Creating a Mathematical model by combining Darcy's law and the Reynolds equation

 Darcy's Law

The equation below describes fluid flow through a permeable medium like fiber bundles.

                    Q=−k                                                                                  (1)

Where:

§     Q = the flow rate

§     K=the permeability of the medium

§     A =the cross-sectional area available for flow

§     = the pressure gradient

Reynolds' Equation

The Reynolds equation for fluid film lubrication in a wedge-shaped area is:

                                -h³-6ηU                                                                       (2)

Where:

§  h is the film thickness.

§  P is the pressure.

§  η is the viscosity of the fluid.

§  U is the pulling speed.

Combining the Laws

We will integrate these laws to describe the infiltration of fiber bundles and the transport of resin melt pressure.

Infiltration of Fiber Bundles (Darcy's Law):

                              Q=−k₀                                                                                  (3)

Where k_0​is the Carman-Kozeny coefficient

Transport of Resin Melt Pressure (Reynolds Equation):

                         -h³-6ηU                                                                             (4)

Defining the Wedge Area and Coordinate System

The wedge area is defined by the film thickness (h), radius (R), center distance (L), and angle (θ). The coordinate system will be used to calculate the melt film thickness, melt pressure, and fluid flow.

Calculating the Parameters

Melt Film Thickness (h)

                                     h(x)=δ(1+x/L)tan(θ)                                                                          (5)

Melt Pressure (P)

To solve for P, we integrate the Reynolds equation with the boundary conditions P(0)=P0P(0) = P(0)=P_0​ and P(L)=P_L

                              P(x)=P0+(6ηU()dx                                                                      (6)

Fluid Flow (q)

                                 q=−k₀                                                                                               (7)