Overview of non-Newtonian boundary layer flows and heat transfer

Overview of non-Newtonian boundary layer flows and heat transfer

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Leye M. Amooa , R. Layi Fagbenleb a Stevens Institute of Technology, Hoboken, NJ, United States, b Mechanical Engineering Department and Center for Petroleum, Energy Economics Law, CPEEL (A John D. and Catherine T. McArthur Center of Excellence), University of Ibadan, Ibadan, Oyo State, Nigeria

14.1 Introduction Non-Newtonian fluids are fluids with a stress that can have a nonlinear and/or temporal dependence on the rate of deformation, unlike Newtonian fluids, which demonstrate a linear dependence. The literature reveals that interest in non-Newtonian fluids has grown since the 1940s and 1950s. Since the majority of raw materials and finished products from the processing industry (food, polymers, emulsions, slurries, etc.) are non-Newtonian fluids, it is becoming increasingly important to understand physical characteristics of these fluids [1]. Since most of the differences among the different categories of non-Newtonian fluids are related to their viscosity, which is a dominant physical property within the boundary layer region, a thorough understanding of the flow in the boundary layer is of considerable importance in a range of chemical and processing applications. The nature of boundary layer flow influences not only the drag at a surface or on an immersed object, but also the rates of heat and mass transfer when temperature or concentration gradients exist. The literature shows that there is a significant amount of research with the goal of understanding non-Newtonian flows through pipes and channels due to its relevance to the applications mentioned previously [2,3]. A limited body of research on external flows of non-Newtonian fluids also exists [4–6]. Section 14.2 of this chapter presents a review of selected research performed in relation to the behavior of non-Newtonian boundary layer flows and laminar heat transfer characteristics in non-Newtonian fluids. A summary of current research efforts is provided in Sect. 14.3, followed by a brief overview of future research prospects in this area in Sect. 14.4.

14.2

Background

The authors in [7] discerned three stages in the development of fluid mechanics. During the first stage, studies were focused on ideal fluids, that is, fluids without viscosity, Applications of Heat, Mass and Fluid Boundary Layers. https://doi.org/10.1016/B978-0-12-817949-9.00022-0 Copyright © 2020 Elsevier Ltd. All rights reserved.

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compressibility, or elasticity, with all remaining material properties kept constant. Such fluids are hypothetical and were used mainly for analysis. Despite the seemingly crude approximations, theories on inviscid and incompressible fluids led to groundbreaking results in many areas of science and engineering (e.g., accurate prediction of lift force, which paradoxically is a viscous effect). The next stage introduced viscous effects. Viscous effects and the equations used to model them were introduced in fluid mechanics studies in the first half the 19th century. These were buttressed by Ludwig Prandtl in 1904, who assumed that viscosity becomes important only in the boundary layer, formed in the direct vicinity of a solid surface. Prandtl recognized that in low-viscosity fluids, viscous effects could be localized to thin layers near boundaries, and subsequently, his student Blasius showed how the mathematics of this approximation could be developed for a flat plate. Hence, the flow domain was decomposed into a region of ideal fluid (far from the surface) and a viscous fluid (close to the surface). This approach is the basis for classical fluid dynamics. Lastly, the third stage, which is still an active area of research, addresses the departure from Newton’s linear law of viscosity. Its importance was appreciated at the beginning of the last century, as many industrial materials could not be accurately described with this simple relation. Two sources of non-Newtonian behavior can be distinguished. On a microscopic level, it is the molecular structure of fluid particles. Spherical and roughly spherical particles produce a Newtonian behavior, whilst the addition of long chains of particles may cause Newton’s approximation to become invalid. On a macroscopic level, mixtures such as emulsions and slurries may become non-Newtonian even though their components are Newtonian. Fluids can exhibit non-Newtonian behavior in several ways. They may be purely viscous, in that the stress depends on the rate of deformation in a nonlinear fashion, but there is no dependence on the history of the deformation. Viscous non-Newtonian fluids may be further classified as dilatant (shear-thickening, e.g., corn starch in water, or Oobleck) and pseudo-plastic (shear-thinning, e.g., nail polish, ketchup). They may be viscoelastic, in that the stress depends in a well-defined way on the history of the deformation; viscoelastic liquids are also called memory fluids and include fluids like lubricants, Silly Putty, and so on. Fluids may be thixotropic, in that the material properties are time-dependent at constant stress or deformation rate. This category includes fluids such as yogurt and a variety of gels. Other major types of non-Newtonian fluids include yield stress fluids (also called Bingham plastic fluids) that do not flow at all until a critical stress level is reached and liquid crystals that are anisotropic at rest [1]. These types of non-Newtonian fluids discussed above are presented in Fig. 14.1. As a consequence of having so many categories of fluids classified as nonNewtonian, one of the primary difficulties in any theoretical analysis of the motion of such fluids has been the lack of any generally acceptable equation of state between the stress tensor and state of flow of the system applicable to all these categories [8]. However, a few attempts have been made to arrive at a generalized formulation to describe the rheological properties of these fluids [9]. The following sections briefly describe the general characteristics of laminar boundary layers in the various categories of non-Newtonian fluids mentioned previously.

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Figure 14.1 The different types of non-Newtonian fluids.

14.2.1 Time-independent non-Newtonian fluid behavior Fluids in which the strain experienced at any point is purely dependent on the current value of the shear stress at that point are considered to be time-independent nonNewtonian fluids. They may be further categorized as the following: • shear-thinning, or pseudoplastic, fluids; • shear-thickening, or dilatant, fluids; and • viscoplastic fluids.

14.2.1.1

Shear-thinning, or pseudo-plastic, fluids

Shear-thinning is the most common type of time-independent non-Newtonian fluid behavior that can be observed. It is also called pseudoplasticity and is characterized by an apparent viscosity, which decreases with increasing shear rate, as shown in Fig. 14.2. The power-law model, or the Ostwald–de Waele model, is one of the more popular models used to describe the behavior of shear-thinning liquids. The apparent viscosity, μ, for the power-law fluid is given by the following relation: μ = mγ˙ ∗n−1 ,

(14.1)

where γ˙ is the shear rate. In this equation, m and n are empirical curve-fitting parameters known as the fluid consistency coefficient and the flow behavior index, respectively. For shear-thinning fluids, the value of n varies from 0 to 1. For values of n greater than 1, the model is applicable to shear-thickening fluids, while n = 1 represents Newtonian fluids. Thus, the smaller the value of n, the greater the degree of shear-thinning. Significant deviations from this model are observed at very low and very high shear rates, which are captured well by other models, such as, the Carreau and Ellis models. The application of boundary layer theory to an Ostwald–de Waele, or power-law, fluid was first described by Schowalter [10] and Acrivos et al. [4], where it was

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Figure 14.2 Shear-thinning behavior.

shown that self-similar solutions exist for the boundary layer flow of a power-law fluid when the flow is of the Falkner–Skan type. Specifically, Schowalter [10] examined the boundary-layer equations of a power-law fluid in the absence of body forces and showed the existence of self-similar solutions when the external velocity is of the form x m , where x is the distance along the surface of the body. Acrivos et al. [4] considered the boundary layer flow of a power-law fluid for the case m = 0, which corresponds to a flow along a flat plate. The continuity and momentum transport equations of a power-law fluid assuming a Cartesian co-ordinate system in dimensional form can be written as follows: ∂u∗ ∂v ∗ ∂w ∗ + + ∗ = 0, ∂x ∗ ∂y ∗ ∂z   ∗ ∂τy∗∗ x ∗ ∂τz∗∗ x ∗ ∂u ∂u∗ ∂u∗ ∂p ∗ ∂τ ∗∗ ∗ ρ u∗ ∗ + v ∗ ∗ + w ∗ ∗ = ∗ + x ∗x + + ∂x ∂y ∂z ∂x ∂x ∂y ∗ ∂z∗ (14.2)   ∗ ∗ ∗ ∗ ∗ ∂τy ∗ y ∗ ∂τz∗∗ y ∗ ∂p ∗ ∂τx ∗ y ∗ ∗ ∂v ∗ ∂v ∗ ∂v ρ u +v +w + + , = ∗+ ∂x ∗ ∂y ∗ ∂z∗ ∂y ∂x ∗ ∂y ∗ ∂z∗   ∂τy∗∗ z∗ ∂τz∗∗ z∗ ∂v ∗ ∂w ∗ ∂w ∗ ∂p ∗ ∂τ ∗∗ ∗ ρ w ∗ ∗ + v ∗ ∗ + w ∗ ∗ = ∗ + x ∗z + + , ∂x ∂y ∂z ∂z ∂x ∂y ∗ ∂z∗ variables (u∗ , v ∗ , w ∗ ) and p ∗ represent the components of fluid velocity and pressure inside the flow, respectively. Variables (x ∗ , y ∗ , z∗ ) are dimensional coordinates of a given location and ρ is the density. The components of the stress tensor are given by τij∗ . In dimensionless form, where L is taken as a typical length, U as typical speed, and ε as a scaling factor that is a function of the Reynolds number, the dimensionless

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variables can be expressed as x∗ , L u∗ u= , U

x=

y∗ , L v∗ v= , U y=

z∗ , L w∗ w= , U

z=

p=

p∗ . ρU 2

(14.3)

Thus, the continuity and momentum transport equations in dimensionless form can be written as ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z   ∂τyx ∂τxx ∂u ∂u ∂p 1 ∂τzx ∂u +v +w =− + + + , u ∂x ∂y ∂z ∂x ρU 2 ∂x ∂y ∂z     ∂τxy ∂τyy ∂τzy 1 ∂p ∂v ∂v ∂v 1 =− ,  u +v +w + + + ∂x ∂y ∂z  ∂y ρU 2 ∂x ∂y ∂z     ∂τxz ∂τyz ∂τzz ∂w ∂w 1 ∂p 1 ∂w +v +w =− + + + .  u ∂x ∂y ∂z  ∂z ρU 2 ∂x ∂y ∂z

(14.4)

At sufficiently large Reynolds numbers, upon further simplification, the equations that describe boundary layer flow in three-dimensional form of a power-law fluid can be written as ∂u ∂v ∂w + + = 0, ∂x ∂y ∂z    n−3 ∂u ∂ 2 u ∂u ∂ 2 u ∂u δu ∂u ∂p ∂u 2 (n − 1) +v +w =− +κ + u ∂x ∂y ∂z δx ∂y ∂y 2 ∂z ∂y∂z ∂y   2   2 2 ∂u ∂ u ∂u ∂ u ∂ 2u ∂u ∂ u + +κ +(n − 1) + 2) , ∂y ∂z∂y ∂z ∂z2 ∂z ∂y 2 ∂z ∂p 0=− , ∂y ∂p 0=− . ∂z (14.5) The following equation defines k:  2  2 ∂u ∂u κ= + . ∂y ∂z

(14.6)

Assuming that the Carreau model of viscosity applies to the fluid, the boundary layer equations would differ slightly from the Ostwald–de Waele form. The dynamic viscosity of a Carreau fluid can be written as n−1  2 μ∗ = μ∞ + (μ0 − μ∞ ) 1 + (K1 γ˙ ∗ )2 ,

(14.7)

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where μ0 is the zero-shear-rate viscosity, μ∞ is the infinite-shear-rate viscosity, γ˙ ∗ is the shear rate, and K1 is the characteristic time constant, sometimes referred to as the relaxation time. Adopting a similar approach to that used in power-law fluids, the boundary layer equations for fluids whose viscosity is governed by the Carreau model can be written as ∂u ∂v + = 0, ∂x ∂y

⎡ ⎤   n−3   δu ∂p ⎣ ∂u 2 ∂u 2 2 ⎦ ∂ 2 u ∂u , +v =− + 1 + C0 1 + n λ 1+ λ u ∂x ∂y δx ∂y ∂y ∂y 2 0=−

∂p . ∂y (14.8)

∞ , and λ is the dimensionless The variable C0 is the viscosity ratio given by μ0μ−μ ∞ equivalent of K1 . Thompson and Snyder [11] examined the boundary-layer flow of a power-law fluid in the presence of fluid injection at the surface with the aim of determining the drag reduction potential of such non-Newtonian fluids. Solutions were found for a range of mass injection rates, the results indicating that the skin-friction coefficient decreases monotonically as the fluid index increases. For a fixed rate of fluid injection, it was shown that the percentage reduction in the skin-friction coefficient was greater for smaller values of the fluid index. Andersson and Toften [12] discussed aspects of obtaining a numerical solution to the laminar boundary layer equations for a power-law fluid. Their work provides a concise review of various techniques for finding solutions for laminar boundary-layer flow of power-law fluids and certain notable shortcomings of these techniques.

14.2.1.2

Shear-thickening, or dilatant, fluids

Dilatant fluids are similar to pseudoplastic systems in that they show no yield stress, but their apparent viscosity increases with increasing shear rate; thus, these fluids are also called shear-thickening fluids. This type of fluid behavior was originally observed in concentrated suspensions. At low shear rates, the liquid lubricates the motion of each particle past other particles and the resulting stresses are consequently small. At higher shear rates, on the other hand, the material expands or dilates slightly (as also observed in the transport of sand dunes) so that there is no longer a sufficient amount of liquid to fill the increased void and prevent direct solid–solid contacts that result in increased friction and higher shear stresses. This mechanism causes the apparent viscosity to rise rapidly with increasing rate of shear. Eqs. (14.2)–(14.8) can also be applied to shear-thickening fluids to understand the behavior of boundary layers in such instances. The term “dilatant” has also been used for all other fluids that exhibit increasing apparent viscosity with increasing rate of shear. Many of these, such as starch pastes,

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are not true suspensions and show no dilation on shearing. The above explanation therefore is not applicable, nevertheless such materials are still commonly referred to as dilatant fluids. Of the time-independent fluids, this subclass has received very little attention; consequently, little reliable data are available.

14.2.1.3

Viscoplastic fluids

This type of fluid behavior is characterized by the existence of a yield stress that must be exceeded before the fluid will deform or flow. Conversely, such a material will deform elastically (or flow en masse like a rigid body) when the externally applied stress is smaller than the yield stress. Once the magnitude of the external stress has exceeded the value of the yield stress, the flow curve may be linear or nonlinear but will not pass through origin, as seen in Fig. 14.1. The flow of muddy rivers is a typical example. Among many viscoplastic fluids, there is a special class called Bingham plastics. For Bingham plastic fluid, the shear stress beyond the yield stress is linearly proportional to the shear rate. If the yield stress approaches zero, the Bingham plastic fluid can be approximately treated as a Newtonian fluid. Mathematically, this model can be represented as: τ = τ0 + μγ˙ when τ ≥ τ0 ; and γ˙ = 0 when τ < τ0 ,

(14.9)

where τ is the shear stress and τ0 is the yield stress, below which the fluid behaves essentially like a rigid body. Regarding the flow of the Bingham fluid, the stress varies in space and time. There can be regions in the fluid where the yield stress is exceeded, and other regions where it is not. The boundaries between the two regions are the yield surfaces. Tracking the yield surfaces as the flow evolves is one of the most complicated problems associated with the Bingham model. The stability of the viscoplastic flows depends on what happens to these nonmaterial surfaces when a sudden perturbation is introduced.

14.2.2 Time-dependent non-Newtonian fluid behavior In practice, apparent viscosities may depend not only on the rate of shear but also on the time for which the fluid has been subjected to shearing, as seen in crude oils, emulsions, and so on. They are further classified based on whether the apparent viscosity decreases or increases with the amount of time they are subjected to shearing, as thixotropic and rheopectic fluids, respectively.

14.2.2.1

Thixotropic and rheopectic fluids

In thixotropic fluids, the shear stress (or apparent viscosity) decreases with increasing time of application of a constant shear rate. The macroscopic rheological properties of many complex fluids depend on their microscopic structure. If the flow curve is measured in a single experiment in which the shear rate is steadily increased at a constant rate from zero to some maximum value and then decreased at the same rate to zero again, a hysteresis loop of the form shown in Fig. 14.3 is obtained; the height, shape,

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Applications of Heat, Mass and Fluid Boundary Layers

and enclosed area of the hysteresis loop depend on the duration of shearing, the rate of increase and decrease of the shear rate, and the past kinematic history of the sample. Thixotropic behavior occurs if, when exposed to high shear rates, the microstructure gradually breaks down and the fluid becomes less viscous, and if, when exposed to low shear rates, the microstructure is gradually rebuilt and the fluid becomes more viscous. This is common for colloidal dispersions. The dynamics of the microstructure may occur on different timescales from the macroscopic flow. Coussot et al. [13] proposed a simple model to describe the rheological properties of a thixotropic fluid as   μ = μo 1 + λn ,   ∂V ∂λ 1 = −α λ, ∂t θ ∂y

(14.10)

where V is the velocity, y is the distance normal to the flow direction, μ0 , n, θ and α are constant parameters for a given fluid, and μ is the apparent viscosity of the thixotropic fluid defined as μ=

τ ∂V ∂y

,

(14.11)

where τ is the shear stress. The degree of jamming of the thixotropic fluid can be represented by a single parameter λ which describes the instantaneous state of fluid structure. This parameter, λ, is also called the degree of jamming of the fluid. Barnes [14] provides an excellent review of thixotropic fluids, considering examples of coal suspensions, oils and lubricants, paints, detergents, clay suspensions, creams and pharmaceutical products, blood, and so on. Barnes also provides a theoretical and mathematical basis for the observed behavior from the point of view of microstructural changes, concluding that more work is needed to develop a comprehensive theory to describe such fluid systems. In rheopectic fluids, an opposite effect is observed, wherein the shear stress (or the apparent viscosity) decreases with increasing time of application of a constant shear rate. As seen in Fig. 14.3, the direction of the hysteresis loop is reversed compared to the thixotropic fluid.

14.2.3 Non-Newtonian laminar boundary layer flows Laminar boundary layers in non-Newtonian flows have been studied over the past half century. Bizzell and Slattery [15] provided a succinct explanation of boundary layer flows of non-Newtonian fluids and proposed an approximate solution of the boundary layer equations. The following subsections briefly describe the popular fundamental studies that have been undertaken to improve our understanding of the behavior of non-Newtonian flows over or through simple geometries such as flat plates, pipes, cylindrical annuli, porous surfaces, and so on.

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Figure 14.3 Behavior of thixotropic and rheopectic fluids, showing the hysteresis loops.

14.2.3.1

Laminar boundary layer flows through pipes

For all fluids entering a small pipe from either a much larger one or from a reservoir, the initial velocity profile is approximately flat, and it undergoes a progressive change until fully developed flow is established, as shown schematically in Fig. 14.4. The thickness of the boundary layer is theoretically zero at the entrance and increases progressively along the tube. The retardation of the fluid in the wall region must be accompanied by a concomitant acceleration in the central region in order to maintain continuity. When the velocity profile has reached its final shape, the flow is fully developed and the boundary layers may be considered to have converged at the centerline. It is customary to define an entry length, Le , as the distance from the inlet at which the centerline velocity is 99% of the velocity of the fully-developed flow. The pressure gradient in this entry region is different from that for fully developed flow and is a function of the initial velocity profile. There are two factors influencing the pressure gradient in the entry region: first, some pressure energy is converted into kinetic energy as the fluid in the central core accelerates, and second, the higher-velocity gradients in the wall region result in greater frictional losses. It is important to estimate both the pressure drop occurring in the region before flow has been fully developed and the extent of this entrance length. This situation is amenable to analysis by repeated use of the mechanical energy balance equation. Dodge and Metzner (1959) indicated that both the entrance length and the extra pressure loss for inelastic fluids were similar to those for Newtonian fluids. In the case of non-Newtonian flow, it is necessary to use an appropriate apparent viscosity. Although the apparent viscosity, μa , is defined in the same way as for a Newtonian fluid, it no longer has the same fundamental significance, and other, equally valid definitions of apparent viscosities may be made. In a flow through a pipe, where the shear stress varies with radial location, the value of μa also varies. The conditions near the pipe wall are most important. A fluid with a yield stress (e.g., Bingham plastic) will flow only if the applied stress (proportional to pressure gradient) exceeds the yield stress. There will be a solid

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Figure 14.4 Development of the boundary layer and velocity profile for laminar flow of a power-law fluid in the entrance region of a pipe.

Figure 14.5 Development of the boundary layer and velocity profile for laminar flow of a Bingham plastic fluid, demonstrating “plug-flow” at the core.

plug-like core flowing in the middle of the pipe where the applied shear stress is less than the yield stress, as shown in Fig. 14.5. Rudman and Blackburn [16] have implemented the non-Newtonian viscosity in their direct numerical simulation (DNS) code and validated for power-law fluids against laminar pipe flow and axisymmetric Taylor–Couette flow, both of which have analytical solutions. For the Herschel–Bulkley model, their code was validated against laminar pipe flow, and apart from ensuring correct viscosity estimates for known shear fields, no additional tasks were performed for the Carreau–Yasuda model. In all laminar simulation cases, the authors reported that numerical and theoretical velocity profiles agreed to within 0.01%, and the code is believed to accurately predict the laminar flow of non-Newtonian fluids with generalized Newtonian rheologies.

14.2.3.2

Laminar boundary layer flow over a flat plate

Yao and Molla [17,18] provided a realistic model to describe the laminar flow of a nonNewtonian power-law fluid over a flat plate. They stated that two widespread mistakes

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appear continuously in papers studying boundary layers involving the traditional twoparameter power-law model of non-Newtonian fluids: (a) failure to recognize that a length scale is associated with the power-law correlation and (b) unrealistic physical results, introduced by the traditional power-law correlation, that viscosity either vanishes or becomes infinite within the limit of a large or small shear rate, respectively. They removed the singularities at the leading edge of the flat plate and numerically solved the boundary layer equations using a simple finite difference method formulation. Hirschhorn et al. [6] provided a detailed analysis of a magneto-hydrodynamic boundary layer slip flow and heat transfer of a power-law fluid over a flat plate.

14.2.3.3

Laminar boundary layer flow over a moving stretched sheet

Investigations of boundary layer flow and heat transfer of viscous fluids over a stretched flat sheet are important in many manufacturing processes, such as polymer extrusion, drawing of copper wires, continuous stretching of plastic films and artificial fibers, hot rolling, wire drawing, glass-fiber, metal extrusion, and metal spinning. Sakiadis [19] initiated the study of the boundary layer flow over a stretched surface moving with a constant velocity and formulated a boundary-layer equation for twodimensional and axisymmetric flows. Although the geometry is similar to a flat plate, since the boundary conditions of the flow over a stretched flat sheet are different from that on a flat plate, the Blasius solution of the laminar boundary layer equations over a flat plate are not valid in this case. Andersson and Kumaran [20] analyzed the nonNewtonian fluid flow over a moving plane sheet with its surface velocity proportional to the distance from the slit raised to an arbitrary power m. Their numerical results showed that the boundary layer thickness decreases monotonically with increasing values of power-law index n.

14.2.3.4

Laminar boundary layer flow over a porous surface

Savins [21] presented a broad overview of non-Newtonian boundary layer flows followed by detailed analyses pertaining to models of porous media such as sand packs and matrices of uniformly packed spheres and woven screen, as well as alundum plugs, sandstone cores, porous metal disks, sintered glasses, and compressed glass wool. A model porous medium is typically unique in geometric morphology, and there are formidable problems in precisely defining the flow conditions existing within any particular porous structure. Abnormal increases in flow resistance that resemble a shear-thickening response is observed in flow of fluids through porous media. This general type of behavior has been observed in porous media flow experiments involving a variety of dilute to moderately concentrated solutions of high-molecular-weight polymers. Flow destabilization or premature departure from laminar-like behavior in dilute polymeric solutions have also been observed in flow experiments with porous media. Geometry-dependent flow behavior in porous media has been detected and seems to be characteristic of certain micellar systems.

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14.2.3.5

Applications of Heat, Mass and Fluid Boundary Layers

Instabilities in non-Newtonian laminar boundary layers

Pearson [22] provided a detailed review of instability in laminar non-Newtonian boundary layers. Scenarios in which essentially non-Newtonian effects are responsible for the instability were separated from cases in which a mere modification of Newtonian analyses is sufficient. Unidirectional confined flow examples such as circular Couette flow (to study Taylor vortices) and cellular convection flow (to study Bernard instabilities) were followed by unconfined flow examples such as jets and filament drawings to explain scenarios in which modifications to Newtonian analysis is sufficient. To explain the dominance of instabilities due to purely non-Newtonian effects, scenarios such as melt fracture, Couette flow and fluidized beds were explained in detail. Pearson [22] concluded that “there are far too many parametric functions needed to describe the constitutive relations for any particular material, which give rise at best to a bewildering number of approximate dimensionless groups characterizing any particular flow” and recommended that further research be undertaken to better understand some of these characteristics of instability.

14.2.4 Heat transfer in non-Newtonian laminar boundary layers When a fluid and the immersed surface are at different temperatures, heat transfer occurs. If the heat transfer rate is small in relation to the thermal capacity of the flowing stream, its temperature remains constant. The surface may remain at a constant temperature, the heat flux at the surface may be maintained constant, or surface conditions may be between these two limits. Because the temperature gradient is highest near the surface and the temperature of the fluid stream is approached asymptotically, a thermal boundary layer may be postulated that covers the region close to the surface and in which the whole of the temperature gradient is assumed to lie. Thus, a momentum and a thermal boundary layer will develop simultaneously whenever the fluid stream and the immersed surface are at different temperatures. The momentum and energy equations are coupled, because the physical properties of non-Newtonian fluids are normally temperature and shear-rate dependent. The resulting governing equations for momentum and heat transfer require numerical solutions. However, if the physical properties of the fluid do not vary significantly over the relevant temperature interval, there is little interaction between the two boundary layers and they may both be assumed to develop independently of one another. The physical properties other than apparent viscosity may be taken as constant for commonly encountered non-Newtonian fluids. In general, the thermal and momentum boundary layers will not correspond. Metzner [2,23] provided an extensive overview of some of the critical aspects of heat transfer in a non-Newtonian laminar boundary layer and the mathematical formulation to describe its features. Specific studies related to thermodynamics and entropy generation in non-Newtonian laminar boundary layers are discussed in the following subsection.

Overview of non-Newtonian boundary layer flows and heat transfer

14.2.4.1

425

Thermodynamic-entropy generation in non-Newtonian boundary layers

As discussed in the previous section, the thermodynamic properties of non-Newtonian fluid systems play an important role in the understanding of the system’s overall behavior. One of the popular analyses undertaken in this direction is the exergy analysis. Exergy analysis relies on the laws of thermodynamics to establish the theoretical limit of an ideal or reversible operation and the extent to which the operation of the given system departs from the ideal. The departure is measured by the calculated quantity called destroyed exergy or irreversibility. This quantity is proportional to the generated entropy according to the well-established Gouy–Stodola theorem. The minimization of entropy generation requires the use of more than thermodynamics—fluid mechanics, heat and mass transfer, materials, constraints, and geometry are also needed in order to establish the relationships between the physical configuration and the destruction of exergy [24]. In the field of heat transfer, the entropy generation minimization method reveals the inherent competition between heat transfer and fluid flow irreversibilities in the optimization of devices subjected to overall constraints. Therefore, the main objective of this method is determining possible ways of minimizing entropy generation. The method used to achieve this purpose is known as entropy generation minimization or thermodynamic optimization. Thermodynamic irreversibility associated with the flow system provides insight into frictional and heat transfer losses in the system. Moreover, the entropy generation is associated with the thermodynamic irreversibilities occurring in the system. Consequently, thermodynamic irreversibility can be quantified through entropy calculations. The second-law analysis of a non-Newtonian fluid over a horizontal surface and through pipes and several other geometries has many significant applications in thermal engineering and industries. One of the fundamental problems of the engineering processes is improving thermal systems during convection in any fluid. The second-law analysis is one of the best tools for improving the performance of the engineering processes. It investigates the irreversibility due to fluid flow and heat transfer in terms of the entropy generation rate [25]. Bejan [26] provided an extensive overview of the application of second-law analysis for the study of entropy generation and how it affects the thermodynamic characteristics of a non-Newtonian laminar boundary layer. Second-law thermodynamic analysis was applied to the study of non-Newtonian fluids with variable thermophysical properties in a circular channel [27]. The study revealed that global entropy generation increases with power-law index and Brinkman number, and where the thermophysical property variation effect causes a decrease in entropy generation. Entropy analysis of mixed convective magnetohydrodynamic flow of a viscoelastic fluid over a stretching surface is considered in [28], showing that both the local and the average entropy generation number increases with increase in the viscoelastic and magnetic parameters. A decrease in average entropy generation is, however, observed and is due to mixed convection. Entropy generation of a non-Newtonian nanofluid from a stretching surface is presented in [29], where the dimensionless Deborah number, thermophoresis number, Eckert number, and Brinkman

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number all have the same increasing effect on the entropy generation number. Another particularly interesting study of entropy generation analysis of non-Newtonian fluid flow over micropatterned surface, considering the effect of slip at the surface, is presented in [30]. The study revealed that entropy generation number decreases with an increasing slip coefficient for shear-thinning and shear-thickening fluids and Newtonian fluids. This study provides some physical insights into the dynamics of boundary layer flows over micropatterned surfaces. Entropy generation serves as a practical tool for the optimization of non-Newtonian normal and nanofluid flows [31]. The study of the non-Newtonian Eyring–Powell nanofluid over a stretching surface shows that entropy generation is an increasing function of the relevant physical parameters such as the Nusselt number. The interested reader may consult other entropy generation works on non-Newtonian fluid flow and heat transfer [32–39].

14.2.5 Non-Newtonian nanofluid boundary layer transfer Nanofluids may have the potential to significantly increase heat transfer rates in a variety of areas such as industrial cooling applications, nuclear reactors, the transportation industry, micro-electromechanical systems, electronics and instrumentation, and biomedical applications. Nanofluids have also been found to possess enhanced thermophysical properties such as thermal conductivity, thermal diffusivity, viscosity, and convective heat transfer coefficients compared to those of base fluids such as oil or water. There is an ever-increasing interest in understanding the behavior of nanofluids, some of which can behave as non-Newtonian fluids due to their microstructure or higher volume fraction loading of nanoparticles. Nanofluids can behave as both Newtonian and non-Newtonian fluids depending on particle loading and higher volume fractions. Heat transfer is directly related to a fluid’s thermophysical properties. Buongiorno [40] presents an excellent review of some of the recent advances made in this direction at MIT. Nanofluids are engineered colloids made of a base fluid and nanoparticles 1–100 nm in size. Nanofluids have higher thermal conductivity and single-phase heat transfer coefficients than their base fluids. In particular, the heat transfer coefficient increases appear to surpass the thermal-conductivity effect, and cannot be predicted by traditional pure-fluid correlations such as the Dittus–Boelter’s correlation. In the nanofluid literature, this behavior is generally attributed to thermal dispersion and intensified turbulence, which are caused by nanoparticle motion. To test the validity of this assumption, Buongiorno [40] considered seven slip mechanisms that can produce a relative velocity between the nanoparticles and the base fluid. These are inertia, Brownian diffusion, thermophoresis, diffusiophoresis, Magnus effect, fluid drainage, and gravity. Buongiorno concluded that of these seven only Brownian diffusion and thermophoresis are important slip mechanisms in nanofluids. Brownian motion occurs in a direction from high to low nanoparticle concentrations, whereas thermophoresis occurs in the direction from hot to cold. The authors in [40] also proposed an alternative explanation for the abnormal heat transfer coefficient increases. The nanofluid properties may vary significantly within the boundary layer

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because of the effects of the temperature gradient and thermophoresis. For a heated fluid, these effects can result in a significant decrease in viscosity within the boundary layer, thus leading to heat transfer enhancement. The physics and understanding of nanofluids continues to be a multidisciplinary effort. Mathematical formulations and computational tools, including magnetohydrodynamic equations, are being considered to solve several of the research problems related to slip flow in non-Newtonian nanofluids [41,42]. This is still nascent, given that the mechanisms governing the enhancements of nanofluids in normal base fluids have yet to be well elucidated. Nonetheless, there are notable theoretical works in this regard, and the reader may further consult the open literature.

14.2.5.1

Extensions of the Merk–Chao–Fagbenle method to non-Newtonian fluids

Series-based methodologies continue to attract research interests. The Merk–Chao– Fagbenle (MCF) method is one such method that has equally found utility for research into non-Newtonian fluids. Lin and Chern [43] were perhaps the first to extend the MCF method to non-Newtonian fluid flow. Later, Kim [44] extended the MCF method to the study of power-law fluids with applications to a wedge and circular cylinder, considering a step change in surface temperature distribution. Chang et al. [45], at the University of Toledo, Ohio, utilized the methodology to study natural convection from two-dimensional and axisymmetric bodies of arbitrary contours, concluding that the technique provides a general, accurate, and relatively simple way to analyze transport phenomena in laminar boundary layers of power-law fluids. Kim and Esseniyi [46] studied forced convection of power-law fluids over a rotating nonisothermal body, remarking that, in addition to obtaining good results, they found that the role of rotation parameter in dilatant fluid flows is less significant than in pseudoplastic flows. Meissner [47], in his Master’s thesis, which later resulted in the published work [48], considered a power-law fluid flow over a flat plate, horizontal circular cylinder, and sphere. The work was the first to apply the MCF method to mixed convection power-law fluids, yielding very meaningful results. Howell et al. [49] also utilized the method, for a power-law fluid considering nonlinear velocity and linearly stretching surface velocity, with freestream velocity taken to be zero. The authors concluded that the MCF method is useful for solving difficult transport problems, where simple transformations and universal functions are used to solve the fundamental differential equations, regardless of geometry. Rao et al. [50] considered the utility of the MCF method with injection and suction at a moving wall, indicating that the convergence of the MCF method is excellent. Last, Shokouhmand and Soleimani [51] considered the effect of viscous dissipation of a power-law fluid using the MCF method. They remarked that, due to the difficulty of finding similarity solutions for non-Newtonian flows over a surface in the presence of injection or suction (there is no similarity solution for most cases),1 the MCF method is particularly appropriate for solving these 1 Similarity solutions for non-Newtonian fluid have been discussed in S.Y. Lee and W.F. Ames, AICHE

Journal, 700–708, 1966.

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problems when the transformation to a simple ordinary differential equation is not possible. In another recent work [52], the authors remark that the thermal boundary layer for pseudoplastic fluids is thicker than that of dilatant fluids, further establishing also that a direct relationship exists between dimensionless temperature and injection parameter or increase in heat generation parameter. With the exception of the authors of [46] and [52] who extended their works to consider dilatant and pseudoplastic fluids, it appears that all studies using the MCF method involved power-law or Ostwald–de Waele fluids. Though power-law fluids constitute the simplest and most useful type of non-Newtonian fluids, many research opportunities arise, and it is instructive to extend the methodology to other types of non-Newtonian fluids (e.g., Sisko, Casson, micropolar, and Jeffrey fluids) to better and further characterize the difficult transport problems therein. Furthermore, the very disciplined MCF method is yet to be employed to exposing the underlying transport phenomena of non-Newtonian nanofluid boundary layer transfer.

14.3 A note on current research status and applications of non-Newtonian fluids This section covers a selection of the current applications of research on nonNewtonian laminar boundary layer and heat transfer. As mentioned in the earlier sections of this chapter, non-Newtonian laminar boundary layer flows are observed in several domains, including but not limited to biological systems, chemical and process engineering systems, geoscience systems, transportation systems, and food and pharmaceutical processing systems. A prominent research example for each of these research domains is described in what follows.

14.3.1 Biological/biomedical systems: vascular fluid dynamics There is considerable evidence that vascular fluid dynamics plays an important role in the development and progression of arterial stenosis, one of the most wide-spread human diseases, that leads to the malfunction of the cardiovascular system. Although the exact mechanisms responsible for the initiation of this phenomenon are not clearly known, it has been established that once a mild stenosis is developed, the resulting flow disorder influences the development of the disease and arterial deformity and changes the regional blood rheology [53]. The assumption of the Newtonian behavior of blood is acceptable for high shear-rate flow, that is, the case of flow through larger arteries. It is not, however, valid when the shear-rate is low, as for the flow in smaller arteries and in the downstream of the stenosis. It has been pointed out that in some disease conditions, blood exhibits remarkable non-Newtonian properties. Thurston [54] has shown conclusively that blood, being a suspension of enumerable cells, possesses significant viscoelastic properties. In most of the investigations relevant to the domain under discussion, the flow is mainly considered in cylindrical pipes of uniform cross-section area. However, it is

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Figure 14.6 Snapshots showing instantaneous RBC distribution from representative simulations for nonstenosed (left column) and 84% stenosed (right column) vessels [56].

well-known that blood vessels bifurcate at frequent intervals and the diameter of the vessels varies with the distance. Mandal [55] has developed a mathematical model in order to study the notable characteristics of non-Newtonian blood flow through flexible tapered arteries in the presence of stenosis subject to the pulsatile pressure gradient. The apparent viscosity of blood is observed to increase by several folds when compared to non-stenosed vessels [56]. An asymmetric distribution of the red blood cells, caused by geometric focusing in stenosed vessels, is observed to play a major role in the enhancement, as seen in Fig. 14.6. With new biomedical applications related to the targeted delivery of drugs, minimally-invasive surgeries, and post-operative healing procedures becoming a critical area of research, several studies are being carried out in this direction [57,58].

14.3.2 Chemical systems: pharmaceutical products The processing of many pharmaceutical products involves non-Newtonian fluids. Xanthan gum and Carbopol are two common additives that cause a fluid to have shear-thinning properties. These fluids have unique mixing properties that have been studied extensively in experiments. When yield-stress shear-thinning fluids are mixed in stirred tanks, a well-known phenomenon that occurs is the formation of well-mixed zones around the impellers (which constitutes boundary layer flow) while the remainder of the fluid remains relatively stagnant. These well-mixed zones, commonly known as caverns, have been characterized using computational fluid dynamics (CFD) for various impellers [59]. These CFD tools can be used effectively to understand some of these complex nonNewtonian flow phenomena. Having a validated model for non-Newtonian fluids is extremely valuable for predicting the behavior of these fluids in larger and/or more complex equipment that cannot be easily validated. Because cavern sizes and nonNewtonian behavior can change when scaling up in ways that cannot be correlated simply to dimensionless numbers, the implementation of validated CFD models may prove to be the most effective way to scale up processes with shear-thinning fluids.

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Computational fluid dynamics may also help in the selection of novel approaches to designing scale-down experiments. Simulations could be used to identify experimental fluid properties with similar flow behavior on small scales, as for large-scale equipment. The most meaningful experiments for process scale-downs may not necessarily use fluids with physical properties identical to those of the final material. With the help of validated CFD simulations, fluids can be selected to recreate the flow patterns in the laboratory that will be formed at large scales.

14.3.3 Food processing systems: processing of tomato ketchup Ketchups are time-independent, non-Newtonian fluids that show a small thixotropy (Bottiglieri et al., 1991). Tomato ketchup obtains its v iscosity from naturally occurring pectic substances in fruits. Their rheological behavior is important during handling, storage, processing and transport of concentra ted suspensi ons in industry. Other factors such as enzymatic degradations, pectin fraction protein interaction, pulp content, Tomato ketchups is a time-independent, non-Newtonian fluid that shows a small thixotropy. Tomato ketchup obtains its viscosity from naturally occurring pectic substances in fruits. The rheological behavior of these fruits is important during handling, storage, processing, and transport of concentrated suspensions in industry [60]. Other factors such as enzymatic degradations, pectin–protein interaction, pulp content, homogenization process, and concentration may also affect the consistency of tomato products. Knowledge of the rheological properties of fluid and semisolid foodstuffs is important to the design of flow processes in quality control, storage, and processing stability, and in understanding and designing texture. Usually, viscosity is considered an important physical property related to the quality of food products. Therefore, reliable and accurate rheological data are necessary for designing and optimizing various food-processing equipment such as pumps, piping, heat exchangers, evaporators, sterilizers, filters, and mixers. Similar challenges exist in designing process flows for plants to produce toothpastes, shampoos, baby food, and several other products that are non-Newtonian in nature.

14.3.4 Geosciences: drilling muds During drilling operations, drilling muds are pumped from a surface mud tank through the drill-pipe (several kilometers in length), through nozzles in the rotating drill-bit, and back to the mud tank through the annular space between the wellbore wall and the drill pipe. Drilling muds have several functions: to support the wellbore wall and prevent its collapse; to prevent ingress of formation fluids (gas and liquid) into the wellbore; to transport rock cuttings to the surface; to minimize settling of the cuttings if circulation is interrupted; to clear the workface; to cool the drill-bit; and to lubricate the drill string [61]. A schematic of this is shown in Fig. 14.7. The underlying challenge to fluid dynamicists has been to calculate the flow-field within the drillstring–wellbore annulus, a situation usually idealized as one of steady isothermal, fully developed laminar flow of a shear-thinning liquid (modeled as a gen-

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Figure 14.7 A schematic of a drillstring–wellbore interaction, in which drill muds are used.

eralized Newtonian fluid) through an annulus consisting of an outer cylinder and an inner cylinder that may be offset (i.e., eccentric) and rotating. Escudier et al. [62] have employed a finite volume method-based numerical scheme to model the fully developed laminar flow of an inelastic shear-thinning power-law fluid through an eccentric annulus with inner cylinder rotation. The authors in [38] used a model generally used for Newtonian flows but with a viscosity term that is coupled to the local shear rate.

14.3.5 Transportation systems: transport of crude oil emulsions Water-in-crude oil emulsions often show a shear-thinning non-Newtonian behavior, meaning that the viscosity reduces as the shear rate increases. The influence of shear rate on the viscosity is observed to increase as the concentration of the dispersed phase increases [63]. The effective viscosity of an emulsion can greatly exceed either the crude or the water single-phase viscosities. The apparent viscosity of these mixtures depends on many factors such as the viscosity of the oil and of the water, water content, temperature, droplet size distribution, amount of solids in the crude oil, and shear rate. Wax content also appears to contribute to the stabilization of emulsions. At temperatures below the wax appearance temperature, wax crystals precipitate and interact with the oil–water interface. As a result, waxy crude-oil shows high emulsion stability at low temperatures. The experiments with several different crude-oil mixtures illustrate the complexity of the pipe flow of oil–water emulsions and the challenges in understanding the coupling of surface chemistry and fluid flow [63].

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14.4 Future directions One of the major open questions regarding non-Newtonian fluids addresses issues of physics, chemistry and engineering more than the mathematics involved. Which equation should be used to model a given fluid? Wilson [63] provided a short list of open mathematical problems related to non-Newtonian fluids and reported that there are several more application-oriented fluid flow problems involving non-Newtonian fluids that require attention. With the availability of high performance computing resources, DNS is becoming one of the popular tools used by researchers to understand some of the critical non-Newtonian boundary layer phenomena. Even though DNS is primarily being used in transitional and turbulent flow scenarios, the DNS codes are generally validated against laminar flow scenarios for which analytical and/or experimental results are available. As discussed in several of the previous sections, there is tremendous scope for pursuing research on non-Newtonian laminar boundary layers as applicable to a variety of applications relevant to the modern-day industry. However, due to the uncertainty associated with the question of which equations should be used for a given application, constructing a generalized predictive mathematical model that captures the complex phenomena underlying the observed flow features is at least challenging, if not impossible.

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