This book serves as an introduction to the continuum mechanics and mathematical modeling of complex fluids in living systems. The form and function of living systems are intimately tied to the nature of surrounding fluid environments, which commonly exhibit nonlinear and history dependent responses to forces and displacements. With ever-increasing capabilities in the visualization and manipulation of biological systems, research on the fundamental phenomena, models, measurements, and analysis of complex fluids has taken a number of exciting directions. In this book, many of the world’s foremost experts explore key topics such as: •Macro- and micro-rheological techniques for measuring the material properties of complex biofluids and the subtleties of data interpretation •Experimental observations and rheology of complex biological materials, including mucus, cell membranes, the cytoskeleton, and blood •The motility of microorganisms in complex fluids and the dynamics of active suspension •Challenges and solutions in the numerical simulation of biologically relevant complex fluid flows
This volume will be accessible to advanced undergraduate and beginning graduate students in engineering, mathematics, biology, and the physical sciences, but will appeal to anyone interested in the intricate and beautiful nature of complex fluids in the context of living systems.

Part I Introduction to Complex Fluids

1 Introduction to Complex Fluids 3

Alexander Morozov and Saverio E. Spagnolie

      1 Introduction 3

      2 Newtonian Fluid Mechanics 4

      2.1 Material (Lagrangian) and Spatial (Eulerian) Variables 5

      2.2 Conservation of Mass 6

      2.3 Conservation of Momentum 7

      2.4 The Cauchy Stress Tensor and the Navier-Stokes Equations 8

      2.5 Dimensional Analysis and the Stokes Equations 12

      3 Generalized Newtonian Fluids 13

      3.1 Shear-Thinning and Shear-Thickening Fluids 14

      3.2 Carreau-Yasuda and Power-Law Fluids 15

      3.3 Mechanical Instability of Extremely Shear-Thinning Fluids 18

      4 Differential Constitutive Equations for Viscoelastic Fluids 20

      4.1 Linear Maxwell Fluids and Kelvin-Voigt Solids 20

      4.2 Objectivity and Convected Derivatives 23

      4.3 Canonical Nonlinear Differential Constitutive Equations 28

      4.4 A Kinetic Theory: The Linear Elastic Dumbbell Model 33

      5 Material Properties of Viscoelastic Fluids 39

      5.1 Normal Stress Differences 39

      5.2 Normal-Stress Measurements 42

      5.3 Other Flows 46

      6 Final Words of Caution: A Health Warning 47

      7 Conclusion 51

      References 51

2 Complex Fluids and Soft Structures in the Human Body 53

Paula A. Vasquez and M. Gregory Forest

      1 Introduction 53

      1.1 Biological Materials in the Human Body 58

      2 Mucus in the Human Body 64

      2.1 Mucus Composition 65

      2.2 Mucus Viscoelasticity 67

      2.3 Respiratory Mucus Clearance 72

      2.4 Diffusion in Mucus 83

      3 Modeling Structure and Dynamics Within a Single Cell: The Mitotic Yeast Spindle 91

      3.1 Modeling Mitosis in Yeast Cells 91

      4 Modeling Cell Motility 101

      References 104

Part II Rheology of Complex Biological Fluids

3 Theoretical Microrheology 113

Roseanna N. Zia and John F. Brady

      1 Introduction 114

      2 Passive Microrheology: Brownian Motion 117

      2.1 Single-Particle Diffusion and the Viscosity of Newtonian Solvents 118

      2.2 Extension to Viscoelastic Fluids: The Generalized Stokes-Einstein Relation 121

      2.3 Validity of the Stokes-Einstein Relation? 123

      2.4 Dual-Probe Microrheology 130

      3 Nonequilibrium Systems: Active Microrheology 132

      3.1 ModelSystem 133

      3.2 Microviscosity 134

      3.3 Force-Induced Diffusion: Microdiffusivity 139

      3.4 A Complete Picture: Microviscosity, Microdiffusivity, and Normal Stresses 143

      3.5 Time-Dependent Flows 147

      3.6 Brownian Dynamics Simulations 149

      4 A "Non-equilibrium Equation of State" 152

      5 Experimental Measurement 153

      6 Summary 154

      References 155

4 Membrane Rheology 159

Arthur A. Evans and Alex 5. Levine

      1 Overview of Membranes and Langmuir Monolayers 160

      2 Membrane Mechanics 165

      3 Dynamical Linear Response 166

      3.1 Flat Membranes 167

      3.2 Curved Surfaces 170

      4 Monolayer Rheology Experiments 172

      4.1 Macroscopic Methods 173

      4.2 Microrheology of Membranes 174

      4.3 The Case of the Missing Modulus 175

      4.4 Submerged Particle Microrheology 177

      5 Open Questions and New Challenges 182

      References 184

5 Rheology and Mechanics of the Cytoskeleton 187

Hamed Hatami-Marbini and Mohammad R.K. Mofrad

      1 Introduction: Intracellular Structure and Composition 187

      2 Cytoskeletal Rheology and Mechanics 192

      2.1 Experimental Studies 194

      2.2 Computational Studies: Continuum and Discrete Descriptions 196

      References 202

6 Experimental Challenges of Shear Rheology: How to Avoid Bad Data 207

Randy H. Ewoldt, Michael T. Johnston, and Lucas M. Caretta

      1 Introduction 207

      2 Background: Material Functions 210

      3 Challenges 214

      3.1 Instrument Specifications 214

      3.2 Instrument Inertia 217

      3.3 Fluid Inertia and Secondary Flows 220

      3.4 Surface Tension Forces 225

      3.5 Free Surface Films 228

      3.6 Slip 229

      3.7 Small Volume and Small Gap 231

      3.8 Other Issues 235

      4 Conclusions 237

      Appendix 238

      References 239

Part III Locomotion and Active Matter

7 Locomotion Through Complex Fluids: An Experimental View 245

Josue Sznitman and Paulo E. Arratia

      1 Introduction 245

      2 Basic Principles: Fluid Dynamics of Swimming at Low Reynolds Number 247

      3 Experiments in Newtonian Fluids 250

      3.1 From Scale-Up Models to Live Microorganisms 250

      3.2 Propulsive Force and Flow Measurements 252

      4 From Newtonian to Complex Fluids 255

      4.1 Swimming in Viscoelastic Fluids: Expectations 257

      5 Experiments in Viscoelastic Fluids 259

      5.1 Scale-Up Experiments 259

      5.2 Experiments with Live Organisms 261

      5.3 Fluid-Assisted Locomotion in Complex Fluids: Artificial Swimmers 272

      6 Conclusions and Outlook 276

      References 278

8 Theory of Locomotion Through Complex Fluids 283

Gwynn J. Elfring and Eric Lauga

      1 Introduction 283

      2 Locomotion in Fluids 284

      2.1 Boundary Motion 284

      2.2 The Lorentz Reciprocal Theorem 287

      2.3 Swimming in Newtonian Fluids 289

      2.4 Small Amplitude Motion 291

      3 Locomotion in Non-Newtonian Fluids 292

      3.1 Small-Amplitude Perturbations 293

      3.2 Slowly Varying Flows 298

      4 Infinite Models 302

      4.1 Taylor Swimming Sheet 303

      4.2 Large-Amplitude Deformations 307

      4.3 Shear-Dependent Viscosity 310

      4.4 Prescribed Forcing 311

      4.5 Two-Fluid Models 312

      4.6 Collective Effects 314

      5 Perspective 315

      References 316

9 Theory of Active Suspensions 319

David Saintillan and Michael J. Shelley

      1 Background 319

      2 A Simple Kinetic Model 325

      2.1 Smoluchowski Equation 325

      2.2 Mean-Field Flow and Active Stress Tensor 327

      2.3 The Conformational Entropy 329

      2.4 Stability of the Uniform Isotropic State 330

      3 Extensions and Applications 333

      3.1 Concentrated Suspensions 333

      3.2 Confinement 337

      3.3 Chemotaxis 340

      3.4 Fluid Viscoelasticity 343

      4 Other Active Fluids 344

      4.1 Microtubules and Motor Proteins 344

      4.2 Chemically Active Particles 348

      5 Outlook 351

      References 351

Part IV Computational Methods

10 Computational Challenges for Simulating Strongly Elastic Flows in Biology 359

Robert D. Guy and Becca Thomases

      1 Strongly Elastic Flows 361

      1.1 Historical Perspective 362

      1.2 Advances from Analysis 363

      1.3 High-Weissenberg Number Problem in the Oldroyd-B Model 364

      1.4 Numerical Approaches 366

      1.5 Molecular Models 369

      1.6 Extensional Flow Simulations 371

      2 Immersed Boundary Methods 376

      2.1 Immersed Boundary Equations 378

      2.2 Explicit-Time Stepping 379

      2.3 Implicit-Time Stepping 380

      3 Locomotion of Undulatory Swimmers 382

      3.1 Swimmer Model 382

      3.2 Swimming Speed 387

      3.3 Time and Space Resolution 389

      3.4 Effect of Increasing Bending Stiffness 391

      3.5 Efficiency of the Implicit-Time Method 392

      4 Conclusions 394

      References 395

11 Cell Distribution and Segregation Phenomena During Blood Flow 399

Amit Kumar and Michael D. Graham

      1 Background 400

      1.1 Blood: Components and Physiological Functions 400

      1.2 Rheology and Nonuniform Flow Phenomena in Blood 402

      1.3 Distribution of Blood Cells During Flow: Cell-Free Layer and Margination 404

      1.4 Effect of Plasma Rheology on Cell Distribution 408

      1.5 Motivation and Goals 410

      2 Problem Formulation and Implementation 411

      2.1 Fluid Flow Problem 411

      2.2 Membrane Mechanics 417

      3 Segregation by Membrane Rigidity: Simulations and Theory 420

      3.1 Boundary Integral Simulations of Binary Suspensions 420

      3.2 Master Equation Model for Binary Suspensions 422

      4 Effect of Polymer Additives 428

      5 Conclusions and Outlook 429

      References 431

Index 437

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