Stress balance

Thus, only the normal Reynolds stresses (i = j) are directly dissipated in a high-Reynolds-number turbulent flow. The shear stresses (i / j), on the other hand, are dissipated indirectly, i.e., the pressure-rate-of-strain tensor first transfers their energy to the normal stresses, where it can be dissipated directly. Without this redistribution of energy, the shear stresses would grow unbounded in a simple shear flow due to the unbalanced production term Vu given by (2.108). This fact is just one illustration of the key role played by 7 ., -in the Reynolds stress balance equation. 

This new particle diameter represents the separation at which the hydrodynamic stress balances with the thermal and thermodynamic stresses ... 

Bubbles and drops tend to deform when subject to external fluid fields until normal and shear stresses balance at the fluid-fluid interface. When compared with the infinite number of shapes possible for solid particles, fluid particles at steady state are severely limited in the number of possibilities since such features as sharp corners or protuberances are precluded by the interfacial force balance. 

For the calculation of the thermal shock-induced stresses, we consider the plate shown in Fig. 15.1 with Young s modulus E, Poisson s ratio v, and coefficient of thermal expansion (CTE) a, initially held at temperature /j. If the top and bottom surfaces of the plate come into sudden contact with a medium of lower temperature T they will cool and try to contract. However, the inner part of the plate initially remains at a higher temperature, which hinders the contraction of the outer surfaces, giving rise to tensile surface stresses balanced by a distribution of compressive stresses at the interior. By contrast, if the surfaces come into contact with a medium of higher temperature Tm, they will try to expand. As the interior will be at a lower temperature, it will constrain the expansion of the surfaces, thus giving rise to compressive surface stresses balanced by a distribution of tensile stresses at the interior. 

Mathematically, this problem bears some resemblance to those considered above. The governing partial differential equation is still Eq. (6), and on the surfaces boundary conditions of constant potential, constant charge density or linear regulation [i.e., Eq. (45)] must be imposed. However, a further constraint arises from the need to satisfy mechanical equilibrium at the interface, and it is this new condition that provides the mathematical relation needed to calculate the interface shape. The equation is the normal component of the surface stress balance, and it is given by [12]... 

Mean flow behaviors and turbulence characteristics of DR solutions differ from those of Newtonian solutions. Topics covered in this section include the maximum DR asymptote (MDRA), mean velocity profiles, diameter scale-up, turbulence intensities, stress balance, and streak spacing in polymer and surfactant DR solutions as well as the HTR-DR relationship for surfactant DR solutions. Numerical simulations for polymer DR are noted and speculations on DR mechanisms are offered. 

Stress Balance with Reduced Reynolds Stress... 

Several investigators have observed lower Reynolds stresses than expected for DR solutions caused by low v and difference in phase between u and v. Recently, several researchers observed zero Reynolds stress profiles for high-DR surfactant solutions, clearly illustrating a major stress deficit in these sys-Fig. 14 is a schematic illustrating the magnitude of the additional term, tdr, needed to satisfy the stress balance at any point for systems with zero Reynolds stresses. 

Fig. 3 shows a simplified compactible filter cake. Darcy s law and a stress balance involving the accumulated friction drag on the particulate structure (Fig. 3) are used to develop basic theory of flow through compactible porous media. 

For somebody the physical meaning of the surface stresses might be intuitive and best explained formulating a stress balance over an infinitesimal Eulerian control volume. 

The deformation of elastic solids occurs because of the stretching of intermo-lecular bonds to a point where internal stresses balance the externally applied stress (11,13). At this point, an equilibrium deformation is established. As there is little motion involved in the stretching of bonds, this occurs rapidly, and the equilibrium deformation is established infinitesimally. Deviations from ideal identity occur whenever the elastic limit of the solid material is exceeded and irreversible sample deformation results, i.e., breakage of chemical bonds (2,14). Irreversible sample deformation leading to fracture forms the basis of tensile testing (11). 

To obtain the normal component, which is generally referred to as the normal-stress balance, wetaketheinnerproductof(2-134) withn. Recalling that T = -pi + Tandf = -pi + t, this gives... 

We have obtained the Young Laplace relationship from normal-stress balance (2-135) by invoking the limiting case of no motion in the fluids. One might be tempted to suppose that the condition of no motion is independent of the interface, e.g., that it is determined by whether there is some source of fluid motion in the bulk-phase fluids away from the interface. However, this is not correct. In fact, if (V n) / constant on the interface (i.e., independent of position on S), the Young-Laplace equation cannot be satisfied, and there must be fluid motion so that the balance of normal stresses includes viscous contributions. Utilizing (2 137), we see that the condition (V n) = constant requires the sum of Rf1 and R2 1 to be constant on S. Examples of surfaces that satisfy this requirement are a sphere, where R = Ih = R (the radius) a circular cylinder, where R = R, R2 = 00 and a flat interface, where Ri = R2 = 00. 

To this point, we have considered only the component of the stress balance (2-134), in the direction normal to the interface. There will be, in general, two tangential components of (2-134), which we obtain by taking the inner product with the two orthogonal unit tangent vectors that are normal to n. If we denote these unit vectors as t (with i = 1 or 2), the so-called shear-stress balances can be written symbolically in the form,... 

If we consider, first, the case with grad vy = 0, we see that the tangential-stress balance requires continuity of the tangential stress. If the fluids are Newtonian, this condition can also be written in terms of the rate of strain, in the form... 

Here, we consider only the simpler situation in which the surfactant is assumed to be relatively dilute so that it is mobile on the interface and contributes a change only in the interfacial tension, without any more complex dynamical or rheological effects. In this case, the boundary conditions derived for a fluid interface still apply. Specifically, the dynamic and kinematic boundary conditions, in the form (2 122) and (2-129), respectively, and the stress balance, in the form (2 134), can still be used. However, the interfacial tension, which appears in the stress balance, now depends on the local concentration of surfactant. We shall discuss how this concentration is defined shortly. First, however, we note that flows involving an interface with surfactant are qualitatively similar to thermocapillary flows. The primary difference is that the concentration distribution of surfactant on the interface is almost always dominated by convection and diffusion within the interface, whereas the... 

The time-dependent fiinction Hit ) is determined by the rate of increase or decrease in the bubble volume. The governing equations and boundary conditions that remain to be satisfied are (1) the radial component of the Navier Stokes equation (2) the kinematic condition, in the form of Eq. (2 129), at the bubble surface and (3) the normal-stress balance, (2 135), at the bubble surface with = 0. Generally, for a gas bubble, the zero-shear-stress condition also must be satisfied at the bubble surface, but xrti = 0, for a purely radial velocity field of the form (4-193), and this condition thus provides no usefirl information for the present problem. 

A more useful alternative is to express (4-201) in terms of the pressure pB(t) inside the bubble rather than p(R. t). To do this, we must use the normal-stress balance, (2-135), at the bubble surface, which takes the form... 

The tangential-stress condition is satisfied automatically for an inviscid fluid (p = 0) because x = 0 in this case, and there is no viscous contribution to the normal-stress balance. 

However, there is no solution of this equation that satisfies the conditions (6 142) atx = 0 andx = 1, as well as the constant-volume constraint (6-143). This tells us that the gravitational contribution to the interface stress balance is insufficient, by itself, to enforce the condition that the interface is pinned at the ends of the thin gap. In fact, the solution of (6-158) is linear inx and thus can only satisfy the constant-volume constraint, (6 143), by falling below the mean height for x < (1/2) (i.e., h < 0) and rising above it for x > (1/2) (i.e., h >0). This makes sense only if the end walls exceed // = <7 in height so that the fluid at x = 1 remains within the container, and then only if the interface is not required to satisfy the conditions (6-142) or any other condition at the end walls. Typically, however, if the interface is not pinned at the corner as required by (6-142), it is required to satisfy a condition that fixes the contact angle between the interface and the end walls, and this condition also cannot be satisfied by the solution of (6-158). 

With the velocity and pressure gradient now evaluated to 0(1), we can use the normal-stress balance at 0(1), namely (6-245a), to determine the shape function h at 0(5). If we differentiate (6-245a) and insert dpl(>)/<)x from (6-250), we see that... 

Now, the drop shape will be non-spherical if this is necessary to satisfy the boundary conditions at its surface, and, specifically, to satisfy the normal-stress balance, (2 135). To determine the condition that leads to small deformations, we can nondimensionalize the boundary conditions using the same characteristic scales that were used for the governing equations, (7 198) and (7-199). The result for the normal-stress balance is... 

A qualitative prediction of the role of the capillary number in drop deformation can be obtained from the normal-stress balance. Now, if the shape is spherical, the capillary term on the right-hand side is a constant, and thus the viscous pressure and stress contributions... 

The remaining step is to obtain the shape function /o. The first step is to evaluate the left-hand side of the normal-stress balance (7-210),... 

Comparing (7-229) with the left-hand side of (7-227), we see that the two terms involving P p) in the normal-stress balance cancel exactly, and (7-227) reduces to the simple form... 

Thus the normal-stress balance is precisely satisfied with fo = 0. The capillary (or interfacial-tension) contribution is simply to produce a jump in pressure equal to n = 2/Ca across the drop surface. This pressure jump is, in fact, precisely the result (2 138) that was derived earlier, simply written in dimensionless terms based on a characteristic pressure,... 

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