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## Über dieses Buch

This book provides analytical solutions to a number of classical problems in transport processes, i.e. in fluid mechanics, heat and mass transfer. Expanding computing power and more efficient numerical methods have increased the importance of computational tools. However, the interpretation of these results is often difficult and the computational results need to be tested against the analytical results, making analytical solutions a valuable commodity. Furthermore, analytical solutions for transport processes provide a much deeper understanding of the physical phenomena involved in a given process than do corresponding numerical solutions. Though this book primarily addresses the needs of researchers and practitioners, it may also be beneficial for graduate students just entering the field.

## Inhaltsverzeichnis

### Chapter 1. The Equations of Change in Fluid Mechanics and Their Analytical Solutions

In our discussion of problems in fluid mechanics, we restrict the theoretical basis to incompressible fluid continua with negligible effects of dissipative heating. The rheological constitutive equation may be the Stokesian law for Newtonian fluids or may involve linear viscoelasticity. We put together the equations of change in fluid mechanics first and then discuss the concepts for solving them analytically. We put together solutions for a selection of problems of interest in transport processes of engineering applications. Extensive discussions of exact solutions of the Navier-Stokes equations and of common errors in finding exact solutions of nonlinear differential equations are found in [6, 7, 12].
Günter Brenn

### Chapter 2. The Equation for the Stokesian Stream Function and Its Solutions

This chapter presents and discusses the equation for the Stokesian stream function. The equation emerges as the one non-zero component of the curl of the two-dimensional momentum equation with the velocity components given as spatial derivatives of the stream function. The stream function is defined such that its derivatives yield a solenoidal velocity field. The analyses of the flows discussed in Part I of this book are based on this function. In view of our search for analytical solutions, we are restricted to laminar two-dimensional flow in simple geometries. The equations of change therefore need no turbulence modelling, the concept of the Stokesian stream function can be applied for representing the flow velocity, and the boundary conditions are easy to formulate and implement analytically in the general solutions. The fluids are treated as incompressible and Newtonian or linear viscoelastic. The linear viscoelastic liquids exhibit a viscosity depending on frequency, but not on shear rate. Furthermore, we restrict this analysis to flows without heat and mass transfer, i.e. we solve the continuity and momentum equations and disregard the influence of viscous dissipation on the energy budget of the flow. We are therefore restricted to flow without viscous heating. Problems of heat and mass transfer are the subjects of Part II of this book.
Günter Brenn

### Chapter 3. Laminar Two-Dimensional Flow

The present chapter discusses flows through structures with solid walls and constant flow cross sections, and flows outside the surfaces of solid bodies in motion, allowing for the formation of two-dimensional velocity fields. The flows may be steady or unsteady. We discuss a selection of classical flows with generic relevance for technical applications, as represented in other books as well [3, 9, 11, 15]. We add the discussion of a flow relevant for the biomechanics of brain injuries. In all cases we start from the stream functions derived in the preceding chapter.
Günter Brenn

### Chapter 4. Lubrication Flow

The lubrication approximation in analysing flow fields makes use of the geometrical properties of the flow field that it is long in the flow direction and narrow in a direction transverse to it. This “slenderness” of the flow field has as a consequence that the orders of magnitude of the velocities and their spatial derivatives in the two coordinate directions are very different. A narrow flow field allows both effects of the inertia of the fluid and derivatives of viscous stress in the main flow direction to be neglected in the momentum balance, so that the flow is dominated by an equilibrium of pressure and viscous forces. What is essentially solved, therefore, are the Stokes equations. We first derive the lubrication approximation and then discuss some flows of this kind with technical relevance.
Günter Brenn

### Chapter 5. Boundary-Layer Flow

The present chapter discusses steady flows of boundary-layer type, which may be described analytically in simple geometries of the flow fields by solutions of the momentum equation in boundary-layer form. The boundary-layer character of the flows is brought about by the superimposition of convective momentum transport in the main flow direction and diffusive (or turbulent) transport in the direction transverse to it. This type of flow occurs not only due to the presence of solid walls, but also in the propagation of momentum by the two mechanisms in free shear layers, submerged free jets, and wakes. All the flows discussed here are treated by the concept of self-similarity. In [2] we find an extensive section by Yarin about self-similar flows, with a large table of solutions including temperature fields.
Günter Brenn

### Chapter 6. Flows with Interfaces

We now look at linear flows with interfaces. These flows and their instability are elementary to the formation of the disperse phase in many gas–liquid two-phase flows, such as sprays and bubbly flows. We are interested in the instability of liquid sheets and jets submerged in another, immiscible fluid. For transport processes across the interface, oscillations of drops and bubbles may have a significant influence. We also look at the behaviour of drops upon impact on a solid substrate. The fluid system is a linear viscoelastic liquid in an outer immiscible host fluid. The correspondence principle allows for the derivation of the equations as for a Newtonian liquid, but with a frequency-dependent viscosity. The ambient medium is a gas for the liquid sheet flow, since the flow of a liquid sheet in an ambient liquid medium does not seem to be of much technical relevance. In the liquid jet and drop cases, we treat the ambient fluid such that it may be either a liquid or a gas.
Günter Brenn

### Chapter 7. The Equations of Change for Heat and Mass Transfer and Their Analytical Solutions

The present chapter puts together the thermal energy equation, with its special form for heat conduction, and the continuity equation for a component of a mixture in its various forms, including the purely diffusive form for equimolar processes. The two subsequent Chaps. 8 and 9 discuss analytical solutions of these equations.
Günter Brenn

### Chapter 8. Heat Transfer

This chapter discusses problems of thermal energy transport which may be solved analytically. The processes may be conductive in nature, involving the diffusion equation as the underlying differential equation for the spatiotemporal evolution of temperature. As far as convective processes are concerned, the intention to have analytical descriptions clearly puts the restriction to laminar flow in simple geometries.
Günter Brenn

### Chapter 9. Mass Transfer

The present chapter discusses problems of mass transport which may be solved analytically. The processes may consist of mass transport by equimolar diffusion, which involves the diffusion equation as the underlying differential equation for the spatiotemporal evolution of the species concentration. As far as convective processes are concerned, the intention to have analytical descriptions clearly puts the restriction to laminar flow in simple geometries.
Günter Brenn

### Backmatter

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