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I was reading the research paper Homogenization of peristaltic flows in piezoelectric porous media and came across the hydrodynamic equation:

$$\mu \nabla^2 v^f -\underline{ \rho_f (\dot{v}^f + w \cdot \nabla) v^f)} = \nabla p -f $$

where $\mu$ is the dynamic viscosity, $v^f$ is the fluid velocity, $w$ is the seepage velocity, $p$ is the pressure, and $f$ is the body force.

What confuses me is the time derivative term of the velocity (the underlined term in the equation above), especially when I compare it with the general form of the Navier Stokes equation:

$$\underline{\rho (\frac{\partial v^f}{\partial t} + (v^f \cdot \nabla) v^f)} = - \nabla p + \mu \nabla^2 v^f + \frac{1}{3} \mu \nabla(\nabla \cdot v^f) + \rho g $$

where $u$ here is the fluid velocity, $\rho$ is its density, $p$ is pressure of fluid, $\mu$ is the dynamic viscosity of fluid, $\rho g$ is the body force (gravity).

Comparing the time derivative terms from both equations (the underlined terms), I don't understand why the seepage velocity is used in the material time derivative in the first equation. The seepage velocity is the velocity of fluid passing between the pores: I don't understand why it is used in the material time derivative that its multiplied by the fluid velocity. Why are the two velocities are used together?

Also, the term of partial time derivative w.r.t. time in the first equation is $\rho_f \dot{v}^f v^f$ where as in the second equation its $\rho \frac{\partial u}{\partial t}$, and so I don't understand why in the first equation we have it multiplied with the fluid velocity.

Note: In both equations, the fluid is a Newtonian fluid.

Quillo
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user134613
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1 Answers1

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Here my guess: the authors are solving fluid equations on a grid moving with the solid matrix, implicitly using an ALE (arbitrary Lagrangian Eulerian) approach.

To add some details, in continuum mechanics there are three main approaches:

  • Lagrangian: every material point is labelled with a coordinate $\mathbf{x}_0$
  • Eulerian: the problem is described using a stationary set of coordinates $\mathbf{x}$
  • arbitrary Lagrangian Eulerian: the problem is described following points labelled with coordinates $\mathbf{x}_b$ in an arbitrary motion.

These approaches are related with derivative of composite functions. As an example, the position in space of the material point labelled with $\mathbf{x}_0$ and the arbitrary point labelled with $\mathbf{x}_b$ can be described by functions \begin{equation} \mathbf{x}(\mathbf{x}_0, t) \qquad , \qquad \mathbf{x} (\mathbf{x}_b,t) \ , \end{equation} and their velocity by \begin{equation} \mathbf{v}_0 = \dfrac{\partial \mathbf{x}}{\partial t}\bigg|_{\mathbf{x}_0}=:\mathbf{u} \qquad , \qquad \mathbf{v}_b = \dfrac{\partial \mathbf{x}}{\partial t}\bigg|_{\mathbf{x}_b} \ . \end{equation} Given a function $f(\mathbf{x},t)$ of the Eulerian coordinates, it can be represented with Lagrangian or arbitrary coordinates as \begin{equation} f(\mathbf{x},t) = f(\mathbf{x}(\mathbf{x}_0,t),t) =: f_0(\mathbf{x}_0,t) \\ f(\mathbf{x},t) = f(\mathbf{x}(\mathbf{x}_b,t),t) =: f_b(\mathbf{x}_b,t) \ , \end{equation} and the time derivatives of these functions are related by \begin{equation} \dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}_0} = \dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}} + \mathbf{u} \cdot \nabla f \\ \dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}_b} = \dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}} + \mathbf{v}_b \cdot \nabla f \ . \end{equation} If we apply these relations to the time derivative of the velocity field, it's possible to write \begin{equation} \mathbf{a} = \dfrac{\partial \mathbf{u}}{\partial t}\bigg|_{\mathbf{x}_0} = \dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}} + ( \mathbf{u} \cdot \nabla ) \mathbf{u} =\dfrac{\partial f}{\partial t}\bigg|_{\mathbf{x}_b} + ( ( \mathbf{u} - \mathbf{v}_b ) \cdot \nabla ) \mathbf{u} \ . \end{equation} If we mean the operator $\dot(f)$ as the time derivative w.r.t. the points of the solid mesh labelled with $\mathbf{x}_b$, $\frac{partial}{\partial t}|_{\mathbf{x}_b}$, we get the expression of the paper for the fluid equation \begin{equation} \dot{\mathbf{u}} + ( ( \mathbf{u} - \mathbf{v}_b ) \cdot \nabla ) \mathbf{u} \ . \end{equation}

basics
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