## Inflow

### Quick Start

Under applied pressure, the viscous adhesive (it might be a coating or hot melt) will flow into the porous fibre bundle. Will the combination of pressure, time and viscosity allow the adhesive to flow sufficiently into and around the fibres?

### Inflow

Under a pressure, P (see below), flow into the fibre web is controlled by the viscosity and the web's porosity ε, two factors S and T that depend on ε, and the permeability coefficient k which depends on the fibre diameter D via:

`S = -4(1-ε)-(1-ε)^2-3-2ln(1-ε)`

`T = ln(1-ε)+(1-(1-ε)^2)/(1+(1-ε)^2)`

`k =- (D^2)/(16(1-ε))(S.T)/(S+T)`

The speed of the advancing fluid in the z direction (through the thickness of the sample) also depends on the pressure, P, and is given by:

`(dz)/(dt) = (Pk(1-ε))/(εηz) = (PD^2)/(16εηz)(S.T)/(S+T)`

From this velocity we can see how the advancing front moves into the sample during your chosen time-scale. An average shear rate (advancing speed divided by fibre diameter) is shown.

And by integrating over time (up to our chosen time, t) we can find how far the fluid has entered the web. The graph shows the initial rapid flow which then slows down as viscous drag increases with depth. Obviously you can't push in more than can be filled by the *Thickness*, h, provided as an input, so everything stops when there's no more to push in. This depth limit depends on the porosity and is calculated simply: `d_"limit"=h/ε`.

The illustration shows how much you are filling the fibres.

What viscosity value should you enter? If you have a hot-melt then you can calculate its temperature from the Melt Thermal app. For particle dispersions you can use the High-Shear Particle app. In either case so you might have to loop back to your viscosity data to see if the viscosity you've chosen matches the shear-dependent viscosity from your rheometer. There's some circularity here - if the viscosity is a bit high, the shear rate will be a bit low, so the viscosity might be even higher. There's nothing an app like this can do to resolve such an issue.

### Pressure

If you are relying on capillary forces alone (plus a minimum Applied pressure of 1 kPa) then you need to calculate the pressure and set P. A common formula that depends on surface tension, γ, contact angle θ and fibre volume `φ=1-ε` is:

`P=(2γ)/r(3φcosθ)/(2(1-φ))`

The total pressure is calculated for you, so you know whether to worry about surface tensions and contact angles or not.