Stokes’ hypothesis

For the kind of boundary-layer problems that are our principal focus here, the dynamic viscosity p will play a very important role and k will be relatively unimportant. The dynamic viscosity primarily governs the behavior of a fluid in shearing flow. The higher the viscosity, the more the fluid resists deformation for a given shear stress. 

This chapter established three important concepts that are essential for the derivation of the conservation equations governing fluid flow. First, the Reynolds transport theorem was developed to relate a system to an Eulerian control volume. The substantial derivative that emerges from the Reynolds transport theorem can be thought of as a generalized time derivative that accommodates local fluid motion. For example, the fluid acceleration vector 

The final objective of this chapter was to develop quantitative relationships between a fluid s strain-rate and stress fields. Expressions for the strain rates were developed in terms of velocities and velocity gradients. Then, using Stokes s postulates, the stress field was found to be proportional to the strain rates and a physical property of the fluid called viscosity. The fact that the stress tensor and strain-rate tensor share the same principal coordinates is an important factor in applying Stokes s postulates. The stress-strain-rate relationships are fundamental to the Navier-Stokes equations, which describe conservation of momentum in fluids. 

1 Begin with the general vector form of the substantial derivative, as stated below  

Using the general vector operations, develop expressions for the acceleration components in each of the three coordinate directions in cartesian coordinates (jc, y, z). Compare the results with the equivalent development in the text for cylindrical coordinates, and discuss the differences. 

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The identity tensor by is zero for i J and unity for i =J. The coefficient X is a material property related to the bulk viscosity, K = X + 2 l/3. There is considerable uncertainty about the value of K. Traditionally, Stokes hypothesis, K = 0, has been invoked, but the vahdity of this hypothesis is doubtful (Slattery, ibid.). For incompressible flow, the value of bulk viscosity is immaterial as Eq. (6-23) reduces to... 

Considering the Stokes hypothesis, evaluate the diagonal invariant of the stress tensor for this flow field. 

The dissipation function, also called viscous dissipation, represents the irreversible conversion of kinetic energy into thermal energy. Since the dynamic viscosity p, is positive and all the terms are squared, the first two terms of the dissipation must be always positive. The bulk viscosity can be negative the Stokes hypothesis (Section 2.11) says that k = —2p/3. It turns out that the necessary condition for the dissipation function to be positive is that... 

Rewrite these equations assuming that the bulk viscosity can be determined from the Stokes hypothesis. 

The temperature, composition, and density are presumed to have only radial variations. The pressure, however, is allowed to vary throughout the flow, but in a very special way as will be derived shortly. Also the magnitude of the pressure variations is assumed to be small compared to the mean thermodynamic pressure. Using these assumptions, and invoking the Stokes hypothesis to give X = —2p./3, we can reduce the mass-continuity and Navier-Stokes equations to the following ... 

Consider the steady flow inside a cylindrical channel, which is described by the two-dimensional axisymmetric continuity and Navier-Stokes equations (as summarized in Section 3.12.2). Assume the Stokes hypothesis to relate the two viscosities, low-speed flow, a perfect gas, and no body forces. The boundary-layer derivation begins at the same starting point as with axisymmetric stagnation flow, Section 6.2. Assuming no circumferential velocity component, the following is a general statement of the Navier-Stokes equations ... 

We will presume a Newtonian fluid and neglect the normally very small bulk viscosity. The statement (3.56) for the stress tensor is then transformed into Stokes formulation, the so-called Stokes hypothesis ... 

The compressible Navier-Stokes equations are the governing conservation laws for mass, momentum, and energy. These laws are written assuming that the fluid is Newtonian and follows the Fourier law of diffusion, so that the stress tensor is a linear function of the velocity gradients and the heat flux vector is proportional to the temperature gradient. Adding Stokes hypothesis, which expresses that the changes of volume do not involve viscosity, the compressible Navier-Stokes equations may be written as... 

Equations 6.33 through 6.35 include factor X, which is only important for compressible fluids. This factor is based on the Stokes hypothesis and is determined as... 

Stokes hypothesis holds Navier-Stokes equations apply... 

Stokes hypothesis holds true Navier-Stokes equations hold true. 

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