Stress tensor physical interpretation

The total stress tensor, T, is thus interpreted physically as the surface forces per surface unit acting through the infinitesimal surface on the surrounding fluid with normal unit vector n directed out of the CV. This means that the total stress tensor by definition acts on the surrounding fluid. The counteracting force on the fluid element (CV) is therefore expressed in terms of the total stress tensor by introducing a minus sign in (1.65). 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Re3molds stress tensor The first and second terms on the RHS denote the production... 

Here the divergence of the stress tensor is the net force per unit volume acting on a fluid element. Note that is not a simple divergence because is a dyadic and not a vector, although the term is interpretable physically as a rate of momentum change. 

Since It IS only the divergence of p,v, that appears in Eq (6 8), one could add to the < > quantity in the last line of Eq (6 8) any solenoidal vector function However, it follows from the physic interpretation after Eq (6.8) that the identification we have made is the correct one This same pomt has been discussed by Irving and Kirkwood (see the Appendix to [6( ) with regard to the stress tensor See also Appendix A of this article... 

As seen, the SPH formulation of the equations of fluid dynamics reduces them to a set of ordinary differential equations (cf. eqn [32]) for the motion of each of the particles within the simulation. Hence, any numerical technique for the solution of coupled ordinary differential equations can be used for their solution. The physical picmre that emerges from these equations is very appealing and closely resembles the interpretation of dissipative particles in DPD. However, SPH does not include thermal fluctuations in the form of a random stress tensor and heat flux as in the classical Landau-Iifshitz theory of hydrodynamic fluctuations. Therefore, the validity of SPH to the study of complex fluids at mesoscopic scales where these fluctuations are important is presently questionable. ... 

The physical interpretation of the terms in the equation is not necessary obvious. The first term on the LHS denotes the rate of accumulation of the kinematic turbulent momentum flux within the control volume. The second term on the LHS denotes the advection of the kinematic turbulent momentum flux by the mean velocity. In other words, the left hand side of the equation constitutes the substantial time derivative of the Reynolds stress tensor v v. The first and second terms on the RHS denote the production of the kinematic turbulent momentum flux by the mean velocity shears. The third term on the RHS denotes the transport of the kinematic momentum flux by turbulent motions (turbulent diffusion). This latter term is unknown and constitutes the well known moment closure problem in turbulence modeling. The fourth and fifth terms on the RHS denote the turbulent transport by the velocity-pressure-gradient correlation terms (pressure diffusion). The sixth term on the RHS denotes the redistribution by the return to isotropy term. In the engineering literature this term is called the pressure-strain correlation, but is nevertheless characterized by its redistributive nature (e.g., [132]). The seventh term on the RHS denotes the molecular diffusion of the turbulent momentum flux. The eighth term on the RHS denotes the viscous dissipation term. This term is often abbreviated by the symbol... 

We recall here the physical interpretation of the stress tensor in Cartesian coordinates [161, pp. 131-132]. Let tj be the stress vector (surface force) representing the force per unit area exerted by the material outside the coordinate surface upon the material inside (where the unit outward normal to this surface is in the direction e ). The component Uj then represents the component of this stress vector at a point on the coordinate surface. For example, if the x coordinate surface has unit outward normal i/ = (1,0,0) then the stress vector at a point on this coordinate surface is simply ti = BiUjUj = eita = (tii, 21, 3i)- A similar interpretation arises for the couple stress tensor. The components tn, 22 and 33 are called the normal stresses or direct stresses and the components ti2, t2i> i3, 3i, 23, 32 are called the shear stresses. 

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