Stress tensor surface forces

Turn now to the work term dW/dt. The stress tensor causes forces on the surfaces of a control volume, through which fluid is moving, and the result is work. 

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. 

Surface waves at an interface between two innniscible fluids involve effects due to gravity (g) and surface tension (a) forces. (In this section, o denotes surface tension and a denotes the stress tensor. The two should not be coiifiised with one another.) In a hydrodynamic approach, the interface is treated as a sharp boundary and the two bulk phases as incompressible. The Navier-Stokes equations for the two bulk phases (balance of macroscopic forces is the mgredient) along with the boundary condition at the interface (surface tension o enters here) are solved for possible hamionic oscillations of the interface of the fomi, exp [-(iu + s)t + i V-.r], where m is the frequency, is the damping coefficient, s tlie 2-d wavevector of the periodic oscillation and. ra 2-d vector parallel to the surface. For a liquid-vapour interface which we consider, away from the critical point, the vapour density is negligible compared to the liquid density and one obtains the hydrodynamic dispersion relation for surface waves + s>tf. The temi gq in the dispersion relation arises from... 

Field variables identified by their magnitude and two associated directions are called second-order tensors (by analogy a scalar is said to be a zero-order tensor and a vector is a first-order tensor). An important example of a second-order tensor is the physical function stress which is a surface force identified by magnitude, direction and orientation of the surface upon which it is acting. Using a mathematical approach a second-order Cartesian tensor is defined as an entity having nine components T/j, i, j = 1, 2, 3, in the Cartesian coordinate system of ol23 which on rotation of the system to ol 2 3 become... 

Fig. 1. Force and stress acting on a parallelepipedic element. Sx and Fx denote a surface element perpendicular to the. v-axis and a force colinear to the. v-axis, respectively. For stress tensor elements <7 the subscripts i and j (x,y or z) relate to the orientations of the force and the surface element, respectively.

In Fig. 20 we show a theoretical dispersion plot using these parameters and a tensile stress = 2.7 x 10 dyn/cm. Due to the symmetry of the modes at X the stress tensor tpy does not affect the surface eigenmodes at this symmetry point. In addition, we have softened the intralayer force constant 4>ii in the first layer by about 10%. With these parameters, we find good agreement between experimental data and theoretical dispersion curves. 

The element ryl of the second-order stress tensor t is the force-per-unit area acting in the x direction on an elemental fluid surface perpendicular to the y direction. The stress tensor is symmetric, and so ryx = tiv. 

The force F and the stress r are both vectors, which are typically represented in components that align with a coordinate system. Since the stress vector at any surface whose orientation is represented by the outward normal n may be determined from the stress tensor, it follows that... 

In general, as seen in Section 2.8.4, the vector of surface forces (per unit volume) on a differential element can be represented as the divergence of the tensor stress field... 

In Chapter 2 considerable effort is devoted to establishing the relationship between the stress tensor and the strain-rate tensor. The normal and shear stresses that act on the surfaces of a fluid particle are found to depend on the velocity field in a definite, but relatively complex, manner (Eqs. 2.140 and 2.180). Therefore, when these expressions for the forces are substituted into the momentum equation, Eq. 3.53, an equation emerges that has velocities (and pressure) as the dependent variables. This is a very important result. If the forces were not explicit functions of the velocity field, then more dependent variables would likely be needed and a larger, more complex system of equations would emerge. In terms of the velocity field, the Navier-Stokes equations are stated as... 

If.we neglect the effect of surface tension, there is only one component of stress tensor a = azz = F/itR2 (F is extending force) independent of z in case of uniaxial tension of a cylinder. 

Here ma is the bulk solid-fluid interaction force, T.s the partial Cauchy stress in the solid, p/ the hydrostatic pressure in the perfect fluid, IIS the second-order stress in the solid, ha the density of partial body forces, ta the partial surface tractions, ts the traction corresponding to the second-order stress tensor in the solid and dvs/dn the directional derivative of v.s. along the outward unit normal n to the boundary cXl of C. 

For an element in equilibrium with no body forces, the equations of equilibrium were obtained by Lame and Clapeyron (1831). Consider the stresses in a cubic element in equilibrium as shown in Fig. 2.3. Denote 7y as a component of the stress tensor T acting on a plane whose normal is in the direction of e and the resulting force is in the direction of ej. In the Cartesian coordinates in Fig. 2.3, the total force on the pair of element surfaces whose normal vectors are in the direction of ex can be given by... 

The surface forces that act on the control volume are due to the stress field in the deforming fluid defined by the stress tensor ji. We discuss the nature of the stress tensor further in the next section at this point, it will suffice to state that ji is a symmetric second-order tensor, which has nine components. It is convenient to divide the stress tensor into two parts ... 

Turning back to Eq. 2.5-6, the surface forces Fs can now be expressed in terms of the total stress tensor n as follows ... 

Surface tractions or contact forces produce a stress field in the fluid element characterized by a stress tensor T. Its negative is interpreted as the diffusive flux of momentum, and x x (—T) is the diffusive flux of angular momentum or torque distribution. If stresses and torques are presumed to be in local equilibrium, the tensor T is easily shown to be symmetric. 

We denote the stress tensor inside the curly brackets in Eq. (13) asT. Equation (13) shows that the solution for the potential and its gradient at the particle surface are all that are required to calculate the force on a particle via the linearized Poisson-Boltzmann equation. 

Electrostatic. In many practical situations, both membrane and solute have net negative charges. Hence, as the solute approaches a pore in the membrane it experiences an electrostatic repulsion. A quantitative theoretical description of this interaction requires solution of the non-linear Poisson-Boltzmann equation for the interacting solute and membrane followed by calculation of the resulting force by integrating the electric stress tensor on the solute surface. Due to the complexity of the geometry... 

The mass forces may be the gravitational force, the force due to the rotational motion of a system, and the Lorentz force that is proportional to the vector product of the molecular velocity of component i and the magnetic field strength. The normal stress tensor a produces a surface force. No shear stresses occur (t = 0) in a fluid, which is in mechanical equilibrium. 

The interaction force P can be calculated by integrating the excess osmotic pressure An and the Maxwell stress tensor T over an arbitrary closed surface 21 enclosing either one of the two interacting particles (Fig. 8.3), which is written as [8]... 

In Sections 1.3.1 and 1.3.2, we discussed the shear stress and the extensional stress in shearing flows and extensional flows, respectively. These are components of the three-dimensional state-of-stress tensor T. The ith row of T is the force per unit area that material exterior to a unit cube exerts on a surface perpendicular to the tth coordinate axis (see Fig. 1-18). In general, if F is the force per unit area acting on a surface perpendicular to an arbitrary outward-directed unit vector, n, then... 

With the aim of relating the force per unit area at a point to the components of the stress tensor at that point, let us consider (3) the tetrahedron of Figure 4.4, in which a force per unit area, /, is applied to the oblique surface AS. The other surfaces of the tetrahedron, ASi, ASj, and AS, respectively perpendicular to the Xi, X2, and X3 coordinate axes, can be obtained from AS from the expressions. 

The analysis of the balance of forces on the lateral surface of the beam carried out on the basis of the preliminary hypothesis indicates that the only nonzero component of the stress tensor is simple extension. In fact, some parts of the transverse section are under tension, and others are under compression, and the two effects combined produce the flexion. According to Figure 17.1, the component jxx of the strain is given by... 

Equation (8.175) is a generalization of Ehrenfest s theorem (Ehrenfest 1927). This theorem relates the forces acting on a subsystem or atom in a molecule to the forces exerted on its surface and to the time derivative of the momentum density mJ(r). It constitutes the quantum analogue of Newton s equation of motion in classical mechanics expressed in terms of a vector current density and a stress tensor, both defined in real space. 

The term T (Q) is the virial of the forces exerted on the surface of the subsystem, a term expressible in terms of the stress tensor previously defined in eqns (8.173) and (6.12),... 

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