1 Fundamental equations
1.1 The Reynolds Transport Theorem
Consider a material volume \(V(t)\):
We are interested in the property \(\Phi\) contained in \(V(t)\), particularly how it changes in time, i.e. its derivative with respect to time: \[ \frac{d}{dt}\int_{V(t)}\Phi dV \]
\[ \frac{d}{dt}\int_{V(t)}\Phi dV = \int_{V(t)} \frac{\partial\Phi}{\partial t} + \nabla\cdot(\vec{u}\phi)\ dV \] and after application of the divergence theorem for the second integral term: \[ \frac{d}{dt}\int_{V(t)}\Phi dV = \int_{V(t)} \frac{\partial\Phi}{\partial t}dV + \int_{S(t)} (\Phi\vec{u})\cdot\vec{n}\ dA \tag{1.1}\]
- \(S(t)\)
- Surface of the volume \(V(t)\)
- \(\vec{x}(t, x)\)
- Potion at time \(t\) of a particle initially at \(\vec{x}\)
- \(\vec{u}(\vec{x},t)\)
- Velocity, \(\vec{u}=\frac{\partial x}{\partial t}\)
- \(\Phi(\vec{x},t)\)
- Property associated with the particle
The two different integral terms in the Reynolds Transport Theorem have two different meanings:
- Intrinsic change
- \(\int_{V(t)} \frac{\partial\Phi}{\partial t}dV\)
- Change due to fluid transport
- \(\int_{S(t)} (\Phi\vec{u})\cdot\vec{n}\ dA\)
1.2 The Convection-Diffusion Equation
Let \(\phi\) be some quantity defined as a “density”, i.e. quantity per unit mass. This could for instance be heat or a concentration of substance. The conservation law is then: \[ \frac{d}{dt}\int_{V(t)}\rho\phi\ dV = \int_{S(t)}\vec{f}\cdot\vec{n}\ dS + \int_{V(t)}q\ dV \]
We then use Fick’s law as a constitutive relation: \[ \vec{f}=k\nabla\phi \]
Now we apply the Reynolds Transfer Theorem (Equation 1.1) by setting \(\Phi=\rho\phi\): \[ \frac{d}{dt}\int_{V(t)}\rho\phi\ dV = \int_{V(t)}\frac{\partial(\rho\phi)}{\partial t}\ dV + \int_{S(t)}(\rho\phi\vec{u})\cdot\vec{n}\ dA \]
1.3 The Navier-Stokes Equations
1.3.1 Conservation of Mass
Let \(\Phi\) be the fluid density \(\rho\). Then: \[ \int_{V(t)} \rho\ dV = \text{Mass of the fluid parcel} \]
We now insert \(\Phi=\rho\) into the Reynolds Transport Theorem (Equation 1.1). \[ \frac{d}{dt} \int_{V(t)}\rho\ dV = \int_{V(t)}\frac{\partial\rho}{\partial t} + \nabla\cdot(\rho\vec{u})\ dV = 0 \]
The equivalency to zero follows from the fact that mass within a fluid parcel (no matter how it deforms) is constant, i.e. conserved. This means that the term \(\frac{d}{dt} \int_{V(t)}\rho\ dV\) which represents the change in mass over time is equal to zero.
If we inspect the right-hand side of the Reynolds Transport Theorem, knowing that it must be zero, we see that consequently the integrand must be zero. From this, we get directly to the general form of the conservation of mass equation, or continuity equation, Equation 1.2 for continuum fluids.
\[ \frac{\partial\rho}{\partial t}+\nabla\cdot(\rho\vec{u})=0 \tag{1.2}\]
If we assume \(\rho\) to be constant in time, i.e. \(\frac{\partial\rho}{\partial t}=0\), we find that the following condition must hold: \[\nabla\cdot\vec{u}=0\] This is the definition of incompressible flow, also known as divergence-free flow. Note that this still allows \(\rho\) to vary in space, meaning different densities at different locations.
1.3.2 Conservation of Momentum
To obtain the momentum equations, we set \(\Phi=\rho\vec{u}\) and substitute into the Reynolds Transport Theorem (Equation 1.1): \[ \frac{d}{dt}\int_{V(t)}\rho u_\alpha\ dV = \int_{V(t)}\frac{\partial(\rho u_\alpha)}{\partial t} + \nabla\cdot(\rho u_\alpha\vec{u})\ dV \] Here, \(\alpha\) denotes the dimension in which we are looking at the equation. For 3D flow, we have \(\alpha=1,2,3\)
The latter term can be rewritten in different forms with \(\beta\) also denoting a dimension: \[ \int_{V(t)}\nabla\cdot(\rho u_\alpha\vec{u})\ dV = \sum_{\beta=1}^{n}\frac{\partial}{\partial x_\beta}(\rho u_\alpha u_\beta) = (\rho u_\alpha u_\beta)_{,\beta} = \nabla\cdot(\rho\vec{u}\otimes\vec{u}) \]
As we look at this equation only in 1, 2 or 3 dimensions, \(n\) can be at most 3.
The term \((\rho u_\alpha u_\beta)_{,\beta}\) is written in the Einstein notation. In essence, it is a convention for a short-form sum over all elements: \[ \sum_{\alpha=1}^n u_i = u_1 + u_2 + u_3 + ... = u_i\ \text{(now in the Einstein sense)} \]
Furthermore, a comma “\(,\)” denotes a partial derivative such that: \[ \frac{\partial a}{\partial b} = a_{,b} \]
The term \(\nabla\cdot(\rho\vec{u}\otimes\vec{u})\) contains the tensor product operator \(\otimes\). Here, it is a tensor product between two vectors. As an example, assume two vectors \(u\) and \(v\) of 3 elements each. Then the result will be a 3x3 matrix: \[ u\otimes v = \begin{bmatrix} u_1 v_1 & u_1 v_2 & u_1 v_3\\ u_2 v_1 & u_2 v_2 & u_2 v_3\\ u_3 v_1 & u_3 v_2 & u_3 v_3 \end{bmatrix} \]
These equations (for which one exists per dimension) can also be expressed in a single equation using the tensor product notation: \[ \frac{d}{dt}\int_{V(t)}\rho \vec{u}\ dV = \int_{V(t)}\frac{\partial(\rho \vec{u})}{\partial t} + \nabla\cdot(\rho\vec{u}\otimes\vec{u})\ dV \tag{1.3}\]
From Newton’s law, we know that \(\frac{d(mv)}{dt}=\sum F\). Rewriting and expanding the force into body and surface forces, we find: \[ \frac{d}{dt}\int_{V(t)}\rho\vec{u}\ dV = \int_{V(t)} \vec{f}^b\rho\ dV + \int_{S(t)}\vec{f}^s\ dA \] where:
- Body forces (forces that act across the entire volume of the fluid parcel)
- \(\int_{V(t)} \vec{f}^b\rho\ dV\)
- Surface forces (forces that only apply at the surface of the fluid parcel)
- \(\int_{S(t)}\vec{f}^s\ dA\)
The Cauchy representation of stress at any material point is defined by a \(n^2\)-element stress tensor: \[ \vec{f}^s = \vec{\vec{\tau}}\cdot\vec{n} \] Consequently, in 3D flow, the stress tensor would be a tensor consisting of 9 quantities.
We then use the divergence theorem to convert the surface integral to a volume integral where we introduce the tensor \(\vec{\vec{\tau}}\): \[ \int_{S(t)}\vec{f}^s\ dA = \int_{S(t)} \vec{\vec{\tau}}\cdot\vec{n}\ dA = \int_{V(t)}\nabla\cdot\vec{\vec{\tau}}\ dV \]
Substituting this result back into Equation 1.3, we get to the momentum conservation equation for fluids Equation 1.4.
\[ \frac{\partial\rho\vec{u}}{\partial t} + \nabla\cdot(\rho\vec{u}\otimes\vec{u}) = \rho \vec{f}^b + \nabla\cdot\vec{\vec{\tau}} \tag{1.4}\]
The tensor \(\vec{\vec{\tau}}\) can be further resolved with a constitutive relation. Constitutive relations relate the stress to a strain rate.
For Newtonian fluids (e.g. water), this is defined by Equation 1.5, where \(pI\) is the pressure multiplied with the unit tensor, \(\mu\) is the dynamic viscosity and \(\vec{\vec{E}}\) is the strain rate tensor:
\[ \vec{\vec{\tau}} = -pI + 2\mu(\vec{\vec{E}}-\frac{1}{3}\Delta\vec{\vec{T}}) \tag{1.5}\]
- Strain rate tensor
- \(\vec{\vec{E}} = \frac{1}{2}(\nabla\vec{u}+\nabla\vec{u}^T)\)
- Tensor \(\vec{\vec{T}}\)
- \(\vec{\vec{T}}=\text{trace}(\vec{\vec{E}})=\nabla\cdot\vec{u}\)
Non-Newtonian fluids like blood or polymers must be modeled with different constitutive relations.
Finally, we can construct the full, compressible Navier-Stokes equations.
In its full compressible form, no simplifications can be made. \[ \frac{\partial(\rho\vec{u})}{\partial t} + \nabla\cdot(\rho\vec{u}\otimes\vec{u}) = -\nabla p + \nabla\cdot\left[2\mu\left(\vec{\vec{E}}-\frac{1}{3}\Delta\vec{\vec{T}}\right)\right] + \rho \vec{f}^b \]
With \(\nabla\cdot\vec{u}=0\), the Navier-Stokes equations reduce to:
\[ \frac{\partial(\rho\vec{u})}{\partial t} + \nabla\cdot(\rho\vec{u}\otimes\vec{u}) = -\nabla p + u\nabla^2\vec{u}+\rho \vec{f}^b \]
Some of the terms in the Navier-Stokes equations have specific meaning:
- Inertial term
- \(\frac{\partial(\rho\vec{u})}{\partial t}\)
- Convective term
- \(\nabla\cdot(\rho\vec{u}\otimes\vec{u})\)
- Pressure gradient term
- \(-\nabla p\)
- Diffusive term
- \(u\nabla^2\vec{u}\)
It may be useful to substitute the quantities in the Navier-Stokes equations with non-dimensional counterparts under use of a characteristic length \(L\), a characteristic velocity \(U\) and a reference density \(\rho_r\).
- Non-dimensional position
- \(\vec{x}' = \frac{\vec{x}}{L}\)
- Non-dimensional velocity
- \(\vec{u}' = \frac{\vec{u}}{U}\)
- Non-dimensional density
- \(\rho' = \frac{\rho}{\rho_r}\)
- Non-dimensional time
- \(t' = U \frac{t}{L}\)
- Non-dimensional pressure
- \(p' = \frac{p}{\rho_rU^2}\)
- Non-dimensional body force
- \((\vec{f}^b)' = \frac{L}{U^2}\vec{f}^b\)
When these non-dimensional quantities are used, the Navier-Stokes equations become:
\[ \frac{\partial(\rho\vec{u}')}{\partial t'} + \nabla\cdot(\rho_r\vec{u}'\otimes\vec{u}') = -\nabla p' + \nabla\cdot\left[\frac{2}{\text{Re}}\left(\vec{\vec{E}}'-\frac{1}{3}\Delta'\vec{\vec{T}}'\right)\right] + \vec{f}^b \]
- Reynolds Number
- \(\text{Re}=\frac{\rho_rUL}{\mu}\)