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Diffstat (limited to 'source/know/concept/lubrication-theory/index.md')
| -rw-r--r-- | source/know/concept/lubrication-theory/index.md | 67 |
1 files changed, 34 insertions, 33 deletions
diff --git a/source/know/concept/lubrication-theory/index.md b/source/know/concept/lubrication-theory/index.md index 4015526..54200d3 100644 --- a/source/know/concept/lubrication-theory/index.md +++ b/source/know/concept/lubrication-theory/index.md @@ -9,8 +9,7 @@ categories: layout: "concept" --- -**Lubricants** are widely used -to reduce friction between two moving surfaces. +**Lubricants** are widely used to reduce friction between two moving surfaces. In fluid mechanics, **lubrication theory** is the study of fluids that are tightly constrained in one dimension, especially those in small gaps between moving surfaces. @@ -34,32 +33,32 @@ $$\begin{aligned} \approx \frac{d^2}{L^2} \mathrm{Re} \end{aligned}$$ -If $$d$$ is small enough compared to $$L$$, -then $$\mathrm{Re}_\mathrm{gap} \ll 1$$. +If $$d$$ is small enough compared to $$L$$, then $$\mathrm{Re}_\mathrm{gap} \ll 1$$. More formally, we need $$d \ll L / \sqrt{\mathrm{Re}}$$, so we are inside the boundary layer, in the realm of the [Prandtl equations](/know/concept/prandtl-equations/). Let $$\mathrm{Re}_\mathrm{gap} \ll 1$$. -We are thus dealing with *Stokes flow*, in which case +We are then dealing with *Stokes flow*, in which case the [Navier-Stokes equations](/know/concept/navier-stokes/equations/) can be reduced to the following *Stokes equations*: $$\begin{aligned} \pdv{p}{x} = \eta \: \Big( \pdvn{2}{v_x}{x} + \pdvn{2}{v_x}{y} \Big) - \qquad \quad + \qquad \qquad \pdv{p}{y} = \eta \: \Big( \pdvn{2}{v_y}{x} + \pdvn{2}{v_y}{y} \Big) \end{aligned}$$ -Let the $$y = 0$$ plane be an infinite flat surface, +Let the $$y = 0$$ plane be an infinite flat surface +(a good approximation because $$d \ll L$$), sliding in the positive $$x$$-direction at a constant velocity $$U$$. -On the other side of the gap, -an arbitrary surface is described by $$h(x)$$. +On the other side of the gap, an arbitrary surface +is described by a height function $$h(x)$$. Since the gap is so narrow, -and the surfaces' movements cause large shear stresses inside, +and the surfaces' movements cause large shear stresses inside it, $$v_y$$ is negligible compared to $$v_x$$. Furthermore, because the gap is so long, we assume that $$\ipdv{v_x}{x}$$ is negligible compared to $$\ipdv{v_x}{y}$$. @@ -68,7 +67,7 @@ This reduces the Stokes equations to: $$\begin{aligned} \pdv{p}{x} = \eta \pdvn{2}{v_x}{y} - \qquad \quad + \qquad \qquad \pdv{p}{y} = 0 \end{aligned}$$ @@ -92,18 +91,20 @@ $$\begin{aligned} \end{aligned}$$ The moving bottom surface drags fluid in the $$x$$-direction -at a volumetric rate $$Q$$, given by: +at a volumetric rate $$Q(x)$$, given by: $$\begin{aligned} - Q - = \int_0^{h(x)} v_x(x, y) \dd{y} - = \bigg[ \frac{p'}{6 \eta} y^3 - \frac{p'}{4 \eta} h y^2 - \frac{U}{2 h} y^2 + U y \bigg]_0^{h} - = - \frac{p'}{12 \eta} h^3 + \frac{U}{2} h + Q(x) + &= \int_0^{h(x)} v_x(x, y) \dd{y} + \\ + &= \bigg[ \frac{p'}{6 \eta} y^3 - \frac{p'}{4 \eta} h y^2 - \frac{U}{2 h} y^2 + U y \bigg]_0^{h} + \\ + &= - \frac{p'}{12 \eta} h^3(x) + \frac{U}{2} h(x) \end{aligned}$$ -Assuming that the lubricant is incompressible, -meaning that the same volume of fluid must be leaving a point as is entering it. -In other words, $$Q$$ is independent of $$x$$, +Let us assume that the lubricant is incompressible, +meaning that the same volume of fluid must be both leaving and entering the gap. +In that case, $$Q$$ is independent of $$x$$, which allows us to write $$p'(x)$$ in terms of measurable constants and the known function $$h(x)$$: @@ -139,7 +140,8 @@ $$\begin{aligned} = - 2 h' \frac{U h - 3 Q}{h^4} \big( 2 h y - 3 y^2 \big) \end{aligned}$$ -Integrating with respect to $$y$$ thus leads to the following transverse velocity $$v_y$$: +Integrating with respect to $$y$$ therefore leads to +the following transverse velocity $$v_y$$: $$\begin{aligned} \boxed{ @@ -148,10 +150,11 @@ $$\begin{aligned} } \end{aligned}$$ -Typically, the lubricant is not in a preexisting pressure differential, -i.e it is not getting pumped through the system. -Although the pressure gradient $$p'$$ need not be zero, -we therefore expect that its integral vanishes: +Usually, the lubricant is not getting pumped through the system. +In that case, although the pressure gradient $$p'$$ need not be zero in all points +(i.e. there may be complex dynamics inside the gap), +we do expect that its integral across the gap vanishes +(because both sides are at the same pressure): $$\begin{aligned} 0 @@ -164,7 +167,7 @@ Isolating this for $$Q$$, and defining $$q$$ as below, yields a simple equation: $$\begin{aligned} Q = \frac{1}{2} U q - \qquad \quad + \qquad \qquad q \equiv \frac{\int_L h^{-2} \dd{x}}{\int_L h^{-3} \dd{x}} \end{aligned}$$ @@ -178,27 +181,25 @@ $$\begin{aligned} &= U \Big( 1 - \frac{y}{h} \Big) \Big( 1 - \frac{3 y (h - q)}{h^2} \Big) \end{aligned}$$ -The first factor is always positive, -but the second can be negative, -if for some $$y$$-values: +The first factors are always positive, but the last one can be negative, +if any $$y$$-values satisfy: $$\begin{aligned} h^2 < 3 y (h - q) - \quad \implies \quad + \qquad \implies \qquad y > \frac{h^2}{3 (h - q)} \end{aligned}$$ -Since $$h > y$$, such $$y$$-values will only exist +Since $$h \le y$$, such $$y$$-values will only exist if $$h$$ is larger than some threshold: $$\begin{aligned} 3 (h - q) > h - \quad \implies \quad + \qquad \implies \qquad h > \frac{3}{2} q \end{aligned}$$ -If this condition is satisfied, -there will be some flow reversal: +If this condition is satisfied, there will be some flow reversal: rather than just getting dragged by the shearing motion, the lubricant instead "rolls" inside the gap. This is confirmed by $$v_y$$: |