Nonlocal response functions for predicting shear flow of strongly inhomogeneous fluids. I. Sinusoidally driven shear and sinusoidally driven inhomogeneity

We present theoretical expressions for the density, strain rate, and shear pressure profiles in strongly inhomogeneous fluids undergoing steady shear flow with periodic boundary conditions. The expressions that we obtain take the form of truncated functional expansions. In these functional expansions, the independent variables are the spatially sinusoidal longitudinal and transverse forces that we apply in nonequilibrium molecular-dynamics simulations. The longitudinal force produces strong density inhomogeneity, and the transverse force produces sinusoidal shear. The functional expansions define new material properties, the response functions, which characterize the system's nonlocal response to the longitudinal force and the transverse force. We find that the sinusoidal longitudinal force, which is mainly responsible for the generation of density inhomogeneity, also modulates the strain rate and shear pressure profiles. Likewise, we find that the sinusoidal transverse force, which is mainly responsible for the generation of sinusoidal shear flow, can also modify the density. These cross couplings between density inhomogeneity and shear flow are also characterized by nonlocal response functions. We conduct nonequilibrium molecular-dynamics simulations to calculate all of the response functions needed to describe the response of the system for weak shear flow in the presence of strong density inhomogeneity up to the third order in the functional expansion. The response functions are then substituted directly into the truncated functional expansions and used to predict the density, velocity, and shear pressure profiles. The results are compared to the directly evaluated profiles from molecular-dynamics simulations, and we find that the predicted profiles from the truncated functional expansions are in excellent agreement with the directly computed density, velocity, and shear pressure profiles.

Figure 1

Schematic representation of the STF and SLF fields. The arrows show the direction of the forces. The length of the arrows, as well as the sinusoidal line, indicate the strength of the force. STF shown for $F^x\left(y\right)=F_1^{x}sin\left(k_{1}y\right)$ and SLF shown for $F^y\left(y\right)=F_2^{y}sin\left(k_{2}y\right)$.

Figure 2

(a) Strain rate and (b) shear pressure response coefficients for $n=1$ calculated over a range of STF field strengths. The axis intercept for the linear fit represents the zero STF field strength extrapolation.

Figure 3

Nonlinear density response to the STF calculated over a range of STF field strengths for $n=1$. The zero-field value of the response function is determined by the axis intercept of the linear fit.

Figure 4

Coefficients for first- and second-order density response functions calculated over a range of SLF field strengths for $m=10$ in the absence of the STF. The zero-field values of the response functions are determined by the axis intercepts of the linear fits.

Figure 5

Bilinear response function coefficients ${\xi }_{1,10}^{xy+}$ (i.e., with STF at $n=1$ and SLF at $m=10$), calculated over a range of SLF field strengths for fixed STF field strength $F_1^x=0.05$.

Figure 6

Zero SLF field strength extrapolated values for the bilinear response function coefficients plotted against STF field strength. $m=6$ (dots), $m=8$ (crosses), $m=10$ (circles), $m=12$ (pluses), and $m=14$ (triangles). Numerical values for the average of each set of four data points for several values of $m$ are shown in Table 2.

Figure 7

Third-order strain rate response function coefficient ${\xi }_{1,10,10}^{xyy0}$ (i.e., STF $n=1$ and SLF $m=10$) calculated over a range of SLF field strengths at fixed STF amplitude $F_1^x=1.0$.

Figure 8

Average of ${\xi }_{1,m,m}^{xyy0}$ calculated for $F_m^y=2.0$, 3.0, 4.0, and 5.0 for each STF field strength $F_1^x=0.05$, 0.10, 0.15, and 0.20. $m=6$ (dots), $m=8$ (crosses), $m=10$ (circles), $m=12$ (pluses), and $m=14$ (triangles). A linear fit is used to determine the zero STF field strength extrapolation. Numerical values for the axis intercepts for a subset of the $m$ values investigated are shown in Table 3.

Figure 9

Example SLF dependence of third-order strain rate response function coefficient ${\xi }_{1,10,10}^{xyy+}$. The results shown are for a single STF field strength $F_1^x=0.05$.

Figure 10

Zero STF field strength extrapolations for third-order response functions ${\xi }_{1,m,m}^{xyy\pm }$ and ${\pi }_{1,m,m}^{xyy\pm }$. $m=6$ (dots), $m=8$ (crosses), and $m=10$ (circles).

Figure 11

Fourier series coefficients for the linear response to the STF determined by the method described in Sec. 4a. (a) Strain-rate response function coefficients ${\xi }_n$, and (b) shear pressure response function coefficients ${\pi }_n$. Also shown is (c) the wave-vector-dependent viscosity, calculated from ${\eta }_n=-{\pi }_n/{\xi }_n$.

Figure 12

Nonlinear shear-induced density response to the STF calculated using the zero-field extrapolation method described in Sec. 4b.

Figure 13

Fourier coefficients for (a) the first- and (b) the second-order density response functions calculated over $m=1,2,\cdots ,40$ using the method described in Sec. 4c. Data represented with filled circles are for nonzero SLF field strength calculations. The plus symbols show the values for $m=6$, 8, 10, 12, and 14 calculated using the extrapolation to zero-field strength.

Figure 14

The bilinear response coefficients calculated over the range $m=1,2,\cdots ,15$ by using a single STF with $n=1$ and extrapolating to zero SLF field strength using values calculated at $F_m^y=1.0$, 2.0, 3.0, 4.0, and 5.0 as described in Sec. 4d. Values from Table 2 are also plotted using plus symbols for ${\xi }_{1,m}^{xy+}$ and ${\pi }_{1,m}^{xy+}$, and crosses for ${\xi }_{1,m}^{xy-}$ and ${\pi }_{1,m}^{xy-}$.

Figure 15

Third-order contributions to the fundamental coefficient ($n=1$) of the velocity and the shear pressure. Contributions are plotted as a function of the SLF field strength for $m=10$ for four different STF field strengths that remain constant on each line. The lines represent quadratic fits.

Figure 16

Third-order strain rate (left) and shear pressure (right) response function Fourier coefficients evaluated by the method described in Sec. 4e plotted over $m=3,4,\cdots ,20$ for $n=1$. Dots represent the data calculated using a single SLF field strength $F_m^y=4.0$ and a zero STF field strength extrapolation. The crosses are the data presented in Table 3 for $m=6$, 8, 10, 12, and 14.

Figure 17

Comparison between density profiles calculated using MD simulation and density profiles predicted using first- and second-order response functions. Parts (a) and (c) show the $y$-space density profiles for $m=6$ and 10, respectively. We use an STF with $n=1$ and $F^x=0.02$. Bold lines are used for the MD simulations and thin lines labeled with symbols are used for the predicted profiles. Parts (b) and (d) show the relative residuals between the simulated and predicted profiles for the $m=6$ and 10 SLFs, respectively, normalized using the zero-wave-vector density.

Figure 18

Convergence of the density response function prediction to the MD simulation profile. Results are shown over a single cycle for the SLF with $m=10$ and $F_{10}^y=4.0$.

Figure 19

Comparison of streaming velocity profiles calculated using MD simulation with response function predictions for $n=1$ and $m=10$. Part (a) shows the $y$-space velocity profiles for STF field strengths $F_1^x=0.08$. Bold lines show the MD simulation results and thin lines labeled with symbols are the response function predictions. Part (b) shows the relative residuals.

Figure 20

Convergence of the velocity series Eq. (40) to the MD profile as we include increasing orders of response. A distinction is made between a series containing the third-order response only at the fundamental velocity wavelength and a series containing all third-order terms.

Figure 21

Comparison between the shear pressure profiles measured using MD simulations and the response function predicted profiles using Eq. (42) for $n=1$ and $m=10$. Results are shown for single STF field strength $F_1^x=0.08$ and single SLF field strength $F_{10}^y=4.0$.

Figure 22

Convergence of the shear pressure Fourier series to the MD simulation profile as we include increasing orders of response.