0.4 Stress in fluids

Topics covered in this chapter

• Cauchy's stress principle and the conservation of momentum
• The stress tensor
• The symmetry of the stress tensor
• Hydrostatic pressure
• Principal axes of stress and the notion of isotropy
• The Stokesian fluid
• Constitutive equations of the Stokesian fluid
• The Newtonian fluid
• Interpretation of the constants $\lambda$ and $\mu$

Reading assignment
Chapter 1 in BSL
Chapter 5 in Aris

The only material property of the fluid we have so far discussed is the density. In the last chapter we introduced the rate of deformation or rate of strain tensor. The distinguishing characteristic between fluids and solids is that fluids can undergo unlimited deformation and yet maintain its integrity. The relation between the rate of deformation tensor and stress tensor is the mechanical constitutive equation of the material. An ideal fluid has a stress tensor that is independent of the rate of deformation, i.e., it has an isotropic component, which is identified as the pressure and has zero viscosity.

Cauchy's stress principle and the conservation of momentum

The forces acting on an element of a continuous medium may be of two kinds. External or body forces , such as gravitation or electromagnetic forces, can be regarded as reaching into the medium and acting throughout the volume. If the external force can be describes as the gradient of a scalar, the force is said to be conservative. Internal or contact forces are to be regarded as acting on an element of volume through its bounding surfaces. If an element of volume has an external-bounding surface, the forces there may be specified, e.g., when a constant pressure is applied over a free surface. If the element is internal, the resultant force is that exerted by the material outside the surface upon that inside. Let $\mathbf{n}$ be the unit outward normal at a point of the surface $S$ and ${\mathbf{t}}_{\left(n\right)}$ the force per unit area exerted there by the material outside $S$ . Then Cauchy's principle asserts that ${\mathbf{t}}_{\left(n\right)}$ is a function of the position $\mathbf{x}$ , the time $t$ , and the orientation $\mathbf{n}$ of the surface element. Thus the total internal force exerted on the volume $V$ through the bounding surface $S$ is

If $\mathbf{f}$ is the external force per unit mass (e.g. if 03 is vertical, gravitation will exert a force $-g{\mathbf{e}}_{\left(3\right)}$ per unit mass or $-\rho g{\mathbf{e}}_{\left(3\right)}$ per unit volume, the total external force will be

The principle of the conservation of linear momentum asserts that the sum of these two forces equals the rate of change of linear momentum of the volume, i.e.,

This is just a generalization of Newton's law of motion, which states that the rate of change of momentum of a particle is equal to the sum of forces acting on it. It has been extended to a volume that contains a number of particles.

From the form of these integral relations we can deduce an important relation. Suppose $V$ is a volume of a given shape with characteristic dimension $d$ . Then the volume of $V$ will be proportional to $d^3$ and the area of $S$ to $d^2$ , with the proportionality constants depending only on the shape. Now let $V$ shrink to a point but preserve its shape, then the volume integrals in the last equation will decrease as $d^3$ but the surface integral will decrease as $d^2$ . It follows that