Cauchy’s equation of motion

Application of the weighted residual method to the solution of incompressible non-Newtonian equations of continuity and motion can be based on a variety of different schemes. Tn what follows general outlines and the formulation of the working equations of these schemes are explained. In these formulations Cauchy s equation of motion, which includes the extra stress derivatives (Equation (1.4)), is used to preseiwe the generality of the derivations. However, velocity and pressure are the only field unknowns which are obtainable from the solution of the equations of continuity and motion. The extra stress in Cauchy s equation of motion is either substituted in terms of velocity gradients or calculated via a viscoelastic constitutive equation in a separate step. 

By considering Cauchy s equations of motion [Eq. (10)], Truesdell derived the theorem of stress means,... 

The individual fluid elements of a flowing fluid are not only displaced in terms of their position but are also deformed under the influence of the normal stresses tu and the shear stresses T (i j)- The deformation velocity depends on the relative movement of the individual points of mass to each other. It is only in the case when the points of mass in a fluid element do not move relatively to each other that the fluid element behaves like a rigid solid and will not be deformed. Therefore a relationship between the velocity field and the deformation, and with that also between the velocity field and the stress tensor must exist. This relationship is required if we wish to express the stress tensor in terms of the velocities in Cauchy s equation of motion. 

In order to solve Cauchy s equation of motion, which is valid for any substance, a further relationship between the stress and strain tensors, or between the stress... 

Putting this expression into Cauchy s equation of motion, (3.48), yields the so-called Navier-Stokes equation... 

Equation (3.90) is the mass balance or continuity equation, (3.91) the momentum balance or Cauchy s equation of motion and (3.92) is the energy balance. As a momentum balance exists for each of the three coordinate directions, j = 1, 2, 3, there are five balance equations in total. The enthalpy form (3.83) is equivalent to the energy balance (3.92). 

This is known as Cauchy s equation of motion. It is clear from our derivation that it is simply the differential form of Newton s second law of mechanics applied to a moving fluid. 

We see that application of the angular acceleration principle does reduce, somewhat, the imbalance between the number of unknowns and equations that derive from the basic principles of mass and momentum conservation. In particular, we have shown that the stress tensor must be symmetric. Complete specification of a symmetric tensor requires only six independent components rather than the full nine that would be required in general for a second-order tensor. Nevertheless, for an incompressible fluid we still have nine apparently independent unknowns and only four independent relationships between them. It is clear that the equations derived up to now - namely, the equation of continuity and Cauchy s equation of motion do not provide enough information to uniquely describe a flow system. Additional relations need to be derived or otherwise obtained. These are the so-called constitutive equations. We shall return to the problem of specifying constitutive equations shortly. First, however, we wish to consider the last available conservation principle, namely, conservation of energy. 

It appears from (2-45) that contributions from any of the terms on the right-hand side will lead to a change in the sum of kinetic and internal energy, but may not contribute separately to one or the other of these energy terms. However, this is not true as we may see by further examination. First, we may note that the Cauchy s equation of motion provides an independent relationship for the rate of change of kinetic energy. In particular, if we take the inner product of (2-32) with u, we obtain... 

Problem 2-12. From Cauchy s equation of motion for the steady flow of an incompressible fluid in the absence of body forces, derive the integral momentum balances for the hydrodynamic force F and torque L on an arbitrary body Sb immersed in the fluid ... 

Equation (10), which establishes the balance of forces at a point within the body, is known as Cauchy s law of motion. 

Although there is no immediately useful information that we can glean from (2-56), we shall see that it provides a constraint on allowable constitutive relationships for T and q. In this sense, it plays a similar role to Newton s second law for angular momentum, which led to the constraint (2 41) that T be symmetric in the absence of body couples. In solving fluid mechanics problems, assuming that the fluid is isothermal, we will use the equation of continuity, (2-5) or (2-20), and the Cauchy equation of motion, (2-32), to determine the velocity field, but the angular momentum principle and the second law of thermodynamics will appear only indirectly as constraints on allowable constitutive forms for T. Similarly, for nonisothermal conditions, we will use (2-5) or (2-20), (2-32), and either (2-51) or... 

Momentum can be transported by convection and conduction. Convection of momentum is due to the bulk flow of the fluid across the surface associated with it is a momentum flux. Conduction of momentum is due to intermolecular forces on each side of the surface. The momentum flux associated with conductive momentum transport is the stress tensor. The general momentum balance equation is also referred to as Cauchy s equation. The Navier-Stokes equations are a special case of the general equation of motion for which the density and viscosity are constant. The well-known Euler equation is again a special case of the general equation of motion it applies to flow systems in which the viscous effects are negligible. 

Newton s second law states that in an inertial frame the rate of linear momentum is equal to the applied force. Here, by applying the second law to a continuum region, we define the Cauchy stress, and derive the equation of motion. 

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