by the total \(f^{'} (\eta )\) value, where \(\eta\) is sufficiently large to be considered far from the plate in terms of the normalized values of \(f^{'} (\eta )\) over each y.

#### In the boundary layer of a semi-infinite plate

*x*-component of the dimensionless flow velocity.

Next, we calculate the streamline. The “vertical shift” is defined as the distance that the streamline shifts perpendicularly away from the semi-infinite plate (Fig. 3b) (Nakayama 2011). This shift also occurs inside the laminar boundary layer. Thus, streamlines can be expressed in terms of the parallel translation or “displacement” within the laminar boundary layer (Fig. 3b). To calculate a dimensionless streamline at a given distance from the plate, the distribution of the dimensionless vertical shift is required (Fig. 3b).

*S*

_{1}=

*S*

_{2}in Fig. 4a (Nakayama 2011). This approach may also be applied to the dimensionless velocity profile of nonlinear fluids (Ishak and Bachok 2009). The distribution of the dimensionless vertical shift in a laminar boundary layer is calculated as follows: for \(f\left( {\eta = \eta_{\infty } } \right) > 0.995\), which is regarded as sufficiently close to the dimensionless flow rate \(f = 1\) at infinite distance, we determine the minimum \(\eta\) at \(f_{\infty }\) for \(f\left( {\eta = \eta_{\infty } } \right) > 0.995\). In this case, the vertical shift \(\eta^{*} (\eta )\) is almost identical to the displacement thickness for \(\eta \ge \eta_{\infty }\).

*S*

_{1}=

*S*

_{2}represents the slashed area of \(\eta = \eta_{\infty }\), \(\eta_{\infty }^{*}\) denotes \(\eta^{*}\) at \(\eta = \eta_{\infty }\), and \(\Delta \eta^{*}\) is the deviation between \(\eta_{\infty }^{*}\) and \(\eta^{*}\) (\(\eta_{1} = \eta_{\infty } - \Delta \eta\)). Thus, Eq. (14) can be rewritten as

Next, we consider how to calculate the dimensionless streamline. In this calculation, we ignore any change in viscosity as a function of *y* (perpendicular to the plate) that is attributable to the nonlinearity of the viscosity and the growth of the boundary layer over time.

*x*-component of the flow rate in the boundary layer, and the growth of the vertical shift is proportional to the flow rate. Let \(\eta^{*} \left( {aX, y} \right)\) represent the displacement distance as of a fluid particle in the direction perpendicular to the plate (i.e., the

*y*-direction), where the dimensionless distance from the plate is \(\eta_{f}\) at distance

*x*=

*aX*. Then, when \(x = aX\), the dimensionless displacement distance \(\eta^{*} (aX, \eta )\) becomes

*y*-axis at \(x = 0\) (original line) and then move because of their own velocity, deforming the original line. The initial conditions are \(x = 0\) at \(t = 0\). At \(t = t_{1}\), the \(x\) coordinate of a given fluid particle is \(X\). Next, the movement of a fluid particle along the streamline passing along \(\eta_{f} (t_{1} )\) at \(x = X\) from \(t = 0\) to \(t = t_{1}\) could be calculated from \(t = 0\; {\text{to}}\; t_{1}\) using an interval of \(\Delta t\). First, we consider position \((x, \eta )\) of a fluid particle and \(f^{'} (\eta )\). The dimensionless distance \(\eta\) is calculated from Eq. (13) for the two models.

Because \(\eta\) is determined, the corresponding \(f^{'} (\eta )\) is also determined. The \(x\) coordinate of the next position is calculated by adding \(f^{'} (\eta )\Delta t\) to the previous \(x\) coordinate. Because a fluid particle moves along the streamline, \(y\), \(\eta\) and \(f^{'} (\eta )\) at the next position can be determined accordingly. The calculation is repeated until \(t = t_{1}\). Thus, \(g(\eta )\), the non-dimensional displacement structure, is calculated from the *x*/*X* ratio obtained from \(t = 0\) to \(t = t_{1}\) and \(\eta_{f}\) is obtained using Eq. (17).

### Calculation of flow time and shear stress

*x*=

*X*

_{ f }. Thus, the time required to generate a deformed streak line is given by

*X*

_{ f }/

*t*

_{ f }. However, Eq. (19) has a singularity at

*n*= 2, where \(t_{f}\) diverges. Similar results were reported by Denier and Dabrowski (2004) and Dabrowski (2009), suggesting infinite number of modal solutions for

*n*= 2. Therefore, we believe that this issue remains unresolved. Based on these considerations, and as stated in the introduction, in this study we considered the case for \(n \le 1\), as we restrict the application of Eq. (19) to fluids with \(n \le 1\).

*y*= 0 could be written as below with the observational displacement

*δ*at

*η*

_{ f }from experiment,

To evaluate the particle relationship between \(y\) and \(\eta\), which is necessary to estimate the time and stress during flow, in this study, we use “80% thickness” instead of “99% thickness,” which is the general definition for the thickness of a boundary layer. This approach reduced both the measurement errors and the computation time. The relevant thickness is the vertical distance between the plate and the *y*-coordinate, where *x* encompasses 80% of the displacement of the fluid particles at infinite distance \(X_{f}\). The quantities \(\eta_{\text{sn}}\) for a semi-infinite plate (or \(\eta_{\text{in}}\) for an infinite plate) and \(\delta\) can be obtained at 80% thickness from the dimensionless displacement-distribution graphs and the experimentally observed displacement structure. Thus, all the parameters in Eq. (19) can be obtained from either experimental results or models, allowing us to calculate the time to form deformed streak lines and the flow stress. \(f^{{\prime \prime }} (\eta )\) at *y* = 0 can be obtained from the calculation of \(f^{'} (\eta )\).

*X*

_{ f }/

*t*is the flow rate.

*n*, the graph is convex. For the results presented in Fig. 5a and b, the displacement structures for

*n*= 1.0 have different curvatures than for other values of

*n*. The velocity profile obtained by Ishak and Bachok (2009) shows similar characteristics, and Dabrowski (2009) reported similar results. However, the models of Zheng et al. (2008) do not exhibit these characteristics. The corresponding plots of \(f^{'} (\eta )\) for various

*n*of the infinite plate model are given in Fig. 5c and is further applied for the \(f^{{\prime \prime }} (\eta )\) used in Eq. (21) to calculate the shear stress.

## Comparison to viscous-fluid experiments

The dynamic parameters of viscous fluid derived above are compared with the experimentally derived values. We consider two examples: The first example is water with *n* = 1, for which the experimental results were reported by Nakayama (2011). The second example is an edible sauce with *n* < 1, which is utilized to illustrate a pseudoplastic fluid. Because no existing examples reflect the deformed structure of the edible sauce, we use the density provided by the manufacturer and other parameters obtained by Miura et al. (2008) for this experiment.

### Newtonian fluid (*n* = 1): water

*n*= 1.0) with a kinematic viscosity coefficient of \(K/\rho = 1.002 \times 10^{ - 6} \;{\text{m}}^{2} / {\text{s}}\) and density of \(\rho = 0.998\;{\text{g}}/{\text{cm}}^{3}\) at 20 °C. We compare the result of the semi-infinite plate model with the experimental results reported by Nakayama (2011), in which the laminar boundary layer was visualized as shown in Fig. 6.

*y*= 0 for

*n*= 1.0 in Fig. 5c, we can obtained the shear stress \(\tau\) from Eq. (22), which would have the values from approximately 1 to 0.1 Pa for a fluid particle moving from 1 mm to 1 cm, represented as an inverse linear relationship shown in Fig. 7a.

### Non-Newtonian fluid (*n* < 1): edible sauce (*n* = 0.262)

Taking the edible sauce with *n* < 1 as an example of a nonlinear fluid, the fluid parameters are \(n = 0.262\) and \(K = 142\;{\text{kg/}}\left( {{\text{m s}}^{1.738} } \right)\) at 20 °C (Miura et al. 2008) and \(\rho = 1.19 \times 10^{3} \;{\text{kg}}/{\text{m}}^{3}\) (provided by the manufacturer). Considering the case for infinite plate with \(\eta_{\text{in}}\) = 1.43 and *n* = 0.262, according to Eq. (21), we could obtain the shear stress as function of time. Considering speed of fluid flow, we can also present the shear stress on the plate as a function of \(X\) as shown in Fig. 7b for various flow rates of 0.2, 0.2, and 1 m/sec. Here, the plot was made using \(f^{{\prime \prime }} (0) = 0.41\) for \(n = 0.262\) according to Fig. 5c.

Through the distribution of the distance to the plate boundary for various flow rates, by the possible estimated flow rate examined in the field, it might give some indications on the shear stress distribution along the boundary of an identified fault for non-Newtonian material. In comparison with Fig. 7a of Newtonian material, the yielding shear stress from Non-Newtonian material is several order higher than the value of Newtonian material in our case of *n* = 0.262. For a certain distance, one-order difference in flow rate yields almost three-order difference in shear stress. And, the shear stresses drop significantly with distance, suggesting very large shear stress close to the plate boundary for non-Newtonian material. The estimation of the shear stress along the boundary using Newtonian material in simulation might give lower estimation of the shear stress.

Comparing the calculated and experimental results, it reveals some technical errors in the estimation of the flow rate and time. One error might come from the difficulty in measuring the distance. Using the hydrogen bubble method, accurately measuring the distance from the tip of the plate in the direction parallel to the plate is difficult because bubbles crowd around the tip of the plate. Another possible error is the finite thickness of the plate because that the streamlines vary as a function of the plate thickness. Fortunately, these errors are not significant enough to compromise the validity of the calculation results.

For a flow with an extremely small Reynolds number, the viscosity of the fluid affects the dimensionless parameters, which renders the meaningless of Reynolds number (Leal 1992). Therefore, these models we derived here needed to be handled with care for the flows with extremely small Reynolds numbers (e.g., *n* < 0.1). Fortunately, most of the ductile deformation features were modeled considering with our equivalent power-law index *n* greater than 0.1 (e.g., Talbot 1999; Brodsky et al. 2009).

## Summary

We derived a series of functions to estimate the time required for the generation of certain dynamic parameters (i.e., the flow rate, shear stress, and deformed structure) based on the geometric information of the boundary layer of the flow. The applications to two models: infinite plate and semi-infinite plate, were represented. For the semi-infinite plate model, a final displacement structure was obtained by computing the flow velocity at every spatial coordinate to obtain the resultant streamline, which is parallel to the plate in an infinite plate model. However, the dynamics of the laminar boundary layer in the direction perpendicular to the plate must be considered when it approaches the boundary. Therefore, the equations, thus, derived in this paper could be applied to estimate the flow velocity, stress and time required for the deformation of streamlines. And, the results could be further compared to the experimental observations. The calculated time required for the streamlines to deform and the flow velocity are consistent with the corresponding experimental results, suggesting the reliability of our derived equations and methodology applied in this paper. Because few experimental examples of the deformation structure in a nonlinear (*n* < 1.0) fluid exist, we used the parameters for edible sauce in our experiment. Although the presented results of our calculations are reasonable, we still suggest that the validity of the model might need to be justified with various fluids. This topic will be the subject of future experimental investigations. We hope it is possible to apply this technique to the observations in the field in the future although a comparable geological feature to the setting of this derived technique might remain some modifications. Once the ductile deformation structure could provide an indication of the relative displacement (e.g., vein across the deformation structure as a marker for displacement), it would be possible to estimate the time required for the formation of the ductile deformation structure and the equivalent ductile fluid flow speed as *U*
_{
∞
}, and, thus, the stress distribution along the displacement. The flow direction as indication above could be considered as a flow parallel to the strongest deformed area. Our model setting is for an asymmetric shear deformation. For a larger scale, the crustal deformation observed by GPS data (Additional file 1: Fig. S3, http://mekira.gsi.go.jp/project/f3_10_5/ja/ondex.html) in Japan islands shows a significant asymmetric shear deformation feature, especially the motion across the right-lateral motion of Median Tectonic Line (MTL) in Shikoku island. It shows that not all of the right-lateral motions were resolved by MTL. Some shear motion remained to the north of MTL (northern regime of Shikoku island), which might be involved with possible viscous motion. The application of our derived technique might give some hints to the understanding of the large-scale asymmetric ductile deformation feature along MTL although more details might need to be addressed as compared to the GPS data through modeling. We hope the derived results in this study could facilitate better understanding on the fault zone dynamics, especially for viscous ductile deformation feature, which is often observed within and nearby fault zones in the field.

## Declarations

### Authors’ contributions

HT carried out the idea and basic theory in the modeling, and paper draft. YS developed the calculation, and paper draft. KM carried out the logical structure of paper, paper writing and submission. All authors read and approved the final manuscript.

### Acknowledgements

We sincerely thank Dr. Yasuki Nakayama of the Future Technology Research Institute and Professor Katsumi Aoyama of Tokai University for their comments regarding the details of the fluid-visualization experiment and for allowing us to include a photograph of their experiment in this paper. We also thank Dr. Takehiko Hiraga, Dr. Yasuko Takei and Dr. Atsuko Namiki for their helpful discussions regarding the formulation of fluid flow. Dr. Kuniyo Kawabata kindly helped with the equations. The help provided by Dr. Norio Shigematsu of the Advanced Industrial Science and Technology is also appreciated. Finally, we thank OTAFUKU SAUCE Co., Ltd., for providing the data regarding the edible sauce.

### Competing interests

The authors declare that they have no competing interests.

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## Authors’ Affiliations

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