Cauchys equation of motion

By introducing the stress vector t according to (3.43) into the momentum equation (3.40) we obtain 

With the help of Gauss law we rearrange the surface integral 

3 Convective heat and mass transfer. Single phase flow 

This equation holds for any volume V (t) and therefore also in the limiting case of V(t) —y 0. From this follows Cauchy s equation of motion2  

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Performing a momentum balance over a differential volume of a homogenous material leads to the Cauchy equation of motion,... 

This relation corresponds to the Cauchy equation of motion in fluid dynamics. 

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... 

Let us now return to the equations of motion for a Newtonian fluid. With the constitutive equation, (2-80) [or (2-81) if the fluid is incompressible], the continuity equation, (2-5) [or (2-20) if the fluid is incompressible], and the Cauchy equation of motion, (2-32), we have achieved a balance between the number of independent variables and the number of equations for an isothermal fluid. If the fluid is not isothermal, we can add the thermal energy equation, (2-52), and the thermal constitutive equation, (2-67), and the system is still fully specified insofar as the balance between independent variables and governing equations is concerned. 

So far, we have simply stated the Cauchy equation of motion and the Newtonian constitutive equations as a set of nine independent equations involving u, T, and p. It is evident in this case, however, that the constitutive equation, (2-80), for the stress [or equivalently (2-86)] can be substituted directly into the Cauchy equation to provide a set of equations that involve only u and p (orp). These combined equations take the form... 

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. 

After the substitution of Cauchy stress via Equation (3.20) and the viscous part of the extra stress in terms of rate of defonnation, the equation of motion is written as... 

These forms of the equation of motion are commonly called the Cauchy momentum equations. For generalized Newtonian fluids we can define the terms of the deviatoric stress tensor as a function of a generalized Newtonian viscosity, p, and the components of the rate of deformation tensor, as described in Table 5.3. 

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... 

In this section, we combine the Cauchy equation and the Newtonian constitutive equation to obtain the Navier-Stokes equation of motion. First, however, we briefly reconsider the notion of pressure in a general, Newtonian fluid. 

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 ... 

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. 

A detailed formulation of the employed 3-D BEM is too extensive and beyond the scope of this paper and can be found in O Brien and Rizos (2005), Rizos (1993), Rizos (2000), Rizos and Karabalis (1994) and Rizos and Loya (2002). The BEM uses the time domain 4th order B-Spline fundamental solutions of the 3-D full space along with higher order spatial discretization of the boundary. The Boundary Integral Equation associated to the Navier-Cauchy governing equations of motion is expressed in a discrete form yielding a system of algebraic equations at step N relating displacements u to forces f at discrete boundary nodes in the BEM model and at discrete time instants tj and Ty, as... 

Please note the identity between the velocity v and the time derivative of u. Furthermore the quantities f, a, q and r stand for the sum of externally applied body forces, the transposed Cauchy stress tensor, the heat flux, and an arbitrary energy production term (e.g. due to latent heat during phase transitions). Eqs. (1-3) are equations of motion for the five unknown fields p, u and e. They are universal, namely material-independent. To solve these equations, the constitutive quantities, viz. heat flux and stress tensor, must be replaced by constitutive equations (cf. subsequent paragraph) q = (T, VT,...) and = (T,u,...). Moreover, up to now no temperature T occur in the balances (1-3). For this reason a caloric state equation, e = e T), must be introduced, which allows for replacing the internal energy e by temperature T. 

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. 

The governing equations that control material responses are given by the mass conservation law (2.97) and the equation of motion (2.104) if no energy conservation is considered. Note that the Cauchy stress is symmetric under the conservation law of moment of linear momentum. Furthermore, if the change of mass density is small (or it may be constant), the equation to be solved is given by (2.104). The unknowns in this equation are the velocity v (or displacement u in the small strain theory) and the stress a, i.e. giving a total of nine, that is, three for v (or u) and six for three components, therefore it cannot be solved, suggesting that we must introduce a relationship between v (or u) and [Pg.40]

If we choose as the summational invariants m, mC and mC, we obtain the Cauchy set of conservation equations named the continuity (2.217), the equation of motion (2.223), and the equation of energy (2.230). The only difference in the final result is that the pressure tensor, p, and the heat flux vector, q, are made up of two parts (i.e., a kinetic and a collisional contribution) ... 

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