Introduction  (Page 3/5)

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Curves, surfaces, and volumes

We will be dealing with regions of space, V , having volume that may be bound by surfaces, S, having area. Regions of the surface may be bound by a closed curve, C , having length.

Surfaces are defined by one relationship between the spatial coordinates.

${x}^{3}=f\left({x}^{1},{x}^{2}\right),\phantom{\rule{0.277778em}{0ex}}or\phantom{\rule{0.277778em}{0ex}}F\left({x}^{1},{x}^{2},{x}^{3}\right)=0,\phantom{\rule{0.277778em}{0ex}}or\phantom{\rule{0.277778em}{0ex}}F\left(x\right)=0$

Alternatively, a pair of surface coordinates, ${u}^{1}$ , ${u}^{2}$ can define a surface.

${x}^{i}={x}^{i}\left({u}^{1},{u}^{2}\right),\phantom{\rule{0.277778em}{0ex}}i=1,2,3\phantom{\rule{0.277778em}{0ex}}or\phantom{\rule{0.277778em}{0ex}}\mathbf{x}=\mathbf{x}\left({u}^{1},{u}^{2}\right)$

Each point on the surface that has continuous first derivatives has associated with it the normal vector, n , a unit vector that is perpendicular or normal to the surface and is outwardly directed if it is a closed surface. Fluid-fluid interfaces need to also be characterized by the mean curvature, H, at each point on the surface to describe the normal component of the momentum balance across the interface. The flux of a vector, f , across a differential element of the surface is denoted as follows, i.e. the normal component of the flux vector multiplied by the differential area.

$\mathbf{f}\mathbf{·}\mathbf{n}\phantom{\rule{0.277778em}{0ex}}da$

Curves are defined by two relationships between the spatial coordinates or by the intersection of two surfaces.

${f}_{1}\left({x}^{1},{x}^{2},{x}^{3}\right)=0\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}{f}_{2}\left({x}^{1},{x}^{2},{x}^{3}\right)=0,\phantom{\rule{0.277778em}{0ex}}or\phantom{\rule{0.277778em}{0ex}}{f}_{1}\left(x\right)=0\phantom{\rule{0.277778em}{0ex}}and\phantom{\rule{0.277778em}{0ex}}{f}_{2}\left(x\right)=0$

Alternatively, a curve in space can be parameterized by a single parameter, such as the distance along the curve, s or time, t .

$\mathbf{x}=\mathbf{x}\left(s\right)$

The tangent vector is a unit vector that is tangent to each point on the curve.

$\mathbf{\tau }=d\mathbf{x}\left(s\right)/ds$

The component of a vector, f , tangent to a differential element of a curve is denoted as follows.

$\mathbf{f}·\mathbf{\tau }\phantom{\rule{0.277778em}{0ex}}ds$

If the parameter along the curve is time, the differential of position with respect to time is the velocity vector and the differential of velocity is acceleration.

$\begin{array}{l}\mathbf{v}=\frac{d\mathbf{x}}{dt}\\ \mathbf{a}=\frac{d\mathbf{v}}{dt}\end{array}$

Coordinate systems

Scalars, vectors, and tensors are physical entities that are independent of the choice of coordinate systems. However, the components of vectors and tensors depend on the choice of coordinate systems. The algebra and calculus of vectors and tensors will be illustrated here with Cartesian coordinate systems but these operations are valid with any coordinate system. The student is suggested to read Aris to learn about curvilinear coordinate systems. Bird, Stewart, and Lightfoot express the components of the relevant vector and tensor equations in Cartesian, cylindrical polar, and spherical polar coordinate systems.

Cartesian coordinates have coordinate axes that have the same direction in the entire space and the coordinate values have the units of length. Curvilinear coordinates, in general, may have coordinate axis that are in different directions at different locations in space and have coordinate values that may not have the units of length, e.g., θ in the cylindrical polar system. If ( ${y}^{1}$ , ${y}^{2}$ , ${y}^{3}$ ) are Cartesian coordinates and ( ${x}^{1}$ , ${x}^{2}$ , ${x}^{3}$ ) are curvilinear coordinates, a differential length is related to the differential of the coordinates by the following relations.

$\begin{array}{c}d{s}^{2}=\sum _{k=1}^{3}d{y}^{k}d{y}^{k}\hfill \\ d{y}^{k}=\frac{\partial {y}^{k}}{\partial {x}^{i}}d{x}^{i}\equiv \frac{\partial {y}^{k}}{\partial {x}^{1}}d{x}^{1}+\frac{\partial {y}^{k}}{\partial {x}^{2}}d{x}^{2}+\frac{\partial {y}^{k}}{\partial {x}^{3}}d{x}^{3}\hfill \\ d{s}^{2}=\sum _{k=1}^{3}\left(\frac{\partial {y}^{k}}{\partial {x}^{i}}d{x}^{i}\right)\left(\frac{\partial {y}^{k}}{\partial {x}^{j}}d{x}^{j}\right)\hfill \\ ={g}_{ij}d{x}_{i}d{x}_{j}\\ {g}_{ij}=\sum _{k=1}^{3}\left(\frac{\partial {y}^{k}}{\partial {x}^{i}}\right)\left(\frac{\partial {y}^{k}}{\partial {x}^{j}}\right)\hfill \end{array}$

where g ij are components of the metric tensor which transforms differential of the coordinates to differential of length. Summation is understood for repeated indices. Calculus in a curvilinear coordinate system will require the metric tensor.

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