# Shell Equation Properties and Models¶

In this section we list all “material-region” specific models and properties associated with GOMA’s extensive shell equation capability. Currently we have specialized shell equations for Reynolds lubrication flow (lubp), open Reynolds film flow (shell_film_H), energy (shell_energy, convection and diffusion, coupled with lubrication), thin porous media (closed cell and open cell), melting and phase change and more. While many of these cards are actual material properties, most are geometry and kinematic related. The most appropriate place for these cards are region/ material files because they are actually boundary conditions and related parameters which arise from the reduction of order (integration through the thin film). For more information, please see the shell-equation tutorial (GT-036).

## Upper Height Function Constants¶

Upper Height Function Constants = {model_name} <floatlist>


### Description / Usage¶

This card takes the specification of the upper-height function for the confined channel lubrication capability, or the lub_p equation. This function specifies the height of the channel versus distance and time. Currently three models for {model_name} are permissible:

 CONSTANT_SPEED This model invokes a squeeze/separation velocity uniformly across the entire material region, viz. the two walls are brought together/apart at a constant rate. This option requires two floating point values the separation velocity (rate) in units of length/time the initial wall separation in units of length An OPTIONAL parameter which scales the addition of an external field called “HEIGHT” which is read in using the External Field or External Pixel Field capabilities. If this field is present, the value of it is added to the height calculated with this model. ROLL_ON This model invokes a squeeze/separation velocity in a hinging-motion along one boundary. The model is best explained with the figure in the technical discussion section. The equation for the gap h as a function of time and the input parameters (floats) is as follows: is x0 in units of length is hlow in units of length is h Δ, in units of length is the verticle separation velocity (if negative then squeeze velocity) in units of length/time is the length of the plate, L.
 ROLL This model is used for a roll coating geometry. This option requires 8 floats: x-coordinate of origin, L. y-coordinate of orgin, L. z-coordinate of origin, L. Direction angle 1 of rotation axis Direction angle 2of rotation axis Direction angle 3of rotation axis rotation speed L/t. FLAT_GRAD_FLAT This model used two arctan functions to mimic a flat region, then a region of constant slope, then another flat region. The transitions between the two regions are curved by the arctan function. This currently on works for changes in the x direction. This option requires five floating point values x location of the first transition (flat to grad) height of the first flat region x location of the second transition (grad to flat) height of the second flat region parameter controlling the curvature of the transitions POLY_TIME This time applies a time-dependent lubrication height in the form of a polynomial. It can take as many arguments as GOMA can handle, and the resulting height function is value of Ci
 JOURNAL This model simulates a journal bearing. It is intended to be run on a cylindrical shell mesh aligned along the z axis and centered at (0,0). It could be extended to be more flexible, but this is all it is currently capable of. The height is defined by h(θ ) = C(1+ε cos(0)) Where C is the mean lubrication height and is the eccentricity of the two cylinders, with the smallest gap in the –y direction. C ε EXTERNAL_FIELD Not recognized. Oddly, this model is invoked with the extra optional float on the CONSTANT_SPEED option.
External Field = HEIGHT Q1 name.exoII (see this card)


### Examples¶

Following is a sample card:

Upper Height Function Constants = CONSTANT_SPEED {v_sq = -0.001} {h_i=0.001}


This results in an upper wall speed of 0.001 in a direction which reduces the gap, which is initial 0.001.

### Technical Discussion¶

The material function model ROLL_ON prescribes the squeezing/separation motion of two non-parallel flate plates about a hinge point, as shown in the figure below.

## Lower Height Function Constants¶

Lower Height Function Constants = {model_name} <floatlist>


### Description / Usage¶

This card takes the specification of the lower-height function for the confined channel lubrication capability, or the lub_p equation. This function specifies the height of the channel versus distance and time. Currently three models for {model_name} are permissible:

 CONSTANT_SPEED This model invokes a squeeze/separation velocity uniformly across the entire material region, viz. the two walls are brought together/apart at a constant rate. This option requires two floating point values the separation velocity (rate) in units of length/time the initial wall separation in units of length An OPTIONAL parameter which scales the addition of an external field called “HEIGHT” which is read in using the External Field or External Pixel Field capabilities. If this field is present, the value of it is added to the height calculated with this model. ROLL_ON This model invokes a squeeze/separation velocity in a hinging-motion along one boundary. The model is best explained with the figure in the technical discussion section. The equation for the gap h as a function of time and the input parameters (floats) is as follows: is x0 in units of length is hlow in units of length is h Δ, in units of length is the verticle separation velocity (if negative then squeeze velocity) in units of length/time is the length of the plate, L.
 ROLL This model is used for a roll coating geometry. This option requires 8 floats: x-coordinate of origin, L. y-coordinate of orgin, L. z-coordinate of origin, L. Direction angle 1 of rotation axis Direction angle 2 of rotation axis Direction angle 3 of rotation axis rotation speed L/t. TABLE {LINEAR | BILINEAR} [integer2] [FILE = filenm] Please see discussion at the beginning of the material properties Chapter 5 for input description and options. Most likely character_string1 will be LOWER_DISTANCE This option is good for inputing table geometry versus distance. Specifically, an arbitrary lower height function model is input as a function of the x-direction coordinate of the Lower Velocity Function model. This option in turn requires the use of SLIDER_POLY_TIME lower velocity function model. See example below.

### Examples¶

Following is a sample card:

Lower Height Function Constants = CONSTANT_SPEED {v_sq = -0.001} {h_i=0.001}


This results in an lower wall speed of 0.001 in a direction which reduces the gap, which is initial 0.001.

In another example:

Lower Height Function Constants = TABLE 2 LOWER_DISTANCE 0 LINEAR FILE=shell.dat

where shell.dat is a table with 2 columns, the first the position, the second the height.

### Technical Discussion¶

The material function model ROLL_ON prescribes the squeezing/separation motion of two non-parallel flate plates about a hinge point, as shown in the figure below.

## Upper Velocity Function Constants¶

Upper Velocity Function Constants = {model_name} <floatlist>


### Description / Usage¶

This card takes the specification of the upper-wall velocity function for the confined channel lubrication capability, or the lub_p equation. This function specifies the velocity of the upper channel wall as a function of time. Currently two models for {model_name} are permissible:

 CONSTANT This model invokes a squeeze/separation velocity uniformly across the entire material region, viz. the two walls are brought together/apart at a constant rate. This option requires two floating point values is the velocity component in the x-direction. L/t. is the velocity component in the y-direction. L/t is the velocity component in the z-direction. L/t (NOTE: this is usually taken as zero as it is set in the Upper Wall Height Function model) ROLL This model invokes a wall velocity which corresponds to a rolling-motion. This model takes nine constants ???? : Roll radius, L. x-coordinate of axis origin, L. y-coordinate of axis orgin, L. z-coordinate of axis origin, L. Direction angle 1 of rotation axis Direction angle 2of rotation axis Direction angle 3of rotation axis Squeeze rate rotation rate TANGENTIAL_ROTATE his model allows a velocity that is always tangential to a shell surface, not necessarily aligned along the coordinate directions. It requires three specifications. First, a vector (v) that is always non-colinear to the normal vector of the shell must be specified. This is used to make unique tangent vectors. The last two specifications are the two tangential components to the velocity. The first velocity is applied in the direction of t1 = v×n. The second velocity is then applied in the t = t ×n direction. vx vy vz velocity in the t1 direction velocity in the t2 direction CIRCLE_MELT Model which allows a converging or diverging height that is like a circle. Also works for melting. - x-location of the circle center (circle is in x-y plane) - radius of circle - minimum height of circle

### Examples¶

Following is a sample card:

Upper Velocity Function Constants = CONSTANT {v_x= -0.001} {vy=0.00} {vz=0}


This card results in an upper wall speed of -0.001 in the x-direction which is tangential to the substrate, thus generating a Couette component to the flow field.

### Technical Discussion¶

For non-curved shell meshes, most of the time they are oriented with the x-, y-, or zplane. This card is aimed at applying a tangential motion to that plane, and so one of the three components is usually zero.

## Lower Velocity Function Constants¶

Lower Velocity Function Constants = {model_name} <floatlist>


### Description / Usage¶

This card takes the specification of the Lower-wall velocity function for the confined channel lubrication capability, or the lub_p equation. This function specifies the velocity of the Lower channel wall as a function of time. Currently two models for {model_name} are permissible:

 CONSTANT This model invokes a squeeze/separation velocity uniformly across the entire material region, viz. the two walls are brought together/apart at a constant rate. This option requires two floating point values is the velocity component in the x-direction. L/t is the velocity component in the y-direction. L/t is the velocity component in the z-direction. L/t (NOTE: this is usually taken as zero as it is set in the Lower Wall Height Function model) SLIDER_POLY_TIME This model implements a spatially-uniform velocity in the x-direction that is specified as a polynomial in time. The value of time may be scaled by a given scaling factor and the polynomial may have an unlimited number of terms. is the time scaling factor are the coefficients in front of the t^(i-2) term
 ROLL This model invokes a wall velocity which corresponds to a rolling-motion. This model takes nine constants ???? : Roll radius, L. x-coordinate of axis origin, L. y-coordinate of axis orgin, L. z-coordinate of axis origin, L. Direction angle 1 of rotation axis Direction angle 2of rotation axis Direction angle 3of rotation axis Squeeze rate. rotation rate TANGENTIAL_ROTATE This model allows a unique specification of tangential motion in a lubrication shell element. Previous implementations allowed specification only in terms of coordinate direction, but this option can be used to rotate a cylinder. Five floats are required x-comnponent of a vector tangential to the shell. This vector must never be normal to the shell. It is then projected onto the shell. y-comnponent of a vector tangential to the shell. z-comnponent of a vector tangential to the shell. U1, or scalar speed of wall velocity in a direction determined by the cross product ot the tangent vector and the normal vector to the shell. (L/t) U2 scalar speed component in direction normal to U1. (L/t)

### Examples¶

Following is a sample card:

Lower Velocity Function Constants = CONSTANT {v_x= -0.001} {vy=0.00} {vz=0}


This card results in an Lower wall speed of -0.001 in the x-direction which is tangential to the substrate, thus generating a Couette component to the flow field.

### Technical Discussion¶

For non-curved shell meshes, most of the time they are oriented with the x-, y-, or zplane. This card is aimed at applying a tangential motion to that plane, and so one of the three components is usually zero.

## Upper Contact Angle¶

Upper Contact Angle = {model_name} <floatlist>


### Description / Usage¶

This card sets contact angle of the liquid phase on the upper-wall for the two-phase capability in the lub_p equation (viz. when using the level-set equation to model the motion of a meniscus in a thin gap, where the in-plan curvature is neglected. Currently one model {model_name} is permissible:

 CONSTANT This model is used to set a constant contact able of the the free surface at the upper wall. Contact angle of less than 90 degrees is considered as nonwetting with respect to the heavier level-set phase. Only one floating point value is required. is the contact angle in degrees.

### Examples¶

Following is a sample card:

Upper Contact Angle = CONSTANT 180.


This card results in an upper wall contact able to 180 degrees, which is perfectly wetting. If the lower wall is given the same angle, then the capillary pressure jump will go as 2/h, where h is the gap.

## Lower Contact Angle¶

Lower Contact Angle = {model_name} <floatlist>


### Description / Usage¶

This card sets contact angle of the liquid phase on the lower-wall for the two-phase capability in the lub_p equation (viz. when using the level-set equation to model the motion of a meniscus in a thin gap, where the in-plan curvature is neglected. Currently one model {model_name} is permissible:

 CONSTANT This model is used to set a constant contact able of the the free surface at the lower wall. Contact angle of less than 90 degrees is considered as nonwetting with respect to the heavier level-set phase. Only one floating point value is required. is the contact angle in degrees.

### Examples¶

Following is a sample card:

Lower Contact Angle = CONSTANT 180.


This card results in an lower wall contact able to 180 degrees, which is perfectly wetting. If the lower wall is given the same angle, then the capillary pressure jump will go as 2/h, where h is the gap.

## Lubrication Fluid Source¶

Lubrication Fluid Source = {model_name} <floatlist>


### Description / Usage¶

This card sets a fluid mass source term in the lub_p equation. Can be used to specify inflow mass fluxes over the entire portion of the lubrication gap in which the lub_p equation is active (over the shell material). This flux might be the result of an injection of fluid, or even melting. Currently two models {model_name} are permissible:

 CONSTANT This model is used to set a constant fluid source in units of velocity. Only one floating point value is required. is the velocity of the fluid source. MELT This model is used to set fluid source in units of Velocity which results from an analytical model of lubricated melt bearing flow due to Stiffler (1959). Three floating point values are required. is load on the slider in units of pressure is the Stiffler delta factor. Unitless but depends on the aspect ratio. is the length of the slider in the direction of the motion.

### Examples¶

Following is a sample card:

Lubrication Fluid Source = CONSTANT 180.


## Lubrication Momentum Source¶

Lubrication Momentum Source = {model_name} <floatlist>


### Description / Usage¶

This card sets a fluid “body force per unit volume” source term in the lub_p equation. This capability can be used to specify a force field over the entire shell area (over the shell material).. Currently two models {model_name} are permissible:

 CONSTANT THIS MODEL NOT IMPLEMENTED AS OF 11/11/2010. This model is used to set a constant fluid momentum source in units of force per unit volume. Only one floating point value is required. is the fluid momentum source in F/L^3. JXB This model is used to set fluid momentum source in units of force per unit volume which comes from externally supplied current density J field and magnetic B fields. These fields are suppled with the external field capability in Goma in a component wise fashion. Please consult the technical discussion below. is scale factor which may be used for nondimensionalization. Typically this is set to 1.0.

### Examples¶

Following is a sample card:

Lubrication Momentum Source = JXB 1.


### Technical Discussion¶

The two vector fields J, the current flux, and B, the magnetic induction, must be supplied to Goma in order to activate this option. At present, these fields must be supplied with the External Field cards, which provide the specific names of nodal variable fields in the EXODUS II files from which the fields are read. The three components of the J field must be called JX_REAL, JY_REAL, and JZ_REAL. Likewise the B field components must be called BX_REAL, BY_REAL, and BZ_REAL. These names are the default names coming from the electromagnetics code like Alegra. Because of the different coordinate convention when using cylindrical components, the fields have been made compatible with those arising from TORO II. It is the interface with TORO that also makes the Lorentz scaling (lsf) necessary so that the fixed set of units in TORO (MKS) can be adjusted to the user-selected units in Goma.