Mathematical Analysis of Material Derivative of Volume and Surface Integrals

To analyze the material derivative of volume and surface integrals, we’ll start by defining the material derivative, also known as the substantial derivative. Then, we’ll examine its application to volume and surface integrals using mathematical notation and concepts from vector calculus.

The Material Derivative

The derivative D/Dt, also known as the material time derivative is in fact, the time derivative of the quantity B for a fixed material point. And the material points are often defined by the initial position vector at t = 0. The position vector of the material point at any time t > 0 is given by:

Any flow variable defined for an arbitrary material point x0 will be defined by B(x₀,t).
Thus the corresponding spatial description will be B[x(x₀,t),t].
From the above description, we can say that there are two types of time derivatives:

The relationship between the two derivatives can be obtained by the application of chain rule:

The time derivatives at fixed spatial locations are called Eulerian time derivatives, and those taken at a fixed material point are called are known as Lagrangian.

Now coming to the material derivative of volume integrals, i.e. Reynold`s Transport Theorem which states that:

The above theorem is nothing but an extension of Leibnitz rule of differentiation when the limits of integral itself are time dependent.

Applying the usual definition of derivative as the limit of “delta t” tending to 0 to the left hand side of the above equation, we get:

In the above equation we now add and subtract the below term:

The equation then becomes:

The second term is just:

And the first term can be re-written as:

Which shows that it is the integral of B(t) over the differential volume element
V_m(t+δt) – V_m(t).

In order to find the above integral, any differential element dA_m of V_m(t) will move a distance (u.n) δt.dA_m in a time interval of time δt. Thus the above volume integral over V_m(t+δt) – V_m(t) will be converted to:

Where A_m(t) is the surface of the material volume.

Hence the final form of the Reynolds Transport Theorem will be:

The further step can be taken by the application of Gauss Divergence Theorem over the area integral of the right-hand side of the above equation to obtain the familiar form of equivalent differential form of the above equation.

Let us take the case of a simple application where we have density ρ(x ,t) instead of the flow variable B(x, t). Then we get by application of Gauss divergence Theorem as said above to the final differential form of mass conservation:

Material Derivative of Surface Integral:

In this section, we are going to apply the same concepts developed above for the case of material derivative of volume integrals to obtain the governing equations for surfactant concentration diffusion equation over an interface of two media.
Here we will use the principle of conservation of surfactant concentration over a material surface area assuming the absence of any sources or sinks, either because of chemical reactions or a flux to or from the surrounding bulk phase liquids. So, we will get:

Where D/Dt is the material derivative for points on the interface and S_m(t) is the surface element on the interface.

Again, following the same steps as described in the above section of Reynold`s Transport Theorem for material volume and keeping in mind to replace the volume integrals with surface integrals and surface integrals with contour integrals, we can obtain an equation for surfactant concentration diffusion equation over an interface of two media:

Where u = velocity of the surfactant particles along the surface of the interface
Γ = concentration of the surfactant over the interface
n_c = unit vector normal to the contour of surface element
n_t = unit vector tangential to the contour of surface element
n = unit vector normal to the surface of interface
C_m(t) = contour containing the surface element

Now n_c can be rewritten as n_c = n_t x n.

So, the second term of the right-hand side of above equation can be rewritten as:

Where dl = elemental tangential vector along the contour of surface element.

So we can see that the integrand inside the integral is nothing but essentially a scalar triple product of three vectors Γu, dl and n which is [Γu dl n].

By using the property of scalar triple product:

Now by the application of Stokes’s Theorem over a closed contour, we can convert the contour integral to the surface integral bounded by the contour, i.e.:

Evaluating the term containing the vector triple product, above contour integral can be re-written as:

So, we can define a new operator in the plane of the interface as:

Hence, the contour integral of the right hand side of the material derivative of the surface integral can be written as:

Hence, the very first equation of this section for the material derivative equation becomes:

Or,

So, the differential form will be:

Where:

If we decompose the velocity of the surfactant in two mutually perpendicular directions, i.e., along the surface of the interface and perpendicular to the surface of the interface, we will get:

So the above governing differential equation becomes:

Hence, in the absence of diffusion, we can see that there are two terms contributing to change in Γ, one is simple convection with the interface velocity us, and the second term known as dilution term. If we add a diffusion contribution, considering the fact that diffusion term arises due to the Brownian motion and can be expressed as gradient of surfactant concentration in analogy with temperature gradient in case of diffusive heat conduction, the final form of surfactant concentration diffusion equation with diffusive term will become:

Essentially, this equation can now be used in a variety of cases like the explanation of the movement of camphor particles on the surface of water, etc.

Conclusion:

The mathematical analysis of the material derivative of volume and surface integrals allows for a deeper understanding of how physical quantities evolve in a moving fluid. By examining the temporal changes in volume and surface integrals of scalar or vector fields, we can gain insights into the dynamics and behavior of fluid flow systems.