I am facing trouble when solving the Navier stokes equation for unsteady flow of viscous incompressible fluid with constant fluid properties. Until now, I read two cases:
- Unsteady flow of viscous incompressible fluid over a suddenly accelerate flat plane (flow over a plane wall suddenly set in motion)
- Unsteady flow of viscous incompressible fluid over an oscillating plate (flow of a viscous incompressible fluid due to an oscillating plane wall)
For the first case, after simplifying the continuity and navier stokes equation, we get, (Let there be an incompressible viscous fluid over a half plane $y=0$ i.e., xz-plane)
$$ \begin{align} \frac{\partial V_x}{\partial t}=\mu \frac{\partial^2 V_x}{\partial y^2}\\ \text{boundary conditions,}\\ V_x=0\quad\text{When }t\leq 0\quad\text{for all }y\\ V_x = U \quad\text{at }y=0\quad\text{When }t>0\\ V_x = 0 \quad\text{at }y=\infty\quad\text{When }t>0 \end{align} $$ To obtain the desired solution, the PDE is reduced to an ODE by the substitutions,
$$\eta=\frac{y}{2\sqrt{\mu t}},\: V_x=U\:f(\eta)$$
Now, for the second case, similarly we get (where the plate be oscillating with a constant amplitude and frequency with velocity $U\cos nt$),
$$ \begin{align} \frac{\partial V_x}{\partial t}=\mu \frac{\partial^2 V_x}{\partial y^2}\\ \text{boundary conditions,}\\ V_x = U\cos nt \quad\text{at }y=0\quad\text{When }t>0\\ V_x = 0 \quad\text{at }y=\infty\quad\text{When }t>0 \end{align} $$ To obtain the desired solution, the PDE is reduced to an ODE by the substitutions,
$$V_x(y,t)=\Re \{\exp(int)f(y)\}$$
I couldn't understand, from where they got those transformations? Or the intuition of those transformations. It will be a great help if anyone help me to figure out this. Without knowing the reason, it seems memorizing without understanding.
I agree that instructors often present similarity solutions as if the only way to come up with them is through either "hand waving" or dimensional analysis. Fortunately there are formal methods to come up with similarity solutions. The problem is that the formal methods get really complicated really quickly and they are difficult to teach in a single lecture. That explains why they are not usually taught, because logistically it is difficult.
The key here is the chain rule. I'll give a simplified example of the thought process that works for the kind of cases cited above.
Suppose there exists a similarity coordinate $\eta = f( t, y )$ that converts a PDE into an ODE. For many cases, $\eta = y / g(t)$, but for complete generality there is no need to assume this (but it will save a lot of time!).
Divide the dependent variable up into a function that depends on the independent variables $t$ and $y$ and a function of the similarity coordinate $\eta$. This usually is pretty easy to figure out based on the boundary conditions.
Apply the chain rule to convert the $t$ and $y$ derivatives into $\eta$ derivatives.
Plug these derivatives into the PDE.
The final step is the most difficult and requires some skill or experience in figuring out what to do. The derivatives are now all written in terms of one variable, the similarity coordinate $\eta$, but the coefficients in front of the derivative terms are still functions of $t$ and $y$. To finish converting the PDE into an ODE, you have to figure out how to make these coefficients equal to functions only of the similarity coordinate $\eta$. That turns the PDE into an ODE. The choices here depend on several factors, including the boundary conditions and what ultimate form you would like the ODE to be in. Without a particular example it is difficult to show this, but this is gist of the process.
This simplified example just demonstrates the overall thought process. For more information about the various formal methods used here, I suggest checking out of the book Similarity analyses of boundary value problems in engineering by Arthur Hansen. This book goes into great detail about the process and includes many examples.