Fluid Dynamic Stress Research Articles

Aortic Valve Stenosis Causes Accumulation of Extracellular Hemoglobin and Systemic Endothelial Dysfunction.

Whether aortic valve stenosis (AS) can adversely affect systemic endothelial function independently of standard modifiable cardiovascular risk factors is unknown. We therefore investigated endothelial and cardiac function in an experimental model of AS mice devoid of standard modifiable cardiovascular risk factors and human cohorts with AS scheduled for transcatheter aortic valve replacement. Endothelial function was determined by flow-mediated dilation using ultrasound. Extracellular hemoglobin (eHb) concentrations and nitric oxide (NO) consumption were determined in blood plasma of mice and humans by ELISA and chemiluminescence. This was complemented by measurements of aortic blood flow using 4-dimensional flow acquisition by magnetic resonance imaging and computational fluid dynamics simulations. The effects of plasma and red blood cell (RBC) suspensions on vascular function were determined in transfer experiments in a murine vasorelaxation bioassay system. In mice, the induction of AS caused systemic endothelial dysfunction. In the presence of normal systolic left ventricular function and mild hypertrophy, the increase in the transvalvular gradient was associated with elevated eryptosis, increased eHb, and increased plasma NO consumption; eHb sequestration by haptoglobin restored endothelial function. Because the aortic valve orifice area in patients with AS decreased, postvalvular mechanical stress in the central ascending aorta increased. This was associated with elevated eHb, circulating RBC-derived microvesicles, eryptotic cells, lower haptoglobin levels without clinically relevant anemia, and consecutive endothelial dysfunction. Transfer experiments demonstrated that reduction of eHb by treatment with haptoglobin or elimination of fluid dynamic stress by transcatheter aortic valve replacement restored endothelial function. In patients with AS and subclinical RBC fragmentation, the remaining circulating RBCs before and after transcatheter aortic valve replacement exhibited intact membrane function, deformability, and resistance to osmotic and hypoxic stress. AS increases postvalvular swirling blood flow in the central ascending aorta, triggering RBC fragmentation with the accumulation of hemoglobin in the plasma. This increases NO consumption in blood, thereby limiting vascular NO bioavailability. Thus, AS itself promotes systemic endothelial dysfunction independent of other established risk factors. Transcatheter aortic valve replacement is capable of limiting NO scavenging and rescuing endothelial function by realigning postvalvular blood flow to near physiological patterns. URL: https://www.clinicaltrials.gov; Unique identifiers: NCT05603520 and NCT01805739.

Enhanced solute transport and steady mechanical stimulation in a novel dynamic perifusion bioreactor increase the efficiency of the in vitro culture of ovarian cortical tissue strips.

Introduction: We report the development and preliminary evaluation of a novel dynamic bioreactor to culture ovarian cortical tissue strips that leverages tissue response to enhanced oxygen transport and adequate mechanical stimulation. In vitro multistep ovarian tissue static culture followed by mature oocyte generation, fertilization, and embryo transfer promises to use the reserve of dormant follicles. Unfortunately, static in vitro culture of ovarian tissue does not promote development of primordial to secondary follicles or sustain follicle viability and thereby limits the number of obtainable mature oocytes. Enhancing oxygen transport to and exerting mechanical stimulation on ovarian tissue in a dynamic bioreactor may more closely mimic the physiological microenvironment and thus promote follicle activation, development, and viability. Materials and Methods: The most transport-effective dynamic bioreactor design was modified using 3D models of medium and oxygen transport to maximize strip perifusion and apply tissue fluid dynamic shear stresses and direct compressive strains to elicit tissue response. Prototypes of the final bioreactor design were manufactured with materials of varying cytocompatibility and assessed by testing the effect of leachables on sperm motility. Effectiveness of the bioreactor culture was characterized against static controls by culturing fresh bovine ovarian tissue strips for 7days at 4.8 × 10-5m/s medium filtration flux in air at -15% maximal total compressive strain and by assessing follicle development, health, and viability. Results and Conclusions: Culture in dynamic bioreactors promoted effective oxygen transport to tissues and stimulated tissues with strains and fluid dynamic shear stresses that, although non-uniform, significantly influenced tissue metabolism. Tissue strip culture in bioreactors made of cytocompatible polypropylene preserved follicle viability and promoted follicle development better than static culture, less so in bioreactors made of cytotoxic ABS-like resin.

Characterisation of particle stress in turbulent impeller flows utilising photo-optical measurements of a flocculation system – PART 1

This study aims at developing an experimental method to characterise the fluid dynamic stress acting on particles in a lab-scale stirred tank reactor. The method is based on a photo-optical inline particle size measurement technique, which is able to capture the complex morphology of a stress-sensitive clay-floc system. The novel image-based analysis uses a Convolutional Neural Network (CNN), which was trained for particle detection. Its output provides a high-quality particle recognition for subsequent size and shape analysis throughout the flocculation process under various experimental conditions. The flocculation system is used to characterise the stress induced by four impeller geometries in the reactor. The obtained changes of floc size and shape allow to evaluate the particle stress in more detail compared to previous studies. The agreement of main findings of the floc breakage process with known literature assesses the developed method to be applied successfully in further studies.

The Impact of Fluid Dynamic Stress in Stirred Bioreactors – The Scale of the Biological Entity: A Personal View

AbstractFrom the start of industrial biotechnology, there has been a perception that biological entities are damaged by stirring, so‐called ‘shear damage'. Often, it was the soft option to explain a loss of performance when it was due to other factors, such as bubble ingestion with proteins or on scale‐up, where tip speed increased when it was due to decreased homogeneity, especially in pH. For many years, poor control and the range of analytical tools available made a more in‐depth explanation difficult; and the concepts of ‘high' and ‘low shear' impellers, now largely disproven, increased it. Here, the size of the biological entity is compared to the Kolmogorov microscale of turbulence leading to a reasonably clear picture emerging. The article starts with the author's introduction to the issue approximately 42 year ago with enzymes, conveniently also the smallest entity; and finishes with the largest, filamentous fungi.

Experimental-numerical method for calculating bending moments in swimming fish shows that fish larvae control undulatory swimming with simple actuation.

Most fish swim with body undulations that result from fluid-structure interactions between the fish's internal tissues and the surrounding water. Gaining insight into these complex fluid-structure interactions is essential to understand how fish swim. To this end, we developed a dedicated experimental-numerical inverse dynamics approach to calculate the lateral bending moment distributions for a large-amplitude undulatory swimmer that moves freely in three-dimensional space. We combined automated motion tracking from multiple synchronised high-speed video sequences, computation of fluid dynamic stresses on the swimmer's body from computational fluid dynamics, and bending moment calculations using these stresses as input for a novel beam model of the body. The bending moment, which represent the system's net actuation, varies over time and along the fish's central axis due to muscle actions, passive tissues, inertia, and fluid dynamics. Our three-dimensional analysis of 113 swimming events of zebrafish larvae ranging in age from 3 to 12 days after fertilisation shows that these bending moment patterns are not only relatively simple but also strikingly similar throughout early development and from fast starts to periodic swimming. This suggests that fish larvae may produce and adjust swimming movements relatively simply, yet effectively, while restructuring their neuromuscular control system throughout their rapid development.

Establishing the scalable manufacture of primary human T-cells in an automated stirred-tank bioreactor.

Advanced cell and gene therapies such as chimeric antigen receptor T-cell immunotherapies (CAR-T), present a novel therapeutic modality for the treatment of acute and chronic conditions including acute lymphoblastic leukemia and non-Hodgkin lymphoma. However, the development of such immunotherapies requires the manufacture of large numbers of T-cells, which remains a major translational and commercial bottleneck due to the manual, small-scale, and often static culturing systems used for their production. Such systems are used because there is an unsubstantiated concern that primary T-cells are shear sensitive, or prefer static conditions, and therefore do not grow as effectively in more scalable, agitated systems, such as stirred-tank bioreactors, as compared with T-flasks and culture bags. In this study, we demonstrate that not only T-cells can be cultivated in an automated stirred-tank bioreactor system (ambr® 250), but that their growth is consistently and significantly better than that in T-flask static culture, with equivalent cell quality. Moreover, we demonstrate that at progressively higher agitation rates over the range studied here, and thereby, higher specific power inputs (P/M W kg-1 ), the higher the final viable T-cell density; that is, a cell density of 4.65 ± 0.24 × 106 viable cells ml-1 obtained at the highest P/M of 74 × 10-4 W kg-1 in comparison with 0.91 ± 0.07 × 106 viable cells ml-1 at the lowest P/M of 3.1 × 10-4 W kg-1 . We posit that this improvement is due to the inability at the lower agitation rates to effectively suspend the Dynabeads®, which are required to activate the T-cells; and that contact between them is improved at the higher agitation rates. Importantly, from the data obtained, there is no indication that T-cells prefer being grown under static conditions or are sensitive to fluid dynamic stresses within a stirred-tank bioreactor system at the agitation speeds investigated. Indeed, the opposite has proven to be the case, whereby, the cells grow better under higher agitation speeds while maintaining their quality. This study is the first demonstration of primary T-cell ex vivo manufacture activated by Dynabeads® in an automated stirred-tank bioreactor system such as the ambr® 250 and the findings have the potential to be applied to multiple other cell candidates for advanced therapy applications.

Computational fluid dynamic and thermal stress analysis of coatings for high‐temperature corrosion protection of aerospace gas turbine blades

AbstractThe current investigation presents detailed finite‐element simulations of coating stress analysis for a three‐dimensional, three‐layered model of a test sample representing a typical gas turbine component. Structural steel, titanium alloy, and silicon carbide are selected for main inner, middle, and outermost layers respectively. ANSYS is used to conduct three types of analysis—static structural, thermal stress analysis, and also computational fluid dynamic erosion (via ANSYS FLUENT). The specified geometry, which corresponds to corrosion test samples exactly, is discretized using a body‐sizing meshing approach, comprising mainly of tetrahedron cells. Refinements were concentrated at the connection points between the layers to shift the focus toward the static effects dissipated between them. A detailed grid independence study is conducted to confirm the accuracy of the selected mesh densities. The momentum and energy equations were solved, and the viscous heating option was applied to represent the improved thermal physics of heat transfer between the layers of the structures. A discrete phase model (DPM) in ANSYS FLUENT was used, which allows for the injection of continuous uniform air particles onto the model, thereby enabling an option for calculating the corrosion factor caused by hot air injection. Extensive visualization of results is provided. The simulations show that ceramic (silicon carbide) when combined with titanium clearly provide good thermal protection; however, the ceramic coating is susceptible to cracking and the titanium coating layer on its own achieves significant thermal resistance. Higher strains are computed for the two‐layer model than the single layer model (thermal case). However even with titanium only present as a coating the maximum equivalent elastic strain is still dangerously close to the lower edge. Only with the three‐layer combined ceramic and titanium coating model is the maximum equivalent strain pushed deeper towards the core central area. Here the desired effect of restricting high stresses to the strongest region of the gas turbine blade model is achieved, whereas in the other two models, lower strains are produced in the core central zones. Generally, the CFD analysis reveals that maximum erosion rates are confined to a local zone on the upper face of the three‐layer system which is in fact the sacrificial layer (ceramic coating). The titanium is not debonded or damaged which is essential for creating a buffer to the actual blade surface and mitigating penetrative corrosive effects. The present analysis may further be generalized to consider three‐dimensional blade geometries and corrosive chemical reaction effects encountered in gas turbine aero‐engines.

Normal fluid stresses are prevalent in rotary ventricular assist devices: A computational fluid dynamics analysis.

Despite the evolution of ventricular assist devices, ventricular assist device patients still suffer from complications due to the damage to blood by fluid dynamic stress. Since rotary ventricular assist devices are assumed to exert mainly shear stress, studies of blood damage are based on shear flow experiments. However, measurements and simulations of cell and protein deformation show normal and shear stresses deform, and potentially damage, cells and proteins differently. The aim was to use computational fluid dynamics to assess the prevalence of normal stress, in comparison with shear stress, in rotary ventricular assist devices. Our calculations showed normal stresses do occur in rotary ventricular assist devices: the fluid volumes experiencing normal stress above 10 Pa were 0.011 mL (0.092%) and 0.027 mL (0.39%) for the HeartWare HVAD and HeartMate II (HMII), and normal stresses over 100 Pa were present. However, the shear stress volumes were up to two orders of magnitude larger than the normal stress volumes. Considering thresholds for red blood cell and von Willebrand factor deformation by normal and shear stresses, the fluid volumes causing deformation by normal stress were between 2.5 and 5 times the size of those causing deformation by shear stress. The exposure times to the individual normal stress deformation regions were around 1 ms. The results clearly show, for the first time, that while blood within rotary ventricular assist devices experiences more shear stress at much higher magnitudes as compared with normal stress, there is sufficient normal stress exposure present to cause deformation of, and potentially damage to, the blood components. This study is the first to quantify the fluid stress components in real blood contacting devices.

Agitation and aeration of stirred-bioreactors for the microcarrier culture of human mesenchymal stem cells and potential implications for large-scale bioprocess development

The impact of agitation rate and sparged aeration on BM-hMSC expansion in conventional stirred tank bioreactors was assessed. It was found that a decrease in impeller speed to below NJS caused sampling difficulties, clumping and an increase to ∼2 NJS decreased the growth rate though an intermediate value of ∼1.3 NJS did not. Additionally, over this range of agitation intensities, cell quality remained unchanged post-harvest suggesting that poor growth performance at the highest speed was due to a failure of the cells to attach efficiently to microcarriers rather than damage to the cells due to fluid dynamic stress. Further it was shown that direct aeration of the culture medium both with and without Pluronic F68 via a sparger at NJS was detrimental to BM-hMSC growth. Again, this reduction in growth seems to be associated with poor attachment rather than cell damage, which due to the mechanism of PluronicTM F68 reducing the cell hydrophobicity and thus the affinity of the BM-hMSCs to attach to the microcarrier, leads to a poorer performance in the presence of the surfactant. Certain post-harvest quality characteristics are also detrimentally impacted compared to headspace aeration. This problem is discussed in terms of the need to facilitate future large-scale process development where headspace aeration at NJS may not be sufficient to meet culture needs at higher cell densities.

Model-based image analysis of a tethered Brownian fibre for shear stress sensing.

The measurement of fluid dynamic shear stress acting on a biologically relevant surface is a challenging problem, particularly in the complex environment of, for example, the vasculature. While an experimental method for the direct detection of wall shear stress via the imaging of a synthetic biology nanorod has recently been developed, the data interpretation so far has been limited to phenomenological random walk modelling, small-angle approximation, and image analysis techniques which do not take into account the production of an image from a three-dimensional subject. In this report, we develop a mathematical and statistical framework to estimate shear stress from rapid imaging sequences based firstly on stochastic modelling of the dynamics of a tethered Brownian fibre in shear flow, and secondly on a novel model-based image analysis, which reconstructs fibre positions by solving the inverse problem of image formation. This framework is tested on experimental data, providing the first mechanistically rational analysis of the novel assay. What follows further develops the established theory for an untethered particle in a semi-dilute suspension, which is of relevance to, for example, the study of Brownian nanowires without flow, and presents new ideas in the field of multi-disciplinary image analysis.

Scale-up agitation criteria for Trichoderma reesei fermentation

Scale-up of aerobic fungal fermentation processes still remains a challenging issue for the biotechnology industry. This difficulty arises due to the complex interactions between operating conditions (agitation, aeration, etc.), the physicochemical state of the broth (viscosity, the dissolved oxygen concentration, etc.) and the biology of fungi (growth, production, morphology, etc.). Because of their size, filamentous fungi are affected by fluid dynamic stresses but quantification of this complex parameter is a difficult task. In general, indirect criteria are used for the effect of fluid dynamic stresses on scale-up (tip speed, power draw or the energy dissipation/circulation function (EDCF)). In order to better understand the impact of such criteria on the fermentation of the fungus Trichoderma reesei, a wide range of agitation conditions has been explored. The morphology of T. reesei fungus, its specific growth rate and the rheological properties of the broth have all been measured both at bench scale (∼2.5L) and for the first time, at a typical commercial scale. These three aspects of the fermentation at both scales were then compared with respect to tip speed, specific power and EDCF. This work has shown that tip speed as a correlator of any of these parameters is totally ineffective whilst the EDCF is clearly the best for extrapolating laboratory data to the commercial scale.

Large-scale generation of differentiated cells to achieve regenerative medicine

The effects of microgravity and fluid dynamic stress on embryoid bodies generated from pluripotent stem cells induce and direct their differentiation. Using this hydrodynamic effect combined with exogenous factors and three-dimensional culture, a new technique has been developed to produce functional, effective, and safe hepatocytes for transplantation. The evolution of this technique will lead to automated production of a large number of differentiated cells and will significantly contribute to regenerative medicine.

Scale-down studies for assessing the impact of different stress parameters on growth and product quality during animal cell culture

Two series of reproducible fed-batch bench scale cultures have been undertaken, one series simulating the impact of spatial variations in pH and nutrients as found at commercial scale on performance, the other, the impact of fluid dynamic stresses associated with agitation. The first was unsuccessful because, somewhat surprisingly, the use of a peristaltic pump to circulate cells and medium through different spatial environments always led to a similar reduction in culture time and resulting product titre compared to uncirculated controls. This fall was sufficient to essentially mask other effects. In the second, even at maximum specific energy dissipation rates up to ∼160 times>with laminar extensional flow and ∼25 times>with turbulent flow compared to typical commercial conditions, no significant effects were observed on cell growth and viability. Most importantly, in all of the cases studied, product quality was unaffected compared to controls. In addition, it is suggested that because of the possibility of cell line specific behaviour and the relationship between damage to entities and the Kolmogorov scale of turbulence, sensitivity to fluid dynamic stresses is best studied in turbulent bench scale bioreactors.

Adaptation of the Spodoptera exigua Se301 insect cell line to grow in serum-free suspended culture. Comparison of SeMNPV productivity in serum-free and serum-containing media

Spodoptera exigua Se301 cells have been successfully adapted to two different commercial serum-free media (SFM; Ex-Cell 420 and Serum-Free Insect Medium-1) by gradually reducing the 10 %-added serum-containing medium content from 100 % to 0 % (v/v) in suspended cultures. Both direct adaptation to a serum-free medium and cell growth in the absence of protective additives against fluid dynamic stress [polyvinyl pyrrolidone and polyvinyl alcohol] and disaggregation [dextran sulfate] proved impossible. Cells grew reproducibly in both SFMs once the serum had been completely removed, although the use of Ex-Cell 420 resulted in higher growth rates and cell densities. Turbulence was sufficiently high to reduce growth rates and final cell densities at the highest Reynolds number investigated, although no clear influence of agitation was observed on virus productivity. Both attached and suspended Se301 cell cultures were successfully infected with the SeMNPV baculovirus. Cells adapted to different conditions (attached or suspended culture, serum-containing or serum-free medium) showed different occlusion bodies productivities at the two multiplicities of infection assayed (0.1 and 0.5).

Enhanced Osteogenesis of Human Alveolar Bone-Derived Mesenchymal Stem Cells for Tooth Tissue Engineering Using Fluid Shear Stress in a Rocking Culture Method

This study instituted a simple approach to stimulate alveolar bone regeneration for tooth tissue engineering by controlling effects of low fluid dynamic shear stress (LFDSS) on growth and differentiation in vitro. Human alveolar bone-derived mesenchymal stem cells (hABMSCs) harvested from human mandibular alveolar bone were cultured with LFDSS to generate cultures containing bone-like formations. To distinguish between osteodifferentiation and bone-like formation, cells were cultured either with or without fluid shear stress. The calcium content and alkaline phosphatase (ALP) activity of hABMSCs were used as indicators of osteogenesis. Cell viability and proliferation after stimulating with LFDSS for 10-60 min/day were higher than with longer stimulations. Mineralized nodules formed when osteoblasts were cultured with an induction medium, a marker of osteogenic differentiation. ALP activity tended to increase after 10 and 60 min/day of stimulation. In addition, LFDSS conditions also increased gene expression of IBSP, RUNX2, COL-I, ALP, OCN, and OPN, as shown by reverse transcriptase-polymerase chain reaction. From the results of a proteomics array, LFDSS groups were intensely expressed with several factors (EGF, HGF, IGF, TGF, and PDGF). Furthermore, CD146 and Stro-1 expression increased in cells treated with 30 min/day and decreased in cells treated with 120 min/day, as determined by cell surface antigen analysis by fluorescence-activated cell-sorting analysis. These results strongly showed that LFDSS at the proper intensity and time enhanced the differentiation and maturation of hABMSCs. In conclusion, an appropriate level of LFDSS can potently and positively modulate proliferation and differentiation in hABMSCs.

A Transient 3D-CFD Model Incorporating Biological Processes for Use in Tissue Engineering

In this article a mathematical model is presented in which the fluid dynamic interaction between the liquid flow in a scaffold and growing cells is simulated. The model is based on a computational fluid dynamic (CFD) model for the representation of the fluid dynamic conditions in the scaffold. It includes furthermore a simple biological growth model based on Michaelis Menten type kinetics for the growth of cells. The model includes biomass, substrate and oxygen as the most important growth limiting components in the system. Furthermore the growth, decay and maintenance respiration of the cells are considered in the model. In a variation of the model the growth of the biomass is influenced by the fluid dynamic induced shear stress level, which the cells are exposed to. In parallel an experimental growth of stem cells has been performed in a 3D perfusion reactor system and the culturing has been stopped after 2, 8 and 13 days. The development of the cells is compared to the simulated growth of cells and it is attempted to draw a conclusion about the impact of the shear stress on the cell growth.

Effect of Fluid Dynamic Stress on Culture Performance in an Industrial CHO mAb Production Process

Effect of Fluid Dynamic Stress on Culture Performance in an Industrial CHO mAb Production Process

Design optimization of a helical endothelial cell culture device

The specific roles of mass transfer and fluid dynamic stresses on endothelial function, important in atherogenesis, are not known. Further, the effects of mass transfer and fluid dynamic stresses are difficult to separate because areas of "abnormal" mass transfer and "abnormal" wall shear stress tend to co-localize (where abnormal is defined as any deviation from the mass transfer rate or wall shear stress present in a long straight artery with the same flow rate and diameter). Our goal was to design a cell culture device which gives maximum separation between areas of abnormal shear stress and areas of abnormal mass transfer. We used design optimization principles to design a helical cell culture device. Using shear stress and mass transfer fields predicted by solving the governing equations, the area of the device which was exposed to low rates of mass transfer and normal levels of wall shear stress was determined. The design optimization method then maximized this area by varying the design variables, resulting in the optimum design. The optimum design had Reynolds number = 50, helical radius = 3.23 and helical pitch = 3.82. The area of the device which was exposed to low rates of mass transfer and regular levels of wall shear stress was about 4.5 times the inlet cross-sectional area of the device or about 5% of the device total internal surface area. An optimum design was successfully determined and the methodology used was shown to be robust. The area of the device which was exposed to low rates of mass transfer and regular levels of wall shear stress occurred in a defined region which should aid further experimental work.

Effects of fluid dynamic stress on fracturing of cell-aggregated tissue during purification for islets of Langerhans transplantation

Among clinical treatments for type 1 diabetes mellitus, the transplantation of islets of Langerhans to the portal vein of the hepar is a commonly used treatment for glucose homeostasis. Islet purification using the density gradient of a solution in a centrifuge separator is required for safety and efficiency. In the purification, the number of tissues to be transplanted is reduced by removing the acinar tissue and gathering the islet from the digest of pancreas. However, the mechanical effects on the fracture of islets in the centrifuge due to fluid dynamic stress are a serious problem in the purification process. In this study, a preliminary experiment using a cylindrical rotating viscometer with a simple geometry is conducted in order to systematically clarify the effect of fluid dynamic stress on the fracture of islets. The effects of fluid dynamic stress on the islet configuration is quantitatively measured for various flow conditions, and a predictive fracture model is developed based on the experimental results. Furthermore, in the practical purification process in the COBE (Gambro BCT), which is widely used in clinical applications, we perform a numerical analysis of the fluid dynamic stress based on Navier–Stokes equations to estimate the stress conditions for islets. Using the fracture model and numerical analysis, the islet fracture characteristics using the COBE are successfully investigated. The results obtained in this study provide crucial information for the purification of islets by centrifuge in practical and clinical applications.

Vapour bubble collapse in isothermal and non-isothermal liquids

The motion of a vapour bubble in a subcooled liquid is studied numerically assuming axial symmetry but allowing the surface to deform under the action of the fluid dynamic stress. The flattening of the bubble in the plane orthogonal to the translational velocity increases the added mass and slows it down, while, at the same time, the decreasing volume tends to increase the velocity. The deformation of the interface also increases the surface area exposed to the incoming cooler liquid. The competition among these opposing processes is subtle and the details of the condensation cannot be captured by simpler models, two of which are considered. In spite of these differences, the estimate of the total collapse time given by a spherical model is close to that of the deforming bubble model for the cases studied. In addition to an isothermal liquid, some examples in which the bubble encounters warmer and colder liquid regions are shown.