Tangential stress balance

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... 

We now consider the hydrodynamic part of the problem, which is described by the Stokes equations (2.1.2). The fluid velocity components satisfy (2.2.2) remote from the drop, and the solution is bounded within the drop. On the interface, the no-flow condition (2.2.6) holds and condition (2.2.7) of continuity of the tangential velocity component must be satisfied. Moreover, the boundary condition of the tangential stress balance is to be used ... 

The estimate of u following from Eq. (42) is ti = 0 VS). This equation should be viewed as an ODE defining parallel flow in the film cross-section. It is solved subject to the no slip condition u XyO) = 0 (in the Galilean frame of the solid support) together with the tangential stress balance condition at the free interface. The latter reduces in the leading order to dzu h) = 0. The solution is elementary ... 

Next, the tangential stress condition (29) is used to obtain the first evolution equation. The axial velocity correction Wi follows from (86) since the only function of r is Wi, and is found in terms of wq and So- This is then substituted along with the expression (88) for Wo into (29) to yield. The leading order contribution of the tangential stress balance is the consistency condition wor = 0. while the next order yields the first desired evolution equation. The second equation follows from the leading order contributions of the kinematic condition (31). The system to be addressed is ... 

The governing equations and boundary conditions have been developed in Section 2.2. The system to be solved consists of the Navier-Stokes and continuity equations (34), (35), the concentration equation in the bulk (36) as well as the boundary conditions (39), (41) and (42) which are the surface mass balance, kinetic flux balance ad tangential stress balance respectively. As reviewed in Section 2.2, most studies have examined the effect of dilute concentrations, i.e. trace amounts of surfactants. At low concentrations the drag is found... 

Condition (128) is the tangential stress balance written as a boundary condition for the vorticity. A final condition is the kinetic fiux balance (41). In the first set of results to be discussed we compute the diflFusion control case (see above) and so take the limit Bi This implies the following equation of state from (41) ... 

In the second step the convective terms are dropped and a Poisson equation for the pressure emerging from the zero divergence of the velocity field, must be solved. In this step, the tangential stress balance evaluates F explicitly. 

Note that as we do not need an explicit expression for the temperature field in the bottom boundary layer, the results do not depend on the type of heat exchange at the solid support. Finally, let us use BC (165) for the tangential stress balance. It becomes... 

One imposes the tangential stress balance at the surface of the bubble ... 

On the other hand, if a liquid film experiences interfacial tension gradients, then the balance of tangential stresses at the interface gives rise to the key relationship... 

Prom this equality it follows that when the surface tension is constant, the tangential stress is continuous over the interface. In fluid mechanics it is thus frequently assumed that the tangential components of velocity are continuous across a phase interface [199]. With this simplification, vj t = Vk,t, the jump momentum balance yields ... 

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,... 

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. 

Statement of the problem. Let us consider a similar problem in which the upper boundary of the channel is free and the surface tension depends linearly on temperature. The balance of tangential stresses on the free surface will then involve the thermocapillary stresses. The corresponding boundary condition has the form... 

The forces that give rise to the phenomena spoken of appear because of the alteration in stresses at the interface between two immiscible fluid phases. For a curved interface there is a difference in pressure between the two fluids given by the Young-Laplace equation. This pressure difference is termed the capillary pressure, and since the normal stress component at the interface must be continuous, then that pressure added to the hydrostatic pressure must balance. A balance can always be achieved under static conditions. In addition, the tangential stress must also be continuous at the interface. However, if there... 

Equation 31a is the steady state form of the kinematic condition which also is used to describe wave motion as discussed on page 595 of Levich s book, Physicochemical Hydrodynamics. One also must equate the normal and tangential components of the forces in each phase at the free surface. Since we consider a gas-liquid interface, we neglect gas phase resistance due to its viscosity and include only the pressure it imposes on the interface on the gas side. Therefore, at y - -h(x) the tangential component of the stress tensor for the liquid phase is equal to the tangential force created by the change in surface tension with temperature in the x direction. Thus the tangential force balance at the interface becomes... 

Since VjT is parallel to Ug, it follows from this equation that Us = 0. It means that the there is no surfactant transfer over the bubble surface, i.e. the surface is fully retarded and the bubble moves like a solid body, which considerably increases the tangential stress at the bubble surface. The balance of forces in the direction of the x axis on the bubble s surface can be written approximately as... 

To close the system of equations for the fluid motion the tangential stress boundary condition and the force balance equation are used. The boundary condition for the balance of the surface excess linear momentum, see equations (8) and (9), takes into account the influence of the surface tension gradient, surface viscosity, and the electric part of the bulk pressure stress tensor. In the lubrication approximation the tangential stress boundary condition at the interface, using Eqs. (17) and (18), is simplified to... 

In the absence of viscous forces, this equation provides a relationship for the change in pressure across a curved interface. It should be noted that pressure jumps across a fluid interface can only sustain normal stress jumps they do not result in a tangential stress jump. Therefore, tangential surface stresses can only be balanced by viscous stresses associated with fluid motion. Consequently, in the absence of other effects, it is not possible to have a static system in the presence of surface tension gradient. A flow induced by a gradient in the surface tension is called Marangoni flow. 

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