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8.6.1 Boundary Conditions Categories

The governing equations that were developed earlier requires some boundary conditions and initial conditions. These conditions described physical situations that are believed or should exist or approximated. These conditions can be categorized by the velocity, pressure, or in more general terms as the shear stress conditions (mostly at the interface). For this discussion, the shear tensor will be separated into two categories, pressure (at the interface direction) and shear stress (perpendicular to the area). A common velocity condition is that the liquid has the same value as the solid interface velocity. In the literature, this condition is referred as the "no slip'' condition. The solid surface is rough thus the liquid participles (or molecules) are slowed to be at the solid surface velocity. This boundary condition was experimentally observed under many conditions yet it is not universal true. The slip condition (as oppose to "no slip'' condition) exist in situations where the scale is very small and the velocity is relatively very small. The slip condition is dealing with a difference in the velocity between the solid (or other material) and the fluid media. The difference between the small scale and the large scale is that the slip can be neglected in the large scale while the slip cannot be neglected in the small scale. In another view, the difference in the velocities vanishes as the scale increases.

Fig. 8.13 Dimensional free surface describing \(\widehat{\pmb{n}}\) and \(\widehat{\pmb{t}}\).

Another condition which affects whether the slip condition exist is how rapidly of the velocity change. The slip condition cannot be ignored in some regions, when the flow is with a strong velocity fluctuations. Mathematically the "no slip'' condition is written as

\[
    \label{dif:eq:noSlipCondV}
    \widehat{\mathbf{t}} \cdot \left( \pmb{U}_{fluid} - \pmb{U}_{boundary} \right)  = 0 \tag{61}
\]

where \(\widehat{\mathbf{n}}\) is referred to the area direction (perpendicular to the area see Figure ??. While this condition (61) is given in a vector form, it is more common to write this condition as a given velocity at a certain point such as

\[
    \label{dif:eq:noSlipCond}
    U(\ell) = U_{\ell} \tag{62}
\]

Note, the "no slip'' condition is applicable to the ideal fluid ("inviscid flows'') because this kind of flow normally deals with large scales. The 'slip' condition is written in similar fashion to equation (61) as

\[
    \label{dif:eq:slipCondV}
    \widehat{\mathbf{t}} \cdot \left( \pmb{U}_{fluid} - \pmb{U}_{boundary} \right)  = f(Q,scale, etc)   \tag{62}
\]
As oppose to a given velocity at particular point, a requirement on the acceleration (velocity) can be given in unknown position.

The condition (61) can be mathematically represented in another way for free surface
conditions. To make sure that all the material is accounted for in the control volume (does not cross the free surface), the relative perpendicular velocity at the interface must be zero. The location of the (free) moving boundary can be given as \(f(\widehat{\pmb{r}},t) =0\) as the equation which describes the bounding surface. The perpendicular relative velocity at the surface must be zero and therefore  
\[
    \label{dif:eq:perpendicularUSurface}
    \dfrac{Df}{Dt} = 0 \quad \mbox{ on the surface } f (\widehat{\pmb{r}},t) = 0                          \tag{63}
\]
This condition is called the kinematic boundary condition. For example, the free surface in the two dimensional case is represented as \(f(t,x,y)\). The condition becomes as
\[
    \label{dif:eq:2perpendicularUSurface}
    0 = \dfrac{\partial f}{\partial t} +
    U_x\, \dfrac{\partial f}{\partial x} +     
    U_y\, \dfrac{\partial f}{\partial y}      \tag{64}
\]
The solution of this condition, sometime, is extremely hard to handle because the location isn't given but the derivative given on unknown location. In this book, this condition will not be discussed (at least not plane to be written). The free surface is a special case of moving surfaces where the surface between two distinct fluids. In reality the interface between these two fluids is not a sharp transition but only approximation (see for the surface theory). There are situations where the transition should be analyzed as a continuous transition between two phases. In other cases, the transition is idealized an almost jump (a few molecules thickness). Furthermore, there are situations where the fluid (above one of the sides) should be considered as weightless material. In these cases the assumptions are that the transition occurs in a sharp line, and the density has a jump while the shear stress are continuous (in some cases continuously approach zero value). While a jump in density does not break any physical laws (at least those present in the solution), the jump in a shear stress (without a jump in density) does break a physical law. A jump in the shear stress creates infinite force on the adjoin thin layer. Off course, this condition cannot be tolerated since infinite velocity (acceleration) is impossible. The jump in shear stress can appear when the density has a jump in density. The jump in the density (between the two fluids) creates a surface tension which offset the jump in the shear stress. This condition is expressed mathematically by equating the shear stress difference to the forces results due to the surface tension. The shear stress difference is
\[
    \label{dif:eq:JumpStress}
    \Delta \boldsymbol{\tau}^{(n)} = 0 =  
    \Delta {\boldsymbol{\tau}^{(n)}}_{\text{ upper surface}} -
    \Delta {\boldsymbol{\tau}^{(n)}}_{\text{ lower surface}} \tag{65}
\]
where the index \((n)\) indicate that shear stress are normal (in the surface area).
If the surface is straight there is no jump in the shear stress. The condition with curved surface are out the scope of this book yet mathematically the condition is given as without explanation as
\[
    \label{dif:eq:bcCurvedSurface}
    \widehat{\pmb{n}}\cdot {\boldsymbol{\tau}^{(n)}} = \sigma \left( \dfrac{1\dfrac{}{}}{R_1} + \dfrac{1}{R_2}\right) \  
    \widehat{\pmb{t}}\cdot {\boldsymbol{\tau}^{(t)}} = - \widehat{\pmb{t}}\cdot  \nabla\sigma   \tag{66}
\]
where \(\widehat{\pmb{n}}\) is the unit normal and \(\widehat{\pmb{t}}\) is a unit tangent to the surface (notice that direction pointed out of the "center'' see Figure ??) and \(R_1\) and \(R_2\) are principal radii. One of results of the free surface condition (or in general, the moving surface condition) is that  integration constant is unknown). In same instances, this constant is determined from the volume conservation. In index notation equation (67) is written  
\[
    \label{dif:eq:bcCurvedSurfaceIndex}
    \tau_{ij}^{(1)}\,n_j  + \sigma \, n_i \left( \dfrac{1}{R_1} + \dfrac{1}{R_2}\right) =   
    \tau_{ij}^{(2)}\,n_j   \tag{67}
\]
where \(1\) is the upper surface and \(2\) is the lower surface. For example in one dimensional  
\[
    \label{dif:eq:nAndT}  
    \begin{array}{rl}
    \widehat{\pmb{n}} = &  \dfrac{\left(-f^\prime{}(x), 1 \right) }{\sqrt{1+ \left(f^\prime{}(x)\right)^2 }} \
    \widehat{\pmb{t}} = & \dfrac{\left(1,f^\prime{}(x) \right) }{\sqrt{1+ \left(f^\prime{}(x)\right)^2 }}  
    \end{array} \tag{68}
\]
the unit vector is given as two vectors in \(x\) and \(y\) and the radius is given by equation (??). The equation is given by  
\[
    \label{dif:eq:1DfreeSurface}
    \dfrac{\partial f}{\partial t} + U_x \dfrac{\partial f}{\partial x} = U_y   \tag{69}
\]

The Pressure Condition

The second condition that commonality prescribed at the interface is the static pressure at a specific location. The static pressure is measured perpendicular to the flow direction. The last condition is similar to the pressure condition of prescribed shear stress or a relationship to it. In this category include the boundary conditions with issues of surface tension which were discussed earlier. It can be noticed that the boundary conditions that involve the surface tension are of the kind where the condition is given on boundary but no at a specific location.

Gravity as Driving Force

The body forces, in general and gravity in a particular, are the condition that given on the flow beside the velocity, shear stress (including the surface tension) and the pressure. The gravity is a common body force which is considered in many fluid mechanics problems. The gravity can be considered as a constant force in most cases (see for dimensional analysis for the reasons).

Shear Stress and Surface Tension as Driving Force

Fig. 8.14 Kerosene lamp.

If the fluid was solid material, pulling the side will pull all the material. In fluid (mostly liquid) shear stress pulling side (surface) will have limited effect and yet sometime is significant and more rarely dominate. Consider, for example, the case shown in Figure 8.14. The shear stress carry the material as if part of the material was a solid material. For example, in the kerosene lamp the burning occurs at the surface of the lamp top and the liquid is at the bottom. The liquid does not move up due the gravity (actually it is against the gravity) but because the surface tension.

Fig. 8.15 Flow in a kendle with a surfece tension gradient.

The physical conditions in Figure 8.14 are used to idealize the flow around an inner rode to understand how to apply the surface tension to the boundary conditions. The fluid surrounds the rode and flows upwards. In that case, the velocity at the surface of the inner rode is zero. The velocity at the outer surface is unknown. The boundary condition at outer surface given by a jump of the shear stress. The outer diameter is depends on the surface tension (the larger surface tension the smaller the liquid diameter). The surface tension is a function of the temperature therefore the gradient in surface tension is result of temperature gradient. In this book, this effect is not discussed. However, somewhere downstream the temperature gradient is insignificant. Even in that case, the surface tension gradient remains. It can be noticed that, under the assumption presented here, there are two principal radii of the flow. One radius toward the center of the rode while the other radius is infinite (approximatly). In that case, the contribution due to the curvature is zero in the direction of the flow (see Figure 8.15). The only (almost) propelling source of the flow is the surface gradient (\(\dfrac{\partial \sigma}{\partial n}\)).

Contributors

  • Dr. Genick Bar-Meir. Permission is granted to copy, distribute and/or modify this document under the terms of the GNU Free Documentation License, Version 1.2 or later or Potto license.