S0659 ANNALS OF GEOPHYSICS, 60, 6, S0659, 2017; doi: 10.4401/ag-7295 Slip of a Two-degree-of-freedom Spring-slider Model in the Presence of Slip-dependent Friction and Viscosity Jeen-Hwa Wang1 1 Institute of Earth Sciences, Academia Sinica, Nangang, Taipei, Taiwan Article history Received October 21, 2016; accepted February 26, 2017. Subject classification: Two-degree-of-freedom spring-slider model, Slip-dependent friction, Viscosity, Stiffness ratio, Phase portrait. ABSTRACT In this study, we study the slip of a two-degree-of-freedom spring-slid- er model, consisting of two sliders, in the presence of slip-dependent friction due to thermal pressurization and viscosity. Simulation results show that seismic coupling between the two sliders is weak when the stiffness ratio, s, of the model is smaller than 5 and/or either Uc1 or Uc2, which are the characteristic displacements of friction law at the two sliders, is smaller than 0.5. The patterns of motions of two sliders yielded by large Uc1 and small Uc2 are opposite to those by small Uc1 and large Uc2. The ratio, φ, of static friction force at slider 2 to that at slider 1 is a factor in influencing the motions of two sliders. Higher φ leads to a longer delay time to trigger the motion of slider 2. Slider 2 cannot move when φ is higher than a critical value which depends on other model parameters. The presence of viscosity between the sliders and moving plate results in increases in duration times and predominant periods of motions of sliders and depresses the generation of an attractor. Viscos- ity results in small amplitudes and low velocities of motions of sliders. 1. Introduction The rupture processes of an earthquake essen- tially consist of three steps: nucleation (or initiation), dynamical propagation, and arrest. It is necessary to study the mechanisms controlling the whole rupture processes. Such processes are very complicated and can be controlled by several factors. The major factors include brittle-ductile fracture rheology [Jeffreys, 1942; Scholz, 1990], pore fluid pressure [Scholz, 1990], and thermal pressurization [Rice, 2006]. Friction is one of the most important factors in controlling earthquake dynamics [Nur, 1978; Dieter- ich, 1979; Ruina, 1983; Knopoff et al., 1992; Rice, 1993; Wang, 1996, 1997, 2000, 2012; Rubin and Ampuero, 2005; Ampuero and Rubin, 2008; Bhattacharya and Rubin, 2014]. The friction coefficient, μ (=0.6–0.8 in general), is defined as the ratio of shear stress,τ, to the effective normal stress, σeff , on the fault plane [Byerlee, 1978]. The frictional force between two contact planes is classically considered to drop from static one to dy- namic one after the two planes move relatively. Indeed, the friction law that has been inferred from laboratory experiments is quite complicated and not completely understood, especially for that on natural faults due to limited observational constraints. This makes the prop- er constitutive law for fault friction an elusive mathe- matical formulation. The commonly-used friction law is the rate- and state-dependent friction law with the state evolution laws [Dieterich, 1972, 1979; Ruina, 1983; Shimamoto 1986]. A detailed description of the law can be found in several articles [e.g., Marone. 1998; Wang, 2002; Bizzarri and Cocco, 2006c; and Bizzari, 2011c]. Several simple friction laws have been taken into ac- count by some researchers, for examples, a velocity-de- pendent, weakening-hardening friction law [Burridge and Knopoff, 1967]; a purely nonlinearly velocity-weak- ening friction law [Carlson and Langer, 1989]; a piece- wise, linearly velocity-dependent weakening-hardening friction law [Wang, 1995, 1996, 2012]; a nonlinearly ve- locity-weakening friction law [Noda et al., 2009]; a dis- placement softening-hardening friction law [Cao and Aki, 1984/85]; and a piecewise, linearly slip-dependent friction law [Ionescu and Campillo, 1999; and Urata et al., 2008]. Cochard and Madariaga [1994] and Madar- iaga and Cochard [1994] assumed that purely veloci- ty-dependent friction models can lead to unphysical phenomena or mathematically ill-posed problems. This means that the purely velocity-dependent friction WANG 2 law is very unstable at low velocities both during the passage of the rupture front and during the possible slip arrest phase. Ohnaka [2003] stressed that purely velocity-dependent friction is in contrast with labo- ratory evidence. On the other hand, Lu et al. [2010] assumed that velocity-weakening friction plays an im- portant role on earthquake ruptures. Bizzarri [2011c] deeply discussed this problem. From theoretical studies, Bizzarri and Cocco [2006a,b,c] revealed that melting of rocks and fault gouge is likely to occur even with the inclusion of the thermal pressurization of pore fluids. Moreover, the dramatic fault weakening at high slip rates predicted by the flash heating of micro-asperity contacts is not able to avert melting [Bizzarri, 2009]. When fluids are present in faults, thermal pressurization can play a sig- nificant role on earthquake rupture and also result in resistance on the fault plane [Sibson, 1973; Fialko, 2004; Bizzari and Cocco, 2006a,b,c; Rice, 2006; Wang, 2009, 2011, 2013, 2016a; Bizzarri, 2010; Bizzarri, 2011a,b]. In this study, a slip-weakening friction law induced by thermal pressurization will be taken into account. Among numerous friction laws, the only constitu- tive law able to avoid the melting is a slip- and veloci- ty-weakening friction law [Sone and Shimamoto, 2009; Bizzarri, 2010], for which the fault weakening is so dra- matic that it cannot be counterbalanced by the resulting enhanced slip velocities. However, both thermal pres- surization of pore fluids and flash heating predict not only a very dramatic stress drop, but also a very high peak in fault slip velocity, so that the final result is that melting temperature is very often exceeded, unless the slipping zone is extremely large [Bizzarri and Cocco, 2006b; Bizzarri, 2009]. Bizzarri [2011a] stressed that when melting occurs, the rheological behavior of the fault zone no longer obeys the Coulomb–Amonton– Mohr formulation, in that a viscous rheology is needed to describe the traction evolution during the ruptures. Jeffreys [1942] first emphasized the importance of viscosity on faulting. Viscosity in a fault can be controlled by frictional melts [Byerlee, 1968]. Tem- perature, pressure, water content, etc., will influence viscosity [Turcotte and Schubert, 1982]. Scholz [1990] suggested that the residual strength of fault-generated friction melts would be high and so present significant viscous resistance to shear. This inhibits continued slip. Spray [1993; 1995] observed that most pseudo- tachylytes [Sibson, 1975] are partial melts possessing low viscosity, and capable of generating a sufficient melt volume to reduce the effective normal stress. Thus, friction melts can act as fault lubricants during co-seismic slip and viscosity decreases with increasing temperature [Spray, 2005]. Rice et al. [2001] discussed the physical basis of rate- and state-dependent friction, including the direct effect in thermally activated pro- cesses allowing creep slippage at asperity contacts on the fault surface. Wang [2007] stressed the importance of viscosity on earthquake ruptures. Wang [2011] as- sumed that quartz plasticity could be formed in the main slip zone of the earthquake when T>300 oC after the fault ruptured. The shear zone with quartz plas- ticity would be localized in a very thin heated layer, for example, 5-mm thick layer. Quartz plasticity could lubricate the fault plane at higher T and yield viscous stresses to resist slip at lower T. Some researchers have already investigated the effect of viscosity in spring- block system [e.g., Shaw, 1994; Yoshino, 1998; Hainzl et al., 1999; Wang, 2016a]. On the other hand, several researchers [Knopoff et al., 1973; Cohen, 1979; Xu and Knopoff, 1994; Knopoff and Ni, 2001] took the viscous effect as a factor in causing seismic radiation to reduce energy during earthquake ruptures. Since the ingredients of an ideal model are only partly understood, a set of equations to describe com- prehensively fault dynamics has not yet been estab- lished. Nevertheless, some models, for instance the crack model and dynamical spring-slider model, have been developed to approach fault dynamics for a long time. Although the frictional effect on earthquake rup- tures has been widely studied as mentioned above, the studies of viscous effect on earthquake ruptures are rare. The viscous effect mentioned in Rice et al. [2001] was an implicit factor which is included within the direct effect of rate- and state-dependent friction law. Wang [2016a] studied frictional and viscous effects on slip of a one-degree-of-freedom spring-slider model associated to a fault. However, the interaction between two sliders related to two faults or two fault segments. In this work, I will explore the effects of slip-weaken- ing friction due to thermal pressurization and viscosi- ty on earthquake ruptures based on a two-degree-of- freedom spring-slider model, which is generally used to approach an earthquake fault, because numerous earthquakes consist of few segments [Galvanetto, 2002; Turcotte, 1992; Dragoni and Santini, 2010, 2011, 2012, 2014; and cited references herein]. For example, Wang [2007] applied this model to study the difference in ground motions between the northern and south- ern segments of the Chelungpu fault, along which the 1999 Chi-Chi, Taiwan, Ms 7.6 earthquake ruptured. In this study, the frictional factors include the charac- teristic displacement of the friction law and the differ- SLIP OF A TWO-BODY MODEL WITH FRICTION AND VISCOSITY 3 ence on static friction forces between two sliders. The viscous effect is represented by an explicit parameter. Included also is the seismic coupling which will be defined below. The effects on the possibility of simul- taneous motions of two sliders, which will lead to a larger-sized event, due to seismic coupling and friction will be studied. The frictional and viscous effects on interaction between two sliders, including the trigger- ing of the second slider by the first one, will be stud- ied. Results will be significant on the understanding of earthquake ruptures. 2. Two-degree-of-freedom Spring-slider Model 2.1 Model The model consists of two sliders of mass mi (i=1, 2) and three spring. One coil spring of strength K links two sliders and each slider is also pulled by a leaf spring of strength Li (i=1, 2) from a moving plate with a constant velocity vP (Figure 2). The moving plate provides the driving force on the sliders. At time t=0, all sliders rest in an equilibrium state. The i-th slider is located at position ui, measured with respect to its in- itial equilibrium position, along the x-axis. Each slider is subjected to a slip- dependent frictional force, Fi(ui) (i=1, 2), where ui is the displacement of the i-th slider, and a velocity-dependent viscous force, Φ(vi), where vi=dui/dt is the velocity of the i-th slider. The equation of motion of the system is: (1a) m1(d 2u1/dt 2)=K(u2-u1)-L1(u1-vpt)-F1(u1)-Φ(v1) (1a) m2(d 2u2/dt 2)=K(u1-u2)-L2(u2-vpt)-F2(u2)-Φ(v2) (1b) It is noted that the total forces in Equation (1) are null when the sliders do not slide, that is, vi=0. For sim- plification, the inertial effect is considered to be equal for the two sliders, i.e., m1=m2=m. Considering the two sliders to be two segments of a single earthquake fault, the coupling between the moving plate and each fault segment should be equal, thus giving L1=L2=L. Equation (1) consists of two processes: The first one is the coupling process between the moving plate and a slider through the leaf spring L. The other one is the generation of “self-stress” [Andrews, 1978], which originates from the joint effect of the coil spring K be- tween two sliders and the leaf spring L. The coil spring K plays a role only in transferring energy from one slid- er to the other; thus, it does not change the total ener- gy of the system. However, the leaf spring L plays two roles: One is the supply of energy to the system from the driving force caused by the moving plate, i.e., the Lvpt term in Equation (1), and the other is the loss of energy from the system. Hence, the leaf spring L can change the total energy in the system. Therefore, the stiffness ratio s=K/L is a significant parameter repre- senting the level of conservation of energy in the sys- tem [Wang, 1995]. In this study, s is taken to represent seismic coupling. Larger s shows that the coupling be- tween two sliders is stronger than that between a slider and the moving plate. This results in a smaller loss of energy through the L spring, thus indicating a high- er level of conservation of energy in the system. Of course, smaller s indicates a lower level of conserva- tion of energy. When L is constant, small K (less than a critical value) can produce an unstable rupture. When K<5. Hence, the two sliders move almost simultaneously when s≥5, for which the system is self-organized. Figure 5 shows the results when s=1. In each plot the solid line is differ- ent from the dotted line, thus showing weak coupling between the two sliders. This phenomenon also exists when s<5. The peak values of A/Amax and V/Vmax of slider 1 come earlier than those of slider 2, while the peak value of U/Umax of the former comes later than that of the latter. The amplitudes of A/Amax V/Vmax, and U/Umax on slider 2 are all larger than those on slid- er 1. Results show the directivity of motions of sliders. Figures 4 and 5 suggest that strong coupling (with large s) between two fault segments is more capable of gen- erating a larger-sized event than week coupling (with small s). Hence, for an earthquake fault consisting of a few segments it is easier to generate a larger-sized event from larger s than from smaller s. Of course, the value of s of generating a larger-sized event increases with the number of sliders [Wang, 1995]. From numerical simu- lations, Wang [1995] obtained a power-law correlation between the b-value of the Gutenberg-Richter frequen- cy-magnitude law and s: b~s-2/3 for the cumulative fre- quency and b~s-1/2 for the discrete frequency. Larger b for smaller s is related to a bigger number of smaller events and smaller b for larger s to a bigger number of larger events. The present result is consistent with his. Figures 5a and 5b reveal that the predominant periods of the two sliders are almost the same, because their values of model parameters are equal. In Figures 4d and 5d, the absolute values of slope at the fixed points are likely both smaller than 1 and thus they can be an attractor. 4.2 Effect due to Uc The effect due to Uc can be seen from Figures 6−10. We examine the upper-bound values of Uc1 and Uc2 for allowing weak coupling between two sliders. In order to see the effect, s=1 is taken into account. Figure 6 shows the simulation results for Uc1=0.5 and Uc2=0.5 when the values of other model parameters are the same as those in Figure 5. In each plot the solid line is almost coincided with the dotted line, thus ex- hibiting strong coupling between the two sliders. Nu- merical tests also show the same phenomenon when Uc1>0.5 and Uc2>0.5. Hence, the two sliders move al- most simultaneously when Uc1≥0.5 and Uc2≥0.5 at the two sliders. In Figure 6d, the fixed point for slider 1 is just the original point and thus it cannot be an attrac- tor; and that for slider 2 is smaller than 1 and thus it can be an attractor. Figure 7 shows the simulation results for Uc1=0.1 and Uc2=0.5 when the values of other model parame- ters are the same as those in Figure 6. In each plot the solid line is different from the dotted line. The peak values of A/Amax and V/Vmax of slider 1 come earlier than those of slider 2, while the peak value of U/Umax of the former comes later than that of the latter. The peak values of A/Amax, V/Vmax, and U/Umax of slider 1 are all lower than those of slider 2. The values of Uc1=0.1 and Uc2=0.5 are equivalent to γ1=10 and γ1=2, respectively. Hence, there is a larger force drop (or stress drop) on slider 1 than on slider 2, thus causing a large acceleration on slider 1 to push it to move. Then, the motion of slider 1 enforces slider 2 to move. Like Figure 5, the directivity effect exists in the present case. In Figure 7d, the absolute values of slope at the fixed points are likely smaller than 1 for slider 1 and larger than 1 for slider 2. Thus, the fixed point for the former can be an attractor, yet not for the latter. Figure 8 shows the simulation results for Uc1=0.5 and Uc2=0.1 when the values of other model param- eters are the same as those in Figures 6 and 7. Obvi- ously, in each plot the solid line is different from the Number of Figure s η1 η2 φ Uc1 Uc2 4 5 0 0 1 0.1 0.1 5 1 0 0 1 0.1 0.1 6 1 0 0 1 0.5 0.5 7 1 0 0 1 0.1 0.5 8 1 0 0 1 0.5 0.1 9 1 0 0 1.05 0.1 0.5 10 1 0 0 1.18 0.1 0.5 11 1 10 10 1 0.1 0.5 12 1 10 2 1 0.1 0.5 13 1 10 20 1 0.1 0.5 Table 1. Values of model parameters used in this study. WANG 8 dotted line. Figure 8 is totally opposite to Figure 7. The peak values of A/Amax and V/Vmax of slider 1 come later than those of slider 2, while the peak value of U/Umax of the former comes earlier than that of the latter. The peak values of A/Amax, V/Vmax, and U/Umax of slider 1 are all higher than those of slider 2. The values of Uc1=0.5 and Uc2=0.1 are equivalent to γ1=2 and γ2=2, respectively. Hence, there is a larger force drop (or stress drop) on slider 2 than on slider 1. Slider 1 moves first and then pushes slider 2 to move. A larger force drop (or stress drop) on slider 2 than on slider 1 makes a faster and bigger increase in acceleration on the former than on the latter. This makes the direc- tivity effect do not exist in the present case. In Figure 8d, the absolute values of slope at the fixed points are likely larger than 1 for slider 1 and smaller than 1 for slider 2. Thus, the fixed point for the latter can be an attractor, yet not for the former. 4.3 Effect due to static frictional force The effect of difference in Fo1 and Fo2 on the mo- tions of two sliders is numerically made based on various values of φ=Fo2/Fo1 when the values of other model parameters are the same as those in Figure 7. It is noted that since slider 1 is considered to be the first one to move, φ must be equal to or larger than 1. Results show that slider 2 cannot move when φ≥1.9. The value of φ=1.9 is the critical one for the present case. Of course, the critical value depends upon the values of other mod- el parameters. For example, it somewhat increases with s. When 1.9>φ≥1.18, the velocity of slider 1 increases, decreases, and reaches the minimum value when the ve- locity of slider 2 reaches its peak value. Then, slider 1 is pulled by slider 2 to move again. The separation of two events of slider 1 increases with φ. When φ<1.18, the ve- locity of both sliders increases, decreases, and becomes zero, and slider 1 cannot move again. Figures 9 and 10 show the simulation results for φ=1.05 and φ=1.18, respectively, when the values of other model parameters are the same as those in Fig- ure 7. Obviously, in each plot the solid line is different from the dotted line. Like Figure 7, the peak values of A/Amax and V/Vmax of slider 1 come earlier than those at slider 2, while the peak value of U/Umax of the for- mer comes later than that of the latter. The peak val- ues of A/Amax, V/Vmax, and U/Umax of slider 1 are all lower than those of slider 2. The reason to cause the results is the same as that for Figure 7. The difference Figure 4. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/ Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=5, η1=0, η2=0, φ=1, Uc1=0.1, and Uc2=0.1. SLIP OF A TWO-BODY MODEL WITH FRICTION AND VISCOSITY 9 Figure 5. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=5, η1=0, η2=0, φ=1, Uc1=0.1, and Uc2=0.1. Figure 6. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=0, η2=0, φ=1, Uc1=0.5, and Uc2=0.5. WANG 10 Figure 7. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=0, η2=0, φ=1, Uc1=0.1, and Uc2=0.5. Figure 8. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=0, η2=0, φ=1, Uc1=0.5, and Uc2=0.1. SLIP OF A TWO-BODY MODEL WITH FRICTION AND VISCOSITY 11 Figure 9. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=0, η2=0, φ=1, Uc1=0.1, and Uc2=0.5. Figure 10. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/Umax) and the phase portrait of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2)r for s=1, η1=0, η2=0, φ=1.18, Uc1=0.1, and Uc2=0.5. WANG 12 Figure 11. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/ Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=10, η2=10, φ=1, Uc1=0.1, and Uc2=0.5.. Figure 12. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (V/ Vmax) and the phase portraits of υ versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=10, η2=2, φ=1, Uc1=0.1, and Uc2=0.5. SLIP OF A TWO-BODY MODEL WITH FRICTION AND VISCOSITY 13 between Figure 7 and Figure 9 as well as Figure 10 is that the increases in A/Amax, V/Vmax, and U/Umax come slightly later for Figures 9 and 10 than for Figure 7, be- cause it takes a longer time to increase the force on slider 2 from driving force due to φ>1 or Fo2>Fo1. Ob- viously, there are two bumps in the temporal variation in velocity of slider 1. In Figure 9d, the absolute values of slope at the fixed points are likely smaller than 1 for slider 1 and larger than 1 for slider 2. Thus, the fixed point for the former can be an attractor, yet not for the latter. In Figure 10d, the absolute values of slope at the fixed points are likely smaller than 1 and thus they can be an attractor. Clearly, φ can influence the generation an attractor in slider 2. Figures 4−10 show that nonlinear slip-depend- ent friction can result in an attractor for chaotic slip in the model. This is similar to the chaotic motions in a two-degree-of-freedom system studied by numerous researches [e.g., Huang and Turcotte, 1990a,b, 1992; de Sousa Viera, 1999; and Abe and Kato, 2013]. 4.4 Effect due to viscosity Figures 11–13 show the simulation results for η1=10 and η2=10, η1=10 and η2=2, and η1=10 and η2=20, respectively, when the values of other model parameters are the same as those in Figure 7. Obvi- ously, in each plot the solid line is different from the dotted line. In Figure 11, the peak values of A/Amax V/ Vmax, and U/Umax of slider 1 come earlier than those of slider 2. The peak values of A/Amax, V/Vmax, and U/Umax of slider 1 are all lower than those of slider 2. The peak values of A/Amax V/Vmax, and U/Umax are all smaller at slider 1 than at slider 2 in Figure 7, yet opposite in Figure 2. This indicates that viscosity can affect the amplitudes of the three quantities. In Figure 12, the peak value of A/Amax comes slightly earlier at slider 1 than at slider 2; while the peak value of V/Vmax comes slightly later at slider 1 than at slider 2. The peak values of A/Amax and V/Vmax of slider 1 are lower than those of slider 2. The value of U/Umax of slider 1 is first similar to and then larger than that of slider 2. In Fig- ure 13, the peak values of A/Amax and V/Vmax of slider 1 come earlier than those at slider 2. The peak values of A/Amax and V/Vmax of slider 1 are larger than those of slider 2. The value of U/Umax of slider 1 is larger than that of slider 2. Unlike Figures 4–10, the duration times in Figures 11–13 become longer due to the viscous effect. Since weak coupling (with s=1) between the two sliders, their motions can be considered to be somewhat in- Figure 13. The time sequences of normalized acceleration (A/Amax), normalized velocity (V/Vmax), and normalized displacement (U/ Umax) and the phase portraits of V versus U of the two sliders (solid line for slider 1 and dashed line for slider 2) for s=1, η1=10, η2=20, φ=1, Uc1=0.1, and Uc2=0.5. WANG 14 dependent. Wang [2016b] studied the viscous effect on the predominant period of a one-degree-of-freedom spring-slider model. Here, a study of the viscous effect on the predominant periods of the two-degree-of-free- dom spring-slider model is conducted. From Equation (1), the natural period of each slider is To=2π(m/L) 1/2 in the absence of friction and viscosity. When the two sliders are linked together, the natural period of each slider must be slightly different from To. When viscosity is present, the natural period is T1=To1/(1-C1 2/4mL) for slider 1 and T2=To2/(1-C2 2/4mL) for slider 2. Obviously, viscosity produces damping and increases the predom- inant period of oscillations of the slider. The system is under-damping, critical damping, and over-damping when Ci 2/4mL<1, Ci 2/4mL=1, and Ci 2/4mL>1, respec- tively. Since To1=To2 in this study, the ratio of T2 to T1 is: T2/T1=[(α- C1 2 )/(α- C2 2 )]1/2 (8) where α=4mL. In Figure 11, T1≈T2 because of C1=C2 from η1=η2. In Figure 12, T1>T2 because of C1>C2 from η1>η2. In Figure 13, T1