Microscale Flow and Heat Transfer: Mathematical Modeling and Flow Physics

Abstract

4.2 Examples Needing a Look Beyond the Navier-StokesEquations This book covers concepts and the latest developments on microscale flow and heat transfer phenomena involving a gas. The book is organised in two parts: the first part focuses on the fluid flow and heat transfer characteristics of gaseous slip flows. The second part presents modelling of such flows using higher-order continuum transport equations. The Navier-Stokes equations based solution is provided to various problems in the slip regime. Several interesting characteristics of slip flows along with useful empirical correlations are documented in the first part of the book. The examples bring out the failure of the conventional equations to adequately describe various phenomena at the microscale. Thereby the readers are introduced to higher order continuum transport (Burnett and Grad) equations, which can potentially overcome these limitations. A clear and easy to follow step by step derivation of the Burnett and Grad equations (superset of the Navier-Stokes equations) is provided in the second part of the book. Analytical solution of these equations, the latest developments in the field, along with scope for future work in this area are also brought out. Presents characteristics of flow in the slip and transition regimes for a clear understanding of microscale flow problems; Provides a derivation of Navier-Stokes equations from microscopic viewpoint; Features a clear and easy to follow step-by-step approach to derive Burnett and Grad equations; Describes a complete compilation of few known exact solutions of the Burnett and Grad equations, along with a discussion of the solution aided with plots; Introduces the variants of the Navier-Stokes, Burnett and Grad equations, including the recently proposed Onsager-Burnett and O13 moment equations. Intro; Preface; Acknowledgments; Contents; Nomenclature; 1 Introduction to Microscale Flows and Mathematical Modelling; 1.1 Introduction; 1.2 Applications of Microscale Flows; 1.2.1 Cooling of Electronic Devices; 1.2.2 Micro-nozzles and Micro-thruster; 1.2.3 Breath Analyser; 1.2.4 Microdevice for Conducting Blood Test; 1.3 Classification of Flow Regimes; 1.4 Characteristics of Microscale Flows; 1.4.1 Rarefaction; 1.4.2 Compressibility; 1.4.3 Thermal Creep; 1.4.4 Viscous Dissipation; 1.4.5 Property Variation; 1.4.6 Axial Conduction; 1.4.7 Conjugate Heat Transfer 1.5 Mathematical Modelling of Microscale Flows1.5.1 The Navier-Stokes Equations; 1.5.2 Limitations of Conventional Equations and Boundary Conditions; 1.5.3 Approach to Modelling Microscale Flows; 1.6 Relevance and Scope of the Book; 2 Microscale Flows; 2.1 Introduction; 2.2 Governing Equations for Fluid Flow; 2.2.1 Tensorial Form; 2.2.2 Constitutive Relations; 2.2.3 Compressible Navier-Stokes Equations; 2.2.4 Incompressible Navier-Stokes Equations; 2.3 Boundary Conditions; 2.3.1 Maxwell's Slip Theory; 2.3.2 Derivation of Higher-Order Slip Boundary Condition; 2.3.3 Alternate Slip Models 2.3.4 Value of Slip Coefficients2.4 Some Exact Solutions; 2.4.1 Couette Flow; 2.4.2 Flow in a Microchannel; 2.4.2.1 Perturbation of Navier-Stokes Equations; 2.4.2.2 Integral Approach; Note on Use of the Equations; 2.4.2.3 Comments; 2.4.3 Flow in a Microtube; 2.4.4 Flow in an Arbitrary Cross Section Microchannel; 2.4.5 Flow in the Annulus of Rotating Sphere and Cylinder; 2.5 Observations on Flow in Straight Passages; 2.5.1 Appearance of Knudsen Minima; 2.5.2 Flow in Rough Microchannel; 2.5.3 Transient Flow in a Capillary; 2.6 Observations on Flow in Complex Passages 2.6.1 Flow in Sudden Expansion/Contraction Microchannel2.6.2 Flow in Diverging/Converging Microchannel; 2.6.3 Flow in a Bend Microchannel; 2.7 Useful Empirical Correlations; 2.7.1 Developing Length in Microtube and Microchannel; 2.7.2 Friction Factor for Microchannel of Various CrossSections; 2.8 Summary; 3 Microscale Heat Transfer; 3.1 Introduction; 3.2 Governing Equations; 3.3 Boundary Conditions; 3.3.1 Derivation of First Order Temperature Jump Condition; 3.4 Some Exact Solutions; 3.4.1 Heat Transfer in Microchannel; 3.4.1.1 Uniform Heat Flux; 3.4.1.2 Constant Wall Temperature Case 3.4.1.3 Parametric Variation3.4.2 Heat Transfer Analysis Through a Micropipe; 3.4.2.1 Constant Wall Temperature Case; 3.4.2.2 Parametric Variation; 3.4.3 Heat Transfer Through a Micro-Annulus; 3.4.3.1 Parametric Variation; 3.5 Observations on Other Effects; 3.5.1 Variation in Thermophysical Properties; 3.5.2 Conduction in the Substrate; 3.5.3 Axial Conduction; 3.5.4 Flow Work and Shear Work; 3.6 Observations from Experiments; 3.7 Application to Knudsen Pump; 3.8 Useful Empirical Correlations; 3.9 Summary; 4 Need for Looking Beyond the Navier-Stokes Equations; 4.1 Introduction