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Slide 1 - Supplementary Conditions Slide 2 - Types of supplementary conditions Slide 3 - Types of supplementary conditions (continued) Slide 4 - Boundary Conditions Slide 5 - Kinematic boundary condition example Slide 6 - Kinematic boundary condition example (continued) Slide 7 - Boundary Conditions (continued) Slide 8 - No-Slip condition - caveats Slide 9 - Boundary Conditions (continued) Slide 10 - Boundary Conditions - inviscid versus viscous fluids Slide 11 - Supplementary Conditions
We have now derived a set of four nonlinear partial-differential equations in four unknowns which govern the conservation of mass and balance of linear momentum for either an inviscid or Newtonian-viscous fluid in a spatial frame of reference.
In order to complete the formulation of a well-posed mathematical problem, we now need to determine the types of supplementary conditions which it will be necessary to specify.
Our governing equations are, in general, functions of the independent variables lowercase bold x and lowercase t.
For inviscid flows governed by the Euler equations, the PDE's are the only first-order in space;
Lowercase rho times capital D lowercase v subscript lowercase i over capital D lowercase t equals rho times lowercase f subscript lowercase i minus lowercase p subscript lowercase i.
For viscous flows governed by the Navier-Stokes equations the PDE's are second-order.
Lowercase rho times capital D lowercase v subscript lowercase i over capital D lowercase t equals lowercase rho times lowercase f subscript lowercase i minus lowercase p subscript lowercase i plus mu times lowercase v subscript lowercase i, comma, lowercase j lowercase j.
Because of the spatial derivatives in these equations, boundary conditions will be required -- these model the interactions between the fluid and adjacent physical systems.
The Navier-Stokes equations, being of higher spatial order than the Euler equations, will require more boundary conditions to formulate a complete problem.
For situations in which the flow is unsteady, initial conditions will also be required -- these specify the state of the flow at some given (arbitrary) instant of time.
Any fluid - If capital F equals 0 is a bounding surface (always consisting of the same fluid particles), then
Capital D capital F over capital D lowercase t equals zero, on capital F equals zero.
This is known as the kinematic boundary condition and expresses the fact a bounding surface is a fluid material surface.
In undergraduate courses, this condition is the familiar no-cross-flow (no-penetration) condition at a solid wall; the above form is more general.
Let's assume we have a bounding surface whose equation is
Lowercase z equals lowercase zeta as a function of lowercase x, lowercase y, and lowercase t, which in turn equals capital F equals lowercase z minus lowercase zeta as a function of lowercase x, lowercase y, and lowercase t, equals zero.
Then
Capital D capital F over capital D lowercase t equals the partial derivative with respect to time, lowercase t, of, open bracket, lowercase z minus lowercase zeta as a function of lowercase x, lowercase y, and lowercase t, closed bracket, plus lowercase u times the partial derivative with respect to lowercase x of, open bracket, lowercase z minus lowercase zeta as a function of lowercase x, lowercase y, and lowercase t, closed bracket, plus lowercase v times the partial derivative with respect to lowercase y of, open bracket, lowercase z minus lowercase zeta as a function of lowercase x, lowercase y, and lowercase t, closed bracket, plus lowercase w times the partial derivative with respect to z of, open bracket, lowercase z minus lowercase zeta as a function of lowercase x, lowercase y, and lowercase t, closed bracket, equals
The negative partial derivative of zeta with respect to time, minus lowercase u times the partial of zeta with respect to lowercase x, minus lowercase v times the partial of zeta with respect to lowercase y, plus lowercase w equals zero.
Therefore, for this case, our kinematic boundary condition may be expressed as
Lowercase w equals the partial derivative of lowercase zeta with respect to time, lowercase t, plus lowercase u times the partial derivative of zeta with respect to lowercase x, plus lowercase v times the partial derivative of zeta with respect to lowercase y, on lowercase z equals the function zeta of lowercase x, lowercase y, and lowercase t, time.
Note that, for a stread-flow case, this would reduce to
Lowercase w equals lowercase u times the partial derivative of zeta with respect to lowercase x, plus lowercase v times the partial derivative of zeta with respect to lowercase y, on the surface lowercase z equals zeta as a function of lowercase x and lowercase y.
and, for the trivial situation with lowercase z equals lowercase c, where lowercase c is a constant, we get, simply,
lowercase w equals zero on lowercase z equals lowercase c.
Viscous fluid - On a solid surface,
Open bracket, lowercase bold v dot lowercase bold t, closed bracket, equals zero.
where the vector t is the tangent vector to the surface (and not the stress vector) and square brackets denote the "jump" in the quantity within across the surface, i.e.,
Open square bracket, open parentheses, closed parentheses, closed square bracket, equals, open parentheses, closed parentheses, superscript plus sign, minus, open parentheses, closed parentheses, superscript minus sign.
[D1]This is just the viscous no-slip condition which states that the tangential speed of the fluid equals that of the solid surface.
The viscous no-slip condition holds for most cases of practical interest.
Exceptions are:
flows of rarefied gases
flows of superfluid helium
flows at porous boundaries
Fluid-fluid interface - On such an interface (in all cases below, lowercase bold t, or lowercase t subscript one, represents the surface-tangent vector).
Open square bracket, lowercase bold v, dot lowercase bold t, closed square bracket, equals zero
This is for the no-slip - viscous fluids.
Capital T subscript lowercase i lowercase j, times lowercase n subscript lowercase j times lowercase t subscript lowercase i equals zero.
This is for continuity of shear stress - viscous fluids.
Capital T subscript lowercase i lowercase j times lowercase n subscript lowercase j times lowercase n subscript lowercase i equals capital K times lowercase sigma.
This is jump in normal stress.
Capital K is the interface curvature and lowercase sigma is the surface tension. We also assume that the surface tension is constant and no other surface forces exist.
We have already remarked that the Euler equations, being of lower spatial order than the Navier-Stokes equations require fewer boundary conditions to be posed.
Do we have a choice of which conditions to impose?
The answer is NO. It turns out that solutions to the Euler equations cannot be used to satisfy the viscous, no-slip condition, requiring that these conditions be abandoned when treating these problems.
We have discussed several types of boundary conditions to be applied to both inviscid and viscous fluids.
Our list is not inclusive, but will suffice for most types of problems we shall encounter in this course.
We have not discussed initial conditions in any detail; these are quite specific to the problem at hand and will be dealt with on a case-by-case basis.
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