Timm Krüger

The University of Edinburgh | UoE · School of Engineering

This book is an introduction to the theory, practice, and implementation of the Lattice Boltzmann (LB) method, a powerful computational fluid dynamics method that is steadily gaining attention due to its simplicity, scalability, extensibility, and simple handling of complex geometries. The book contains chapters on the method's background, fundamen...

The formation of pairs and trains of particles in inertial microfluidics is an important consideration for device design and applications, such as particle focusing and separation. We study the formation and stability of linear and staggered pairs of nearly rigid spherical particles of different sizes in a pressure-driven flow through a straight du...

The distribution of red blood cells (RBCs) in the microcirculation determines the oxygen delivery and solute transport to tissues. This process relies on the partitioning of RBCs at successive bifurcations throughout the microvascular network and it is known since the last century that RBCs partition disproportionately to the fractional blood flow...

The tumour microenvironment is abnormal and one of its consequences is that blood vessels are compressed. Vessel compression correlates with reduced survival rates, while decompression of vessels improves tissue oxygenation as well as increases survival rates. Vessel compression contributes, at a single vascular bifurcation, to the increase of hete...

Inertial particle microfluidics (IPMF) is an emerging technology for the manipulation and separation of microparticles and biological cells. Since the flow physics of IPMF is complex and experimental studies are often time-consuming or costly, computer simulations can offer complementary insights. In this tutorial review, we provide a guide for res...

The tumour microvasculature is abnormal, and as a consequence oxygen and drug transport to the tumour tissue is impaired. The abnormal microvasculature contributes to tumour tissue hypoxia, as well as to varying drug penetration depth in the tumour. Many anti-cancer treatments require the presence of oxygen to be fully efficacious, however the ques...

Deterministic lateral displacement (DLD) is a microfluidic method for accurately separating particles by size or deformability. Recent efforts to operate DLD devices in the inertial, rather than in the Stokes flow regime, have been hindered by a loss of separation efficiency and difficulty predicting the separation behavior. One factor contributing...

The tumour microenvironment is abnormal and one of its consequences is that blood vessels are compressed. Vessel compression correlates with reduced survival rates, while decompression of vessels improves tissue oxygenation as well as increases survival rates. Vessel compression contributes, at a single vascular bifurcation, to the increase of hete...

The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media and microfluidic arrays, this solution is important for many real-world systems. We asymptotically determine the flow around a general rectangular doubly-periodic array of infi...

Rosetting, the formation of red blood cell aggregates, is a life-threatening condition in malaria tropica and not yet fully understood. We study rosette stability using a set of microfluidic stenotic channels, with varied narrowing angle and erythrocytes of blood groups O and A. We find reduced ability of a rosette to pass a stenosis without disrup...

The formation of pairs of particles or cells of different types in microfluidic channels can be desired or detrimental in healthcare applications. It is still unclear what role softness heterogeneity plays in the formation of these particle pairs. We use an in-house lattice-Boltzmann-immersed-boundary-finite-element solver to simulate a pair of par...

Deterministic lateral displacement (DLD) is a microfluidic method for accurately separating particles by size or deformability. Recent efforts to operate DLD devices in the inertial, rather than in the Stokes, flow regime have been hindered by a loss of separation efficiency and difficulty predicting the separation behaviour. One factor contributin...

Inertial particle microfluidics (IPMF) is an emerging technology for the manipulation and separation of microparticles and biological cells. Since the flow physics of IPMF is complex and experimental studies are often time-consuming or costly, computer simulations can offer complementary insights. In this tutorial review, we provide a guide for res...

The formation of pairs of particles or cells of different types in microfluidic channels can be desired or detrimental in healthcare applications. It is still unclear what role softness heterogeneity plays in the formation of these particle pairs. We use an in-house lattice-Boltzmann-immersed-boundary-finite-element solver to simulate a pair of par...

The study of flow and particle dynamics in microfluidic cross-slot channels is of high relevance for lab-on-a-chip applications. In this work, we investigate the dynamics of a rigid spherical particle in a cross-slot junction for a channel height-to-width ratio of 0.6 and at a Reynolds number of 120 for which a steady vortex exists in the junction...

The slow viscous flow through a doubly-periodic array of cylinders does not have an analytical solution. However, as a reduced model for the flow within fibrous porous media, this solution is important for many real-world systems. We asymptotically determine the flow around a doubly-periodic array of infinite slender cylinders, by placing doubly-pe...

The study of flow and particle dynamics in microfluidic cross-slot channels is of high relevance for lab-on-a-chip applications. In this work we investigate the dynamics of a rigid spherical particle in a cross-slot junction for a channel height-to-width ratio of 0.6 and at a Reynolds number of 120 for which a steady vortex exists in the junction a...

The formation of pairs and trains of particles in inertial microfluidics is an important consideration for device design and applications, such as particle focussing and separation. We study the formation and stability of linear and staggered pairs of nearly rigid spherical particles of different sizes in a pressure-driven flow through a straight d...

The dynamics of blood flow in the smallest vessels and passages of the human body, where the cellular character of blood becomes prominent, plays a dominant role in the transport and exchange of solutes. Recent studies have revealed that the microhaemodynamics of a vascular network is underpinned by its interconnected structure, and certain structu...

The distribution of red blood cells (RBCs) in the microcirculation determines the oxygen delivery and solute transport to tissues. It is known since the last century that RBCs partition at bifurcations disproportionately to the fractional blood flow rate, leading to heterogeneity of the hematocrit (i.e. volume fraction of RBCs in blood) in microves...

The dynamics of blood flow in the smallest vessels and passages of the human body, where the cellular character of blood becomes prominent, plays a dominant role in the transport and exchange of solutes. Recent studies have revealed that the micro-haemodynamics of a vascular network is underpinned by its interconnected structure, and certain struct...

We investigate the formation and stability of a pair of identical soft capsules in channel flow under mild inertia. We employ a combination of the lattice Boltzmann, finite element and immersed boundary methods to simulate the elastic particles in flow. Validation tests show excellent agreement with numerical results obtained by other research grou...

Rosetting, the formation of red blood cell aggregates, is a life-threatening condition in Malaria tropica and not yet fully understood. We study rosette stability using a set of microfluidic stenotic channels, with varied narrowing angle and erythrocytes of blood groups 0 and A. We find reduced ability of a rosette to pass a stenosis without disrup...

The placenta is a vital interface between the mother and her developing fetus. Micro-haemodynamics of the placenta, where the particulate nature of blood flow cannot be ignored, mediates the relationship between the organ’s structure and its function. However, the placenta’s complex architecture and its relation to pregnancy pathologies remain poor...

The phase shift between pressure and wall shear stress (WSS) has been associated with vascular diseases such as atherosclerosis and aneurysms. The present study aims to understand the effects of geometry and flow properties on the phase shift under the stiff wall assumption, using an immersed-boundary-lattice-Boltzmann method. For pulsatile flow in...

We investigate the formation and stability of a pair of identical soft capsules in channel flow under mild inertia. We employ a combination of the lattice Boltzmann, finite element and immersed boundary methods to simulate the elastic particles in flow. Validation tests show excellent agreement with numerical results obtained by other research grou...

Sprouting angiogenesis is an essential vascularization mechanism consisting of sprouting and remodelling. The remodelling phase is driven by rearrangements of endothelial cells (ECs) within the post-sprouting vascular plexus. Prior work has uncovered how ECs polarize and migrate in response to flow-induced wall shear stress (WSS). However, the ques...

Significance In tumors, tissue oxygen heterogeneity leading to the appearance of hypoxic regions is linked to poor prognosis and reduces the efficiency of therapeutic treatment. Following previous reports that vessel compression, caused by the tumor, leads to tumor tissue oxygen heterogeneity, we formulate a computational model to investigate the m...

Host immune response plays a critical role in disease manifestation and severity. In article number 2006123, Win Sen Kuan, Jongyoon Han, and co‐workers present a whole blood biophysical immune response profiling tool using deterministic lateral displacement microfluidics. The speed and accuracy of immune response stratification demonstrated in an e...

Disease manifestation and severity from acute infections are often due to hyper‐aggressive host immune responses which change within minutes. Current methods for early diagnosis of infections focus on detecting low abundance pathogens, which are time‐consuming, of low sensitivity, and do not reflect the severity of the pathophysiology appropriately...

We present a thermodynamically consistent model of a ternary fluid interacting with elastic membranes. Following a free-energy modeling approach for the fluid phases, we derive the governing equations for the dynamics of the ternary fluid flow and membranes. We also provide the numerical framework for simulating such fluid-structure interaction pro...

Blood is a vital soft matter, and its normal circulation in the human body relies on the distribution of red blood cells (RBCs) at successive bifurcations. Understanding how RBCs are partitioned at bifurcations is key for the optimisation of microfluidic devices as well as for devising novel strategies for diagnosis and treatment of blood-related d...

The tumour microenvironment is abnormal and associated with tumour tissue hypoxia, immunosuppression, and poor response to treatment. One important abnormality present in tumours is vessel compression. Vessel decompression has been shown to increase survival rates in animal models via enhanced and more homogeneous oxygenation. However, our knowledg...

Significance Oxygen heterogeneity in solid tumors is recognized as a limiting factor for therapeutic efficacy. This heterogeneity arises from the abnormal tumor vascular structure. We investigate the role that anomalies in red blood cell transport plays in establishing oxygen heterogeneity in tumor tissue. We introduce a metric to characterize tumo...

The advent of microfluidics in the 1990s promised a revolution in multiple industries from healthcare to chemical processing. Deterministic lateral displacement (DLD) is a continuous-flow micro- fluidic particle separation method discovered in 2004 that has been applied successfully and widely to the separation of blood cells, yeast, spores, bacter...

Spiral microchannels have shown promising results for separation applications. Hydrodynamic particle–particle interactions are a known factor strongly influencing focusing behaviors in inertial devices, with recent work highlighting how the performance of bidisperse mixtures is altered when compared with pure components in square channels. This phe...

Sprouting angiogenesis is an essential vascularisation mechanism consisting of sprouting and remodelling. The remodelling phase is driven by rearrangements of endothelial cells (ECs) within the post-sprouting vascular plexus. Prior work has uncovered how ECs polarise and migrate in response to flow-induced wall shear stress (WSS). However, the ques...

Microfluidic technologies are commonly used for the manipulation of red blood cell (RBC) suspensions and analyses of flow-mediated biomechanics. To enhance the performance of microfluidic devices, understanding the dynamics of the suspensions processed within is crucial. We report novel aspects of the spatio-temporal dynamics of RBC suspensions flo...

Deterministic lateral displacement (DLD) is a microfluidic method of separating particles by size. DLD relies on precise flow patterns to deliver high-resolution particle separation. These patterns determine which particles are displaced laterally, and which follow the flow direction. Prior research has demonstrated that the lateral array boundarie...

Among all existing microfluidic systems, inertial microfluidics has experienced massive growth in many applications such as fractionation of blood cells, isolation of bacteria, and enrichment of cancer cells with some commercially successful products. Despite the immense interest in these devices, their design is based on phenomenological rules tha...

Blood flow in an artery is a fluid-structure interaction problem. It is widely accepted that aneurysm formation, enlargement and failure are associated with wall shear stress (WSS) which is exerted by flowing blood on the aneurysmal wall. To date, the combined effect of aneurysm size and wall elasticity on intra-aneurysm (IA) flow characteristics,...

Soft particles at fluid interfaces play an important role in many aspects of our daily life, such as the food industry, paints and coatings, and medical applications. Analytical methods are not capable of describing the emergent effects of the complex dynamics of suspensions of many soft particles, whereas experiments typically either only capture...

Microfluidic technologies are commonly used for the manipulation of red blood cell (RBC) suspensions and analyses of flow-mediated biomechanics. To maximise the usability of microfluidic devices, understanding the dynamics of the suspensions processed within is crucial. We report novel aspects of the spatio-temporal dynamics of an RBC suspension fl...

Blood flow in an artery is a fluid-structure interaction problem. It is widely accepted that aneurysm formation, enlargement and failure are associated with wall shear stress (WSS) which is exerted by flowing blood on the aneurysmal wall. To date, most of the computational studies of this problem assume rigid walls. In particular, in the case of si...

We present a thermodynamically consistent model of a ternary fluid interacting with elastic membranes. Following a free-energy modelling approach and taking into account the thermodynamics laws, we derive the equations governing the ternary fluid flow and dynamics of the membranes. We also provide the numerical framework for simulating such fluid-s...

Oxygen heterogeneity in solid tumours is recognised as a limiting factor for therapeutic efficacy. This heterogeneity arises from the abnormal vascular structure of the tumour, but the precise mechanisms linking abnormal structure and compromised oxygen transport are only partially understood. In this paper, we investigate the role that RBC transpo...

Soft particles at fluid interfaces play an important role in many aspects of our daily life, such as the food industry, paints and coatings, and medical applications. Analytical methods are not capable of describing the emergent effects of the complex dynamics of suspensions of many soft particles, whereas experiments typically either only capture...

Es war eine entscheidende intellektuelle Leistung der Physiker und Mathematiker des 19. Jahrhunderts, die elektrischen und magnetischen Phänomene in einer gemeinsamen Theorie, der Elektrodynamik, zu vereinigen und dabei dem Begriff von physikalischen Feldern zu einem Durchbruch zu verhelfen. Im Folgenden werden wir diese Vereinigung aber vorerst wi...

Obwohl bei elektrischen Strömen elektrische Ladungsträger in Bewegung sind, ergeben sich oft stationäre Situationen, bei denen die Phänomene des Magnetismus mit zeitunabhängigen Feldgleichungen beschrieben werden können. Dabei ist es natürlich wichtig, dass nicht die Bewegung der einzelnen Elementarladungen verfolgt wird, sondern Ströme durch Versc...

Die Elektrodynamik, die fundamentale Theorie, die elektrische und magnetische Felder miteinander untrennbar verknüpft, ist mit ihren vielfältigen Anwendungen aus unserer Zivilisation heutzutage nicht mehr wegzudenken. Bis zum Anfang des 19. Jahrhunderts kannte die Physik aber nur eine qualitative Phänomenologie von nicht miteinander in Beziehung ge...

Nach der Betrachtung der Grundgleichungen der Elektrodynamik und elektrostatischer Probleme in Kap. 1 und 2 werden wir nun einige mathematische Methoden kennenlernen, die nicht nur für die Elektrodynamik, sondern auch für viele andere Gebiete der theoretischen Physik große Bedeutung haben.

Das Licht der Sterne erreicht uns aus großen Entfernungen durch das Vakuum des Weltalls; sehr lange war aber nicht klar, was Licht überhaupt ist und wie es sich durch das Vakuum ausbreiten kann. Maxwell äußerte bereits kurz nach Aufstellen seiner Gleichungen die Vermutung, dass es „elektromagnetische Wellen“ gäbe und Licht eine solche sei. Im Jahre...

Ende des 19. Jahrhunderts begann Heinrich Hertz (1857–1894), kurz zuvor zum Professor für Experimentalphysik am Polytechnikum in Karlsruhe berufen, mit elektromagnetischen Schwingungen in Spulen zu experimentieren. Es war bereits bekannt, dass es bei Spulen, die an einer Stelle unterbrochen sind, an dieser Stelle zu Funkenüberschlägen kommen kann.

Bd. 4, der sich mit Thermodynamik befasst, gleicht viel mehr einer Wendeltreppe als einem geradlinigen Fortschreiten. Die Axiome der Thermodynamik wurden bereits in Kap. 1 eingeführt und als Abstraktionen physikalischer Erfahrung begründet. Weitere Axiome sind seitdem nicht dazugekommen, stattdessen haben wir sie vertieft: In Kap. 2 durch die stati...

In diesem Kapitel erweitern wir die bisherigen Überlegungen auf quantale Systeme. In Abschn. 5.1 und 5.2 zeigen wir, dass die grundlegenden Konzepte der bisher anhand klassischer Systeme entwickelten Thermodynamik und der statistischen Physik direkt auf quantale Systeme übertragbar sind, wenn folgende Ersetzungen vorgenommen werden: An die Stelle d...

Die Optik beschäftigt sich speziell mit der Ausbreitung von Licht (die meisten ihrer Ergebnisse sind aber auch auf andere elektromagnetische Wellen übertragbar). In vielen Fällen genügt es dabei, davon auszugehen, dass Licht sich in Strahlen ausbreitet; der Wellencharakter des Lichtes kann vernachlässigt werden.

Wir beginnen nun gewissermaßen von Neuem. In Kap. 1 haben wir uns auf die phänomenologische Thermodynamik beschränkt, die bewusst auf jede Kenntnis der sehr vielen mikroskopischen Zustände verzichtet, aus denen ein makroskopischer Zustand zusammengesetzt sein mag. Dieser Zugang kam historisch zuerst, weil er beschritten werden konnte, lange bevor s...

Die Maxwell-Gleichungen, wie wir sie bis jetzt diskutiert haben, sind fundamentale, auf mikroskopischem Niveau gültige Grundgleichungen. Sobald man es mit makroskopischen Körpern zu tun hat, sind die tatsächlichen auf atomarer oder molekularer Ebene vorliegenden Ladungsverteilungen natürlich viel zu komplex, als dass man direkt mit ihnen Berechnung...

Schon vor der Formulierung der speziellen Relativitätstheorie durch Einstein hatten Lorentz und Poincaré das Transformationsverhalten elektromagnetischer Felder unter einem Wechsel des Inertialsystems herausgefunden und waren dabei auf die Effekte der Lorentz-Kontraktion und sogar der Zeitdilatation gestoßen. Einsteins spezielle Relativitätstheorie...

In der Mechanik (Bd. 1, Kap. 5 und 7) wurden zwei relativ abstrakte, aber auch sehr allgemeine Formalismen hergeleitet, mittels derer die Bewegungsgleichungen für ein gegebenes mechanisches System bestimmt werden können: der Lagrange- und der Hamilton-Formalismus. Diese sollen nun auch auf die Elektrodynamik erweitert werden.

Dieses Kapitel führt die wesentlichen Konzepte der Thermodynamik und ihre Axiome auf eine Weise ein, die keinen Bezug zur mikroskopischen Natur der Materie nimmt. Es folgt damit in Abschn. 1.1 zum einen der historischen Entwicklung, welche die Thermodynamik ausgehend von den Begriffen „warm“ und „kalt“ über das Bedürfnis, Wärmekraftmaschinen zu ver...

In Kap. 1 wurde die Thermodynamik zunächst phänomenologisch begründet, d. h. aufgrund solcher Beobachtungen, die mit den Erfahrungen von Temperatur und Wärme verbunden sind. Wir haben dabei die Temperatur als Zustandsgröße eingeführt, den ersten Hauptsatz formuliert und haben nachvollzogen, wie man ausgehend von der grundlegenden Erfahrung irrevers...

Bisher wurden fast nur exakt lösbare Systeme besprochen; allerdings wurde auch schon mehrfach darauf hingewiesen, dass viele Probleme nur näherungsweise lösbar sind. In der Quantenmechanik werden daher diverse Näherungsverfahren verwendet; das wichtigste, die Störungstheorie, soll in diesem Kapitel besprochen werden. Weitere Verfahren werden in Kap...

In Kap. 8 wurde das nichtrelativistische Wasserstoffatom ausführlich behandelt. Nach Abspaltung der Schwerpunktsbewegung vereinfacht es sich auf ein exakt lösbares Einkörperproblem. Berücksichtigt man allerdings die relativistische Spin-Bahn-Kopplung oder wird ein äußeres Feld angelegt, so können die Energieniveaus und Eigenfunktionen des Wassersto...

Symmetrien – man spricht von Raumzeitsymmetrien, wenn auch die Raumzeitkoordinaten transformiert werden, und sonst von inneren Symmetrien – spielen in der Physik eine herausragende Rolle. Beispiele von Raumzeitsymmetrien sind Spiegelungen, Translationen oder Drehungen im Raum. Innere Symmetrien sind z. B. die Ladungsumkehr oder die verallgemeinerte...

Streuexperimente sind ein wichtiges Werkzeug zur Untersuchung von physikalischen Objekten, z. B. Festkörpern, Molekülen, Atomen, Atomkernen und Elementarteilchen. Man kann die Streuung von Teilchen oder von Strahlung an Objekten benutzen, um die Struktur dieser Objekte zu untersuchen und besser zu verstehen. Die Berechnung und Analyse solcher Streu...

Wasserstoffähnliche Ionen sind Atome, die so weit ionisiert sind, dass sie nur noch ein einzelnes Elektron besitzen. Das klassische Einelektronsystem ist der Wasserstoff selbst. Hier ist das Elektron über die Coulomb-Wechselwirkung an ein Proton gebunden. Die beiden Wasserstoffisotope Deuterium \({}^{2}_{1}\)H oder Tritium \({}^{3}_{1}\)H sind weit...

Ein Verständnis der modernen Naturwissenschaften ist ohne Kenntnis der Quantenmechanik, deren Grundgleichungen das Verhalten mikroskopischer Objekte beschreibt, unmöglich. Ohne Quantenmechanik gäbe es kein Verständnis der Eigenschaften von Molekülen, Atomen, Atomkernen und Elementarteilchen oder von Halbleitern und Lasern. Ohne Quantenmechanik könn...

In der Quantentheorie gibt es keine klare Trennung zwischen System und Messapparatur, und die zeitliche Entwicklung eines Quantensystems ist nicht mehr deterministisch, sobald es in Wechselwirkung mit der Umgebung tritt. Nur ein abgeschlossenes und von allen äußeren Einflüssen isoliertes System verändert sich in exakt vorhersagbarer Weise. In diese...

In Kap. 2 wurde bereits kurz angesprochen, wie elektromagnetische Felder in der Quantenmechanik zu berücksichtigen sind. In diesem Kapitel werden wir darauf nun genauer eingehen und sowohl allgemeine Prinzipien als auch spezielle Beispiele dazu diskutieren; insbesondere werden wir in Abschn. 9.1 zunächst die Bedeutung der Potenziale untersuchen, di...

In diesem Kapitel untersuchen wir Lösungen der stationären Schrödinger-Gleichung für eindime