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=  Squeeze Flow  =

Introduction
Squeeze flow (also called squeezing flow, squeezing film flow, or squeeze flow theory) is a type of flow where a material is pressed out or deformed between two parallel plates or objects. First measured in 1874 by Josef Stefan, squeeze flow describes the outward movement of a droplet of material, its area of contact with the plate surfaces, and the effects of internal and external factors such as temperature, viscoelasticity, and heterogeneity of the material. Several squeeze flow models exist to describe Newtonian and non-Newtonian fluids. Squeeze flow applications exist across numerous scientific and engineering disciplines including rheometry, welding engineering, and materials science.

Definitions
Conservation of mass (expressed as the continuity equation), the Navier-Stokes equations for conservation of momentum, and the Reynolds number provide the foundations for calculating and modeling squeeze flow. Boundary conditions for such calculations include assumptions of an incompressible fluid, a two-dimensional system, neglecting of body forces, and neglecting of inertial forces.

Relating applied force to material thickness:

$$F=-\frac{4*L^3*\mu*W}{h^3}{dh \over dt}$$

For most calculations, applied force is constant.

Newtonian fluids
Several equations accurately model Newtonian droplet sizes under different initial conditions.

Measurement of a single asperity, or jagged surface edge, allows for measurement of a very specific cross-section of a droplet. To measure macro-level squeeze flow effects, two models exist for the most common surfaces: circular and rectangular plate squeeze flows.

Single asperity
For single asperity squeeze flow:

$$\frac{h_0}{h}=\left (1+\frac{5*F*t*h_0^2}{4*\mu*W*L_0^3}\right )^{1/5}$$

Where $$h_0$$ is the initial height of the droplet, $$h$$ is the final height of the droplet, $$F$$ is the applied squeezing force, $$t$$ is the squeezing time, $$\mu$$ is the fluid viscosity, $$W$$ is the width of the assumed rectangular plate, and $$L_0$$ is the initial length of the droplet.

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.

Circular plate
For circular plate squeeze flow:

$$\frac{h_0}{h}=\left (1+\frac{16*F*t*h_0^2}{3*\pi*\mu*R^4}\right )^{1/2}$$

$$R$$ is the radius of the circular plate.

Rectangular plate
For rectangular plate squeeze flow:

$$\frac{h_0}{h}=\left (1+\frac{F*t*h_0^2}{2*\mu*W*L^3}\right )^{1/2}$$

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

Non-Newtonian fluids
Simplifying calculations for Newtonian fluids allows for basic analysis of squeeze flow, but many polymers are actually non-Newtonian fluids that exhibit viscoelastic characteristics under deformation. One such model in wide use is for a power law fluid, while other hypothetical models exist for Bingham fluids based on variations in yield stress calculations.

Power law fluid
For squeeze flow in a power law fluid:

$$\frac{h_0}{h}=\left ( 1+t*(\frac{2n+3}{4n+2})(\frac{(4*h_0*L_0)^{n+1}*F*(n+2))}{(2*L_0)^{2n+3}*W*m})^{1/n}\right )^{n/2n+3}$$

Where $$m$$ (or $$K$$) is the flow consistency index and $$n$$ is the dimensionless flow behavior index.

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. Some studies propose 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.

$$\tau = \begin{cases} \eta_2*{du \over dy}+\tau_1, & \text{if }\tau\geq\tau_1 \\ \eta_1*{du \over dy}, & \text{if }\tau<\tau_1 \end{cases}$$

Where $$\eta_2$$ is the known viscosity of the Bingham fluid, $$\eta_1$$ is the "paradoxical" viscosity of the solid-like state, and $$\tau_1$$ is the biviscosity region stress. To determine this new stress:

$$\tau_0=\tau_1(1-\epsilon)$$

Where $$\tau_0$$ is the yield stress and $$\epsilon=\frac{\eta_2}{\eta_1}$$ is the dimensionless viscosity ratio. If $$\epsilon=1$$, the fluid exhibits Newtonian behavior; as $$\epsilon\rightarrow0$$, the Bingham model applies.

Applications
Squeeze flow application is prevalent in several science and engineering fields. Modeling and experimentation assist with understanding the complexities of squeeze flow during processes such as rheological testing, hot plate welding, and composite material joining.

Rheological testing
Squeeze flow rheometry allows for evaluation of polymers under wide ranges of temperatures, shear rates, and flow indexes. The Williams parallel plate plastometer provides analysis for high viscosity materials such as rubber and glass, cure times for epoxy resins, and fiber-filled suspension flows.

Hot plate welding
During conventional hot plate welding, a successful joining phase depends on proper maintenance of squeeze flow to ensure that pressure and temperature create an ideal weld. Excessive pressure causes squeeze out of valuable material and weakens the bond, while failure to allow cooling to room temperature creates weak, brittle welds that crack or break completely during use.

Composite material joining
Prevalent in the aerospace and automotive industries, composites serve as expensive, yet mechanically strong, materials in the construction of several types of aircraft and vehicles. Typically composed of thermosetting polymers, thermoplastics may become a cheaper analog to permit increased manufacturing of these stronger materials through their ability to melt. Characterization and testing of thermoplastic composites allow for study of fiber orientations within the melt and final products. Fiber strand length and size show significant effect on material strength, while fiber orientation must align along the load direction to achieve the same properties as thermosetting composites.