As I know, boundary-layer is formulated for semi-infinite flat plate with zero incidence (the Blasius ode). If the incidence of plate is not zero then it is named wedge flow, mathematically modeled as the Falkner-Skan ode (taking into account wedge angle β).
However, my concern is Navier-Stokes equations for slanted plate (?).
How do we modify Navier-Stokes equations If we slant the plate (over which the fluid flows)?.
Also it would be cool, if you hint to derivation of Falkner-Skan based on Blasius. I only want to know, how velocity components are transformed from flat plate to slanted plate (plate with incidence).
@Filippo Maria Denaro.
Standard Navier-Stokes equations for 2D flows, are originally formulated in terms of velocity partial derivatives in (x-y) as independent spatial coordinates.
By modify, I mean how to transform from the Cartesian (x-y) coordinate, to normal-tangential (n-t) coordinate, where n is the coordinate normal to the slanted plate and t is the coordinate along the slanted plate. Fluid is assumed to flow along t.
plate incidence or slant angle is θ.
What you wrote is not correct. The Navier-Stokes equations are formulated in a general way, valid for any reference system. For example, the density equation
d rho/dt + Div (rho * v) =0
is valid in general and can be specifyed for Cartesian, cilindric, spherical reference system.
You problem is straightforward, you have only a rotation of an angle of the x-y system in t-n' where you can write the equations. For example, the unit vector in tangential direction is t=i tx + j ty.
Consider that a very simple way is to use the x-y system and set an angle for the velocity vector at the inlet
@Filippo Maria Denaro. Am I allowed to use Cartesian Navier-Stokes equations for flat plate, and only use the transformation, gn=-g cos θ and gt=-g sin θ, to study the effect of slope angle θ?.
I have solved the Navier-Stokes equations for viscous flow over flat plate, as thoroughly scalar (not vector field). Now I need to get the solution, if the plate is slanted with the angle θ.
Look at the image, i've attached to this comment.
Question: Am I allowed to use the Cartesian (x-y) Navier-Stokes equations of the case a), for the case c), only with transforming, the gravity in n and t directions as; gn=gy.new=-g cos θ and gt=gx.new=-g sin θ.
It means is it allowed to generalize the Cartesian (x-y) Navier-Stokes equations of the case a), to the case b) and then c), only with transforming the gravity and without any change in velocity derivative components of the case a). It means only transforming gravity, then viewing t as x, and n as y.
Are the velocities also required to be transformed in rotation?.
Dear Saeb AmirAhmadi Chomachar,
I think your problem is same as boundary layer flow past an inclined plate problem. It is one of the classical problems. You can read "Numerical Investigation on Boundary Layer Flow of a Nanofluid Towards an Inclined Plate with Convective Boundary: Boungiorno Nanofluid Model" and the references there for further details.
@Basavarajappa Mahanthesh. Thank you.
I am interested in the reference you cited, but unfortunately, the paper is not open access. Is there any way to get the paper online?.
Article Numerical Investigation on Boundary Layer Flow of a Nanoflui...
@Basavarajappa Mahanthesh. Thanks a lot. It is very helpful. Regards, Saeb.
I'm ready to collaborate with if you want by solving your présent pb by spectral method.
@Abderrahim Wakif. Thank you for the interest. I am not looking for spectral method, as a solution. Regards, AmirAhmadi.
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