Surface gradient

In vector analysis , the surface gradient denotes a differential operator similar to the gradient . The gradient is formed along a surface.

definition

For an area in a scalar field , the surface gradient is defined as ${\displaystyle S}$ ${\displaystyle u}$

${\displaystyle \nabla _{S}u=\nabla u-(\mathbf {\hat {n}} \cdot \nabla u)\mathbf {\hat {n}} }$.

The normal unit vector denotes the surface. The surface gradient thus represents the usual gradient without the portion normal to the surface. It is therefore tangential to the surface. The surface gradient can also be interpreted as an orthogonal projection of the gradient onto the surface. ${\displaystyle \mathbf {\hat {n}} }$

In tensor analysis , the surface gradient is often defined as:

${\displaystyle \nabla _{s}=P\cdot \nabla }$

with the area projection tensor . ${\displaystyle P}$

However, this gradient can also be defined more generally.

Be a scalar field . If the tangential vector field with a scalar product is applied to any vector , then the surface gradient of a scalar field is defined as follows: ${\displaystyle \phi }$ ${\displaystyle \nabla _{(\sigma )}\phi }$ ${\displaystyle \mathbf {c} }$

${\displaystyle \nabla _{(\sigma )}\phi \cdot \mathbf {c} :=\lim _{s\to 0}{\frac {1}{s}}\left\{\phi \left(y^{1}+sc^{1},y^{2}+sc^{2}\right)-\phi \left(y^{1},y^{2}\right)\right\}}$

Let be a spatial field and be a second level tensor field . Then this tensor field transforms any tangential vector field in all points on the surface and the surface gradient of a spatial field is defined as follows: ${\displaystyle \mathbf {v} }$ ${\displaystyle \nabla _{(\sigma )}\mathbf {v} }$ ${\displaystyle \mathbf {c} }$

${\displaystyle \nabla _{(\sigma )}\mathbf {v} \cdot \mathbf {c} :=\lim _{s\to 0}{\frac {1}{s}}\left[\mathbf {v} \left(y^{1}+sc^{1},y^{2}+sc^{2}\right)-\mathbf {v} \left(y^{1},y^{2}\right)\right]}$

Let be a tensor field 2nd level and its a tensor field 3rd level. Then this tensor field transforms any tangential vector field in all points on the surface and the surface gradient of a tensor field 2nd level is defined as follows: ${\displaystyle \mathbf {A} }$ ${\displaystyle \nabla _{(\sigma )}\mathbf {A} }$ ${\displaystyle \mathbf {c} }$

${\displaystyle \nabla _{(\sigma )}\mathbf {A} \cdot \mathbf {c} :=\lim _{s\to 0}{\frac {1}{s}}\left[\mathbf {A} \left(y^{1}+sc^{1},y^{2}+sc^{2}\right)-\mathbf {A} \left(y^{1},y^{2}\right)\right]}$

Individual evidence

1. ↑ R. Shankar Subramanian, Boundary Conditions in Fluid Mechanics (PDF; 34 kB).
2. ↑ Efstathios Michaelides, Clayton T. Crowe, John D. Schwarzkopf: Multiphase Flow Handbook . 2nd Edition. CRC Press, 2016, ISBN 978-1-4987-0100-6 , pp. 940 .
3. ↑ John C. Slattery Leonard Sagis Eun-Suok Oh: Interfacial Transport Phenomena . Springer International Publishing AG, Boston, ISBN 978-0-387-38442-9 , pp. 624 ff .
4. ↑ John C. Slattery Leonard Sagis Eun-Suok Oh: Interfacial Transport Phenomena . Springer International Publishing AG, Boston, ISBN 978-0-387-38442-9 , pp. 647 ff .
5. ↑ John C. Slattery Leonard Sagis Eun-Suok Oh: Interfacial Transport Phenomena . Springer International Publishing AG, Boston, ISBN 978-0-387-38442-9 , pp. 660 ff .