The solution of the equations

A change of coordinate can be applied to Eqs. (4.22),(4.23) in order to obtain a simpler form:

$$\tilde{y} \rightarrow l(\tilde{y})=\int_0^{\tilde{y}} \tilde{n}(y')dy'$$ (4.24)

It follows that when $y$ spans the range $[0,L_y]$, the coordinate $l$ spans the range $[0,\sigma/L_x]$. With this change of coordinate it happens that

$$\frac{d}{d\tilde{y}} \rightarrow \tilde{n}(l)\frac{d}{dl}$$ (4.25)

and the first equation (4.22) reads:

$$\frac{d}{dl}(\tilde{n}(l)\tilde{T}(l))=-1$$ (4.26)

from which is immediate to see that

$$H=\tilde{n}(l)\tilde{T}(l)+l$$ (4.27)

is a constant, i.e. $\frac{d}{dl}H=0$. This is equivalent to observe that

$$p(y)-g\int_0^yn(y')dy'$$ (4.28)

is constant which is nothing more than the Bernoulli theorem for a fluid in the gravitational field with the density depending upon the height.

The relation (4.28) is verified by the model simulated in this work in the Fig. fig:h for almost all the height of the container, apart of the boundary layer near the bottom driving wall.

Figure: Plot of $H$, defined in the text, versus $l$, for three different simulations of the Inclined Plane Model: two cases are with the stochastic wall ($N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$, $T_w=150$ and $T_w=250$), while the third case is with the periodic wall ($N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$, $f_w=80\pi$, $A_w=0.1$

Using the coordinate $l$ introduced in (4.25) and the elimination of $\tilde{n}(l)$ using the recognized constant, that is

$$\tilde{n}(l)=\frac{H-l}{\tilde{T}(l)}$$ (4.29)

the second equation (4.23), after some simplifications, and after a second change of coordinate $l \rightarrow s(l)=H-l$, becomes:

$$\frac{\alpha(r) s}{\tilde{T}(s)^{1/2}} \frac{d^2}{ds^2}\tilde{T}(... ...rac{\beta(r)}{\tilde{T}(s)^{1/2}}\frac{d}{ds}\tilde{T}(s)-s\tilde{T}(s)^{1/2}=0$$ (4.30)

where $\alpha(r)=(A(r)-B(r))/C(r)$, $\beta(r)=(A(r)-\frac{1}{2}B(r))/(C(r))$ are numerically checked to be positive ($\alpha$ is positive for values of $r$ not too low, about $r>0.3$) and are divergent in the limit $r \rightarrow 1$.

The correspondence with the solution of Brey et al. [43] is given by:

The equation (4.31) become a linear equation in $\tilde{T}(s)$ as soon as the change of variable $z(s)=\tilde{T}(s)^{1/2}$ is performed:

$$2\alpha(r)s\frac{d^2}{ds^2}z(s)+2\beta(r)\frac{d}{ds}z(s)-sz(s)=0$$ (4.32)

giving the solution:

$$z(s)=\mathcal{A}s^{-\nu(r)}I_{\nu(r)}(s/\sqrt{2\alpha})+\mathcal{B}s^{-\nu(r)}K_{\nu(r)}(s/\sqrt{2\alpha})$$ (4.33)

where $I_\nu$ and $K_\nu$ are the modified Bessel functions of the first kind and the second kind respectively, $\nu(r)=B(r)/(4(A(r)-B(r)))$ is real and positive for all the values of $r$ greater than the zero of the function $A(r)-B(r)$ (about $r \simeq 0.3$), with $\nu(1)=0$, while $\mathcal{A}$ and $\mathcal{B}$ are constants that must be determined with assigning the boundary conditions.

Then we can derive the expressions for $\tilde{T}(l)$ and $\tilde{n}(l)$:

$$\tilde{T}(l)=(H-l)^{-2\nu(r)}(\mathcal{A}I_{\nu(r)}((H-l)/\sqrt{2\alpha(r)})+\mathcal{B}K_{\nu(r)}((H-l)/\sqrt{2\alpha(r)}))^2$$ (4.34)

$$\tilde{n}(l)=\frac{(H-l)^{1+2\nu(r)}}{(\mathcal{A}J_{\nu(r)}((H-l)/\sqrt{2\alpha(r)})+\mathcal{B}N_{\nu(r)}((H-l)/\sqrt{2\alpha(r)}))^2}$$ (4.35)

To calculate the expressions of $\tilde{T}$ and $\tilde{n}$ as a function of the original coordinate $\tilde{y}$ one needs to solve the equation

$$\frac{d}{dl}\tilde{y}(l)=\frac{1}{\tilde{n}(l)}$$ (4.36)

putting in it the solution (4.36). However one can obtain a comparison with the numerical simulations using the new coordinate $l$. For a discussion of the boundary conditions needed to eliminate the constants $H$, $\mathcal{A}$ and $\mathcal{B}$ we refer the reader to the paper of Brey et al.[43]. In this paper the authors show that the solution fit very well a large region in the bulk but cannot work on the boundary regions near the vibrating bottom and near the open surface. The authors show also that the minimum of the temperature is compatible with the proposed equations.

Figure: Profiles of dimensionless hydrodynamic fields $\overline {n}$, $\overline {v}_y$ and $\overline {T}$ versus the dimensionless height $y/\sigma _B$, for the Inclined Plane Model with the stochastic wall at temperature $T_w=250$. $N=5000$, $N_w \approx 180$, $r=0.7$, $r_w=0.7$, $g_e=-1$.