The Physics of Non-Newtonian Fluids M. Grogan 10. April 2006 [1][4] R. Fox, A. McDonald, P. Pritchard, Introduction to Fluid Mechanics, 6 th Ed. 2004 Wiley, United States of America. p3, p27. [2],[3] B Schutz, A first course in General Relativity. 2004 Cambridge University Press, United Kingdom. p89, p36. [5][8][12] C. Pozirikidis, Fluid Dynamics Theory, Computation and Numerical Simulation. 2001 Kluwer Academic Publishers, Boston/Dordrecht/London. p205, p281, p253. [6] A. Majda, A. Bertozzi, Vorticity and Incompressible Flow. 2002 Cambridge University Press, United Kingdom. p2. [7][9] http://en.wikipedia.org/wiki/ Navier-Stokes_equations / Non- Newtonian_fluid [10] G. Astarita, G Marrucci, Principles of Non-Newtonian Fluid Mechanics. 1974 McGraw-Hill Book Company, United Kingdom. p45 [11] Non-Newtonian Fluid Dynamics Reseach Group homepage http://web.mit.edu/nnf [13] http://stokes.uscd.edu/c_pozrikidis/fdlib Introduction: Newtonian Fluids, such as water, are encountered daily and are well understood by people from many disciplines. Non-Newtonian fluids are less common and less easy to understand, so they have not been studied as much. For this paper I am going to revisit the principles of Newtonian fluids in some detail before trying to understand any others, then I will talk about how non-newtonian fluids differ and the different types of Non- Newtonian fluids. Finally I will be talking about current experiments and looking at some future research in the field. 1. Ideal Fluids: An ideal fluid has no density gradient, that is, the density is constant regardless of the position in the fluid. This leads to the idea that fluids are incompressible, since a small compression would lead to an increased density around the site of compression. In reality almost all fluids are slightly compressible, but this turns out to be not such a bad generalization for liquids. To clarify what exactly I mean by this we need to form a definition of the word 'fluid' which in everyday language has become synonymous with 'liquid' but scientifically there is a difference. A fluid is a substance that deforms continuously under the application of a shear stress, no matter how small the stress may be [1]. Hence, our accurate definition of a fluid actually encompasses liquids and gases however our definition of an ideal fluid definitely does not. This is clear to visualize when you imagine trying to compress a gas, an easy thing to do since in a gas molecules are spread far apart. As you squash it together the molecules get closer together, thus the density is changing and - since the change is not instantaneous throughout the fluid - there are local density gradients built up. For this paper I will not be dealing with gases as non-newtonian fluids because we have different gas laws to describe the motions of gases outside of hydrodynamics. General Relativity Aside: Another interesting application of this definition of fluid is that in General Relativity we have motivation for calling a continuum of bodies such as a galaxy a fluid [2]. These galaxies actually obey some of the equations which have been developed from hydrodynamics, including a generalization of the stress tensor which we will see shortly.
2. A Recap of Tensors: In mathematics a tensor is the generalization of a vector to an object with an arbitrary number of indices's. Similar to a matrix, and in fact 0,2 tensors can be written in matrix form, tensors are useful for describing physical situations which have large numbers of simultaneous or differential equations. One nice thing about tensors is that they are set up so that they have a physical meaning outside of the co-ordinate system which you are working in. This has particularly useful applications in General Relativity, where the co-ordinate systems can be warped and therefore hard to physically interpret [3], but it is also advantageous in hydrodynamics as it is sometimes convenient to work in cylindrical polar coordinates but we want our results to have the same physical meaning in equivalent situations. In particular, when describing flows it is particularly useful for us to have a tensor which describes the stress on the fluid in all directions within the fluid. This is the stress tensor, and it is fully specified by 9 components [4]; = xx xy xz yy yz zx zy zz (1) where we have differentiated between the normal and shear stress by the use of and respectively. The definition of stress is the force per unit area, but in a fluid we are interested in the component of stress as the area tends to 0, given by; F n nn = lim (2) A n 0 A n and similarly F t tn = lim (3) A n 0 A n 3. Newtonian Fluids: Now that we have seen the stress tensor, I will attempt to introduce where it comes from in terms of the material properties of the fluid and our physical intuition. For this argument it is going to be easiest to work in Cartesian coordinates and we are going to look at the forces on a particle surface, coming from the other particles touching that surface. This situation is shown in figure 1, with a small vector force which has components in general in the normal and orthogonal directions to the surface. z x δa Figure 1 The small force δf of one fluid particle on another acts over a small area δa with components in both the normal direction to the surface and the tangential direction. However, having decided to work in Cartesian coordinates it is desirable to express the area vector and the force vector in terms of their components in the x,y and z directions. Taking the definitions of (2) and (3), it is clear that this is enough to give us all 9 components of the stress tensor as each area term has a finite limit for each force term. Thus our understanding of these stresses is that the = df y da x (4) component is a stress on the y plane in the x direction as shown in figure 2. z particle surface y y δa y δf Figure 2 The force and stress on a small element of fluid. x δf x
The stress inside a liquid arises from how difficult it is to slide the layers of particles past each other, which is proportional to how much the liquid is being deformed by the force. We also need one other condition to work out values of the stress tensor for a liquid, and that is the non-slip condition. Actually the non-slip condition is a generalization, and liquids can take a partial slip condition instead but for this I will just be looking at non-slip[5]. A liquid at a solid boundary is not capable of slipping on that boundary, which is the condition that leads to Newtonian flow. The situation shown in figure 3 is a very simple example of how we can use the non-slip condition to work out the stress tensor. Force: δf Velocity: δu δl Initial fluid location, time t δα δy Location of fluid after t+δt solids, F =m du, hence the name Newtonian dt fluids. Fluids which do not obey this law, but some other relationship between the stress and the derivative of velocity are called Non-Newtonian fluids. The Navier-Stokes Equations Aside: The Navier-Stokes equations are used to describe all incompressible flows in terms of the derivative of internal pressure, including the onset of turbulence. They arise from the conservation of momentum for a continuum, however they are not totally understood. In fact, the Navier-Stokes existence and smoothness is one of the Millennium Prize Problems of the Clay Mathematics Institute, who have offered $1,000,000 for progress in understanding this phenomenon [6][7]. The differential nature of the Navier-Stokes equations make them ideally suited to computational work, in particular in the case of Newtonian fluids [8]. δx Figure 3 Two parallel plates deform a fluid with an applied force on the top plate, shown in the case of a 1D flow. Liquid at the top plate which must obey the non-slip condition is moved along with the plate with a rate of deformation given by the rate of change of velocity: deformation rate= du (5) dy if the plate motion is only in the x direction. If we then examine our definition of a fluid we can see that there must be some relation between this and the shear stress. In fact, in a Newtonian fluid we find that the shear stress is proportional to the rate of change of velocity by; = du (6) dy where the proportionality constant, is what we define as the liquid viscosity. The link between this equation for the stress acting on liquids is exactly analogous to the equation for 4. Non-Newtonian Fluids: Non-Newtonian fluids can be grouped into different categories depending on how the stress tensor depends on the derivative of the velocity. Probably the most common type of non- Newtonian fluid are the power law fluids, which have the form =k du n dy (7) where n is called the behavior index and k is the consistency index. Figure 4 shows a plot of the deformation rate with shear stress for the cases n<1 and n>1. When a shear stress is applied the Pseudoplastic fluids, with n<1, become easier to deform. Dilatant fluids are the opposite, so when a shear stress is applied they become more 'solid' [9]. A solid suspended in a liquid is an example of a dilatant fluid, as you stir the mixture it gets harder and harder because the liquid is locked into the gaps between molecules and is not free to flow. An example of a pseudoplastic liquid is a polymer melt, which becomes thinner when more force is applied [10].
Figure 4 The deformation rate of a fluid plotted with the applied shear stress. For all power law non-newtonian fluids, we can adapt the equation to look more like that of a Newtonian fluid and from this we can observe some more interesting classes of materials, similar to dilatant and pseudoplastic but with subtly different properties. The stress can be rewritten as =k du dy n 1 du du = (8) dy dy where the coefficient is called the apparent viscosity. However, for some materials the apparent viscosity is a function of time. Thus for a constant applied stress, the material can behave in different ways depending on how long the stress is applied for. In a thixotrophic material the apparent viscosity decreases with time under a constant applied force. An example of a thixotropic material is non-drip paint, which becomes thin after being stirred for a time, but does not run on the wall when it is brushed on. A rheopectic fluid is the opposite case, its apparent viscosity becomes higher with time for a constant applied stress. An example of one rheopectic fluid which is easy to make at home 1 is discussed in the aside 1 Or not as the case may be, since Custard Powder is not so widely available in the USA! Custard Powder Aside: Like much in the subject of fluids, while simulations require powerful computers and ingenious code, the actual demonstrations can be accomplished in any normal kitchen using a little ingenuity and the correct materials. Custard powder, when mixed with water to the right concentration, becomes a rheopectic fluid. Pour it out in your hand and it seems to be a yellow liquid with a viscosity somewhere around olive oil. However by rolling the solution around with your other hand, you can make it into a ball which feels 'solid' as you are rolling it. When you stop moving it around it relaxes back into the liquid form. Bingham plastics are strictly speaking another class of non-newtonian fluids, but they are somewhat closer to Newtonian fluids than what we have previously looked at. Bingham plastics behave as a solid for some small range of applied stress up to a critical yield point, after which they behave as a Newtonian fluid. This is actually one of the most useful classes of non- Newtonian fluid, and one such example of a Bingham plastic is toothpaste, which on its way out of the tube is forced out like a 'solid' but then moves about in your mouth and cleans your teeth as a 'liquid'. Current Research: The Non-Newtonian Fluid Dynamics Research Group based at Massachusetts Institute of Technology (NNF at MIT) is an active researcher into a wide variety of different non- Newtonian fluids. Two of the areas which they research into are magnetic fluids and open siphons. Magnetic fluids were first invented in the 1960's and are the worlds first successful nanotechnology. They consist of tiny magnets suspended in a liquid, which can be somewhat controlled in shape and motion using other magnets. One of the current applications the group is looking into is the use of magnetic fluids as rotary rings for hard disc drives, which could make drives smaller and more efficient. The open siphon is a remarkable property of non-newtonian fluids. Once liquid has started
to flow, it can continue to flow along the same path keeping its shape even if it is not bound by anything. For a siphon this means that once you start sucking the liquid up you no longer need to be in contact with the surface of the liquid to keep sucking up liquid. This, along with an example of a magnetic fluid, is shown in figure 5 [11]. (a) equations can be approximated using computers by discretizing the differential operator and then using an algorithm to propagate a solution forward in time and space. Fluid flows can be visualized by plotting lines of constant velocity, the characteristic streamlines of the flow. Since this is still a new field, most software libraries tend to only deal with Newtonian flows at the moment, such as the FDLIB library available on the web for UNIX and Linux systems [13]. A similar version of this is available for windows and mac users with Fortran compilers. This library uses the Fortran 77 programming language and its own algorithms to compute various flows, especially focusing on the onset of turbulence, as shown in figure 6. There is a lot of potential for using computers to simulate non-newtonian flows for fluids which have strong industrial and scientific applications. (b) Figure 5 (a) A magnetic fluid arranged into the initials of the research group at MIT. (b) an open siphon, liquid is rising into the syringe with no solid connection between the syringe and the liquid surface. 5. Computational Research in Fluid Dynamics [12]: As previously mentioned, the nature of the Navier-Stokes equations makes fluid flows ideally suited for computational physics. I will now clarify this and give some examples of resources for computation of flows. The Navier- Stokes equation is a second order differential equation which is valid for continuous space, but is difficult to solve analytically because generally it is a non-linear equation. However, differential Figure 6 A simulation of a turbulent flow using the FDLIB routine flow1d. Conclusions: Non-Newtonian fluids, with their counter intuitive behavior, are something that we encounter in everyday life - much of the time without realizing. Their strangeness could have many novel applications to go with the countless applications we already use them in. Such application however requires us to have a more complete understanding of how they will behave in certain situations, and this is where computational fluid dynamics is a useful tool. Research into these fluids exist at a cross-roads between the disciplines of Mechanical Engineering, Physics and Computational Physics, and progress is sure to come from all three.