Tangents and normals in determining the kinematic of fluid flow

What arw the significances of tangents and normals in determining the kinematic of fluid flow?

what arw the significances of tangents and normals in determining the kinematic of fluid flow?

There is this cool thing called Google. Maybe you've heard of it. If you make the cut, you may be given access to it;):D.



I am not going to go over the calculus of all this, but here is a basic example of flux of the flow field.

Suppose we have a portion of the surface z=1-x^{2}-y^{2}. Call this portion, say, {\sigma}.

Find this portion of the above surface that lies above the xy-plane and say it is oriented upwards. We can find the flux, {\Phi}, of the flow field F(x,y,z)=xi+yj+zk across \sigma.

{\Phi}=\int\int_{R}(F\cdot n)\sqrt{\left( \frac{{\partial}z}{{\partial}x} \right)^{2}+\left( \frac{{\partial}z}{{\partial}y}\right)^{2}+1} \;\ dA

R is the projection of the surface, sigma, on the xy-plane. Since it is oriented upwards, we can write:

{\Phi}=\int\int_{R}F\cdot \left(-\frac{{\partial}z}{{\partial}x}i-\frac{{\partial}z}{{\partial}y}j+k \right)dA

Using polar coordinates:

=\int_{0}^{2\pi}\int_{0}^{1}(r^{2}+1)r \;\ drd{\theta}

=\frac{3\pi}{2}

When we canceled the radical, that comes from a theorem.

\int\int_{\sigma}F\cdot n dS=\pm\int\int_{R}F\cdot {\nabla}G \;\ dA

where the + is used if sigma has positive orientation and - if it is negative orientation.

This is a little bit oin the topic, which barely scratches the surface.

Hope it helps a little. Calculus texts will probably touch on the topic in the latter sections under surface integrals, flux, divergence theorem, Stokes' Theorem, etc.

But try googling or finding a book on the topic. Your question is painted with a broad brush.

There is this cool thing called Google. Maybe you've heard of it. If you make the cut, you may be given access to it.
Believe me I have try it and thanks for your answer but it not the answer for my question ;)

The kinematics of a flow describes the motion of the fluid without taking
Into account the forces that cause this motion

Normals show direction because they are perpendicular to the tangent at a point.

Here is a link to a power point that may help a little:

http://www.esm.vt.edu/~dtmook/AOE5104_ONLINE/Class%20Notes/07_Class_StaticsKinematics.pdf

This site is being rather slow this morning, so I am going to log off for a while.