### 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).

For viscous materials, the relationship between the shear rate and the shear stress, *τ*, over an infinite area is generally expressed as (Metzner and Reed 1955)

$$\tau = K\left( {\frac{\partial u}{\partial y}} \right)^{n} ,$$

(1)

where *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.

We consider the infinite plate model, where the fluid is initially stationary on an infinite plate. At a given point in time, fluid begins to move in a direction parallel to the plate surface at the fixed velocity \(U_{\infty }\) (Fig. 1a) at far distance. As the fluid travels along the surface of the plate, the laminar boundary layer gradually becomes thicker. The fluid flow rate changed accordingly when it approaches the boundary. For the semi-infinite plate model, fluid flows at a fixed rate \(U_{\infty }\) in the direction parallel to a stationary semi-infinite plate (Fig. 1b) at far distance. A boundary layer is generated when the fluid contacts the edge of the plate while the rate of fluid flow changed accordingly. The disturbance of the flow that occurs at the edge of a semi-infinite plate is ignored, and the laminar boundary layer is assumed to be stable. Geologically, the moving fluid might not be homogenous over the space, and our model here represents the first order approximation to the geological structure.

Ishak and Bachok (2009) used Eq. (1) and the following equations to describe the dimensionless velocity profile of flow in a laminar boundary layer at a semi-infinite plate. The equations describing the boundary layer for *x*-component flow, *u*, and *y*-component flow, *v*, with density, *ρ*, are

$$\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$$

(2)

$$u\frac{\partial u}{\partial x} + v\frac{\partial u}{\partial y} = \frac{1}{\rho }\frac{\partial \tau }{\partial y}.$$

(3)

The dimensionless distance in the *y*-direction, *η*, is

$$\eta = \left( {\frac{Re}{x/L}} \right)^{{\left( {\frac{1}{n + 1}} \right)}} \frac{y}{L},$$

(4)

where *Re* is the Reynolds number and *L* is the characteristic length. The dimensional stream function, *ψ*, is

$$\psi = LU_{\infty } \left( {\frac{x/L}{Re}} \right)^{{\left( {\frac{1}{n + 1}} \right)}} f(\eta ),$$

(5)

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

$$u = \frac{\partial \psi }{\partial y} , \quad v = - \frac{\partial \psi }{\partial x}.$$

(6)

The Reynolds number is given by

$$Re = \frac{{\rho U_{\infty }^{2 - n} L^{ n} }}{K}.$$

(7)

From these equations, Ishak and Bachok (2009) obtained

$$\left( {\left| {f^{{\prime \prime }} (\eta )} \right|^{n - 1} f^{{\prime \prime }} (\eta )} \right)^{'} + \frac{1}{n + 1}f\left( \eta \right)f^{{\prime \prime }} (\eta ) = 0.$$

(8)

We solved Eq. (8) by using the Runge–Kutta–Fehlberg method and the shooting technique, to obtain the velocity profile, \(f^{'} (\eta )\) as the *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.

Ishak and Bachok (2009) use Eqs. (4) and (5) to determine a dimensionless distance, *η*, in the *y*-direction. By combining these equations, we obtain

$$\eta = \left[ {\frac{{\rho U_{\infty }^{{\left( {2 - n} \right)}} }}{xK}} \right]^{{\left( {\frac{1}{n + 1}} \right)}} y.$$

(9)

We substitute \(x = U_{\infty } t\) into Eq. (9) to obtain

$$\eta = \left[ {\frac{{\rho U_{\infty }^{{\left( {1 - n} \right)}} }}{tK}} \right]^{{\left( {\frac{1}{n + 1}} \right)}} y,$$

(10)

which leads to

$$\frac{y}{ \eta } \propto t^{{\left( {\frac{1}{n + 1}} \right)}} .$$

(11)

Equation (11) indicates that when the dimensionless distance perpendicular to the plate is \(\eta_{{t_{1} }}\) at \(t = t_{1}\), at \(t_{2} = at_{1}\), we have

$$\eta \left( {t_{2} , y} \right) = \left( {\frac{{t_{2} }}{{t_{1} }}} \right)^{{ - \left( {\frac{1}{n + 1}} \right)}} \eta \left( {t_{1} , y} \right),$$

(12)

where *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

$$\eta \left( {x, y} \right)|_{{t = t_{1} }} = \left( {\frac{x}{X}} \right)^{{ - \left( {\frac{1}{n + 1}} \right)}} \eta_{f} ,$$

(13)

where \(\eta_{f} = \eta \left( {X, y} \right)|_{{t = t_{2} }} .\) By applying \(\eta\) to the dimensionless velocity profile shown in Fig. 2, as given by Ishak and Bachok (2009), we obtain the dimensionless flow rate \(f^{'} (\eta )\) for the infinite plate model.

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

A streak line is defined as a line drawn by connecting the points through which fluid particles flow at time t. Because of viscosity, the fluid velocity is lower in the boundary layer than in regions closer to the plate. This streak line becomes more deformed with time. The displacement of each fluid particle that is a certain distance from the plate at each moment is affected as soon as the particle passes over the tip of the plate or as soon as the fluid starts moving (Fig. 3a). In this case, to examine the spatial elements, we ignore the dynamics of the laminar boundary layer. We use Eq. (13) to determine \(\eta\) at point \((x,y)\) in the fluid. By using \(\eta\) in the distribution of the dimensionless velocity profile, \(f(\eta )\), we obtain the *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).

The vertical shift is determined as a function of \(y\) in areas *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 \(\eta \ge \eta_{\infty }\).

Next, we estimate the vertical shift of the boundary layer. Considering the fluid as an aggregation of fluid particles, the vertical shift for each fluid particle is determined by the flow rate in the domain that is close to the plate, and the nonlinear vertical shift \(\eta^{*} (\eta )\) from the plate is defined based on the points in areas \({\text{A}} = {\text{B}}\) in Fig. 4b. The distribution of the vertical shift is then calculated using the recurrence technique. We consider the case \(\eta = \eta_{\infty } - \Delta \eta\), which is infinitesimally close to the plate (separated by distance \(\Delta y\)). The displacement distance of a fluid particle is determined exclusively by the flow rate of the fluid particle at \(\Delta y\). Thus, for \(\eta_{1} = \eta_{\infty } - \Delta \eta\), to balance the two areas B and A in Fig. 4b, the approximate equation becomes

$${\text{B}} = S_{1} - f^{'} \left( {\eta_{\infty }^{*} } \right)\Delta \eta^{*} + \left\{ {f^{'} \left( {\eta_{\infty }^{*} } \right) - f^{'} \left( {\eta_{\infty }^{*} - \Delta \eta^{*} } \right)} \right\}\frac{{\Delta \eta^{*} }}{2}\quad \sim\;S_{2} - \left\{ {f^{'} \left( {\eta_{\infty }^{*} } \right) - f^{'} \left( {\eta_{\infty }^{*} - \Delta \eta^{*} } \right)} \right\}\frac{{2\eta - \Delta \eta - 2\eta_{\infty }^{*} }}{2}\quad + \left\{ {2f^{'} \left( {\eta_{\infty } - \Delta \eta } \right) - f^{'} \left( {\eta_{\infty }^{*} } \right) - f^{'} \left( {\eta_{\infty }^{*} - \Delta \eta^{*} } \right)} \right\}\frac{{\Delta \eta^{*} }}{2} = A ,$$

(14)

where *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

$$\left. {\eta^{*} } \right|_{\eta - \Delta \eta } \sim\frac{{f^{\prime}\left( \eta \right) - f^{\prime}\left( {\eta - \Delta \eta } \right)}}{{f^{\prime}\left( {\eta - \Delta \eta } \right)}} \frac{{2\eta - \Delta \eta - 2\eta_{{}}^{*} }}{2} ,$$

(15)

for a generalized \(\eta .\) Considering the linear relationship between \(f^{\prime}(\eta )\) and \(\eta\), we use the approximation \(\eta^{*} (\eta ) = \eta /2\) near the plate.

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.

Therefore, the displacement is determined only by the *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

$$\eta^{*} \left( {aX, y} \right) = a^{{\frac{1}{n + 1}}} \eta^{*} \left( {X, y} \right).$$

(16)

Equation (16) describes the shape of the dimensionless streamlines. Using the dimensionless vertical shift \(\eta^{*} \left( {X, y} \right)\), we find

$$\eta^{*} \left( {x, \eta } \right) = \left( {1 - \frac{x}{X}} \right)^{{\left( {\frac{1}{n + 1}} \right)}} \eta^{*} \left( {X, \eta } \right).$$

(17)

As described above, the fluid flow is treated as a migration of fluid particles along dimensionless streamlines in the dimensionless profile of the flow rate. Thus, the distribution g \((\eta )\) of the dimensionless displacement can also be determined. At \(t = 0\), fluid particles are arranged parallel to the *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

To calculate both the flow time and shear stress, we first use the displacement \(x = X_{f}\) at \(t = t_{f}\) and calculate the time required to generate a dimensionless deformed streak line. In the next section, this time is compared with the experimentally determined time to validate the model. Because \(X_{f} = U_{\infty } t_{f}\), Eq. (9) becomes

$$\eta_{f} = \left[ {\frac{{\rho X_{f}^{{\left( {1 - n} \right)}} }}{{t_{f}^{2 - n} K}}} \right]^{{\left( {\frac{1}{n + 1}} \right)}} y,$$

(18)

for *x* = *X*
_{
f
}. Thus, the time required to generate a deformed streak line is given by

$$t_{f} = \left[ {\left( {\frac{y}{{\eta_{f} }}} \right)^{{\left( {n + 1} \right)}} \frac{\rho }{K}X_{f}^{{\left( {1 - n} \right)}} } \right]^{{\left( {\frac{1}{2 - n}} \right)}} ,$$

(19)

for both the infinite and semi-infinite models. Equation (19) shows that once the physical properties \((n, K,\rho )\), \(\left( {\frac{y}{{\eta_{f} }}} \right)\) and \(X_{f}\) of the fluid are known, the deformation time \(t_{f}\) can be obtained, and thus, \(U_{\infty }\) can be determined as \(U_{\infty }\) = *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 \(n \le 1\).

Next, we describe the calculation of the shear stress based on the profile of the deformed streak lines observed in the boundary layer in the fluid. In a boundary layer, the shear rate of fluid \(f^{{\prime \prime }} (\eta )\) at *y* = 0 could be written as below with the observational displacement *δ* at *η*
_{
f
} from experiment,

$$\frac{\partial u}{\partial y}|_{y = 0} = U_{\infty } \frac{{\partial f^{'} (\eta )}}{\partial \eta }\frac{\partial \eta }{\partial y}|_{y = 0} = U_{\infty } f^{\prime \prime } (\eta )\frac{{\eta_{f} }}{\delta } .$$

(20)

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 \(f^{'} (\eta )\).

Thus, using \(f^{{\prime \prime }} (\eta )\; {\text{at}} \;y = 0\), which is estimated by the shooting technique, and Eqs. (1) and (20), we determine the total stress for the infinite plate model \(\left( {\eta_{f} = \eta_{\text{in}} } \right)\). The result is

$$\begin{aligned} \tau_{\text{total}} (t) & = K\left[ {U_{\infty } f^{''} (\eta )\frac{{\eta_{\text{in}} }}{\delta }} \right]^{n} = K\left[ {U_{\infty } f^{''} (\eta )\left[ {\frac{{\rho X_{f}^{(1 - n)} K^{{\frac{1}{n}}} }}{{t^{(2 - n)} }}} \right]^{{\left( {\frac{1}{1 + n}} \right)}} } \right]^{n} \\ & = \left[ {f^{''} (\eta )\left[ {\frac{{\rho X_{f}^{2} K^{{\frac{1}{n}}} }}{{t^{3} }}} \right]^{{\frac{1}{1 + n}}} } \right]^{n} = \left[ {f^{''} (\eta )\left[ {\frac{{\rho U^{3} K^{{\frac{1}{n}}} }}{{X_{f} }}} \right]^{{\frac{1}{1 + n}}} } \right]^{n} , \\ \end{aligned}$$

(21)

where \(U_{\infty }\) = *X*
_{
f
}/*t* is the flow rate.

The result for the semi-infinite plate model \(\left( {y_{f} = y_{\text{sn}} } \right)\) is

$$\tau_{\text{total}} (x) = K\left[ {U_{\infty } f^{{\prime \prime }} (\eta )\frac{{\eta_{\text{sn}} }}{\delta }} \right]^{n} = K\left[ {U_{\infty } f^{{\prime \prime }} (\eta )\left\{ {\frac{{\rho U_{\infty }^{{\left( {1 - n} \right)}} }}{tK}} \right\}^{{\left( {\frac{1}{1 + n}} \right)}} } \right]^{n} \quad = \left[ {f^{{\prime \prime }} (\eta )\left\{ {\frac{{\rho U_{\infty } K^{{\frac{1}{n}}} }}{x}} \right\}^{{\frac{1}{1 + n}}} } \right]^{n} .$$

(22)

The results of the calculations of the dimensionless displacement structure for the infinite and semi-infinite plate models using the equations derived above are shown in Fig. 5a and b, respectively. For any *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.