### 物理代写|流体力学代写Fluid Mechanics代考|CHNG2801

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• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 物理代写|流体力学代写Fluid Mechanics代考|Mathematical description of an interfacial layer

In reality, there are, in fact, multiple structures of interfacial and membrane layers. We have chosen to limit ourselves to layers that can be described by continuous families of surfaces that can be deformed over time $(S)$. The interfacial layer (dilated) is bordered on either side by two specific surfaces of this family: a lower surface $\left(S^{-}\right)$that separates it from the continuous medium below, and an upper surface $\left(S^{+}\right)$that separates it from the continuous medium above 1 . The laws of state and, more generally, the constitutive laws of the medium of the interfacial layer and the laws of the adjacent volumic media can be similar or very different.

The present modeling is carried out by acknowledging the balance laws of the physical properties and the constitutive laws of each material medium. However, the main objective will be to establish interface laws by passing from the microscopic description of the interfacial layer, which has a certain thickness, to the macroscopic description of the interface, which is a surface without thickness.

This operation thus involves a change in scale and an integration throughout the thickness of the interfacial zone.

However, it is possible to describe the interfacial zones in curvilinear coordinate systems, where the continuous families of surfaces $(S)$ that are deformable over time will be coordinate surfaces. This can be considered as a real meshing of the interfacial zone – recall that this is dilated in thickness – which can be used to numerically solve the problem, but will mainly be used here for analytical purposes.

In the field of the numerical simulation of fluid mechanics governed by Navier-Stokes equations, orthogonal meshes associated with “finite differences” methods are often used. Indeed, in a large number of problems, walls are represented by curves that constitute essential information. Thus, it would be dangerous to try and account for this information through a simple succession of “staircase steps”, so much so that we are naturally led to using curvilinear orthogonal meshes. These can be done by using a conformal analytical transformation. In the general case, where the shape of the walls is numerically defined, a specific program develops the orthogonal mesh that will be used to compute the flow, and defines all of the elements of the corresponding metric (Huffenus 1969). Figure $1.5$ gives some examples of such meshes (Renaud-Assemat 2011). The meshes are generated for planar or revolution 2D flows as unicity problems in the case of $3 \mathrm{D}$ calculations.
Let us recall that these are external or internal calculations with fixed or mobile limits that are deformable, such as the surfaces of bubbles, drops or contact surfaces, often modeled by spline functions. In this regard, we must mention the work of Ryskin and Leal $(1983,1984)$, Duraiswami and Prosperetti (1992) and Kervella et al. (2012).

## 物理代写|流体力学代写Fluid Mechanics代考|Normal gradient and tangential gradient

Let $f(\mathbf{x}, t)$ be a function that is continuous and derivable, taking values at any point in a volume $(V)$ and at any instant $t$. We can consider, in a given Cartesian location, with the coordinates $x, y, z$, the partial derivatives of $f$ with respect to space and time: $\partial f / \partial t$ and $\partial f / \partial x, \partial f / \partial y, \partial f / \partial z$ forming the gradient vector of $f$ (usually denoted by $\operatorname{grad}(f)$ or $\nabla f$, pronounced as nabla $f$ ).

The vector $\mathbf{N}$ denotes the unit normal to a surface $(S)$ and $f(\mathbf{x}, t)$, a function of space and time, taking values at any point on the surface $(S)$ in the volume $(V)$. The orientation of this normal is a priori arbitrary. It is sometimes determined by the physics of the problem.

Let us accept the existence of the partial derivatives of $f$ at any point in $(S)$. As in the volume, we find the partial derivatives $\partial f / \partial t$ and $\nabla f$, but the gradient vector can be decomposed into a normal component and a tangential component.

The normal gradient of $f$ is written as $\partial f / \partial N=\nabla_{\perp} f=(\mathbf{N} \cdot \nabla f) \mathbf{N}$ and the tangential gradient or parallel gradient can be defined as $\nabla_{| /} f=(\mathbf{1}-\mathbf{N} \otimes \mathbf{N}) \cdot \nabla f$, where 1 is the unit tensor. Of course, we find $\nabla f \equiv \nabla_{i /} f+\nabla_{\perp} f$.

These definitions are valid for a tensor of any order of the function $f(\mathbf{x}, t)$, especially if $f$ is a scalar, a vector or a second-order tensor.

In particular, we can consider the field of unit normals $\mathbf{N}(\mathbf{x}, t)$ to the surface $(S)$, defined from the surface equations, as a function $f(\mathbf{x}, t)$.

With $\mathbf{X}$ being a vector and $\mathbf{N}$ the unit normal vector to the surface, we use the following notations:
$$\mathbf{X}{|}=(\mathbf{1}-\mathbf{N} \otimes \mathbf{N}) \cdot \mathbf{X}, \mathbf{X}{\perp}=(\mathbf{N} \otimes \mathbf{N}) \cdot \mathbf{X}, X_{\perp}=\mathbf{N} \cdot \mathbf{X}$$

## 物理代写|流体力学代写Fluid Mechanics代考|Normal gradient and tangential gradient

$$\mathbf{X} \mid=(\mathbf{1}-\mathbf{N} \otimes \mathbf{N}) \cdot \mathbf{X}, \mathbf{X} \perp=(\mathbf{N} \otimes \mathbf{N}) \cdot \mathbf{X}, X_{\perp}=\mathbf{N} \cdot \mathbf{X}$$

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