2 Cycle frequency of diffusion-driven nucleation

Figure 3. Principle sketch of the growth of a surface nucleus and the detachment of a bubble in a fluid flow.

We consider a crevice that is filled with non-condensable gas surrounded by a flowing, supersaturated liquid, as sketched in figure 3. Due to the supersaturation of the liquid there is a mass flux of gas out of the liquid into the surface nucleus. According to Fick’s law (Fick Reference Fick1855), the mass flux of gas that diffuses into the surface nucleus, ${\dot{m}}$ , is proportional to the concentration gradient at the liquid–gas interface,

(2.1) $$\begin{eqnarray}{\dot{m}}=-M{\mathcal{D}}\int _{A}\unicode[STIX]{x1D735}c\boldsymbol{\cdot }\boldsymbol{n}\,\text{d}A,\end{eqnarray}$$

with concentration $c$ measured in $\text{mol}~\text{m}^{-3}$ , diffusion coefficient ${\mathcal{D}}$ , molar mass $M$ , surface area of the nucleus $A$ and the surface normal vector $\boldsymbol{n}$ . In an equilibrium the concentration of gas that is solved in the liquid increases linearly with the pressure, as described by Henry’s law (Henry Reference Henry1803),

(2.2) $$\begin{eqnarray}c=p{\mathcal{H}},\end{eqnarray}$$

with pressure $p$ and Henry coefficient ${\mathcal{H}}$ accordingly measured in $\text{mol}~(\text{m}^{3}~\text{Pa})^{-1}$ . Neglecting pressure differences due to surface tension, a surface nucleus that is surrounded by a liquid in a concentration equilibrium will not show diffusion-driven nucleation since the concentration gradient is zero. Taking surface tension into account leads to the stability discussion referred to above.

In the case of a pressure decrease the liquid becomes supersaturated. The supersaturation is defined as

(2.3) $$\begin{eqnarray}\unicode[STIX]{x1D701}:=\frac{c_{\infty }}{c_{N}}-1.\end{eqnarray}$$

$c_{\infty }$ is the concentration of gas in the liquid and $c_{N}=p_{N}{\mathcal{H}}$ is the local saturation concentration at the surface nucleus, which is determined by the partial pressure of non-condensable gas inside the surface nucleus $p_{N}$ . A concentration distribution develops.

The surface nucleus grows and when reaching a critical size a part of the surface nucleus detaches as a bubble with mass $m_{B}$ and diameter $d_{B}$ , see figure 3. Depending on its size, this bubble might serve as cavitation nucleus when carried into the low-pressure region of the flow, cf. Blake (Reference Blake1949). A specific amount of gas, approximately the same for every event, remains in the surface nucleus and the process repeats. From daily life we know that this process will last as long as the liquid is supersaturated. In technical flows where cavitation (or degassing) can be observed there is always ‘new’ liquid that flows past the surface nucleus. This keeps the process alive for an arbitrarily long time since the average saturation concentration in the system is higher than the saturation concentration in the low-pressure region. Hence, the liquid in the low-pressure region is supersaturated. Active surface nuclei, i.e. surface nuclei that produce bubbles, are an inexhaustible source for bubbles that might act as cavitation nuclei. Then, stabilisation theories of free-stream nuclei (Fox & Herzfeld Reference Fox and Herzfeld1954; Yount et al. Reference Yount, Gillary and Hoffman1984) fade into the background.

The growth of the surface nucleus is relatively slow while the bubble detachment occurs rapidly. The nucleation rate $f$ , the number of produced bubbles per second (see figure 3), can therefore be calculated with

(2.4) $$\begin{eqnarray}f=\frac{{\dot{m}}}{m_{B}}.\end{eqnarray}$$

This relation will guide us in the following and helps us to derive a physical model that describes the nucleation process based on axiomatic equations. The mass flux ${\dot{m}}$ is treated in § 2.1 and the mass of the detaching bubbles $m_{B}$ in § 2.2. In our analysis we execute an elementary discussion rather than a full mathematical and physical treatment of the problem. The focus of the paper is to identify the main influencing variables and to draw general conclusions about diffusion-driven nucleation.

2.1 Growth phase of the nucleus

Figure 4. Sketch of the advection–diffusion problem at a surface nucleus. Evolution of a concentration field with concentration boundary layer thickness $\unicode[STIX]{x1D6FF}_{c}$ .

The mass flux ${\dot{m}}$ that diffuses into a surface nucleus can be obtained by formulating a plane generic problem, see figure 4. As already mentioned, the concentration gradient at the liquid–gas interface is needed to calculate the mass flux. Solving the diffusion problem for the growth phase will guide as to $Sh\approx 0.66Pe^{1/3}$ for the dimensionless mass flux of gas that diffuses into the surface nucleus (definitions follow in (2.6)).

In the most general case, the concentration field in the liquid satisfies the transient advection–diffusion equation

(2.5) $$\begin{eqnarray}\frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}t}+\boldsymbol{u}\boldsymbol{\cdot }\unicode[STIX]{x1D735}c=\unicode[STIX]{x1D735}\boldsymbol{\cdot }({\mathcal{D}}\unicode[STIX]{x1D735}c),\end{eqnarray}$$

where $\boldsymbol{u}$ is the velocity field of the fluid flow. We assume a stationary process, a negligible flow in the $y$ direction, negligible diffusion in the $x$ direction and a constant diffusion coefficient ${\mathcal{D}}=\text{const}$ . Near the solid surface the velocity increases linearly with the wall distance $u=\dot{\unicode[STIX]{x1D6FE}}y$ as long as the crevices are much smaller than the typical length of the device (diameter of a pipe, height of a gap). The velocity profile $u=\dot{\unicode[STIX]{x1D6FE}}y$ is also valid in turbulent flows if the typical size of the crevice $d$ is smaller than the viscous length $l_{\unicode[STIX]{x1D708}}:=\unicode[STIX]{x1D708}/\sqrt{\unicode[STIX]{x1D70F}_{w}/\unicode[STIX]{x1D71A}}$ , with viscosity $\unicode[STIX]{x1D708}$ and density $\unicode[STIX]{x1D71A}$ of the fluid and wall shear stress $\unicode[STIX]{x1D70F}_{w}$ . The former approach is called the Lévêque approximation and is a common practice for solving heat and mass transfer problems, e.g. the asymmetric Graetz problem (Edwards & Newman Reference Edwards and Newman1985). If $\dot{\unicode[STIX]{x1D6FE}}$ is the relevant kinematic quantity near the solid wall and near the crevice, the suitable definitions of Sherwood number and Péclet number are

(2.6a,b ) $$\begin{eqnarray}Sh:=\frac{{\dot{m}}}{c_{N}\unicode[STIX]{x1D701}M{\mathcal{D}}d},\quad Pe:=\frac{\dot{\unicode[STIX]{x1D6FE}}d^{2}}{{\mathcal{D}}}.\end{eqnarray}$$

The length $d$ denotes the diameter of a circular crevice having the same surface area as a hypothetical rectangular nucleus with side length $d_{N}=d\sqrt{\unicode[STIX]{x03C0}}/2$ . For $Pe\gg 1$ the boundary layer approximation (Schlichting & Gersten Reference Schlichting and Gersten2006) applied to (2.5) gives

(2.7) $$\begin{eqnarray}\dot{\unicode[STIX]{x1D6FE}}y\frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}x}={\mathcal{D}}\frac{\unicode[STIX]{x2202}^{2}c}{\unicode[STIX]{x2202}y^{2}},\end{eqnarray}$$

with the boundary conditions $c(x=0,y)=c(x,y\rightarrow \infty )=c_{\infty }$ and $c(0<x<d_{N},y=0)=c_{N}$ . Using $c_{+}:=(c-c_{N})/(c_{\infty }-c_{N})$ , $x_{+}:=x{\mathcal{D}}/(\dot{\unicode[STIX]{x1D6FE}}d_{N}^{3})$ , $d_{N+}:={\mathcal{D}}/(\dot{\unicode[STIX]{x1D6FE}}d_{N}^{2})$ and $y_{+}:=y/d_{N}$ yields the dimensionless formulation

(2.8) $$\begin{eqnarray}y_{+}\frac{\unicode[STIX]{x2202}c_{+}}{\unicode[STIX]{x2202}x_{+}}=\frac{\unicode[STIX]{x2202}^{2}c_{+}}{\unicode[STIX]{x2202}y_{+}^{2}},\end{eqnarray}$$

with $c_{+}(x_{+}=0,y_{+})=c_{+}(x_{+},y_{+}\rightarrow \infty )=1$ and $c_{+}(0<x_{+}<d_{N+},y_{+}=0)=0$ . With the similarity variable $\unicode[STIX]{x1D702}:=y_{+}/x_{+}^{1/3}$ , i.e. $c_{+}=c_{+}(\unicode[STIX]{x1D702})$ , the boundary value problem is equivalent to

(2.9) $$\begin{eqnarray}\frac{\text{d}^{2}c_{+}}{\text{d}\unicode[STIX]{x1D702}^{2}}+\frac{1}{3}\unicode[STIX]{x1D702}^{2}\frac{\text{d}c_{+}}{\text{d}\unicode[STIX]{x1D702}}=0,\end{eqnarray}$$

with $c_{+}(\unicode[STIX]{x1D702}=0)=0$ and $c_{+}(\unicode[STIX]{x1D702}\rightarrow \infty )=1$ . The transformation $\unicode[STIX]{x1D707}:=\unicode[STIX]{x1D702}^{3}/9$ yields the solution

(2.10) $$\begin{eqnarray}c_{+}(\unicode[STIX]{x1D707})=\frac{\unicode[STIX]{x0393}(1/3)-\unicode[STIX]{x0393}(1/3,\unicode[STIX]{x1D707})}{\unicode[STIX]{x0393}(1/3)}.\end{eqnarray}$$

The series expansion of the incomplete Gamma function $\unicode[STIX]{x0393}(1/3,\unicode[STIX]{x1D707})$ gives the expanded solution

(2.11) $$\begin{eqnarray}c_{+}(\unicode[STIX]{x1D707})=\frac{3\unicode[STIX]{x1D707}^{1/3}-3/4\unicode[STIX]{x1D707}^{4/3}+O(\unicode[STIX]{x1D707}^{7/3})}{\unicode[STIX]{x0393}(1/3)}.\end{eqnarray}$$

Considering only the leading term results in the approximate solution for the region near to the surface nucleus,

(2.12) $$\begin{eqnarray}\frac{c(x,y)}{c_{N}}\approx 1+\unicode[STIX]{x1D701}\frac{3^{1/3}}{\unicode[STIX]{x0393}(1/3)}\left(\frac{\dot{\unicode[STIX]{x1D6FE}}d_{N}^{2}}{{\mathcal{D}}}\right)^{1/3}\left(\frac{x}{d_{N}}\right)^{-1/3}\frac{y}{d_{N}}.\end{eqnarray}$$

As a result, for $u=\dot{\unicode[STIX]{x1D6FE}}y$ , the concentration field and the mass flux ${\dot{m}}$ are proportional to $Pe^{1/3}$ . The latter is given by the surface integral

(2.13) $$\begin{eqnarray}{\dot{m}}={\mathcal{D}}Md_{N}\int _{0}^{d_{N}}\left.\frac{\unicode[STIX]{x2202}c}{\unicode[STIX]{x2202}y}\right|_{y=0}\,\text{d}x\approx {\mathcal{D}}Md\unicode[STIX]{x1D701}c_{N}\frac{3^{4/3}\unicode[STIX]{x03C0}^{5/6}}{2^{8/3}\unicode[STIX]{x0393}(1/3)}Pe^{1/3}\end{eqnarray}$$

or dimensionless

(2.14) $$\begin{eqnarray}Sh\approx 0.66Pe^{1/3}.\end{eqnarray}$$

In our derivation we assume the no-slip condition at the solid wall and expect the velocity field to be unaffected when flowing over the surface nucleus. This does not imply the no-slip condition at the liquid–gas interface, cf. § 2.2. If the flow is better described by a uniform flow profile, $u(y)=U_{\infty }$ , one gets the solution $Sh=\unicode[STIX]{x03C0}^{1/4}2^{-1/2}(U_{\infty }\text{d}/{\mathcal{D}})^{1/2}\approx 0.94Pe^{\prime 1/2}$ with the velocity of the undisturbed flow $U_{\infty }$ (Higbie Reference Higbie1935). This expression is an upper bound for the mass flux. Parkin & Kermeen (Reference Parkin and Kermeen1963) considered the growth of gas bubbles in the boundary layer of a submerged body due to convective air diffusion and get a similar solution. They additionally take diffusion in the flow direction into account. The work of Parkin and Kermeen is one of the few works that consider the intensification of the diffusion mass flux due to convective diffusion in the context of hydrodynamic cavitation.

The results $Sh\approx 0.66Pe^{1/3}$ and $Sh\approx 0.94Pe^{\prime 1/2}$ can be compared to the academic case studied by Epstein & Plesset (Reference Epstein and Plesset1950) mentioned before. The steady state solution for the mass flux that diffuses into a resting spherical bubble with surface area $\unicode[STIX]{x03C0}d^{2}/4$ is $Sh=\unicode[STIX]{x03C0}$ . A comparison of the three solutions shows that advection of course intensifies the mass flux of gas into the surface nucleus. It may be observed that the constant value for the steady case can be larger than the results of the convective approaches in the case of small Péclet numbers. The reason is that the previous assumption of negligible diffusion in the $x$ direction is only valid for $Pe\gg 1$ or $Pe^{\prime }\gg 1$ , respectively. Thus, a comparison of the three cases only makes sense for large Péclet numbers, which usually is fulfilled in cavitating flows.

In the following we use the derived relation $Sh\propto Pe^{1/3}$ , which implies ${\dot{m}}\propto \dot{\unicode[STIX]{x1D6FE}}^{1/3}$ in dimensional quantities. Following equation (2.4), both factors ${\dot{m}}$ and $1/m_{B}$ determine the nucleation rate. As will be seen later, $1/m_{B}$ also follows a power law with the shear rate, showing an exponent even greater than $1/3$ . Thus, the influence of the shear rate on the mass of the bubbles is greater than its influence on the mass flux. An upper bound for the nucleation rate can be determined by using the relation $Sh\propto Pe^{1/2}$ and thus ${\dot{m}}\propto \dot{\unicode[STIX]{x1D6FE}}^{1/2}$ .

The analysis of the diffusion mass flux has to be understood as an elementary discussion of the problem rather than an in-depth examination of all aspects. By assuming a flat interface we neglect the influence of surface tension on the concentration at the liquid–gas interface (smaller mass flux) and the increase of the surface area of the nucleus during its growth (larger mass flux). These effects counteract each other. We also neglect the interaction of the growing surface nucleus with the concentration boundary layer that probably leads to a smaller concentration boundary layer thickness and thus to higher mass fluxes. In total we expect the model equation (2.14) to be a lower bound for the actually occurring diffusion mass flux. A quantification of the mentioned effects is hardly possible on the basis of our approach. A numerical solution of the transient advection–diffusion equation coupled with calculations of the bubble growth and the velocity profile could deliver more detailed results in the future. Nevertheless, our approach captures the relevant physical content of the problem, as the comparison with experimental results in § 4 demonstrates.

2.2 Bubble detachment

Figure 5. Sketch of three types of bubble detachment.

A surface nucleus sitting in the wall of a container in a quiescent liquid, e.g. drinking glass or at a particle in the liquid bulk, grows until it reaches a critical size and a bubble detaches. The buoyancy force overcomes the capillary force. This case has been investigated extensively, cf. Fritz (Reference Fritz1935), Jones et al. (Reference Jones, Evans and Galvin1999), Liger-Belair (Reference Liger-Belair2005). In the present paper we consider the detachment of bubbles from micro crevices in a fluid flow and investigate the influence of the related forces.

We investigate three different cases and end up with three relations for the dimensionless bubble diameter of the form

(2.15a,b ) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}:=\frac{d_{B}}{d},\quad \unicode[STIX]{x1D6FF}=\unicode[STIX]{x1D6FF}(We),\end{eqnarray}$$

with Weber number

(2.16) $$\begin{eqnarray}We:=\frac{\unicode[STIX]{x1D71A}\dot{\unicode[STIX]{x1D6FE}}^{2}d^{3}}{S}.\end{eqnarray}$$

In contrast to the static case, the buoyancy force only plays a minor role, especially when crevices on a microscale are considered. The capillary force is balanced by a dynamic force proportional to $\unicode[STIX]{x1D71A}U^{2}b^{2}$ with a characteristic velocity  $U$ and a characteristic length  $b$ . The dynamic force is made up of two effects: form drag and shear lift due to an unsymmetrical flow around the surface nucleus. Depending on the flow situation, the characteristic length $b$ is the characteristic length of the device $L$ (e.g. diameter of a pipe or height of a gap), the diameter of the crevice $d$ , the bubble diameter $d_{B}$ or a typical length of a surface roughness element $k$ describing the microscopic geometry of the edge of the crevice, see figure 5. The detaching bubbles are characterised by their diameter $d_{B}$ . In the case of spherical bubbles $d_{B}$ is the bubble diameter. In the case of non-spherical bubbles $d_{B}$ is the equivalent diameter of a sphere with the same volume, $d_{B}=(6V_{B}/\unicode[STIX]{x03C0})^{1/3}$ with bubble volume $V_{B}$ . We further assume that the surface nucleus has a mobile surface, resulting in negligible friction at the liquid–gas interface. Thus, the viscous force acting on the surface nucleus is considered to be small. The concept of mobile surfaces has been studied extensively in the context of rising gas bubbles (Lochiel & Calderbank Reference Lochiel and Calderbank1964; Clift, Grace & Weber Reference Clift, Grace and Weber1978; Duineveld Reference Duineveld1995; Peters & Els Reference Peters and Els2012). If the liquid–gas interface is immobilised, e.g. by surfactants or organic material, the viscous force has to be taken into account.

Our analysis of bubble detachment from surface nuclei is based on two competing forces, the dynamic force and the capillary force. It should be noted that there are other major research areas in which the detachment of bubbles is of importance: (i) bubble detachment from wall orifices in liquid cross-flows due to gas injection (e.g. Clift et al. Reference Clift, Grace and Weber1978; Nahra & Kamonati Reference Nahra and Kamonati2003; Duhar & Colin Reference Duhar and Colin2006) and (ii) bubble detachment in flow boiling (e.g. Chen, Pan & Ren Reference Chen, Pan and Ren2012). The main difference to our approach is the need to consider additional forces in the above-mentioned cases, see cited works. In the case of bubble detachment from wall orifices an additional force proportional to the squared gas injection flow rate, called the momentum flux force, has to be considered. In the case of flow boiling a growth force of the surface nucleus as well as an added-mass force have to be taken into account. Both forces come into play because of the rapid expansion of the surface nucleus due to the evaporation of the liquid. Especially the growth force can be dominant, cf. Chen et al. (Reference Chen, Pan and Ren2012). In our case the growth rate of the surface nucleus is relatively small and can be neglected.

To complete our picture of the diffusion-driven nucleation we consider the three cases sketched in figure 5.

Case I: The size of the detaching bubble is of the same order of magnitude as the surface nucleus and the characteristic length of the device, $d_{B}\sim d\sim L$ . Thus $d_{B}$ depends on both the size of the surface nucleus and the characteristic length of the device. This case has been investigated by Peters & Honza (Reference Peters and Honza2014) and Groß, Ludwig & Pelz (Reference Groß, Ludwig and Pelz2015) in a laminar radial gap flow. In laminar flows the typical velocity is proportional to the product of wall shear rate and gap height, $U\propto \dot{\unicode[STIX]{x1D6FE}}L$ . The cross-section area, where the dynamic force is acting, is $b^{2}\propto dL$ . The volume of the detaching bubble is $d_{B}^{3}\propto d^{2}L$ . Thus the dynamic force is proportional to $\unicode[STIX]{x1D71A}\dot{\unicode[STIX]{x1D6FE}}^{2}d^{3}d_{B}^{9}/d^{8}$ . With the capillary force being proportional to $Sd$ one obtains the specific form of equation (2.15)

(2.17) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{3}=(\unicode[STIX]{x1D716}_{I}We)^{-1/3},\end{eqnarray}$$

where $\unicode[STIX]{x1D716}_{I}$ is a dimensionless constant.

Case II: The size of detaching bubble is of the same order of magnitude as the size of the crevice, and both are significantly smaller than the characteristic length of the device, $d_{B}\sim d\ll L$ . Thus the size of the bubble only depends on the size of the crevice. With the typical velocity $U\propto \dot{\unicode[STIX]{x1D6FE}}d_{B}$ , the cross-section area $b^{2}\propto d_{B}^{2}$ , and the capillary force being proportional to $Sd$ , one gets

(2.18) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{3}=(\unicode[STIX]{x1D716}_{II}We)^{-3/4},\end{eqnarray}$$

with the dimensionless constant $\unicode[STIX]{x1D716}_{II}$ . As we will show later, this case describes the most frequently observed mechanism in our experiments with water.

Case III: The detaching bubble is smaller than the surface nucleus and the characteristic length of the device, $d_{B}\ll d$ and $d_{B}\ll L$ . In this case the bubble diameter $d_{B}$ cannot be a function of these lengths. The bubble size depends on the microscopic characteristics of the crevice represented by length $k$ , see figure 5. We observed this detachment mechanism only at crevices with relatively large diameters. For example, this detachment mechanism has been observed in a previous study with silicone oil (Groß et al. Reference Groß, Ludwig and Pelz2016) as well as in the streak cavitation experiments conducted in Lausanne, cf. figure 2 and Guennoun et al. (Reference Guennoun, Farhat, Bouziad and Avellan2003), Guennoun (Reference Guennoun2006). The typical velocity and the cross-section area are the same as in case II. However, the capillary force does not act over the hole crevice but rather on a length proportional to $d_{B}$ . Hence we find

(2.19) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{3}=(\unicode[STIX]{x1D716}_{III}We)^{-1},\end{eqnarray}$$

with the dimensionless constant $\unicode[STIX]{x1D716}_{III}$ .

For cavitating flows in pumps, turbines, nozzles or on propellers, cases II and III are of particular importance, whereas case I could be of importance for cavitation in journal bearings, narrow gaps of seals, as well as in microfluidic devices or biofluidic situations.

To complete the analysis, the bubble mass is required. For an ideal gas with the constitutive equation $p_{B}=\unicode[STIX]{x1D71A}_{B}RT_{B}$ one obtains

(2.20) $$\begin{eqnarray}m_{B}=\frac{\unicode[STIX]{x03C0}}{6}\frac{p_{B}d_{B}^{3}}{RT_{B}}=\frac{\unicode[STIX]{x03C0}}{6}\frac{p_{B}d^{3}}{RT}\unicode[STIX]{x1D6FF}^{3},\end{eqnarray}$$

with the temperature of the bubble being equal to the temperature of the surrounding liquid, $T_{B}=T$ . The initial relation $f={\dot{m}}/m_{B}$ together with (2.13), Henry’s law $c=p{\mathcal{H}}$ and the ideal gas constant ${\mathcal{R}}=RM$ yields the Strouhal number relation

(2.21) $$\begin{eqnarray}Sr:=\frac{fd^{2}}{{\mathcal{D}}}\approx 1.26\unicode[STIX]{x1D6EC}\frac{c_{\infty }/{\mathcal{H}}-p_{N}}{p_{B}}Pe^{1/3}\unicode[STIX]{x1D6FF}^{-3},\end{eqnarray}$$

with

(2.22) $$\begin{eqnarray}\unicode[STIX]{x1D6EC}:={\mathcal{R}}T{\mathcal{H}}\end{eqnarray}$$

being the dimensionless solubility of gas in the liquid. For $p_{B}\approx p_{N}$ this reduces to

(2.23) $$\begin{eqnarray}Sr\approx 1.26\unicode[STIX]{x1D6EC}\unicode[STIX]{x1D701}Pe^{1/3}\unicode[STIX]{x1D6FF}^{-3},\end{eqnarray}$$

with

(2.24) $$\begin{eqnarray}\displaystyle \unicode[STIX]{x1D6FF}^{3}(We)=\left\{\begin{array}{@{}ll@{}}(\unicode[STIX]{x1D716}_{I}We)^{-1/3}\quad & \text{for case I},\\ (\unicode[STIX]{x1D716}_{II}We)^{-3/4}\quad & \text{for case II, and}\\ (\unicode[STIX]{x1D716}_{III}We)^{-1}\quad & \text{for case III.}\end{array}\right. & & \displaystyle\end{eqnarray}$$

The absolute values of the dimensionless constants $\unicode[STIX]{x1D716}_{I}$ , $\unicode[STIX]{x1D716}_{II}$ and $\unicode[STIX]{x1D716}_{III}$ , and the limits of validity are not known. One has to keep in mind that our analysis is based on simple concepts for the capillary force and the dynamic force. We do not consider contact line dynamics, the influence of contact angles and the role of the edge of the crevices for the detachment process. The length over which the capillary force is acting might differ from the length we assume. For a case II detachment the capillary force is calculated with the circumference of the crevice, which refers to an ideal situation, i.e. a symmetrical surface nucleus adhered to the edge of the crevice. To completely solve these issues, more experimental and numerical investigations on contact line dynamics, the role of the contact angles and the micro-structure of the crevices are needed, especially for large shear rates. In other research fields, e.g. heterogeneous nucleation in acoustic cavitation (Atchley & Prosperetti Reference Atchley and Prosperetti1989; Borkent et al. Reference Borkent, Gekle, Prosperetti and Lohse2009), the growth of surface nano-bubbles (Lohse & Zhang Reference Lohse and Zhang2015) and detachment of vapour bubbles of smooth surfaces in flow boiling (Klausner et al. Reference Klausner, Mei, Bernhard and Zeng1993; Chen et al. Reference Chen, Pan and Ren2012), the role of contact line dynamics and contact angle are understood far more and ought to be applied to the situation discussed here. In addition we do not consider the effect of the shape of the surface nucleus on the drag and lift coefficients, which are hidden in the dimensionless constants as well. Taking all these issues into account, we cannot yet specify the values of the dimensionless constant in advance. In the following we validate the asymptotic relations of (2.23) experimentally.

3 Experimental set-up

Experiments on cavitation are often carried out in complex test facilities since the results are to be applied on likewise complex machines such as pumps, propellers or valves. Due to the complexity and size of these test facilities, i.e. cavitation tunnels, there often is a limited accessibility to the working section, resulting in extensive experimental procedures and time-consuming conversion work. Usually the focus is on large-scale cavitation phenomena such as sheet and cloud cavitation, cf. Pelz et al. (Reference Pelz, Keil and Ludwig2014, Reference Pelz, Keil and Groß2017). Without a doubt, small-scale phenomena such as nucleation are difficult to study in these experimental set-ups. The experimental set-up we use for our nucleation experiments is kept simple so that a good accessibility (mechanical and optical) is ensured and we can easily control the experimental parameters. In technical applications a bubble might show an exponential growth (cavitation) right after its detachment from a surface nucleus. The almost simultaneous occurrence of diffusion-driven nucleation and cavitation, cf. figure 2, complicates both the execution of the experiments and the analysis of the collected data. For this reason it is advantageous to separate these two phenomena. In fact, in our experiments the pressure is way above vapour pressure so that cavitation cannot be observed. The formation of bubbles can only be caused by diffusion.

Figure 6. Sketch of experimental set-up.

The focus of the experiment is on the measurement of the nucleation rates, the growth of the surface nuclei and the detachment process by means of high-speed visualisation. Figure 6 shows a sketch of the experimental set-up, consisting of two tanks with a capacity of 50 l each, the test section, piping, gas supply, various measuring instruments, the high-speed camera system and a submicron filtering unit to filter out small particles (not shown in the sketch). The fluid flow is driven by a pressure difference between the two tanks that can be adjusted using compressed air. The liquid, usually tap water, flows from tank 1 through the piping and the test section, where the surface nuclei are located, into tank 2. We measure the pressure in both tanks, the pressure in the test section, the oxygen content and the temperature of the liquid in tank 1. The flow rate is measured with a magneto-inductive flow meter. High-speed visualisations are conducted using an IDT Motion Pro Y7 S3 camera with a resolution of $1920\times 1080$  pixels and a maximum frame rate of 10 600 f.p.s and stroboscopic lighting. It is essential to use a long-distance microscope to visualise the micrometre-sized crevices and the detaching bubbles. To eliminate motion blur of the moving bubbles it is necessary to use exposure times of microseconds, which is in conflict with sufficient lighting. Thus, the measurement of the nucleation rate is much easier and more precise than the measurement of the diameters of the moving bubbles. In order to make a statement about the bubble size possible, the size of the surface nuclei is analysed immediately prior to the detachment.

The test section we use is a pipe with a rectangular cross-section, see figure 7. The rectangular pipe has a width of $w=30~\text{mm}$ and an adjustable gap height $L$ . Distance plates are inserted into the test section to change the gap height between 1.8 and 3.8 mm in discrete steps. We achieve flow rates of up to $10~\text{l}~\text{min}^{-1}$ . By closing a control valve the flow rate can be set to zero. We expect the velocity profile to be laminar. Thus, the shear rate at the wall is $\dot{\unicode[STIX]{x1D6FE}}=6Q/L^{2}w$ , with flow rate $Q$ . We achieve shear rates of the order of magnitude of $10^{2}~\text{s}^{-1}$ – $10^{3}~\text{s}^{-1}$ and supersaturations of the order of magnitude of 0.1–1. In comparison to technical relevant flows in pumps, turbines, propellers or valves, the shear rates are still relatively small.

Figure 7. Sketch of pipe with rectangular cross-section. Blind holes etched in silicon wafer work as surface nuclei. The high-speed camera is mounted either in the top ( $x$ – $z$ -plane) or the side view ( $x$ – $y$ -plane) perspective.

Since the lid and the sidewalls are made of acrylic glass, high-speed visualisations from the top and side view perspective are possible. The observed crevices are cylindrical blind holes with diameters of $25~\unicode[STIX]{x03BC}\text{m}$ , $50~\unicode[STIX]{x03BC}\text{m}$ and $100~\unicode[STIX]{x03BC}\text{m}$ etched into square-shaped silicon wafers with a edge length of 10 mm. The depth of the blind holes is approximately $20~\unicode[STIX]{x03BC}\text{m}$ , while the silicon wafer has a thickness of 0.5 mm. The silicon wafers were produced and kindly given to us by Dr David Fernandez Rivas from the Mesoscale Chemical Systems group at University of Twente, Netherlands. See Fernandez Rivas et al. (Reference Fernandez Rivas, Prosperetti, Zijlstra, Lohse and Gardeniers2010) and Zijlstra et al. (Reference Zijlstra, Fernandez Rivas, Gardeniers, Versluis and Lohse2015) for further information about the production. The ratio of gap height of the test section to diameter of the surface nuclei is at least 18. The maximum ratio is 152. Thus, we expect the nucleation process not to be influenced by the height of the gap. The silicon wafers are mounted on a specimen that is inserted into the test section.

Before starting a series of measurements one has to ensure that the silicon wafers are very clean and dust particles have been removed. The crevices are filled with gas by simply filling the test section with liquid, cf. Bankoff (Reference Bankoff1958). Small amounts of gas are entrapped in the crevices and work as surface nuclei. The saturation of the liquid in tank 1 is adjusted by injecting compressed air at the bottom of the tank. The measurement of the oxygen content is used to ensure that the liquid is saturated at a specific pressure. One of the most important qualities of the set-up is that the supersaturation in the test section can be varied while the other parameters can be kept constant. The pressure in the test section is used to determine the pressure of non-condensable gas in the surface nuclei ( $p_{N}$ ), while the pressure in tank 1 corresponds to the saturation pressure of the liquid ( $p_{\infty }$ ), which is controlled with an oxygen sensor. The supersaturation of the liquid in the test section can be determined with $\unicode[STIX]{x1D701}:=p_{\infty }/p_{N}-1$ , cf. equation (2.3). Increasing the pressure level of the set-up and keeping the pressure difference between the tanks constant, leads to a constant flow rate (for a given gap height) while the supersaturation changes. On the other hand it is possible to keep the supersaturation constant while the flow rate changes. If the pressure level is changed rapidly, the liquid in tank 1 is still saturated at the saturation pressure that has been adjusted before. Keeping the pressure in the test section constant results in a constant supersaturation and a variable flow rate that depends on the pressure difference between the tanks.

4 Experimental results and discussion

Figure 8. Growth of a surface nucleus and detachment of a bubble at a crevice with $d=25~\unicode[STIX]{x03BC}\text{m}$ in water from the side view perspective. The flow is from left to right. The time interval between the images is 500 ms. The surface nucleus mirrors in the silicon wafer.

To get an extended understanding of the nucleation process we investigate the nucleation rates and the volume of detaching bubbles by means of high-speed visualisation. It will be seen that the model for a case II bubble detachment, equation (2.18), and the Strouhal number relation, equation (2.23), are well validated by the experimental findings.

By way of example figure 8 shows a sequence of the growth of a surface nucleus in water and the detachment of a bubble at a micro crevice with a diameter of $d=25~\unicode[STIX]{x03BC}\text{m}$ in a fluid flow with a shear rate of $2026~\text{s}^{-1}$ . During its growth the surface nuclei forms a spherical cap that bends over the edge of the crevice with increasing size. Prior to its detachment the surface nucleus is obviously larger than the crevice. After the detachment (not visible in the pictures) the process repeats. For the shown experiment the nucleation rate is 0.22 Hz. The figure is a typical recording that is used for the evaluation and analysis of nucleation rates and bubble detachment.

Figure 9. Dimensionless bubble volume $\unicode[STIX]{x1D6FF}^{3}$ versus Weber number $We:=\unicode[STIX]{x1D71A}\dot{\unicode[STIX]{x1D6FE}}^{2}d^{3}/S$ . The solid lines are calculated with equation (2.18). Grey symbols are based on measurements of the size of the detaching bubbles. Black symbols are calculated with equation (4.1) based on nucleation rate measurements. Different symbols mark different crevice diameters: $25~\unicode[STIX]{x03BC}\text{m}$ filled circle, $50~\unicode[STIX]{x03BC}\text{m}$ open circle, $100~\unicode[STIX]{x03BC}\text{m}$ square. The ratio of crevice diameter to gap height is always smaller than $1/18$ . The liquid is water with the properties $\unicode[STIX]{x1D71A}=998~\text{kg}~\text{m}^{-3}$ , ${\mathcal{D}}=2\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ , $S=70\times 10^{-3}~\text{kg}~\text{s}^{-2}$ and ${\mathcal{H}}=7.4\times 10^{-6}~\text{mol}~(\text{m}^{3}~\text{Pa})^{-1}$ .

Using the recordings from the side view perspective, it is possible to determine the volume of the surface nucleus immediately prior to the bubble detachment. Assuming that the volume of the detached bubble equals the part of the surface nucleus that protrudes into the water prior to the detachment, cf. figure 8, frame 9, allows the determination of the dimensionless bubble volume $\unicode[STIX]{x1D6FF}^{3}$ . The measurement of the bubble size of course may suffer from measurement uncertainties. The bubble volume is determined using a two-dimensional projection of the surface nucleus. Deformations in the third dimension cannot be considered. Prior to the detachment the surface nucleus is in motion and shows deformations of the shape, which makes the determination of the bubble volume difficult in some cases, especially in the case of large shear rates. Furthermore, the resolution of the images complicates an exact determination of the bubble volume. Based on our evaluation, measuring uncertainties of 10 %–15 % of the bubble diameter $\unicode[STIX]{x1D6FF}$ are reasonable values. In addition, large fluctuations between the individual measurements have to be considered. So far, it is not known which parameters are responsible for the scattering of the data.

The grey symbols in figure 9 represent the measured dimensionless bubble volume $\unicode[STIX]{x1D6FF}^{3}$ plotted against Weber number $We:=\unicode[STIX]{x1D71A}\dot{\unicode[STIX]{x1D6FE}}^{2}d^{3}/S$ for different crevice diameters. The different symbols mark different crevice diameters that have been evaluated. As one expects, the dimensionless bubble volume decreases with increasing Weber number, and thus increasing shear rate. In the measurements we observed case II detachments. The measured bubble volumes are in the range of $\unicode[STIX]{x1D6FF}^{3}=1$ – $1\times 10^{3}$ , i.e. a range of $\unicode[STIX]{x1D6FF}=1$ –10 for the bubble diameter.

In addition to size measurements, we also measured the nucleation rate $f$ . Our experiments showed that there is a nonlinear dependence of the nucleation rate on the shear $\dot{\unicode[STIX]{x1D6FE}}$ and a linear dependence on the supersaturation $\unicode[STIX]{x1D701}$ . We observed that the nucleation rate is very constant over time. We recorded at least 15 bubble detachments at each operation point so that we can calculate a reasonable standard error. Our measurements show that the 95 % confidence interval of the nucleation rate is smaller than 10 % of the measured value of the nucleation rate, usually better.

In figure 9 an additional and independent representation of the dimensionless bubble volume $\unicode[STIX]{x1D6FF}^{3}$ is presented. The black symbols in figure 9 are deduced from measurements of the nucleation rate $f$ (or Strouhal number $Sr$ ) which can be measured with smaller measuring uncertainty than the bubble size. We rearrange equation (2.23), calculate

(4.1) $$\begin{eqnarray}\unicode[STIX]{x1D6FF}^{3}=\frac{1.26\unicode[STIX]{x1D701}\unicode[STIX]{x1D6EC}Pe^{1/3}}{Sr},\end{eqnarray}$$

and plot it against Weber number $We=\unicode[STIX]{x1D71A}\dot{\unicode[STIX]{x1D6FE}}^{2}d^{3}/S$ . As one expects, the calculated dimensionless bubble volume also decreases with increasing Weber number. Together, both data sets, i.e. bubble size and nucleation rate measurements, underline that our model is a reasonable approach describing diffusion-driven nucleation from surface nuclei.

The graph shows that the data of the calculated dimensionless bubble volume approach the relation $\unicode[STIX]{x1D6FF}^{3}=(\unicode[STIX]{x1D716}_{II}We)^{-3/4}$ with $\unicode[STIX]{x1D716}_{II}=5\times 10^{2}$ . Based on the bubble size measurements, $\unicode[STIX]{x1D716}_{II}=5\times 10^{1}$ is a more reliable value. The main reason for the discrepancies between the two data sets are (i) large (systematic) measurement uncertainties of the bubble size measurements, which cannot be determined further so far (assumption that bubble size equals size of protruding part of surface nucleus, size measurement based on two-dimensional projection, moving surface nucleus) and (ii) uncertainties in the mass flux model. As discussed in § 2.1, the calculated diffusion mass flux can be interpreted as a lower bound. The surface area of the surface nuclei is obviously larger than the surface area of the crevices, cf. figure 8. Furthermore, the influence of the protruding surface nucleus on the thickness of the concentration boundary layer is neglected. An underestimation of the diffusion mass flux leads to smaller values of the calculated dimensionless bubble diameter, and thus to a higher value of $\unicode[STIX]{x1D716}_{II}$ , cf. equation (4.1). Both mentioned aspects could easily sum up to an order of magnitude in the mass flux, and thus in the calculated dimensionless bubble volume. It should be noted that for a constant Weber number a factor of ten in the dimensionless constants leads to a reduction of the bubble volume by a factor of 0.17 and to a reduction of the bubble diameter by a factor of 0.56.

For a symmetrical surface nucleus that is adhered to the edge of the crevice one would expect $\unicode[STIX]{x1D716}_{II}$ to be of the order of unity, since it represents the detachment criterion, which says that the dynamic force has to overcome the capillary force, which is calculated with the circumference of the crevice. This refers to an ideal situation. As mentioned in § 2.2, our detachment model does not contain all aspects of the detachment process, i.e. contact line dynamics, surface energy issues, the role of the edge of the crevice for bubble detachment, as well as the shape-dependent drag and lift coefficients. The detachment model has to be understood as a rough approach. High-speed visualisations of the growth of surface nuclei show that the surface nucleus bends over the edge of the crevice and is far from being symmetrical, cf. figure 8. The deformation of the surface nucleus depends on both its size and the shear rate at the wall. Large values of $\unicode[STIX]{x1D716}_{II}$ as shown in figure 9 indicate an overestimation of the capillary force. The length over which the capillary force is acting might differ from the length we assume, i.e. the diameter of the crevice. If this length can be determined somehow, $\unicode[STIX]{x1D716}_{II}$ can be scaled with $\unicode[STIX]{x1D716}_{II}^{\ast }=\unicode[STIX]{x1D716}_{II}d^{\ast }/d$ , with $d^{\ast }$ being the relevant length.

In our experiments we observe nucleation rates of the order of magnitude of 0.1 Hz–1 Hz in the investigated range of shear rates. At first glance, these frequencies seem to be very low. One has to keep in mind that shear rates in cavitating flows can be substantially higher, i.e. $10^{4}~\text{s}^{-1}$ – $10^{5}~\text{s}^{-1}$ , cf. § 5. On the other hand, crevices in technical applications can be at least one order of magnitude smaller than in our experiments, depending on the production process (turning, milling, grinding, casting). Diameters of $d\approx 1~\unicode[STIX]{x03BC}\text{m}$ are reasonable in technical applications. If the model assumptions hold true also for higher shear rates, the results can be used to make statements on the nucleation rate also for these cases. On the basis of figure 9 one can evaluate a Strouhal number of $Sr=0.1$ for a Weber number of approximately $We\approx 10^{-3}$ . In the case of a moderate supersaturation of $\unicode[STIX]{x1D701}=1$ and ${\mathcal{D}}=2\times 10^{-9}~\text{m}^{2}~\text{s}^{-1}$ , this is related to a nucleation rate of $f=200~\text{Hz}$ for a crevice diameter of $d=1~\unicode[STIX]{x03BC}\text{m}$ . The required shear rate for this to happen is $\dot{\unicode[STIX]{x1D6FE}}=2.6\times 10^{5}~\text{s}^{-1}$ .

This example shows that the results presented can indeed be applied to technical flow problems. Of course, the transfer of the model to higher shear rates might be crucial since it has been validated only for smaller shear rates. So far, we see now restrictions that the model holds true for these cases. This has to be discussed in future investigations. In § 5 the model is applied to two cases of practical interest.

4.1 Bubble detachment as a Plateau–Rayleigh instability

Figure 10. Detachment of a bubble with volume $V_{B}$ from a surface nucleus as a consequence of a Plateau–Rayleigh instability from the side view perspective. The time interval between the images is 2 ms. The flow is from left to right. The liquid is silicone oil with a kinematic viscosity of $20\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ and a capillary constant of $21\times 10^{-3}~\text{kg}~\text{s}^{-2}$ . The surface nucleus and the detached bubble mirrors in the polished surface of the specimen.

In Groß et al. (Reference Groß, Ludwig and Pelz2016) we presented experimental results gained in a similar experimental set-up using the same test section. In these experiments we used silicone oil with a kinematic viscosity of $20\times 10^{-6}~\text{m}^{2}~\text{s}^{-1}$ instead of water. The surface tension of the used silicone oil is $21\times 10^{-3}~\text{kg}~\text{s}^{-2}$ and the density is $945~\text{kg}~\text{m}^{-3}$ . Relatively large blind holes with a diameter of 0.8 mm worked as nucleation sites. The ratios of gap height to the diameter of the surface nucleus are $L/d=1.25$ and 3.75, respectively. Due to these small ratios, the fluid flow and thus the bubble detachment mechanism are influenced strongly. Nevertheless, we observed a detachment mechanism that might be of particular importance for some applications.

Figure 10 shows twelve high-speed images of the detachment of a bubble from a surface nucleus in the side view perspective, cf. Groß et al. (Reference Groß, Ludwig and Pelz2016). The images illustrate that the detachment process is a consequence of a Plateau–Rayleigh instability (Rayleigh Reference Rayleigh1879). The detachment works as follows. The initial situation is a surface nucleus that expands over the edge of the bore and forms a cylindrical body. Small perturbations with different wavelengths that occur due to the fluid flow deform the cylindrical part of the nucleus. Deformations with wavelengths smaller than a critical wavelength increase the surface of the nucleus. Wavelengths larger than the critical wavelength lead to a reduction of the surface. Regarding to Rayleigh’s work principle, the nucleus is stable to deformations that increase the surface since work would have to be done. Deformations that decrease the surface will not be damped but increased because the system attempts to reach an energetically favourable state. For a cylindrical body with diameter $d_{c}$ with an infinite length the critical wavelength is its circumference $\unicode[STIX]{x03C0}d_{c}$ (Rayleigh Reference Rayleigh1879).

The largest wavelength that occurs approximately equals the length of the cylindrical part of the nucleus. The length of the nucleus, and thus the largest wavelength, increases while the surface nucleus is growing. The nucleus starts to oscillate up and down as soon as the critical length is reached. Since the perturbations are not attenuated, the cylindrical part of the surface nucleus deforms so strongly that it constricts and a bubble detaches. The Plateau–Rayleigh instability is a geometrical criterion. Of course, the bubble shape and the expansion over the edge of the bore are results of the acting forces.

Figure 11. (a) Volume of detached bubble versus shear rate. (b) Dimensionless bubble volume $\unicode[STIX]{x1D6FF}^{3}$ versus Weber number. The solid line is calculated with equation (2.19) and $\unicode[STIX]{x1D716}_{III}=0.01$ . (c) Volume of detached bubble versus diameter of the cylindrical part of the surface nucleus. The solid line marks the critical volume $V_{B}=\unicode[STIX]{x03C0}^{2}d_{c}^{3}/4$ . The open circles mark measurements with a gap height of 3 mm. The filled circles mark measurements with a gap height of 1 mm. The liquid is silicone oil with the properties $\unicode[STIX]{x1D71A}=945~\text{kg}~\text{m}^{-3}$ and $S=21\times 10^{-3}~\text{kg}~\text{s}^{-2}$ .

Figure 11 shows how the volume of the detached bubbles depends on the shear rate (graph a) as well as on the cylindrical part of the surface nucleus (graph c). As one expects, the volume of the detached bubbles decreases with increasing shear rate. The graphs illustrate that the detachment process depends only marginally on the gap height. Figure 11(a) also confirms the validity $Sr\propto WePe^{1/3}$ , equation (2.23) case III. The model implies $m_{B}\propto V_{B}\propto \dot{\unicode[STIX]{x1D6FE}}^{-2}$ , which can also be found in the graph. At first glance this finding is somehow surprising, since the detachment mechanism described here is not driven by forces but rather a stability limit based on surface energy. In fact, the growth of the nucleus and the formation of the shape follow the same physical law as the detachment mechanism described in § 2.2. The only difference is the trigger that causes the detachment (balance of forces or Rayleigh–Plateau instability). To underline this finding, figure 11(b) shows the dimensionless bubble volume $\unicode[STIX]{x1D6FF}^{3}$ plotted against Weber number. The solid line is calculated with equation (2.19) and the dimensionless constant is determined as $\unicode[STIX]{x1D716}_{III}=0.01$ . One clearly sees that the experimental results are well described by the asymptote $\unicode[STIX]{x1D6FF}^{3}=(\unicode[STIX]{x1D716}_{III}We)^{-1}$ .

It should be noted that the value of $\unicode[STIX]{x1D716}_{III}$ is very small compared to the values of $\unicode[STIX]{x1D716}_{II}$ discussed in the previous section. This is due to an overestimation of the dynamic force in the present case. The surface nucleus shows an ‘streamlined’ shape leading to a reduction of drag. Hence, it is not surprising that $\unicode[STIX]{x1D716}_{III}$ has a relatively small value.

Figure 11(c) illustrates how the volume of the detached bubbles increases with increasing diameter of the cylindrical part of the nucleus. As mentioned before, the surface nucleus becomes unstable when it reaches a critical size, and perturbations with wavelengths larger than its diameter occur. The straightforward approach to calculate the volume of the detached bubbles is to calculate the volume of a cylinder whose length equals its circumference, $V_{B}=\unicode[STIX]{x03C0}^{2}d_{c}^{3}/4$ . The solid line in figure 11(c) represents this calculation. The volume of the detached bubble is proportional to the cube of the diameter of the cylindrical part. The measured values lie only slightly above the solid line.

Our experiments with water and silicone oil demonstrate that there are a variety of detachment mechanisms that might occur, cf. figures 8 and 10. At first sight, the Plateau–Rayleigh instability seems not to be of practical importance, especially for water flows. However, there is experimental evidence that this detachment mechanism also occurs in hydrodynamic cavitation. Guennoun (Reference Guennoun2006) observed the detachment of bubbles from a surface nucleus attached to a surface irregularity (see figure 4.7 of his work). The detachment behaviour looks very similar to the behaviour described here (Martijn van Rijsbergen, personal communication, February 2, 2016).

5 Application to technical flow problems

In the previous sections we demonstrated that nucleation rates can be calculated on the basis of the approach $f={\dot{m}}/m_{B}$ . We showed that the three correlations $Sr\propto \unicode[STIX]{x1D701}We^{1/3}Pe^{1/3}$ , $Sr\propto \unicode[STIX]{x1D701}We^{3/4}Pe^{1/3}$ and $Sr\propto \unicode[STIX]{x1D701}WePe^{1/3}$ are useful to calculate nucleation rates occurring at micro crevices. To underline the importance of our findings we discuss our approach using examples of practical interest.

First, we want to take a look on the streak cavitation experiments conducted in Lausanne, cf. figure 2. The theoretical findings indicate that the shear rate plays a crucial role in the described nucleation process. The shear rate on a hydrofoil can be determined using the theory for the evolution of the boundary layer thickness on a flat plate, $\unicode[STIX]{x1D6FF}_{U}=5x/Re_{x}^{1/2}$ with coordinate $x$ , Reynolds number $Re_{x}=U_{\infty }x/\unicode[STIX]{x1D708}$ and velocity of the undisturbed flow $U_{\infty }$ (Schlichting & Gersten Reference Schlichting and Gersten2006). Using $\dot{\unicode[STIX]{x1D6FE}}\approx U_{\infty }/\unicode[STIX]{x1D6FF}_{U}$ we get $\dot{\unicode[STIX]{x1D6FE}}\approx 1/5\sqrt{U_{\infty }^{3}/(x\unicode[STIX]{x1D708})}$ . Guennoun et al. (Reference Guennoun, Farhat, Bouziad and Avellan2003) conducted experiments with $U_{\infty }=20~\text{m}~\text{s}^{-1}$ and observed nucleation at $x=1~\text{mm}$ . At the nucleation site the shear rate has a value of $5.7\times 10^{5}~\text{s}^{-1}$ , and is thus two orders of magnitude larger than the shear rates observed in our experiments. Guennoun observed nucleation rates of up to 5 kHz. Using $f={\dot{m}}/m_{B}$ and the correlation for the mass flux equation (2.13) one obtains a bubble diameter of $d_{B}\approx 18.1~\unicode[STIX]{x03BC}\text{m}$ for a moderate supersaturation of $\unicode[STIX]{x1D701}=1$ . The bubble diameter increases with $\unicode[STIX]{x1D701}^{1/3}$ ( ${\mathcal{R}}=8.314~\text{J}~(\text{mol}~\text{K})^{-1}$ , $T=293.15~\text{K}$ , ${\mathcal{H}}=7.4\times 10^{-6}~\text{mol}~(\text{m}^{3}~\text{Pa})^{-1}$ ). On the basis of (2.23) and assuming a case III detachment we obtain $\unicode[STIX]{x1D716}_{III}=3.67\times 10^{-2}$ with $We=4.6\times 10^{6}$ . This value is surprisingly close to the value of $\unicode[STIX]{x1D716}_{III}=10^{-2}$ determined in the previous section. In the light of this finding a case III detachment seems to be unlikely in the case of micro crevices, due to the large required shear rates.

A second practical application of our model is a plausibility check of Euler–Lagrange simulations of cavitating flows where surface nucleation is considered. Recent computational fluid dynamics (CFD) studies on sheet and cloud cavitation underline the relevance of nucleation in the context of hydrodynamic cavitation. Chahine and his group (e.g. Hsiao, Ma & Chahine Reference Hsiao, Ma and Chahine2014; Chahine Reference Chahine2015; Ma, Hsiao & Chahine Reference Ma, Hsiao and Chahine2015) investigated sheet and cloud cavitation using an Euler–Lagrange model where cavitation nuclei and their bubble dynamics are considered. The nuclei are created at nucleation spots on the surface of the investigated objects. The authors used a nucleation rate of 22 kHz in their simulations. The model works without any empirical mass exchange models that are non-physical and need to be calibrated. Although this approach is computationally expensive in comparison to other cavitation models (e.g. Schnerr & Sauer Reference Schnerr and Sauer.2001; Singhal et al. Reference Singhal, Li, Athavale and Jiang2001), it is physically more consistent for representing the processes of sheet and cloud cavitation. Since there is only a little influence of the bubble radius on the numerical results, the predefined initial bubble radius is set to be $10~\unicode[STIX]{x03BC}\text{m}$ . The nucleation rate and the initial nuclei radius used in the simulations are not experimentally validated. One has to keep in mind that the size of the nucleation spots are of the order of magnitude of the surface roughness or the size of surface irregularities (e.g. shrinkage cavities), and thus of orders of magnitude from 0.1 to $10~\unicode[STIX]{x03BC}\text{m}$ . It would be very unlikely that the diameters of the detaching bubbles are orders of magnitude larger than the size of the nucleation spots for the large shear rates and large nucleation rates occurring in cavitating flows. We can use our model to check the predefined bubble diameter on plausibility.

Assuming a nucleation spot with $d=10~\unicode[STIX]{x03BC}\text{m}$ we can use equation (2.13) to calculate a shear rate of $1.2\times 10^{18}~\text{s}^{-1}$ for a moderate supersaturation of 1. This shear rate is unphysically large. We now use equation (2.23) with $\unicode[STIX]{x1D716}_{II}=5\times 10^{2}$ , cf. figure 9, and calculate the shear rate for a case II detachment to be $\dot{\unicode[STIX]{x1D6FE}}\approx 1.3\times 10^{6}~\text{s}^{-1}$ , which is still large (the properties of the liquid are listed above). The Weber number then is $We\approx 25.3$ . Equation (2.18) then gives a bubble diameter of $d_{B}\approx 0.9~\unicode[STIX]{x03BC}\text{m}$ , which is small but a more reliable value than $20~\unicode[STIX]{x03BC}\text{m}$ . In summary, diffusion-driven nucleation of bubbles with a diameter of $20~\unicode[STIX]{x03BC}\text{m}$ with a frequency of 22 kHz seems implausible in the light of our findings.

In our model we only consider single surface nuclei sitting in a well-defined crevice. In technical applications there are an uncountable number of nucleation spots located on a surface with a variety of geometries, e.g. steps (Pelz et al. Reference Pelz, Keil and Ludwig2014), surface irregularities or roughness elements (van Rijsbergen & van Terwisga Reference van Rijsbergen and van Terwisga2011). In Pelz et al. (Reference Pelz, Keil and Groß2017) we discussed the nucleation process in the context of sheet and cloud cavitation originating from an artificial roughness. Taking into consideration that free-stream nuclei ‘activate’ the artificial roughness and form a large liquid–gas interface, one can conclude that diffusion-driven nucleation is responsible for the production of the vast number of bubbles forming the sheet cavity. Large shear rates together with a sufficiently large liquid–gas interface are the key elements here.

Of course, the process of diffusion-driven nucleation in technical flows is a very complex phenomenon. Different nucleation sites on surfaces could interact with each other, and could combine and form a larger surface nucleus. In addition, detaching bubbles could influence each other or could cause the bubble detachment at surface nuclei further downstream. The nucleation process is influenced by an usually unknown local supersaturation and might be affected by the wettability of the surface. Furthermore, the surface tension and the vapour pressure of the liquid could play an important role as well as turbulence and vortex formation. This work can serve as a first step for further investigations on diffusion-driven nucleation in the context of hydrodynamic cavitation. In the light of the findings presented, the role of diffusion must be reassessed.