Single asperity

For single asperity squeeze flow:

${\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {5*F*t*h_{0}^{2}}{4*\eta *W*L_{0}^{3}}}\right)^{1/5}}$

Where ${\displaystyle 2h_{0}}$ is the initial height of the droplet, ${\displaystyle 2h}$ is the final height of the droplet, ${\displaystyle F}$ is the applied squeezing force, ${\displaystyle t}$ is the squeezing time, ${\displaystyle \eta }$ is the fluid viscosity, ${\displaystyle W}$ is the width of the assumed rectangular plate, and ${\displaystyle 2L_{0}}$ is the initial length of the droplet.^[2]

Based on conservation of mass calculations, the droplet width is inversely proportional to droplet height; as the width increases, the height decreases in response to squeezing forces.^[2]

Circular plate squeeze flow diagram at initial and follow-on conditions; circular plates (gray), droplet (hash marked ellipse between both plates).

Circular plate

For circular plate squeeze flow:

${\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {16*F*t*h_{0}^{2}}{3*\pi *\eta *R^{4}}}\right)^{1/2}}$

${\displaystyle R}$ is the radius of the circular plate.^[2]

Rectangular plate

Rectangular plate squeeze flow diagram at initial and follow-on conditions; rectangular plates (gray), droplet (green).

For rectangular plate squeeze flow:

${\displaystyle {\frac {h_{0}}{h}}=\left(1+{\frac {F*t*h_{0}^{2}}{2*\mu *W*L^{3}}}\right)^{1/2}}$

These calculations assume a melt layer that has a length much larger than the sample width and thickness.^[2]

Power law fluid

For squeeze flow in a power law fluid:

${\displaystyle {\frac {h_{0}}{h}}=\left(1+t*({\frac {2n+3}{4n+2}})({\frac {(4*h_{0}*L_{0})^{n+1}*F*(n+2)}{(2*L_{0})^{2n+3}*W*m}})^{1/n}\right)^{n/2n+3}}$

Where ${\displaystyle m}$ (or ${\displaystyle K}$) is the flow consistency index and ${\displaystyle n}$ is the dimensionless flow behavior index.^[2]

${\displaystyle m=m_{0}*exp\left({\frac {-E_{a}}{R*T}}\right)}$

Where ${\displaystyle m}$ is the flow consistency index, ${\displaystyle m_{0}}$is the initial flow consistency index, ${\displaystyle E_{a}}$is the activation energy, ${\displaystyle R}$is the universal gas constant, and ${\displaystyle T}$is the absolute temperature.^[2]

During experimentation to determine the accuracy of the power law fluid model, observations showed that modeling slow squeeze flow generated inaccurate power law constants (${\displaystyle m}$and ${\displaystyle n}$) using a standard viscometer, and fast squeeze flow demonstrated that polymers may exhibit better lubrication than current constitutive models will predict.^[4] The current empirical model for power law fluids is relatively accurate for modeling inelastic flows, but certain kinematic flow assumptions and incomplete understanding of polymeric lubrication properties tend to provide inaccurate modeling of power law fluids.^[4]

Bingham fluid

Bingham fluids exhibit uncommon characteristics during squeeze flow. While undergoing compression, Bingham fluids should fail to move and act as a solid until achieving a yield stress; however, as the parallel plates move closer together, the fluid shows some radial movement. One study proposes a “biviscosity” model where the Bingham fluid retains some unyielded regions that maintain solid-like properties, while other regions yield and allow for some compression and outward movement.^[3]

${\displaystyle \tau ={\begin{cases}\eta _{2}*{du \over dy}+\tau _{1},&{\text{if }}\tau \geq \tau _{1}\\\eta _{1}*{du \over dy},&{\text{if }}\tau <\tau _{1}\end{cases}}}$

Where ${\displaystyle \eta _{2}}$ is the known viscosity of the Bingham fluid, ${\displaystyle \eta _{1}}$ is the "paradoxical" viscosity of the solid-like state, and ${\displaystyle \tau _{1}}$ is the biviscosity region stress.^[3] To determine this new stress:

${\displaystyle \tau _{0}=\tau _{1}(1-\epsilon )}$

Where ${\displaystyle \tau _{0}}$ is the yield stress and ${\displaystyle \epsilon ={\frac {\eta _{2}}{\eta _{1}}}}$ is the dimensionless viscosity ratio. If ${\displaystyle \epsilon =1}$, the fluid exhibits Newtonian behavior; as ${\displaystyle \epsilon \rightarrow 0}$, the Bingham model applies.^[3]