Fluid–Structure Interaction Effects on Developing Complex Non-Newtonian Flows Within Flexible Tubes

Abstract

Complex non-Newtonian glues are widely used in electrical vehicle (EV) manufacturing plants. In this paper, we focus on initial transient and compressibility issues which are closely associated with high pressure, boundary conditions, and flexible tubes, as well as their respective fluid–structure interaction effects. Both thixotropic and power law non-Newtonian nearly compressible fluid models have been employed to couple with flexible tubes with two different sets of material properties, namely, Young’s modulus and density. In addition to thick-wall cylindrical pressure vessel solutions, different pressure and velocity boundary conditions have also been studied with the consideration of initial transient and steady solutions for acoustic models. Moreover, the radial direction displacement distributions through the tube wall thickness and axial directions compare well within 4 to 9 percentage points with theoretical solutions of thick-wall cylinders under internal and external pressures. Finally, inverse optimization methods have been employed for the calibration of key parameters in comparison with experimental and computational results.

1. Background and Objectives

Modern finite element methods (FEM) and computational fluid dynamics (CFD) programs have enabled engineers to extensively use simulation tools in the design and manufacturing stages [1,2]. As a result, more complex fluid and solid models are directly utilized in the design and analysis phase for a better understanding of various complex engineering systems. For the steady delivery of internal fluids, major and minor friction losses provide design engineers with rough estimates of required pressure drops for a range of volume flow rates; moreover, specific in-depth studies are still needed for specific components of the delivery systems, for instance, the needle valve and the hoses with thermal protective layers [3,4,5]. Moreover, the study of the initial transients with respect to different rheological properties and fluid–structure interaction effects is very much needed [6,7]. For this type of transient viscous fluid with extremely high dynamic viscosities, the complex rheological properties play more significant roles than convection, along with turbulent effects [8,9]. In this paper, through further studies of the complex non-Newtonian polymers utilized in EV manufacturing plants and their respective delivery systems, the importance of effective hierarchical computational models is demonstrated in the comparison of analytical, experimental, and computational models [10,11].

Earlier studies of the controlled dispensing of non-Newtonian glues utilized in EV manufacturing plants are presented in Refs. [12,13]. As shown in Figure 1, a bucket of the non-Newtonian glue within the dispense system is hydraulically actuated. Because of the extreme dynamic viscosity of the non-Newtonian fluid, very high pressure within the range of a few hundred times that of the atmospheric pressure must be introduced for intermittent dispensing. Moreover, due to the glue’s temperature dependence regarding rheological properties, the T-shape dispensing gun connects the reservoir through hoses with extensive thermal protective layers. Moreover, the needle valve within the gun is actuated by a hydraulic piston [5]. In this study, we focus on the interaction between the non-Newtonian, nearly compressible fluid with flexible tubes and hierarchical modeling strategies similar to that of complex structures for more physical insights about complex fluid dispensing systems [11]. Similar to the study on water hammering, we demonstrate that a weak shock, the so-called pressure wave or acoustic signals, can also be carried through the fluid, which is assumed to be an almost compressible or nearly incompressible medium [5,14], yet the added mass, added stiffness, and the viscosity, in this case, provide a fluid delivery system with excessive overall damping. As reported in Refs. [12,13,15], numerous tests have confirmed that given a level of inlet and outlet pressure drop, the flow rate or the average axial velocity tends to approach the steady solution in a way similar to that of a resistor and inductor/capacitor (RL/RC) circuits or Kelvin/Maxwell viscoelastic behaviors. Moreover, because of the extreme viscosity, the glue dispensing through the T-shape gun, the servo-actuated needle valve, and flexible tubes exhibits similar initial transient behaviors. Hence, the precision control of the dispensing system for such non-Newtonian, nearly incompressible fluids demands a better understanding of the developing fluid characteristics. Finally, inverse optimization methods were implemented for the accurate predictions of one, two, or three system parameters.

2. Complex Rheological Properties

In EV manufacturing plants, various glues are utilized for sealing and other applications. The complex non-Newtonian rheological properties have presented tremendous challenges in the precision control of dispensing systems, in particular, with the considerations of variable speeds, durations, and flow rates [16,17]. In our earlier studies of such non-Newtonian fluid models, a Bingham non-Newtonian model with a peak dynamic viscosity of $\mathrm{90,000} \mathrm{Pa}·\mathrm{s},$ a minimum dynamic viscosity of $30 \mathrm{Pa}·\mathrm{s},$ and a critical shear rate of $0.0019 1/\mathrm{s}$ has been employed. The rheology of the polymer material in question is very much shear- and strain- rate-dependent [18,19]. In the computational studies presented in this paper, we employ both the power law and thixotropic non-Newtonian, almost compressible, or nearly incompressible, rheological models as depicted in Figure 2.

In engineering design, it is common to relate the shear stress ${\sigma }_s$ with the shear rate $\dot{\gamma }$. There are, in general, two types of non-Newtonian fluids, namely, shear thinning and shear thickening, based on the tabulated relationship between the dynamic viscosity $\mathit{\mu }$ and the shear rate $\dot{\gamma }$, utilizing the definition of the dynamic viscosity $\mathit{\mu }$ for fluid. The stress tensor ${\tau }_{ij}$ can be expressed as $-p{\delta }_{ij}+2\mathit{\mu }{e}_{ij}$, where the kinematic viscosity is defined as ${\displaystyle \nu =\mathit{\mu }/\rho }$ with the density $\rho$, and the shear strain rate tensor $e_{ij}$ can be denoted as $\left(\partial v_i/\partial x_j+\partial v_j/\partial x_i\right)/2$. For the shear-thinning non-Newtonian fluids of interest to us, we employ a power law model, ${\sigma }_s=A{\left(\dot{\gamma }\right)}^n$, in which the equivalent dynamic viscosity $\mathit{\mu }$ can be expressed as

$${\displaystyle \mathit{\mu }=\min \left(A{\dot{\gamma }}^a,{\mathit{\mu }}_o\right),\mathrm{with} a=n-1<0,}$$

(1)

where A and a, which are also expressed as $a=n-1$ in some of the literature, are constants, and $\dot{\gamma }$ is the effective deformation rate or shear rate defined as ${\displaystyle \sqrt{e_{ij}{e}_{ij}}/2.}$

Based on the experimental data presented in Ref. [13], we have identified the constant $A=3307.0$ and the power $a=-0.6129$. In this study, we also adopt a pragmatic approach with a simple thixotropic, almost compressible non-Newtonian fluid model, as presented in Equation (2), which exhibits an initial dynamic viscosity ${\mathit{\mu }}_i$ with an initial relaxation time ${\tau }_i$ and a final dynamic viscosity ${\mathit{\mu }}_f$ with a final relaxation time ${\tau }_f.$ Note that in Equation (2), key parameters can be fine tuned or optimized with the inverse optimization procedure in Refs. [12,20]. This important data analysis approach can be very effective in linking computational, analytical, and experimental evidence directly with needed recommendations for design and manufacturing. Furthermore, based on experimental evidence of black glue, we employ ${\tau }_i={\tau }_f=50 \mathrm{ms}$. The final dynamic viscosity level ${\mathit{\mu }}_f$ is adjusted. For the pipe diameter $D=5 \mathrm{mm}$ for the volume flow rate $\dot{V}=5 \mathrm{cc}/\mathrm{s}$, the average axial velocity $\overline{v}$ is around $0.2546 \mathrm{m}/\mathrm{s}$, the equivalent shear rate is $101.86 1/\mathrm{s}$, and the corresponding initial dynamic viscosity ${\mathit{\mu }}_i$ is about $446.79 \mathrm{Pa}·\mathrm{s}$, whereas the final dynamic viscosity ${\mathit{\mu }}_f$ is about $76.59 \mathrm{Pa}·\mathrm{s}.$

$${\displaystyle \mathit{\mu }\left(t\right)={\mathit{\mu }}_f(1-e^{-t/{\tau }_f})+{\mathit{\mu }}_i{e}^{-t/{\tau }_i}.}$$

(2)

As confirmed in Figure 2, within the shear rate according to the operation conditions, the thixotropic material does exhibit the so-called shear thinning effects for high shear rates, as well as the so-called shear thickening effects towards extremely high shear rates. In this paper, in addition to the thixotropic non-Newtonian properties illustrated in Equation (2), we employ the density $\rho =1450 \mathrm{k}\mathrm{g}/{\mathrm{m}}^3$ and the compressibility ${\displaystyle \beta =1.2\times {10}^{-9} 1/\mathrm{P}\mathrm{a}.}$ Notice that the compressibility $\beta$, which is also the reciprocal of the bulk modulus $\kappa$, is defined as $-1/V\partial V/\partial p$ or $-1/v\partial v/\partial p$, with the volume V, the specific volume v, or the reciprocal of the density $\rho$, and the pressure is p. Assume that the initial pressure and density for this newly proposed thixotropic non-Newtonian, almost compressible model are $p_o$ and ${\rho }_o$, such that we have

$${\displaystyle p\left(t\right)=p_o+{\int }_{{\rho }_o}^{\rho }\frac{\beta }{\rho }d\rho \mathrm{and} \rho \left(t\right)={\rho }_o+{\int }_{{\rho }_o}^{\rho }\frac{1}{c^2}\frac{\beta }{\rho }d\rho ,}$$

(3)

where the wave speed c is assumed to be constant, defined as ${\displaystyle c^2=\kappa /\rho =dp/d\rho .}$

3. Dry Structure Vibrations

Due to the excessively high pressure utilized in the dispensing of the thixotropic non-Newtonian, almost compressible fluid in EV manufacturing plants, which, in some cases, can exceed $200 \mathrm{b}\mathrm{a}\mathrm{r}$ or $2940 \mathrm{p}\mathrm{s}\mathrm{i},$ the thickness-to-diameter ratio of the tube is more than $1/20$. For instance, in the example elaborated in this study, the wall thickness is $2 \mathrm{mm}$, and the tube diameter is merely $5 \mathrm{mm}$. Therefore, it is necessary to introduce the so-called thick-wall cylindrical pressure vessel model. Consider a differential element, as shown in Figure 3; we have the force equilibrium in the tangential or $\theta$ direction,

$${\displaystyle {\sigma }_{\theta }dr+\rho a_{\theta }rdrd\theta =dr({\sigma }_{\theta }+d{\sigma }_{\theta }),}$$

(4)

along with the force equilibrium in the radial or r direction,

$${\displaystyle {\sigma }_{\theta }drd\theta +\rho a_{r}rdrd\theta +{\sigma }_{r}rd\theta =(r+dr)({\sigma }_r+d{\sigma }_r)d\theta ,}$$

(5)

where we have $cosd\theta /2\simeq 1$ and $sind\theta /2\simeq d\theta /2.$

Moreover, we can simplify the governing equation in both tangential and radial directions as

$${\displaystyle \rho a_{\theta }=\frac{1}{r}\frac{\partial {\sigma }_{\theta }}{\partial \theta } \mathrm{a}\mathrm{n}\mathrm{d} \rho a_r=\frac{{\sigma }_r-{\sigma }_{\theta }}{r}+\frac{\partial {\sigma }_r}{\partial r},}$$

(6)

where $a_r$ and $a_{\theta }$ stand for accelerations in the radial and tangential directions.

As a special case, within an axial directional cross-section, for the static or quasi-static force equilibrium with $a_{\theta }=0$ and $a_r=0$, we have

$${\displaystyle \frac{1}{r}\frac{\partial {\sigma }_{\theta }}{\partial \theta }=0 \mathrm{a}\mathrm{n}\mathrm{d} \frac{{\sigma }_r-{\sigma }_{\theta }}{r}+\frac{\partial {\sigma }_r}{\partial r}=0.}$$

(7)

In fact, Equation (7) implies an axisymmetric nature, namely, no dependence on the polar angle $\theta$. Hence, we have the definition of the so-called hoop strain ${ϵ}_{\theta }$, a normal strain in the tangential or $\theta$ direction, defined as ${\displaystyle \left[2\pi \right(r+u)-2\pi r]/2\pi r=u/r,}$ where u stands for the radial direction expansion displacement, which is only dependent on the radial position marked by the radius r. Likewise, we can define the radial normal strain ${ϵ}_r$ as ${\displaystyle {ϵ}_r=du/dr}$.

For linear elastic materials, Hook’s law yields

$${\displaystyle {ϵ}_r=\frac{1}{E}({\sigma }_r-\nu {\sigma }_{\theta }) \mathrm{a}\mathrm{n}\mathrm{d} {ϵ}_{\theta }=\frac{1}{E}({\sigma }_{\theta }-\nu {\sigma }_r),}$$

(8)

where E and $\nu$ stand for Young’s modulus and Poisson’s ratio.

The linear system equations in Equation (8) yield

$$\begin{array}{ccccc}\hfill {\displaystyle {\sigma }_r}& =& {\displaystyle \frac{E}{1-{\nu }^2}(\nu {ϵ}_{\theta }+{ϵ}_r)}\hfill & =& {\displaystyle \frac{E}{1-{\nu }^2}(\nu \frac{u}{r}+\frac{du}{dr}),}\hfill \\ \hfill {\displaystyle {\sigma }_{\theta }}& =& {\displaystyle \frac{E}{1-{\nu }^2}({ϵ}_{\theta }+\nu {ϵ}_r)}\hfill & =& {\displaystyle \frac{E}{1-{\nu }^2}(\frac{u}{r}+\nu \frac{du}{dr}).}\hfill \end{array}$$

(9)

Thus, in combining Equations (7) and (9), we derive the following Cauchy–Euler equation with respect to the radial displacement $u\left(r\right)$,

$${\displaystyle r^2\frac{d^{2}u}{d{r}^2}+r\frac{du}{dr}-u=0.}$$

(10)

It is very important to observe the consistency of the units in Equation (10), which implies the characteristic solutions in the form of an algebraic or power function $r^n$. If we insert the trial solution $u\left(r\right)=C{r}^n$, we have the following characteristic equation:

$$n(n-1)+n-1=0, \mathrm{or} n^2=1.$$

Hence, we have the two characteristic solutions to Equation (10), namely, $C_1{r}^1$ and $C_2{r}^{-1}$, and as a consequence,

$$\begin{array}{ccccc}\hfill {\displaystyle {\sigma }_r}& =& {\displaystyle \frac{E}{1-{\nu }^2}(\nu \frac{u}{r}+\frac{du}{dr})}\hfill & =& {\displaystyle \frac{E}{1-{\nu }^2}(D_1-\frac{D_2}{r^2}),}\hfill \\ \hfill {\displaystyle {\sigma }_{\theta }}& =& {\displaystyle \frac{E}{1-{\nu }^2}(\frac{u}{r}+\nu \frac{du}{dr})}\hfill & =& {\displaystyle \frac{E}{1-{\nu }^2}(D_1+\frac{D_2}{r^2});}\hfill \end{array}$$

(11)

with ${\displaystyle D_1=(1+\nu )C_1}$ and ${\displaystyle D_2=(1-\nu )C_2}$.

Finally, the normal stresses ${\sigma }_r$ and ${\sigma }_{\theta }$ within the z-direction cross-section can be simply expressed as

$${\displaystyle {\sigma }_r=\frac{E}{1-{\nu }^2}(D_1-\frac{D_2}{r^2}) \mathrm{a}\mathrm{n}\mathrm{d} {\sigma }_{\theta }=\frac{E}{1-{\nu }^2}(D_1+\frac{D_2}{r^2}).}$$

(12)

With the internal and external pressure boundary conditions, we have ${\sigma }_r=-p_i$ at $r=r_i$ and ${\sigma }_r=-p_o$ at $r=r_o$; we derive

$$\begin{array}{ccc}\hfill {\displaystyle {\sigma }_r}& =& {\displaystyle \frac{p_i{r}_i^2-p_o{r}_o^2}{r_o^2-r_i^2}-\frac{(p_i-p_o)r_i^2{r}_o^2}{r^2(r_o^2-r_i^2)};}\hfill \\ \hfill {\displaystyle {\sigma }_{\theta }}& =& {\displaystyle \frac{p_i{r}_i^2-p_o{r}_o^2}{r_o^2-r_i^2}+\frac{(p_i-p_o)r_i^2{r}_o^2}{r^2(r_o^2-r_i^2)}.}\hfill \end{array}$$

(13)

Using Equation (8), we have the following radial direction displacement distribution:

$$\begin{array}{ccccc}\hfill {\displaystyle u}& =& {\displaystyle \frac{r{D}_1}{1+\nu }+\frac{1}{r}\frac{D_2}{1-\nu }}\hfill & =& {\displaystyle \frac{1-\nu }{E}r\frac{p_i{r}_i^2-p_o{r}_o^2}{r_o^2-r_i^2}+\frac{1+\nu }{E}\frac{1}{r}\frac{(p_i-p_o)r_i^2{r}_o^2}{r_o^2-r_i^2};}\hfill \\ \hfill {\displaystyle \frac{du}{dr}}& =& {\displaystyle \frac{D_1}{1+\nu }-\frac{1}{r^2}\frac{D_2}{1-\nu }}\hfill & =& {\displaystyle \frac{1-\nu }{E}\frac{p_i{r}_i^2-p_o{r}_o^2}{r_o^2-r_i^2}-\frac{1+\nu }{E}\frac{1}{r^2}\frac{(p_i-p_o)r_i^2{r}_o^2}{r_o^2-r_i^2},}\hfill \end{array}$$

(14)

where the two constants $D_1$ and $D_2$ are expressed as

$${\displaystyle D_1=\frac{1-{\nu }^2}{E}\frac{p_i{r}_i^2-p_o{r}_o^2}{r_o^2-r_i^2} \mathrm{and} D_2=\frac{1-{\nu }^2}{E}\frac{(p_i-p_o)r_i^2{r}_o^2}{r_o^2-r_i^2}.}$$

In the physical models presented in this paper, the pipe diameter D is $5 \mathrm{mm}$, the pipe length L is $100 \mathrm{mm}$, and the pipe thickness d is $2 \mathrm{mm}$. The length-to-diameter ratio $L/D$ is 20; thus, it is different from the immediate neighborhood around both end supports. The radial displacements of these dry structures should resemble those illustrated in Figure 4 and Figure 5. Figure 4 depicts a comparison between a finite element solution of a long structure and an infinitely long, thick-wall cylindrical pressure vessel theoretical solution. The results other than the immediate neighborhood of the fixed ends compare well within 4 to 9 percentage points. Furthermore, as shown in Figure 5, the dry structure response, in which a steady inlet-to-outlet pressure distribution is applied to the interior of the tube, matches well with the quasi-static expansion of the flexible tube. In this paper, we assume the same Poisson’s ratio of $\nu =0.3$ for both steel and aluminum, whereas the Young’s modulus for the aluminum is set at $67 \mathrm{GPa}$ and $200 \mathrm{GPa}$ for steel. Moreover, in the example chosen for this study, the length of the tube is $100 \mathrm{mm}$, which is 20 times the diameter. Thus, other than the inlet and outlet structural supports with $u=0$, the radial expansion characterized by the displacement $u\left(r\right)$ depends on the axial position via the linear pressure distribution from the inlet to the outlet. As shown in Figure 6, the radial direction displacement for the aluminum tube is nearly three times that of the steel tube, just as their respective Young’s modulus suggests. Moreover, the dynamic response of the flexible tube, as depicted in Figure 6, indicates that the response frequency f is around $246.2 \mathrm{k}\mathrm{H}\mathrm{z}$, namely, the natural frequency $\omega$ is around $1.55\times {10}^6 \mathrm{r}\mathrm{a}\mathrm{d}/\mathrm{s},$ which will be compared with the estimate of the acoustic frequencies.

4. Acoustic Ranges

In this section, we focus on the respective acoustic ranges of the thick-wall tube. Unlike electromagnetic waves, which can propagate through a vacuum, sound, as a form of mechanical vibration, must have a compressible continuum for its waves to propagate, with its energy carried by partly kinematic and partly potential energies [21]. The key assumption of the transmission of sound in air or any other continuum is to ignore the shear effects [22,23]. Consequently, the mass conservation is represented by the relationship between the pressure p and the volumetric strain $\mathsf{\nabla }·\mathbf{u}$, where $\mathbf{u}$ represents the displacement vector of the medium. Thus, for any acoustic medium, we have

$${\displaystyle \rho \frac{d^2\mathbf{u}}{d{t}^2}=\mathsf{\nabla }\mathbf{\sigma }+\mathbf{f},}$$

(15)

where $\mathbf{f}$ stands for the body force, and the stress tensor $\mathbf{\sigma }$ can be simply expressed as $-p\mathbf{I}$.

In practice, for solids, the bulk modules $\kappa$ can be expressed as ${\displaystyle \kappa =\frac{E}{3(1-2\nu )}}$, a function of Young’s modulus E and Poisson’s ratio $\nu$. It is important to recognize that acoustic waves do not trigger large motions or displacements. Hence, derived from Newton’s Second Law of Motion, we have ${\displaystyle \rho \ddot{\mathbf{u}}=-\mathsf{\nabla }p.}$ Furthermore, we also have the pressure and the volumetric strain relationship

$$p=-\kappa {ϵ}_v=-\kappa \mathsf{\nabla }·\mathbf{u}.$$

(16)

Employing Equations (15) and (16), we derive the following key equations:

$${\displaystyle \frac{{\partial }^{2}p}{\partial t^2}=c^2{\nabla }^{2}p \mathrm{and} \frac{{\partial }^2\mathbf{u}}{\partial t^2}=c^2{\nabla }^2\mathbf{u},}$$

(17)

where the wave speed c is expressed as $\sqrt{\kappa /\rho }$ with the bulk modulus $\kappa$ and the density $\rho$.

For the cylindrical pipe, as illustrated in Figure 7, it is convenient to express Equation (17) in the cylindrical coordinate system. Using the separation of variables method, we have $p(r,\theta ,z,t)=\varphi \left(r\right)\Theta \left(\theta \right)Z\left(z\right)T\left(t\right).$ By substituting the pressure expression into the governing equation in Equation (17), we can derive

$${\displaystyle \frac{1}{r\varphi }\frac{d}{dr}\left(r\frac{d\varphi }{dr}\right)+\frac{1}{r^2}\frac{{\Theta }^{″}}{\Theta }+\frac{Z^{″}}{Z}=\frac{1}{c^2}\frac{\ddot{T}}{T}.}$$

For the z-direction or the axial direction with the two fixed surfaces at $z=0$ and $z=L$, the special solution can be expressed as ${\displaystyle Z\left(z\right)=ccos\frac{n\pi }{L}z.}$ With respect to the r-direction or the radial direction, we have a constant pressure or, rather, zero gauge pressure at $r=D/2=R=r_i$ and $r=R+d=r_o$. Notice that the steady solution with two different pressures on the internal and external cylindrical surface, namely, $r=r_i$ and $r=r_o$, will be combined with the transient solutions, for which the gauge pressures at $r=r_i$ and $r=r_o$ are zero, as depicted in Figure 7. Consequently, the normal gradient of the displacement, velocity, and acceleration is zero based on Equation (16). For the polar angle $\theta$, a periodic function with a period of $2\pi$ is assigned; therefore, we have

$${\displaystyle \Theta \left(\theta \right)=d_{1}cos\frac{2m\pi }{2\pi }\theta +d_{2}sin\frac{2m\pi }{2\pi }\theta =d_{1}cosm\theta +d_{2}sinm\theta ,}$$

where $d_1$ and $d_2$ are constants dependent on boundary conditions.

Therefore, when combining the special functions in the axial or z and the polar or $\theta$ directions, we have

$${\displaystyle \frac{{\Theta }^{″}}{\Theta }=-m^2,\frac{Z^{″}}{Z}=-{\left(\frac{n\pi }{L}\right)}^2, \mathrm{and} \frac{\ddot{T}}{T}=-{\omega }^2,}$$

(18)

with the modes m and n in $\theta$ and z directions, respectively.

Finally, the solution in the radial or r direction is governed by the following Bessel function

$${\displaystyle r^2\frac{d^2\varphi }{d{r}^2}+r\frac{d\varphi }{dr}+({\alpha }_n^2{r}^2-m^2)\varphi =0,}$$

(19)

with

$${\alpha }_n^2={\displaystyle \frac{{\omega }^2}{c^2}}-{\displaystyle \frac{n^2{\pi }^2}{L^2}}.$$

(20)

Note that the solution to Equation (19) is a combination of a typical Bessel function of the first kind of order m, namely $J_m\left({\alpha }_{n}r\right)$, and a typical Bessel function of the second kind of order m, namely, $Y_m\left({\alpha }_{n}r\right)$. In this dispensing system, because of the extremely high pressure introduced to overcome the excessive viscosity, the tube or pipe resembles a typical thick wall cylindrical pressure vessel with $r_i\le r\le r_o.$ Thus, the special solution can be expressed as a combination of a Bessel function of the first kind ${\displaystyle J_m\left({\alpha }_{n}r\right)}$ and a Bessel function of the second kind ${\displaystyle Y_m\left({\alpha }_{n}r\right)}$. From Equation (18), we have $T\left(t\right)=a_{1}cos\left({\omega }_{mn}t\right)+a_{2}sin\left({\omega }_{mn}t\right)$, where $a_1$ and $a_2$ are constants that are dependent on the initial and boundary conditions, and the natural frequency ${\omega }_{mn}$ will be determined by the boundary conditions of $\varphi \left(r\right)$, namely $\varphi \left(R\right)=\varphi (R+d)=0$. Notice that nontrivial solutions for $c_1$ and $c_2$ must exist for

$$\begin{array}{ccc}\hfill c_1{J}_m\left({\alpha }_{n}R\right)+c_2{Y}_m\left({\alpha }_{n}R\right)& =& 0,\hfill \\ \hfill c_1{J}_m\left({\alpha }_n(R+d)\right)+c_2{Y}_m\left({\alpha }_n(R+d)\right)& =& 0,\hfill \end{array}$$

(21)

We must have the determinant of the coefficient matrix equal to zero, namely

$${\displaystyle g\left({\alpha }_{mn}\right)=J_m\left({\alpha }_{mn}R\right)Y_m\left({\alpha }_{mn}(R+d)\right)-J_m\left({\alpha }_{mn}(R+d)\right)Y_m\left({\alpha }_{mn}R\right)=0.}$$

(22)

As depicted in Figure 8, with respect to the mode number $m=0,1,2,3$, we have the first group of characteristic solutions ${\alpha }_{nm}$ as $1564.2$, $1590.9$, $1668.4,$ and $1789.61/\mathrm{m}$, respectively, and the second group of characteristic solutions ${\alpha }_{nm}$ as $3138.2$, $3152.1$, $3193.3$, and $3261.51/\mathrm{m}$. Notice that for the geometries selected in this study, the separations of these acoustic frequency ranges are not significant. Furthermore, using Equation (20), we have

$${\displaystyle k_{mn}^2=\frac{{\omega }_{mn}^2}{c^2}={\alpha }_{mn}^2+\frac{n^2{\pi }^2}{L^2},}$$

(23)

where the wave number is defined as ${\displaystyle k_{mn}=2\pi /{\lambda }_{mn},}$ with the wavelength ${\lambda }_{mn}.$

Thus, we have the first four lowest natural frequencies ${\omega }_{10},$${\omega }_{11},$${\omega }_{12},$ and ${\omega }_{13},$ listed as follows:

$$\begin{array}{ccccccc}\hfill {\displaystyle k_{00}}& =& {\displaystyle \frac{{\omega }_{00}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{00}^2+\frac{0^2{\pi }^2}{L^2}}}\hfill & =& 1564.21/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{01}}& =& {\displaystyle \frac{{\omega }_{01}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{01}^2+\frac{1^2{\pi }^2}{L^2}}}\hfill & =& 1564.51/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{02}}& =& {\displaystyle \frac{{\omega }_{02}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{02}^2+\frac{2^2{\pi }^2}{L^2}}}\hfill & =& 1565.51/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{03}}& =& {\displaystyle \frac{{\omega }_{03}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{03}^2+\frac{3^2{\pi }^2}{L^2}}}\hfill & =& 1567.01/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{10}}& =& {\displaystyle \frac{{\omega }_{10}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{10}^2+\frac{0^2{\pi }^2}{L^2}}}\hfill & =& 1590.91/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{11}}& =& {\displaystyle \frac{{\omega }_{11}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{11}^2+\frac{1^2{\pi }^2}{L^2}}}\hfill & =& 1591.21/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{12}}& =& {\displaystyle \frac{{\omega }_{12}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{12}^2+\frac{2^2{\pi }^2}{L^2}}}\hfill & =& 1592.11/\mathrm{m},\hfill \\ \hfill {\displaystyle k_{13}}& =& {\displaystyle \frac{{\omega }_{13}}{c}}\hfill & =& {\displaystyle \sqrt{{\alpha }_{13}^2+\frac{3^2{\pi }^2}{L^2}}}\hfill & =& 1593.71/\mathrm{m}.\hfill \end{array}$$

(24)

Finally, we have the dominant wave length listed as

$${\displaystyle {\lambda }_{10}=3.949 \mathrm{m}\mathrm{m}, {\lambda }_{11}=3.949 \mathrm{m}\mathrm{m}, {\lambda }_{12}=3.946 \mathrm{m}\mathrm{m}, \mathrm{and} {\lambda }_{13}=3.943 \mathrm{m}\mathrm{m}.}$$

For the aluminum pipe, the bulk modulus $\kappa$ is roughly $55.83 \mathrm{GPa}$, whereas for the steel pipe, the bulk modulus $\kappa$ is nearly $166.67 \mathrm{GPa}.$ In this work, we also adopt a density for aluminum at $2700 \mathrm{kg}/{\mathrm{m}}^3$, whereas the density for steel is $7800 \mathrm{kg}/{\mathrm{m}}^3.$ Hence, the wave speeds c for the aluminum and steel tubes are $4622.5 \mathrm{m}/\mathrm{s}$ and $4547.4 \mathrm{m}/\mathrm{s},$ respectively, which confirms the simulation results depicted in Figure 6, namely, the dry structure wave length for the aluminum tube is nearly the same as that of the steel tube. Finally, by employing Equation (24), we can estimate the acoustic range natural frequency $\omega$ to be around $7.2345\times {10}^6 \mathrm{rad}/\sec$ or the frequency f to be around $1.1514 \mathrm{MHz}$ for the aluminum tube, and we can estimate $7.3539\times {10}^6 \mathrm{rad}/\sec$ or the frequency f to be around $1.1704 \mathrm{MHz}$ for the steel tube. Based on the transient responses of both dry structures, as depicted in Figure 6, the structural frequency f is around $245.5 \mathrm{KHz}.$ It is clear that, in comparison, the acoustic frequencies are much higher than the structural frequencies; we focused our attention on structural vibrations for the full-fledged fluid–structure interaction (FSI) analyses.

5. FSI Effects and Developing Flow Transients

In the study of the dispensing of thixotropic non-Newtonian, almost compressible fluids, we have established that a significant inlet and outlet pressure difference is needed to squeeze out the super viscous polymers [13]. With the consideration of initial transients, namely, the inlet flow rate is elevated to the steady level within a short period, in this paper, we adopt a thixotropic model, as depicted in Figure 2. In the series of FSI simulations using Bentley Systems ADINA FSI software, we employed a time step size of $\Delta t=0.2 \mu \mathrm{s}$, along with a ramp size of $0.4 \mu \mathrm{s}.$ As demonstrated in Figure 9, Figure 10, Figure 11, Figure 12 and Figure 13, the dry structure exhibits much less damping in comparison with FSI systems with internal viscous fluids. Moreover, in so-called dry structural models, the same pressure distribution from the inlet to the outlet established in the steady solution tends to introduce a linear structural dynamic response that satisfies the structural boundary conditions, which are fixed at both ends. Because the high aspect ratio is $L/D=20$, for the bulk of the tube, the radial direction displacements still resemble those from the thick-wall cylindrical pressure vessel, as shown in Figure 4, Figure 5 and Figure 6.

It is clear that in our Bentley Systems ADINA FSI models, which are closer to the physical realities, the structural deformation will follow a dynamic response from the inlet to the outlet, yet with significantly more damping in comparison with their respective dry structures, namely, with the damping effects of the very viscous thixotropic non-Newtonian, almost compressible internal fluid. As discussed in the previous section, since the aluminum density-to-Young’s modulus ratio is nearly the same as the steel density-to-Young’s modulus, even though the aluminum Young’s modulus is about one third of that of steel, the wave number k seems to be the same for both aluminum and steel pipes, as confirmed in Figure 11 and Figure 13. Since the wave propagation speeds for both aluminum and steel pipes are similar, the natural frequencies of the dry structures are also the same, as confirmed in Figure 6. Clearly, due to the damping of the viscous fluid, the transient structural solutions will eventually approach those of dry structures, as shown in Figure 6, Figure 11 and Figure 13.

For the precise delivery of these very complicated thixotropic non-Newtonian, almost compressible fluids, the wave speed, which is dependent on the bulk modulus or the reciprocal of the compressibility, as well as the fluid–structure interaction (FSI) effects, can play an important role. Moreover, as shown in Figure 14 and Figure 15, although both compressible fluid models and full-fledged FSI models yield oscillatory pressure differences and average axial velocity during the developing flow phase, the frequencies have significantly increased, whereas the damping ratios stay roughly the same. It is not difficult to conclude that the added stiffness part of the FSI effects seems to be more prominent, which matches with our physical intuitions, namely, both aluminum and steel tubes are very strong, and the rigid tube assumption is reasonable. Moreover, as shown in Figure 14 and Figure 15, both FSI and CFD analyses produce accurate solutions oscillating around the steady theoretical solutions, as elaborated in our earlier reports and Refs. [12,13].

Furthermore, as demonstrated in Figure 14 and Figure 15, comparing the pressure and the average velocity for the CFD case using rigid tubes with those of deformable tubes, it is clear that the slight deformation of the pipe is actually very conducive to the axial motion of the internal fluid. In fact, for biological systems such as the cardiovascular and digestive systems, the motion of the tube wall plays a significant role in conveying fluids and solids. This is an interesting point that is worth noting from a design and manufacturing perspective. In addition to the variabilities of dynamic viscosity, it is also important to estimate the effects of the pipe length for these non-Newtonian fluids, a process similar to the establishment of major friction loss. For the cases presented in Figure 14 and Figure 15, the flow rate is $2\times {10}^{-6} {\mathrm{m}}^3/\mathrm{s}$ or $2 \mathrm{c}\mathrm{c}/\mathrm{s},$ which has an average axial velocity of $0.102 \mathrm{m}/\mathrm{s}$ for the pipe with a diameter of $D=5 \mathrm{mm}.$ Thus, it takes about $0.98\mathrm{s}$ for the Lagrangian fluid particle to move from the inlet to the outlet. Overall, the full-fledged FSI models are depicted in Figure 9 and Figure 10. To further digest these results, the dry structure dynamic results are also compared with the FSI results in Figure 11 and Figure 13. It is clear that the internal fluid contributed significantly to the damping effects.

As shown in Figure 14, the oscillation of both the average axial velocity and the pressure difference between the inlet and the outlet around their respective steady state values tends to dissipate and diminish within the first few milliseconds of the initial transients. This time scale provides important guidelines for EV manufacturing dispensing systems with respect to how fast the activation and closing of the servo motor and the needle valve are. To further the study on the compressibility effect within dispensing systems, we can also investigate the pressure wave and the velocity propagation within the rigid tube. Notice that the Dirichlet pressure boundary at the outlet corresponds to the Neumann velocity boundary with a zero-velocity axial derivative, whereas the Dirichlet velocity boundary at the inlet corresponds to the Neuman pressure boundary with a zero-pressure axial derivative, as elaborated in Refs. [12,13].

Finally, as suggested in Figure 15, for a given flow rate, the inlet and outlet pressure drop for such non-Newtonian internal fluids over time behaves very much like a mechanical vibration system with a critical damping ratio. To characterize the system parameters, such as relaxation time, based on the experimental or computational results, in this study, we employed the so-called inverse optimization method with Newton–Raphson iterative procedures. A good validation of the inverse optimization method is the viscoelastic dynamic model of a downhole sucker rod and its system parameters, which yield almost identical results in comparison with the respective experimental measures as reported in Ref. [12]. The key viscosity effects are damping for both the inlet and outlet pressure drop and the axial velocity transients, as illustrated in Figure 11 and Figure 13. Again, as the dynamic viscosity reaches a critical level, both the inlet and outlet pressure difference and the axial velocity will approach steady solutions, following a similar exponential growth pattern as reported in Ref. [13]. This information is crucial for the design of control algorithms for the dispensing system of many different types of complex fluids.

6. Inverse Optimization Method

Based on documented experimental, computational, and theoretical studies, for an internal viscous fluid with a certain level of compressibility, if the inlet pressure elevation is suddenly introduced, there exists a pressure pulse propagating through the system, a so-called water hammer, as well as an exponential growth of the steady inlet and outlet pressure drop (matching the flow rate, characterized with the so-called relaxation time), which is dependent on rheological properties and pipe geometries [24,25]. Moreover, as suggested in Figure 14 and Figure 15, as the effective dynamic viscosity increases, there exists a similar relaxation time for the developing stage of the inlet and outlet pressure difference, as well as the volume flow rate. In this section, a so-called inverse optimization method, as documented in Refs. [12,26,27], is adopted for the identification of key system parameters.

6.1. Two-Parameter Model

After the initial transient due to the sudden activation of the motor, unlike that of water hammer effects, the inlet and outlet pressure drop follows an exponential growth, similar to RC and RL circuits or viscoelastic materials, to reach the required steady solution value. In fact, such a new relaxation time can be captured using the so-called inverse optimization method. Based on the first four modes of Bessel functions of the first kind, as presented in Refs. [13,25,28], by utilizing the boundary conditions ${\displaystyle J_m\left(\frac{R}{\sqrt{\nu {\tau }_m}}\right)=0}$, we have the first four modes for the relaxation time ${\tau }_m$, with $1\le m\le 4$,

$${\displaystyle \frac{R}{\sqrt{\nu {\tau }_1}}=3.832;\frac{R}{\sqrt{\nu {\tau }_2}}=5.136;\frac{R}{\sqrt{\nu {\tau }_3}}=6.380; \mathrm{a}\mathrm{n}\mathrm{d} \frac{R}{\sqrt{\nu {\tau }_4}}=7.016.}$$

Based on discussions in Refs. [13,25], the fourth mode is derived from the first-order Bessel function of the first kind, not the fourth-order Bessel function of the first kind. More importantly, the scaling factors are selected as the ratio of the relaxation times $a={\tau }_1/{\tau }_2=1.796$; $b={\tau }_1/{\tau }_3=2.772$; and $c={\tau }_1/{\tau }_4=3.352$.

Many complex systems in engineering practices are impossible to characterize with a simple physical and mathematical model; therefore, an implicit, matrix-free iterative method is very useful in providing better guidance for the optimal operation conditions when only the input and output data are available. In order to identify these intrinsic properties C and $\tau$, we set up an inverse engineering problem to minimize the difference measured by the following variational indicator:

$$\begin{array}{ccc}\hfill {\displaystyle E}& =& {\displaystyle \sum _{n=1}^N\frac{1}{2}{\left[p_n-C(1-e^{-\frac{t_n}{\tau }})-(1-C)(1-e^{-\frac{a{t}_n}{\tau }})\right]}^2,}\hfill \\ \hfill {\displaystyle }& =& {\displaystyle \sum _{n=1}^N\frac{1}{2}{\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+(1-C)e^{-\frac{a{t}_n}{\tau }}\right]}^2,}\hfill \end{array}$$

where N represents the number of time stations, $p_n$ is the normalized pressure drop from the inlet to the outlet, normalized with the peak value.

For the minimized error, cost function, or variational indicator E, we have

$${\displaystyle \frac{\partial E}{\partial C}=0 \mathrm{a}\mathrm{n}\mathrm{d} \frac{\partial E}{\partial \tau }=0.}$$

Thus, we have

$$\begin{array}{ccccc}\hfill {\displaystyle f_1}& =& {\displaystyle \frac{\partial E}{\partial C}}\hfill & =& {\displaystyle \sum _{n=1}^N\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+(1-C)e^{-\frac{a{t}_n}{\tau }}\right](e^{-\frac{t_n}{\tau }}-e^{-\frac{a{t}_n}{\tau }}),}\hfill \\ \hfill {\displaystyle f_2}& =& {\displaystyle \frac{\partial E}{\partial \tau }}\hfill & =& {\displaystyle \sum _{n=1}^N\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+(1-C)e^{-\frac{a{t}_n}{\tau }}\right]\left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{(1-C)a{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}\right].}\hfill \end{array}$$

To solve this nonlinear set of equations, we must employ the Newton–Raphson iterative procedures with judiciously chosen starting values. The nonlinear and implicit governing equation about the unknown vector $\mathbf{x}=<C,\tau >$ can be rewritten as

$$\mathbf{f}\left(\mathbf{x}\right)=\mathbf{R},$$

(25)

where the given right-hand-side vector is $\mathbf{R}=<0,0>$.

In order to ensure the success of the Newton–Raphson iterative procedures, the initial guess must be within close proximity of the actual solution $\mathbf{x}$. In engineering practice, we often start with values based on experimental, mathematical, and physical insights. With such an educated guess ${\mathbf{x}}^o=<c^o,{\tau }^o>$ that is not too far from the converged solution, the Jacobian matrix can be defined and evaluated. Assume we have all the information before the $k$th iteration, namely, ${\mathbf{x}}^{k-1}$ and the corresponding $\mathbf{f}\left({\mathbf{x}}^{k-1}\right)$, as well as the so-called Jacobian matrix $\mathbf{J}\left({\mathbf{x}}^{k-1}\right)$ with all entities $J_{ij}$, defined as

$$\begin{array}{ccc}\hfill {\displaystyle J_{11}=\frac{\partial f_1}{\partial x_1}}& =& {\displaystyle \sum _{n=1}^N{(e^{-\frac{t_n}{\tau }}-e^{-\frac{a{t}_n}{\tau }})}^2,}\hfill \\ \hfill {\displaystyle J_{22}=\frac{\partial f_2}{\partial x_2}}& =& {\displaystyle \sum _{n=1}^N\left\{{\left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{(1-C)a{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}\right]}^2+\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+(1-C)e^{-\frac{a{t}_n}{\tau }}\right]\right.}\hfill \\ & & {\displaystyle \left.\left[C{\left(\frac{t_n}{{\tau }^2}\right)}^2{e}^{-\frac{t_n}{\tau }}+(1-C){\left(\frac{a{t}_n}{{\tau }^2}\right)}^2{e}^{-\frac{a{t}_n}{\tau }}-\frac{2C{t}_n}{{\tau }^3}{e}^{-\frac{t_n}{\tau }}-\frac{2(1-C)a{t}_n}{{\tau }^3}{e}^{-\frac{a{t}_n}{\tau }}\right]\right\}.}\hfill \end{array}$$

Similarly, the off-diagonal terms for the Jacobian matrix can also be elaborated as

$$\begin{array}{ccc}\hfill {\displaystyle J_{21}=J_{12}=\frac{\partial f_1}{\partial x_2}}& =& {\displaystyle \sum _{n=1}^N\left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{(1-C)a{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}\right](e^{-\frac{t_n}{\tau }}-e^{-\frac{a{t}_n}{\tau }})}\hfill \\ & +& {\displaystyle \sum _{n=1}^N\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+(1-C)e^{-\frac{a{t}_n}{\tau }}\right](\frac{t_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}-\frac{a{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}).}\hfill \end{array}$$

Note that in this inverse optimization method, the Jacobian matrix is, indeed, symmetric. Finally, the set of nonlinear equations has been changed into a series of incremental linear systems of equations for the unknown increments $\Delta {\mathbf{x}}^k$, defined as follows:

$$\mathbf{J}\left({\mathbf{x}}^{k-1}\right)\Delta {\mathbf{x}}^k=-\mathbf{f}\left({\mathbf{x}}^{k-1}\right),$$

(26)

with the following update:

$${\mathbf{x}}^k={\mathbf{x}}^{k-1}+\Delta {\mathbf{x}}^k.$$

(27)

The iteration identified in Equation (27) will stop with a relative incremental error $ϵ$ that is smaller than the prescribed small number ${ϵ}_o$,

$${\displaystyle \frac{\parallel \Delta {\mathbf{x}}^k\parallel }{\parallel {\mathbf{x}}^o\parallel }=ϵ\le {ϵ}_o.}$$

(28)

With the initial values $C^o=-0.4$, and the relaxation time $\tau =0.04 \mathrm{s}$, as listed in

Table 1. Convergence of the relaxation time $\tau$ and the coefficient C, measured by the relative incremental error $ϵ$, as defined in Equation (28).

No.	Iteration #1	Iteration #2	Iteration #3
$ϵ$	0.287367618320438	0.596245002498978	0.040706688260272
No.	Iteration #4	Iteration #5	Iteration #6
$ϵ$	8.935517076451765 × 10−4	2.098009223697768 × 10−7	2.486180601677040 × 10−14

, the converged relaxation time $\tau$ is $0.037512731850914 \mathrm{s}$ with the coefficient $C=-0.508403427098258.$ As depicted in Figure 16, the results with two-parameter inverse optimization procedures match fairly well with the actual measurement.

6.2. Three-Parameter Model

In order to improve the accuracy of the inverse optimization method, we can utilize multiple relaxation times ${\tau }_1$, ${\tau }_2$, and ${\tau }_3$ and their respective coefficients C and D by minimizing the difference measured by the following variational indicator:

$$\begin{array}{ccc}\hfill {\displaystyle E}& =& {\displaystyle \sum _{n=1}^N\frac{1}{2}{\left[p_n-C(1-e^{-\frac{t_n}{\tau }})-D(1-e^{-\frac{a{t}_n}{\tau }})-(1-C-D)(1-e^{-\frac{b{t}_n}{\tau }})\right]}^2,}\hfill \\ \hfill {\displaystyle }& =& {\displaystyle \sum _{n=1}^N\frac{1}{2}{\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right]}^2,}\hfill \end{array}$$

where a and b are given scaling factors for ${\tau }_2$ and ${\tau }_3$ with respect to ${\tau }_1$, N represents the number of time stations, and $p_n$ is the normalized pressure drop from the inlet to the outlet normalized with the peak value.

Again, for the minimized error, cost function, or variational indicator E, we have

$${\displaystyle \frac{\partial E}{\partial C}=0,\frac{\partial E}{\partial D}=0, \mathrm{a}\mathrm{n}\mathrm{d} \frac{\partial E}{\partial \tau }=0.}$$

(29)

Thus, we have

$$\begin{array}{ccccc}\hfill {\displaystyle f_1}& =& {\displaystyle \frac{\partial E}{\partial C}}\hfill & =& {\displaystyle \sum _{n=1}^N\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right](e^{-\frac{t_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }}),}\hfill \end{array}$$

$$\begin{array}{ccccc}\hfill {\displaystyle f_2}& =& {\displaystyle \frac{\partial E}{\partial D}}\hfill & =& {\displaystyle \sum _{n=1}^N\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right](e^{-\frac{a{t}_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }}),}\hfill \end{array}$$

$$\begin{array}{ccccc}\hfill {\displaystyle f_3}& =& {\displaystyle \frac{\partial E}{\partial \tau }}\hfill & =& {\displaystyle \sum _{n=1}^N\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right]}\hfill \\ & & & & {\displaystyle \left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{Da{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}+\frac{(1-C-D)b{t}_n}{{\tau }^2}{e}^{-\frac{b{t}_n}{\tau }}\right].}\hfill \end{array}$$

Similarly, we must introduce the Newton–Raphson iterative procedures to obtain the solution of the nonlinear and implicit set of equations in Equation (29). Again, the Newton–Raphson iterative method is introduced for this type of nonlinear set of equations. The nonlinear and implicit governing equation about the unknown vector $\mathbf{x}=<C,D,\tau >$ can be rewritten with the given right-hand-side vector $\mathbf{R}=<0,0,0>$. It is very important to point out that the initial guess must be fairly close to the actual solution for the unknown $\mathbf{x}$ to ensure the convergence of the Newton–Raphson iteration scheme.

Again, in practice, we often start with a few tryouts and narrow down the true solution neighborhood. With an educated guess of the initial set of parameters ${\mathbf{x}}^o=<C^o,D^o,{\tau }^o>$ that is not too far from the converged solution, the Jacobian matrix can be defined and evaluated. Assume that we have all the information before the $k$th iteration, namely, ${\mathbf{x}}^{k-1}$ and the corresponding $\mathbf{f}\left({\mathbf{x}}^{k-1}\right)$, as well as the so-called Jacobian matrix $\mathbf{J}\left({\mathbf{x}}^{k-1}\right)$ with all entities $J_{ij}$, defined as

$$\begin{array}{ccc}\hfill {\displaystyle J_{11}=\frac{\partial f_1}{\partial x_1}}& =& {\displaystyle \sum _{n=1}^N{(e^{-\frac{t_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }})}^2,}\hfill \\ \hfill {\displaystyle J_{12}=\frac{\partial f_1}{\partial x_2}}& =& {\displaystyle \sum _{n=1}^N(e^{-\frac{t_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }})(e^{-\frac{a{t}_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }}),}\hfill \\ \hfill {\displaystyle J_{22}=\frac{\partial f_2}{\partial x_2}}& =& {\displaystyle \sum _{n=1}^N{(e^{-\frac{a{t}_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }})}^2;}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle J_{13}=\frac{\partial f_1}{\partial x_3}}& =& {\displaystyle \sum _{n=1}^N\left\{\left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{Da{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}+\frac{(1-C-D)b{t}_n}{{\tau }^2}{e}^{-\frac{b{t}_n}{\tau }}\right](e^{-\frac{t_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }})\right.}\hfill \\ & +& {\displaystyle \left.\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right](\frac{t_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}-\frac{b{t}_n}{{\tau }^2}{e}^{-\frac{b{t}_n}{\tau }})\right\},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle J_{23}=\frac{\partial f_2}{\partial x_3}}& =& {\displaystyle \sum _{n=1}^N\left\{\left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{Da{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}+\frac{(1-C-D)b{t}_n}{{\tau }^2}{e}^{-\frac{b{t}_n}{\tau }}\right](e^{-\frac{a{t}_n}{\tau }}-e^{-\frac{b{t}_n}{\tau }})\right.}\hfill \\ & +& {\displaystyle \left.\left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right](\frac{a{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}-\frac{b{t}_n}{{\tau }^2}{e}^{-\frac{b{t}_n}{\tau }})\right\},}\hfill \end{array}$$

$$\begin{array}{ccc}\hfill {\displaystyle J_{33}=\frac{\partial f_3}{\partial x_3}}& =& {\displaystyle \sum _{n=1}^N\left\{{\left[\frac{C{t}_n}{{\tau }^2}{e}^{-\frac{t_n}{\tau }}+\frac{Da{t}_n}{{\tau }^2}{e}^{-\frac{a{t}_n}{\tau }}+\frac{(1-C-D)b{t}_n}{{\tau }^2}{e}^{-\frac{b{t}_n}{\tau }}\right]}^2\right.}\hfill \\ & +& {\displaystyle \left[p_n-1+C{e}^{-\frac{t_n}{\tau }}+D{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D)e^{-\frac{b{t}_n}{\tau }}\right]}\hfill \\ & & {\displaystyle \left[C{\left(\frac{t_n}{{\tau }^2}\right)}^2{e}^{-\frac{t_n}{\tau }}+D{\left(\frac{a{t}_n}{{\tau }^2}\right)}^2{e}^{-\frac{a{t}_n}{\tau }}+(1-C-D){\left(\frac{b{t}_n}{{\tau }^2}\right)}^2{e}^{-\frac{b{t}_n}{\tau }}\right.}\hfill \\ & -& {\displaystyle \left.\left.\frac{2C{t}_n}{{\tau }^3}{e}^{-\frac{t_n}{\tau }}-\frac{2Da{t}_n}{{\tau }^3}{e}^{-\frac{a{t}_n}{\tau }}-\frac{2(1-C-D)b{t}_n}{{\tau }^3}{e}^{-\frac{b{t}_n}{\tau }}\right]\right\}.}\hfill \end{array}$$

Note that it is straightforward to confirm that the Jacobian matrix is, indeed, symmetric. After the solution of the following incremental linear system of equations for the unknown $\Delta {\mathbf{x}}^k$, the incremental update is performed until the relative incremental error $ϵ$ is smaller than the prescribed small number ${ϵ}_o$. With the initial values $C^o=-0.2$ and $D^o=0.5$ and the relaxation time ${\tau }^o=0.05 \mathrm{s}$, the converged solutions, as depicted in

Table 2. Convergence of the relaxation time $\tau$ and the coefficients C and D, measured by the relative incremental error $ϵ$, as defined in Equation (28).

No.	Iteration #1	Iteration #2	Iteration #3
$ϵ$	0.317819057139488	0.364465973372030	0.031165495401609
No.	Iteration #4	Iteration #5	Iteration #6
$ϵ$	0.003325789492923	4.819821424742624 × 10−6	3.126703602521533 × 10−11

, are $C=-0.397052683274086$ and $D=0.729607435427227$, and the relaxation time is $\tau =0.046464134342923 \mathrm{s}$. As shown in Figure 16, the match with the actual measurement is much closer than the results with two-parameter inverse optimization procedures, as depicted in Figure 16. Finally, we must point out that this method is not omnipotent. Specific adjustment with augmented Lagrangian functions might be of use for complex systems.

7. Conclusions

In engineering practices, various glues are introduced for sealing and other applications. The complex compressible and thixotropic non-Newtonian rheological properties have presented tremendous challenges in the precision control of dispensing systems with a variety of speed, duration, and flow rate requirements. In this study, fluid–structure interaction effects are analyzed and validated using the available theoretical solutions. Responses within both aluminum and steel tubes are compared for an almost compressible, non-Newtonian fluid with both power law and thixotropic dynamic viscosities. Finally, to match with experimental evidence and simulation results with high dynamic viscosities, an inverse optimization method was developed and implemented. It is demonstrated that the predicted solutions with two- and three-parameter exponential growth models become progressively closer to the experimental observation. With a better understanding of the non-Newtonian rheological behaviors of a specific polymer under different temperature ranges, the reliable and stable delivery of a dispensing system utilized in EV manufacturing plants, which, in this case, consists of servo motors, a doser/meter, thermal insulated hoses, and a needle valve nozzle, can be improved and optimized. Finally, this type of inverse optimization methods does show promise in the identification of key system parameters with either experimental or computational solutions. However, we must point out that this method is not omnipotent. Specific adjustment with augmented Lagrangian functions might be of use for complex systems.