flows in V-shaped channels and on curved surfaces

This second finding relies on a generalized analysis of nonmodal stability due to the nonnormality of the governing perturbation operator. Robust numerical computation of the 3D scalar potential field of the Galileon gravity cube model at solar system scales.

Motivation

This thesis therefore develops new analytical models for a variety of relevant aspects of the MEP propellant management system. This thesis focuses on the propellant delivery aspect of the MEP system, in particular the capillary flow in the groove network that delivers liquid propellant from the reservoir to the emitters.

Figure 1.1: Schematic of single emitter on microfluidic electrospray propulsion (MEP) thruster.

Organization

Second, to support JPL's efforts to directly detect a “fifth force,” we introduce numerical methods for computing the scalar Cubic Galileon Gravity (CGG) field at solar system scales. Robust numerical computation of the 3D scalar potential field of the cubic Galileon gravity model at solar system scales.

Conservation laws and the geometry of surface tension

Setup

Without loss of generality, we can choose the parameterization ⃗σ such that the collector velocity, ujM =∂tσj, is always normal to the collector. For now, we assume that the manifold is unbounded (manifolds with edges will be discussed later in Section 2.1.6).

S XLT(2)XL

Integral quantities: Mass and momentum

TΞj+TΞtujcn¯jdA, (2.5) where ¯j is the outward pointing normal on the surface of Vc (not to be confused with nˆj, the upward pointing normal on the manifold), and ujc is the local velocity of the control volume. We now have to be very careful with the time derivative of the integral on the manifold, since the area of ​​the manifold can change with time.

Figure 2.2: Cross-section of a control volume V c (t) enclosing a section A c (t) of a manifold with a buffer region of ε on each side

Integral quantities: Angular momentum

HereTji is a force in direction acting on the upper surface normal toj; similarly for Tjion the bottom (although the direction is reversed). And each side has a Tij term, which applies force in the j direction to the planes normal to i.

Manifold-centered coordinates

The surface gradient of a vector is therefore the manifold-bounded covariant derivative of this vector plus an. an additional piece normal to the manifold that is proportional to the other basic shape. Now let's calculate the surface gradient of the rank 2 tensor, which also lives exclusively on the manifold:.

Local conservation equations

Fluids

Thus, the left side of equation (2.35a) describes the change in density on the manifold due to changes in the area of ​​the manifold and convection on the manifold, while the right side describes the transfer of mass between the manifold and the bounding fluids (2) and (1 ). Equation (2.35b) describes the transfer of normal momentum between fluids (2) and (1); Note in particular that the manifold can actually transfer momentum normally to itself.

Fluids and kinematic condition

The convective derivative has a multiplying factor of 2; one half is due to fluid convection in the manifold, while the other half is due to the change in direction of the normal vector over time. Equation (2.35c) describes the conservation of tangential momentum; the left-hand side and the last term on the right-hand side simply form the Cauchy momentum equation on the manifold itself; the remaining terms describe the tangential momentum transfer between the manifold and the confining fluids (2) and (1).

Fluids, kinematic condition, and massless interface

• Manifolds with edges (contact lines)
• Non-normality and generalized linear stability analysis
• Autonomous systems
• Nonautonomous systems
• Nonlinear stability analysis with Lyapunov’s direct method

The spectral abscissa of L, denoted βmax, is the real part of the eigenvalue of L with maximum real part, that is, βmax ≡ max re eigL. The spectral abscissa provides a lower bound on the maximum growth of u:. It turns out that the initial condition u(0) that induces the maximum growth rate at t = 0 (i.e. the mode corresponding to the numerical abscissa) need not be the same as the start.

Figure 2.4: A tubular control volume V c of radius ε enclosing a section L c of a contact line, where three manifolds (M (1) , M (2) , M (3) ) meet

Capillary flow in V-grooves

Background and motivation

• Closed capillaries: Static equilibrium
• Closed capillaries: dynamics

The interface balance equation (2.39) then implies pext−(pext−ρgz)|z=zg = 2κmγ ≈2γcosθ/r, where zg is the height of the liquid in the straw. This differential equation was first order in time and second order in z, the axial variable, describing the height of the fluid in the groove as a nonlinear diffusion process.

Figure 4.1: Two types of propellant management devices (PMDs) often found in satellite fuel tanks and used for passively routing propellant within open grooved channels

Model for free surface capillary flow in slender open V-grooves

• Slender limit form of the hydrodynamic equations
• Boundary conditions at the liquid interface
• Interface midline equation H(Z, T ) for capillary flow in slender open V-grooves The fact that the interface shape can only be a segment of a circle, and is therefore independent

This then requires that the cross-sectional shape of the liquid interface be described by a curve of constant curvature. Note also that since R(α, θ)b and Γ(α, θ) are non-negative functions for systems obeying the Concus-Finn condition, the fluid flow direction specified by equation (4.23) is strictly defined by the sign of the local interface slope∂H/∂Z in the absence of gravity.

Table 4.1: Characteristic scalings (lower case) and nondimensional variables (uppercase) used to describe dimensionless system shown in Figure 4.3.

Notable solutions

• Stationary states for time-independent Dirichlet, Neumann, and volume condi- tions
• Self-similar spreading and draining solutions with fixed boundary pressure Previous studies have delineated the conditions leading to existence and uniqueness of self-similar
• Self-similar converging and receding solutions with fixed boundary flux
• Capillary rise against gravity

A simple scaling analysis of equation (4.29) reveals that self-similar solutions are possible whenever T << L2/H ~O(L/ε). To find such solutions, it is convenient to expand equation (4.29) and rewrite it in the form 2When gravity in the Z-direction is included [see equation (4.28)], the only admissible self-similar constant is β = 1/3.

Figure 4.5: Representative stationary solutions subject to Dirichlet conditions H S (Z 1 ) = H 1 = 1.0 and H S (Z 2 ) = H 2 = 0.01, 0.33, 0.67, 1.00, and 1.33 for the range Z ∈ [Z 1 , Z 2 ] = [0, 3.0].

Discussion .1 Limitations.1Limitations

• Rounded V-grooves (U-grooves)

In the V-groove model, the contact angle is only relevant for the movement of the contact line up and down along the side walls of the groove (there is no contact line in the direction of flow). The value b= ln(ϵ−1), where ϵ is the ratio between the slip length and the characteristic interface length by which the angle is measured. Despite the complexity of these models, their dynamics remain qualitatively similar to those of the V-groove.

Introduction

Nonlinear stability of steady states

• Boundedness of C and dryout
• Aside: Exponential stability of the porous medium equation

The particular F defined in equation (5.4) turns out to be compatible with the boundary conditions of interest and has the necessary properties to demonstrate the stability of the system, as outlined below. Note that we used the Dirichlet or Neumann boundary conditions to get rid of the boundary term, [(R/3)(∂R/∂Z)]ZZ2. With the exception of this situation, all stationary states, regardless of the boundary conditions considered above, are exponentially stable.

Generalized linear stability of self-similar states

• Generalized stability of volume non-conserving self-similar solutions
• Numerical results and comparison to analytic bounds

It can be shown that L has no positive eigenvalues ​​for non-terminating solutions of S, i.e. the spectral abscissa βmax[L]<0 (this fact will be proved later). Recall from Section 3.1 that the fastest growth rate of perturbations is given by the numerical abscissa, ωmax, which is the least stable eigenvalue of the transient operator (L+L†)/2. Note that the linear stability of the uniform state S(η) = 1, first shown by Weislogel (2001), is relatively easy to see.

Figure 5.1: Representative self-similar solution S(η) (blue solid line) for advancing state

Conclusion

Influence of attractive van der Waals interactions on the optimal excitations in thermocapillary spreading. A Lyapunov functional for the evolution of solutions to the porous medium equation towards self-similarity. ELECTRIFIED CAPILLARY FLOW FROM A PERFECT CONDUCTIVE FILM IN A SLIMMER V-GROOVE CHANNEL: STABLE, SELF-SIMILAR, AND.

Introduction

We consider only electric fields that are so weak that the sensitivity to the fluid thickness of the capillary pressure at the interface is greater than the Maxwell pressure. Also note that, due to the slender limit assumption, the pressure will be assumed to be constant across the cross-section of the groove. We will first outline the derivation of the general equation of motion for a fluid in a V-groove with an electric field, without assumptions about the electrical properties of the fluid.

V-groove model with Maxwell stress

• Key model assumptions
• Bulk equations
• Flux
• Combined equation

Furthermore, the thickness of the fluid is assumed to be much smaller than the length scale of axial variations. A smoothness condition for the transverse variation across the groove of the external electric field will also be required, details of However, due to the slender boundary and slowly varying electric field assumptions, the axial curvature of the interface is assumed not to affect the electric field.

Figure 6.1: Diagrams of V-groove flow system with electric field. (a) Schematic of a perfectly conducting, wetting liquid film (0 ≤ θ < π/2) flowing within a slender open triangular groove with constant cross-section and perfectly conducting walls

Electric field distribution in conducting groove with thin fluid

• Electric field slow variation assumptions
• Electric field in a 2D wedge

Γ(α, θ) must be calculated numerically by solving the Poisson equation for the velocity along the stream, equation (6.13), in the fluid cross-section (Figure 6.1 b); the result is shown in Figure 6.3 (c). If the slow-varying condition holds, then the electric potential in the vacuum region of the groove can be a non-dimensionalized angle. More specifically, let ψouter(β) be the electric field distribution on a circular section at a distance b from the corner of the slot (i.e. the field distribution at the top of the slot), where β is the angular coordinate (see Figure 6.4) and if ψouter is the cosine transform of ψouter with wavenumber k, it is assumed that |ψouter0 /ψkouter| > (d/b)2(k−1) ∀k.

Figure 6.4: Cross-sectional schematic of the system depicted in Figure 6.1 (b), with quantities relevant to the electric field solution in a 2D wedge emphasized

Equation of motion for perfectly conducting thin film in a V-groove

• Estimate of MEP threshold values

Therefore, we expect that there is a threshold thickness (which depends on the strength of the applied field), above which the fluid in the groove is unstable. The constraint on the characteristic electric potential at the top of the groove to ensure that the liquid remains in the well-positioned regime is then. In the special case of a constant electric field, χ(Z) =χ0 [note that χ0 is so named simply to show that it is a constant; not to refer to χ(Z = 0)], the equation of motion Equation (6.43) can be rewritten as.

Analysis of equation of motion .1 Effect of electric field on flux.1Effect of electric field on flux

• Numerical analysis
• Stationary solutions
• Symmetry analysis and self-similar solution

Clearly, (HA3 −HB3) and −(HAm+4−HBm+4) have opposite signs and therefore the electric field lowers the flux given the same fluid thickness boundary conditions. In the case of a constant electric field strength χ(Z) =χ0, the self-similar equation (6.58) is simplified to Each plot uses a different set of initial slope conditions ∂ηS to construct a representative set of self-similar solutions for the relevant electric field strength χ0 and inner groove half-angle α.

Figure 6.5: Representative stationary solutions of flow of conducting liquids in V-grooves with constant applied electric field, with internal groove half angle α = 30 ◦ , and Dirichlet fluid thick-ness (H) boundary conditions, according to Equation (6.57

Stability analysis

• Nonlinear stability analysis: Stationary, quiescent state
• Nonlinear stability analysis: Stationary interface with non-zero fluid flux and constant electric field
• Generalized linear stability analysis: Self-similar states

But the thin film equation on a flat plate under an electric field is given by (Kim et al., 1992). Following the methodology of Chapter 5, we perform a generalized linear stability analysis of self-similar states with a constant electric field χ(Z) =χ0. The slightly greater stability of narrow (small α) grooves is probably due to the faster decay of the electric field within those grooves.

Numerical validation

• Numerical validation methodology

For the parameter range considered, the error in the approximation of the flux factor Γ compared to the numerically calculated flux factor Γnum. Q/H4 and in the approximation of the cross-sectional area AHb 2 compared to the numerically calculated Anum. The plots in the bottom row of Figure 6.14 show the Maxwell pressure as a fraction of the total pressure.

Figure 6.11: Cross-sectional schematic of numerical test system, based on the more general system depicted in Figure 6.1, but with the addition of a flat electrode at y = b

Discussion

This assumption was necessary to make the analytical approximation of the electric field tractable. The assumption that the walls of the grooves are perfect conductors determined the form of electricity. If the walls of the grooves are assumed to be perfect insulators, a problem arises at sharp angles of the liquid section, as the electric field there would approach a singularity.

Appendix: Additional plots

• Stationary plots
• Self-similar plots

And it will still be the case that, for fixed pressure boundary conditions, the presence of an electric field increases the flux. Because the Maxwell pressure scales as Hm (metm > 0) and the capillary pressure as H−1, then the relative strength of the electric field is smaller as H is smaller. Because the Maxwell pressure scales asSm (metm >0) and the capillary pressure asS−1, then the relative strength of the electric field is smaller than Sis.

Introduction

In the analysis of this work, the inertial terms are negligible, so the Dean effect is also negligible. However, as with the results of Chadwick (1985) and Wang (2012), we will find that the curvature of the V-groove causes the flow to change by changing the flow direction. In this work, the shape of the fluid interface will be affected by the curvature of the groove, but this effect will be due to surface tension rather than inertia.

Derivation of equations of motion .1 Assumptions and method of derivation.1Assumptions and method of derivation

• Coordinates and curvature Backbone coordinates
• Covariant notation
• Fluid equations in curvilinear coordinates
• Slender limit of fluid equations
• Boundary conditions
• Transport equation
• Cross-sectional interface shape
• Pressure
• Flux computation
• Integrated flux
• Final reduced equation Putting all the pieces together,

Extending the triad beyond the spine leads in the most natural way to the definition of the coordinate system. On the right-hand side of the equation, the correction is the additional contribution to the diffusion W due to the curved geometry. Given the cut of the groove cross-section in the plane ξ-υ, Ω(s, t), the current across this slice is RΩ(s,t)⃗u·νdξdυ, where νˆ is the unit normal to the slice.

Figure 7.2: The corner, or backbone of the V-groove is described by a space curve, with local coordinates { ξ,ˆ υ,ˆ ˆ s}

Results

• Steady state solutions

We first compare steady states in grooves of different constant curvature, with Dirichlet pressure (P) boundary conditions. In particular, the pressure in grooves with positive spine curvature is greater than that in straight grooves, which in turn is greater than that in negative-curvature grooves. It will be further shown in the next section that, with identical boundary conditions, the flux in grooves with positive backbone curvature is greater than that in grooves with straight or negative backbone curvature.