In the process of synthetic fiber spinning, a filament of polymer is stretched, usually at high speed, in order to achieve a fiber with a very small cross-section area. For fibers with a circular cross-section, the die shape is not an issue. Other fibers may have a more complex shape, presumably for the optical properties that can be produced. The tri-lobal fiber has again a well-known level of symmetry. For the purpose of generality, a fiber cross-section that has no symmetry is considered.

Figure 4.1: Non-symmetric Die Cross-section for a Tri-lobal Fiber

Figure 4.1: Non-symmetric Die Cross-section for a Tri-lobal Fiber, displays the cross section of the die used for the production of the tri-lobal fiber. In the present case, the length of all three lobes, measured from the center to the tip, are of 0.24, 0.27 and 0.30 mm, respectively for the vertical, the horizontal and the sloped lobe. The lobes width is 0.12 mm, while the curvature radius is 0.06 mm. In addition, the angular distribution of the lobes further enhances the (purposely selected) non-symmetric geometric shape.

This example is designed to illustrate the setup of a viscoelastic flow, and at the same time, to show the effects of viscoelasticity on the fiber shape. For describing the rheology of the melt, the Oldroyd-B model is purposely selected, which is known to exhibit a strong transient elongation viscosity. In a single mode context, the model involves three parameters: the relaxation time, the zero-shear viscosity, and the viscosity of a purely Newtonian component (solvent). For the relaxation time, the value 0.02 s is chosen. The value may appear low, but it is of a similar order of magnitude as the residence time of a fluid particle in the calculation domain. By doing so, a visible development of viscoelastic effects is allowed. For both viscosity factors, the values 80 and 20 Pa.s are selected, respectively.

The three-dimensional calculation domain is shown in Figure 4.2: Non-symmetric Tri-lobal Fiber Spinning: Calculation Domain and Boundaries, along with the named selections for all of the boundaries. Note that zone and boundary names are intuitive, and this helps for setting up the case. Other names are certainly possible, preferably as long as you can unambiguously identify the several entities, for limiting the risk of mistakes. A flow rate Q is imposed at the inlet, slip conditions are applied on the die wall, free surface conditions are imposed on the free jet, while a take-up force is imposed at the outlet. It is also important to keep in mind that the geometric shape of the jet is a priori unknown. In the initial configuration, the cross section of the extruded fiber coincides with that of the die, while the jet would be straight.

Figure 4.2: Non-symmetric Tri-lobal Fiber Spinning: Calculation Domain and Boundaries

With SI units, the following numerical values are selected for the various data involved. A flow rate of 10^-8 m³/s is imposed at the inlet while a take-up force of -10^-3 N is applied at the exit of the computational domain. Eventually, slipping obeying Navier's law is assumed along the die wall, where the coefficient is set to 10⁷ Pa.s/m

This example demonstrates a simple simulation for the shaping of a rubber tire. A relatively simple geometric model is considered for the rubber assembly and for the mold. The assembly consists of four layers: the reinforced carcass, the sidewalls, the reinforcement belts, and the tread. For all four layers, a simple rheological model is considered for the rubber matrix. The carcass is reinforced with fabric cords while the reinforcement belts are merged into a single belt reinforced with two sets of wires. For modeling the fabric cords and wires, an orthotropic fluid model is used.

The concept of reinforced or orthotropic fluid model consists of an isotropic fluid matrix, which is reinforced by means of one or several systems of cords or wires. This reinforcement structure confers a specific orthotropic viscosity to the matrix: since the deformation is hindered along the direction of orthotropy. When deformations of the fluid system are significant, it may be necessary to update the orientation of the reinforcement system accordingly.

For the present purpose, tread patterns are omitted, and a simple mold geometry is proposed to enable a relatively fast simulation for a complex industrial application. In Figure 4.5: Problem Description: The Open Mold (a) and the Initial Assembly (b), both mold parts and the assembly are displayed in their respective initial configurations.. A cross section is made in the assembly for easily identifying the four layers considered. It is here assumed that the assembly is completed before being placed into the mold.

Figure 4.5: Problem Description: The Open Mold (a) and the Initial Assembly (b)

The mold and assembly are displayed with different scales. The four layers presently considered are the carcass (grey), the sidewalls (orange), the reinforcement belts (blue) and the tread (green).

In terms of reinforcements, fabric cords are embedded into the carcass along the axial direction. Two sets of wires are considered in the reinforcement belts with a well-defined initial relative angle. During shaping, the distribution of relative angle will change along with the deformation of the assembly. This mechanism is often referred to as pantographing.

In the present application, the outer diameter of the tire, the height and the width are of 736, 155 and 218 mm, respectively. Following the ETRTO norm (European Tyre and Rim Technical Organization) used in the rubber tire industry, the present tire model would be assigned the classification number 215/70R17.

For computational reasons, only a quarter of the geometry shown in Figure 4.5: Problem Description: The Open Mold (a) and the Initial Assembly (b) is considered and is displayed in Figure 4.6: Problem Description - One Quarter Geometry.

Figure 4.6: Problem Description - One Quarter Geometry

A quarter of the geometry displays the open mold and the assembly, both mold and assembly are displayed with the same scale.

For all four layers considered here, the same rubber matrix is selected. Since the primary focus is the prediction of pantographing that occurs during tire molding, a simple Newtonian model is purposely selected for describing the fluid behavior. The rubber is characterized by a constant (Newtonian) viscosity of 10⁵ Pa.s and a density of 1200 kg/m³.

The fabric cords in the carcass are initially oriented along the axis of the cylinder, and a reinforcement magnitude of 10⁷ Pa.s is considered. The sets of cords in the reinforcement belts have an initial orientation of +40° and ‑40° with respect to the traveling direction, respectively. A reinforcement magnitude of 10⁷ Pa.s is also considered.

The process starts by shaping a pre-tire: this is performed by applying an axial motion to both extremities of the assembly via an appropriate time-dependent velocity boundary condition. For this, velocities of ‑0.028 and +0.028 m/s are assigned respectively at the top and at the bottom ring, and which linearly decreases to 0 within the time interval [2.95, 3.05].

Upon completion of this phase, the pre-tire is virtually inserted into the mold, which is then closed. Both molds are initially at a distance of 0.4 m with each other. Mold motion is imposed within the time interval [3.5, 4.05], with velocities of ‑0.4 and +0.4 m/s assigned respectively to the upper and lower mold within the time interval [3.55, 4.0] and with a linear variation from and to 0, respectively within the intervals [3.5, 3.55] and [4.0, 4.05]. For the contact treatment, default values of 10¹² are kept for the penalty and slipping coefficients while the penetration accuracy is set to 10^‑3 m.

Together with these velocity conditions, a slight pre-blowing pressure is applied on the inner surface of the assembly to prevent possible buckling. Subsequently the blowing pressure is applied in order to shape the tire. More precisely, a pre-blowing pressure of 20,000 Pa is applied until time t=2.5 s. During the time interval [2.6, 4.7], there is zero pressure and eventually, the shaping pressure increases from 0 up to 7 10⁵ Pa within the time interval [4.7, 5.8], and keeps this value afterwards.

From the modeling point of view, four sub-domains are defined for the layers of the assembly, and two additional domains are created for the molds. Interface conditions are defined between successive layers of the assembly. Free surface boundary conditions with a time-dependent inflation pressure are considered for the inner surface of the assembly, while free surface conditions with contact are considered for the outer surface. At both extremities of the assembly, half of the area is treated as a free surface, while a time-dependent velocity is imposed on the other half.

The initial calculation domain for the assembly is discretized by means of 16000 elements (all layers considered). Both molds are discretized by means of about 6000 elements each. Since relatively large deformations are expected, adaptive meshing will be invoked after each sequence of 5 steps, this is set by default. A minimum element size of 0.01 m is selected.

IF(time <= 2.95, -0.028, 
    IF(time >= 3.05, 0., 
        -0.028+(0.+0.028)/(3.05-2.95)*(time-2.95)
      )
  )

IF(time <= 2.95, 0.028, 
    IF(time >= 3.05, 0., 
        0.028+(0.-0.028)/(3.05-2.95)*(time-2.95)
      )
  )

IF( time <= 3.5, 0., 
    IF( time <= 3.55, 0.+(-0.4-0.)/(3.55-3.5)*(time-3.5), 
        IF( time <= 4., -0.4,
            IF( time <= 4.05, -0.4+(0.+0.4)/(4.05-4.)*(time-4.),
                0.
              )
          )
      )
  )

IF( time <= 3.5, 0., 
    IF( time <= 3.55, 0.+(0.4-0.)/(3.55-3.5)*(time-3.5), 
        IF( time <= 4., 0.4,
            IF( time <= 4.05, 0.4+(0.-0.4)/(4.05-4.)*(time-4.),
                0.
              )
          )
      )
  )

IF( time <= 2.5, 0.2e5, 
    IF( time <= 2.6, 0.2e5+(0.-0.2e5)/(2.6-2.5)*(time-2.5), 
        IF( time <= 4.7, 0.,
            IF( time <= 5.8, 0.+(7.e5-0.)/(5.8-4.7)*(time-4.7),
                7.e5
              )
          )
      )
  )

This example illustrates a non-isothermal flow of a Newtonian liquid using a non-conformal mesh technique. This technique allows you to create a connected simulation domain out of geometrically coincident but disconnected parts. This approach facilitates the meshing of the geometry as different parts can be meshed independently.

Figure 4.10: Sub-domains for the 3D Non-conformal Mesh

The geometry representing the three-dimensional calculation domain is depicted in Figure 4.10: Sub-domains for the 3D Non-conformal Mesh and consists of two disconnected sub-domains: the predie (in blue) and the die (in orange).

Figure 4.11: Boundaries for the 3D Non-conformal Mesh

The different named selections and boundaries are shown in Figure 4.11: Boundaries for the 3D Non-conformal Mesh. The inlet is dark blue, the wall is light blue, the predieexit is yellow, the wall2 boundary is light orange, the dieexit in green, and the mesh is visible on dieentry Note that zone and boundary names are intuitive, and this helps for setting up the case.

Figure 4.12: Non-conformal Mesh on the Exit of the Predie and on the Entry of the Die

Figure 4.12: Non-conformal Mesh on the Exit of the Predie and on the Entry of the Die displays the mesh on the exit of the predie (in yellow) and on the entry of the die (in thick blue), clearly demonstrating a non-conformal nature.

A flow rate Q is imposed at the inlet, no slip conditions are applied on the predie and die walls, an outlet is defined on the die outlet.

With SI units, the following numerical values are selected for the various data involved. A flow rate of 3 10^-6 m³/s is imposed at the inlet. A temperature of 373.15 K is imposed at the inlet and along the wall of the predie. A heat flux by convection is imposed along the wall of the die with a transfer coefficient of 50 W/m^2 K and a convection temperature of 353.15 K. A vanishing velocity is imposed along the walls of the predie and the die. The temperature is initialized to 373.15 K on the whole calculation domain.

Foamed polyethylene is extruded to obtain very light insulation parts. Polyethylene with dissolved gas enters through the inlet section and flows through the converging section and the die land. Next, throughout the extrudate, the material is free to deform. The physical foaming process occurs as soon as the mixture leaves the die or at least when the pressure is low enough for allowing the gas bubbles to grow and the dissolved gas to migrate towards the microscopic bubbles.

The Arefmanesh model ([1]) is used for predicting the bubble growth. The model assumes a purely viscous material, and that the time scale of gas diffusion is longer than that of bubble expansion. The model is such that all physics is lumped in a single transport equation that dictates the bubble radius R:

(4–2)

where p_g is the gas pressure and p is the matrix pressure. The parameter may play the role of surface tension and hence prevent a possibly infinite bubble growth. The quantity is a macroscopic viscosity of the foam. The power index a may add some non-linearities to the model but is set to 1 in this example. Eventually, parameter b is a booster if the foaming is not as large as expected: it is also set to 1 in this example.

On the other hand, the density ρ of the mixture is given by:

(4–3)

where ρ_p is the density of the polymer matrix (without blowing agent, that is, with R equivalent to 0), and N is the number of cells per unit volume of gas/polymer mixture. Also, while foaming progresses, the macroscopic viscosity of our mixture decreases since the volume increase results only from gas expansion. Hence, with the bubble growth, the workspace assumes that the zero-shear viscosity of the mixture decreases in the same way as the density:

(4–4)

where is the zero-shear viscosity of the polymer matrix (without blowing agent, that is, with R equivalent to 0).

The model does not necessarily predict an equilibrium bubble size: gas pressure decreases with increasing bubble radius but does not vanish. In actual cases, however, bubbles do not grow infinitely, since only a finite supply of gas is available for their growth, and bubbles attain equilibrium once the gas gets depleted in the polymer. Also, surface tension may eventually dominate over the remaining gas pressure.

In this example, this model is to be used for the determination of die lips shape used for the extrusion of foamed profiles. For this, a 3D profile is considered and is displayed in Figure 4.19: Required Extrudate Profile . The objective is to calculate the die shape lips which will enable the extrusion of the required profile. The primary ingredient of the present example is physical foaming. In such an extrusion flow process, gas is dissolved in the polymer matrix, and will start to expand as soon as the polymer leaves the die, or at least as soon as the pressure in the polymer is low enough for the gas bubbles to expand. In order to achieve this, die lands are usually very short.

Figure 4.19: Required Extrudate Profile

Figure 4.20: Calculation Domain and Boundaries illustrates the initial computational domain. It consists of two subdomains, a short subdomain for the die land and a long subdomain for the extrudate. Four boundaries are considered: the inlet, the die wall, the extrudate surface and the exit of the calculation domain. The corresponding boundary conditions are a volumetric flow rate of 1.4e-5 m³ /s at the inlet, partial slipping along the die wall and obeying Navier's law with a coefficient of 1e7 kg/m² s, free surface conditions on the extrudate surface, and vanishing forces at the exit. Additional geometric information can be useful: the box surrounding the entire computational domain has the dimensions 80.5 mm x 50 mm x 212 mm, the length of the die land is 12 mm, the depth of the left- and right-hand grooves are 12 mm and 9.5 mm, respectively, while the width of the three horns are from left to right of 11 mm, 12 mm, and 12 mm.

Since physical foaming is being considered, a mixture of a polymer matrix and a (massless) blowing agent or gas is also to be considered. The viscosity of the polymer matrix obeys the Bird-Carreau law given by

(4–5)

where = 17450 Pa s, = 1 s and n = 0.45. Also, the density ρ_p of the polymer matrix is 920 kg/m³. Physical foaming originates from the inflation of a sufficiently large number of initially tiny gas bubbles under a pressure higher than the surrounding pressure in the matrix. One can assume that 2000 gas bubbles with an initial radius of 0.027 mm are injected into each volume unit (mm³) of the matrix (leading to the initial astronomic number of 2e12 cells per m³).

Figure 4.20: Calculation Domain and Boundaries

This example uses the Extrusion template, with the goal set to Determine die lip shape, in order to generate most of the objects needed for the simulation. For the fluid material, you should specify the density and the viscosity law of the polymer matrix, in addition to the foaming properties described above (initial gas radius, initial gas pressure, number of cells, etc.).

Initially, for the fluid zone, you specify the fluid domain and activate the foaming.

Next, you specify the boundary conditions as for any extrusion simulation. Note that you should also provide the initial bubble radius at the inlet condition. In the extrudate-exit object, all the edges have already been fixed (as an effect of the template's goal (of determining the die lip shape).

From there, you must update data for the mesh deformations (for the constant die section and extrudate) and delete the adaptive-die-section object (created by the template, but not required in this example).

Lastly, you should also activate the convergence strategies in order to handle all non-linearities present in this simulation: free surfaces and moving interfaces, viscosity and slip, and foaming.

This example demonstrates the simulation of the isothermal filling of a simple 2D part using the Volume of Fluid (VOF) technique. As a reminder, this technique offers a means of simulating a fluid flow with one or several free surfaces and is popular for injection and filling-type problems. The VOF technique uses a time-dependent approach, which is applied on a fixed mesh, that is, the kinematic condition is not tracked, and the simulation domain is not deformed or remeshed.

Theoretically, a drawback of such a technique is that the location where the zero-force condition is applied does not correspond to a region on which natural boundary conditions apply. Consequently, additional approximations are required, beyond those typically employed as part of the finite element method. Because of the explicit nature of VOF algorithms, a so-called Courant type of limitation always occurs, and the time step is limited to a fraction of the time it takes to transport information through an individual element (or cell). It is common to require hundreds of time steps to compute an entire simulation. Because each of these time steps are performed on a fixed domain with an inexpensive numerical technique, a VOF model can still require less CPU time than a simulation involving moving or deforming fluid domain, such as the Arbitrary Lagrangian-Eulerian (ALE) approach. The VOF model implemented here is intrinsically more robust than the ALE approach when modeling free surfaces that merge, separate, or are convoluted, and allows you to simulate problems that a deformation-based technique simply could not attack (for example, a complex cavity filling).

The volume of fluid method tracks a liquid fluid on a domain using material property variable, namely the fluid fraction. This variable is used to identify where the fluid is present, and is governed by a transport equation.

In the current example, consider the following material properties for the polymer, with ρ = 1000 kg/m³; η = 1000 Pa s. Inertia is taken into account. The geometry (see Figure 4.25: Geometry of the Test Sample ) is similar to a test sample generally used in traction tests. The width is 2cm and the total length is 20.5 cm. The cavity is empty at the start of the simulation. The inlet of the computation domain is Boundary1 with a volume flow of 1.5e-04 m³/s is imposed. The outlet is Boundary4 where a zero pressure is applied: it corresponds to a vent from where the air escapes from the cavity in the actual process (but not simulated) and from where some polymer will escape in the last stage of filling. The walls are defined by Boundary2 and Boundary3 where one can assume no slippage along the walls. The filling of the sample should last less than 20 seconds. The simulation will stop as soon as the cavity is filled.

Figure 4.25: Geometry of the Test Sample

This example illustrates the Fluid Solid Interaction capability of the Fluent Materials Processing workspace by modeling the deformation of an extrusion die. Some dies do indeed deform, especially when extruding thin polymer sheet.

The geometry is depicted in Figure 4.28: Zones and consists of three zones - diechannel for the fluid in the die, extrudate for the free jet and soliddie for the steel die itself.

Figure 4.28: Zones

The different boundaries are shown in Figure 4.29: Boundaries .

Figure 4.29: Boundaries

A flow rate Q is imposed at the inlet of the die channel, no slip conditions are applied on the wall of the die channel, an extrudate-exit is defined on the end of the free jet, symmetry conditions are imposed on the vertical and the horizontal planes of symmetry and a free border is defined on the steel die surface.

With SI units, the following numerical values are selected for the various data involved.

A flow rate of 3 x 10^-7 m³/s is imposed at the inlet.

The viscosity of the Newtonian fluid is 10000 Pa s.

The Young modulus of the steel die is 2.1 x 10¹¹ Pa, while its Poisson coefficient is 0.3

Note that there is the assumption that inertia and gravity forces can be neglected.

Internal flows of high-viscosity materials are often characterized by important pressure drops and temperature rise, the latter originating from viscous heating. The simulation of such internal flows can therefore be useful, for example, in order to evaluate the stresses that will be applied to a given flow device. This example demonstrates the thermal simulation of the internal flow of a high-viscosity fluid into a distributor with one inlet and four outlets. The pressure field is examined and one will note that the four outlet flow rates will not be identical, despite the apparent symmetry of the flow. Indeed, under well-known conditions, it can reasonably be assumed that the distribution of material is well balanced. However, geometric effects may affect the temperature distribution, and consequently, the ideal balance of material distribution is no longer guaranteed.

Figure 4.36: Example of a Internal Flow

First consider the bifurcations channel as represented in Figure 4.36: Example of a Internal Flow , that consists of one inlet (I) and four outlets (O1 to O4). The box surrounding the geometry has the dimensions 88 x 26 x 4 mm. In particular, the radius of the channel is 2 mm. The path along the central geometric line is identical from the inlet to each outlet. However, when considering the paths close to the wall, differences are found, affecting the global flow, and consequently, the flow rates at the individual outlets.

One can assume that the fluid obeys the behavior of a generalized Newtonian material, and its shear-rate and temperature dependences of the viscosity are respectively described by means of the Bird-Carreau and Arrhenius laws, with the following parameters:

Fluid enters through the inlet at a rate of 1.5x10^‑6 m³/s, and with a temperature of 453.15 K. Standard outflow boundary conditions are selected at all four exits. Slipping is imposed along the wall, and it obeys the asymptotic law characterized by upper shear stress. Temperature dependence is defined on the upper shear stress value and is described by the Arrhenius law. One can write the following relationship for the tangential force as a function of the tangential velocity and the temperature T:

(4–6)

In this relationship, you can identify the asymptotic law for the dependence with respect to the tangential velocity: it actually bounds the tangential stress. You can also see the Arrhenius function for the temperature dependence, that suggests a decrease of the upper tangential stress limit when temperature increases. The following numerical values are selected for the several parameters:

k = 10⁵ Pa, = 0.002 m/s

= 3000 K, and = 453.15 K

This example describes how to setup and solve a multi-layer film casting simulation where the rheology of the fluid layers is governed by multi-mode Double-Convected Pom-Pom models [DCPP] and the thermal effects are considered.

Figure 4.39: Example of a Multi-layer Film Casting

As shown in Figure 4.39: Example of a Multi-layer Film Casting ., the calculation domain is a square whose side length is 1 m. The other quantities are provided in the MKS system of units. As can be seen at first, next to velocity and thickness unknowns, temperature field is also calculated, since entry temperature and cooling are incorporated into the model. Here, it is considered that both fluid layers exit the slit die with an average velocity (V) of 0.1 m/s, and with a temperature (T) of 473.15 K. The thickness of both layers (h₁, h₂) is of 0.002 m and 0.001 m, respectively. Along the outlet section, a normal velocity of 0.8m/s is imposed. To help convergence, an expression is defined to make this velocity to progressively increase from 0.1 to 0.8 m/s.

A specific attribute of the DCPP model is that it involves the simultaneous calculation of macromolecular orientation and stretch. These quantities obey first-order differential equations and thus require boundary conditions at the inlet of the calculation domain, which describe the melt state when leaving the slit die of the film casting process. The present situation considers that the individual layers exhibit a moderate amount of anisotropy (S₁, S₂) and stretch (Λ₁, Λ₂) at the slit die exit.

The rheology of both fluid layers is described respectively with two and three modes. The linear and non-linear data for both modes are listed in Table 4.2: Properties of Both Fluids: Material Parameters for the Multi-mode DCPP Model. For both fluid layers, a purely viscous component () is added to the extra-stress tensor. Since cooling is involved in the present calculation, the temperature dependence of the rheological properties is invoked, as well as other relevant parameters. They are listed in Table 4.3: Properties of Both Fluids: Parameters for Momentum and Energy Equation.

Table 4.2: Properties of Both Fluids: Material Parameters for the Multi-mode DCPP Model

Mode	Layer 1		Layer 1
	1	2	1	2	3
G [Pa]	601	60.1	402	40.2	4.02
[s]	2.1	0.21	2.2	0.22	0.022
[s]	0.81	0.081	0.82	0.082	0.0082
q	4.1	2.1	4.1	2.2	1.2
	0.151	0.251	0.252	0.262	0.272
[Pa s]	20.1		20.2

Table 4.3: Properties of Both Fluids: Parameters for Momentum and Energy Equation

	Layer 1	Layer 1
Density: [kg/m3]	1000	1100
Heat Conductivity: k W/m/°C]	0.5	0.55
Heat Capacity: [J/kg/°C]	2000	2300
Activation Energy / R:	4000	5000
Ref. Temperature: [°C]	473.15	473.15

For performing the numerical simulation, the mesh shown in Figure 4.40: Film Casting: Finite Element Mesh is used, and it is characterized by 180 elements, with small elements in the vicinity of the starting point of the free border.

Figure 4.40: Film Casting: Finite Element Mesh

Streamline upwinding is used for integrating the convective terms in the constitutive and energy equations, as well as for integrating the kinematic equation of the free border. As shown in Figure 4.39: Example of a Multi-layer Film Casting , evolution schemes are applied on the take-up velocity as well as on the heat transfer coefficient. A linear function is applied to the heat transfer coefficient, while a ramp function is applied on the take-up velocity which increases from 0.1 to 0.8:

(4–7)

Stretching and anisotropy boundary conditions

For the case of the DCPP model used within a film casting simulation, care is required when imposing the boundary conditions for the viscoelastic variables at the inlet of the calculation domain. The inlet is identified as the border where the thickness condition is imposed. Here, instead of imposing values for all components of the orientation tensor, anisotropy described in terms of orientation and factor is given, as well as stretching amplitude. Imposing boundary conditions for the viscoelastic unknowns (orientation and stretching) is optional. If conditions are imposed, three data are required:

By default, the anisotropy direction is along the x-axis, the anisotropy factor is 0, while the stretching is 1. This corresponds to a situation where the melt has not undergone deformation in the die before being extended in the film drawing. Actually, the flow in a die is dominated by shear, which may certainly affect the orientation tensor, while stretching will remain close to unity. Therefore, a typical inlet boundary condition for film casting would involve a non-zero anisotropy factor and a unit stretching. You can change this according to your needs.

On the basis of the value entered for the anisotropy direction and anisotropy factors, the value of the orientation tensor S at the inlet can be evaluated and assigned as S_aa=(2s+1)/3, S_bb=S_zz=(1-s)/3 (where a is the selected direction of anisotropy, while b is perpendicular to it in the plane of the film). The component S_xy=0. If s=0 (isotropic), all three components of S are set to 1/3. If s=1 (100% anisotropy), then S_aa=1, while the other components vanish. Any value between 0 and 1 is valid.

For the stretching variable, recommended values should be at least equal to unity. Values lower than unity are of course permitted but would probably not match the physics undergone in the die.

When a multi-layer system is defined, inlet boundary conditions can be defined independently for each layer, as it is already the case for the thickness. In the case of multi-mode model for a given layer, all modes will undergo the same selected inlet conditions.

Stress condition at the inlet of the calculation domain

It is worth mentioning that the solver exhibits better performances when reasonable inlet boundary conditions are selected for the viscoelastic unknowns of the DCPP model in film casting. Some situations have been reported to perform better when stresses are not imposed at the inlet of the calculation domain. In particular, this is the case when the rheology does not involve any additional purely viscous component.