Chapter 8 Shear Stress in Laminar Flow

Main Topics Introduction Newton's Viscosity Relation- The coefficient of Viscosity Rheology of Non-Newtonian Fluids Viscosity Shear Stress in Multi-dimensional Laminar Flows of a Newtonian Fluid The `No Slip' Condition

8.0 Introduction In the analysis of fluid flow thus far, shear stress has been mentioned, but it has not been related to the fluid or flow properties. We shall now investigate this relation for laminar flow. The shear stress acting on a fluid depends upon the type of flow that exists. In so called laminar flow, the fluid flows in smooth layers or lamina, and the shear stress is the result of the microscopic action (which is unobservable) of the molecules. Turbulent flow is characterized by the large scale, observable fluctuations in fluid and flow properties, and the shear stress is the result of these fluctuations.

8.1 Newton's Viscosity Relation- The coefficient of Viscosity A very important property will be introduced as a consequence of Newton's viscosity law. For a laminar flow whereby fluid particles move in straight, parallel lines, this law states that for certain fluids, called Newtonian fluids, the shear stress on an interface tangent to the direction of flow is proportional to the distance rate of change of velocity in a direction normal to the interface. Mathematically, this is stated as

8.1 Newton's Viscosity Relation- The coefficient of Viscosity Inserting the coefficient of proportionality into the above expression, leads to the result, (Note that viscous momentum transfer is in the direction of the negative velocity gradient, that is, the momentum tends to go in the direction of decreasing velocity. A velocity gradient can thus be thought of as a driving force for momentum transfer.) μ is called the coefficient of viscosity (or simply viscosity), having the dimension of (M / L / θ).

8.1 Newton's Viscosity Relation- The coefficient of Viscosity Figure (8.1.1) Derivation of (eq.8.1.4) The velocity and shear stress profiles in a fluid flowing between two parallel plates is illustrated in Fig.8.1.2

8.2 Rheology of Non-Newtonian Fluids The science of the deformation and flow of materials is often called rheology; an important branch of this subject concerns the behaviour of non-Newtonian fluids. Figure 8.2.1 All gases are Newtonian. All liquids for which we can write a simple chemical formula are Newtonian, such as water, benzene, ethyl alcohol, carbon tetrachloride, and hexane. Most solutions of simple molecules are Newtonian, such as aqueous solutions of inorganic salts and of sugar. Generally, non-Newtonian fluids are complex mixtures: slurries, pastes, gels, polymer solutions, etc. Most non-Newtonian fluids are composed of molecules or particles that are much larger than water molecules, such as the sand grains in a mud or the collagen molecules in gelatin, which are thousands or millions of times larger than water molecules.

8.2 Rheology of Non-Newtonian Fluids 8.2.1. Time-independent Fluids- Bingham plastic fluids: These are the simplest because, as shown in Fig.8.2.1, they differ from Newtonian only in that the linear relationship does not go through the origin. A finite shear stress τy (called yield stress) in N/m2 is needed to initiate flow. Examples of fluids with a yield stress are bread dough, margarine, and toothpaste. 8.2.2. Time-independent Fluids- Pseudo plastic fluids: The majority of non-Newtonian fluids are in this category and include polymer solutions or melts, greases, starch suspensions, mayonnaise, biological fluids, blood, detergent slurries, dispersion media in certain pharmaceuticals, and paints. Pseudo plastic fluids show a viscosity that decreases with increasing velocity gradient.

8.2 Rheology of Non-Newtonian Fluids Non-Newtonian fluid can generally be represented by a power- law equation (sometimes called Ostwald-de Waele equation). where K is the consistency index in N-sn/m2 or lbf-sn/ft2, and n is the flow behaviour index, which is dimensionless. The term K (dvx / dy)n-1 is sometimes called the apparent viscosity. It decreases with increasing shear rate. 8.2.3. Time-independent Fluids- Dilatant fluids: These fluids show an increase in apparent viscosity with increasing shear rate. Solutions showing dilatancy are some corn flour-sugar solutions, starch in water, and some solutions containing high concentrations of powder in water.

8.2 Rheology of Non-Newtonian Fluids 8.2.4. Time-dependent Fluids: For these fluids, the shear stress is a function of time at constant shear rate. For a rheopectic fluid, the shear stress increases with time at constant shear rate; for a thixotropic fluid, it decreases with time. Thixotropic fluids exhibit a reversible decrease in shear stress with time at a constant rate of shear. Examples include some food materials, and paints. Rheopectic fluids exhibit a reversible increase in shear stress with time at a constant rate of shear. Examples are bentonite clay suspensions.

8.3 Viscosity Figure 8.3.1 Derivation of (eq.8.3.5) Equation (8.3.5) indicates that μ is independent of pressure for a gas. This has been shown experimentally, to be essentially true for pressure up to approximately 10 atm (under constant temperature). Viscosity of gases will increase with an increase in temperature. However, the viscosity of liquid, generally will show an opposite trend, that is the viscosity decreases at a high rate with an increase in temperature.

8.4 Shear Stress in Multi-dimensional Laminar Flows of a Newtonian Fluid Figure 8.4.1 Derivation of (eq.8.4.1) Stokes’ Law of Viscosity for Rectangular coordinates, Eq.(8.4.2) and (8.4.3)

8.5 The ‘No Slip’ Condition An important feature of the behaviour of all real fluids (Newtonian and non-Newtonian) is the tendency to cling to any solid flow boundary; all the available experimental evidence indicates that, for conditions in which the continuum concept holds, there is no slip between a solid surface and the fluid next to it. That is the layer of fluid adjacent to the boundary has zero velocity relative to the boundary. When the boundary is a stationary wall, the layer of fluid next to the wall is at rest. If the boundary or wall is moving, the layer of fluid moves at the velocity of the boundary. Hence, it is called no slip condition.

Points to remember Newton’s law of viscosity (2D) and Stoke’s law (3D) are meant for laminar flow. The negative sign in the expression is in accord with the heat and mass transfer processes as such, we are emphasizing the momentum is transferred in the direction of reducing velocity. Bingham plastic fluids require a stress that can overcome the yield stress such that it will behave as Newtonian fluid. The majority of non-Newtonian fluids include polymer solutions or melts, greases, starch suspensions, mayonnaise, biological fluids, blood, detergent slurries, dispersion media in certain pharmaceuticals, and paints.

Points to remember Pseudo plastic fluids show a viscosity that decreases with increasing velocity gradient. Dilatant plastic fluids show a viscosity that increases with increasing velocity gradient. For most gases, viscosity increases with an increasing temperature. For most liquid, viscosity will decrease with an increasing temperature.