What is the Free Boundary Conditions?
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I am trying to simulate liquid film evaporation with free boundary conditions (in cartesian coordinates) and my boundary conditions are thus:
∂h∂x=0,(1)" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂h∂x=0,(1)∂h∂x=0,(1)
∂2h∂x2=0,(2)" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂2h∂x2=0,(2)∂2h∂x2=0,(2)
∂3h∂x3=0.(3)" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂3h∂x3=0.(3)∂3h∂x3=0.(3)
However, I need only two of the above three conditions to satisfy my 4th order non-linear partial differential equation for film thickness, which looks something like.
∂h∂t+h3∂3h∂x3+...=0" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂h∂t+h3∂3h∂x3+...=0∂h∂t+h3∂3h∂x3+...=0
My question is: what does a combination of 1st and 2nd derivative conditions mean and what does a combination of 2nd and 3rd derivatives mean?
If I apply (1) and (2), does it mean that slope and curvature are zero and if I apply (1) and (3), does it mean that slope and shear stress are zero (from analogies of bending beams etc.)
∂h∂x=0,(1)" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂h∂x=0,(1)∂h∂x=0,(1)
∂2h∂x2=0,(2)" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂2h∂x2=0,(2)∂2h∂x2=0,(2)
∂3h∂x3=0.(3)" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂3h∂x3=0.(3)∂3h∂x3=0.(3)
However, I need only two of the above three conditions to satisfy my 4th order non-linear partial differential equation for film thickness, which looks something like.
∂h∂t+h3∂3h∂x3+...=0" role="presentation" style="box-sizing: inherit; margin: 0px; padding: 0px; border: 0px; font-style: normal; font-variant: inherit; font-weight: normal; font-stretch: inherit; line-height: normal; font-family: inherit; font-size: 15px; vertical-align: baseline; display: inline; text-indent: 0px; text-align: center; text-transform: none; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;">∂h∂t+h3∂3h∂x3+...=0∂h∂t+h3∂3h∂x3+...=0
My question is: what does a combination of 1st and 2nd derivative conditions mean and what does a combination of 2nd and 3rd derivatives mean?
If I apply (1) and (2), does it mean that slope and curvature are zero and if I apply (1) and (3), does it mean that slope and shear stress are zero (from analogies of bending beams etc.)
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What is the Free Boundary Conditions?
❱ In a free boundary problem what is known or given to you is the PDE that needs to solve, boundary data, and an additional `free boundary condition’, but what is not known.
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