Question

1. Below is a form of the advection-diffusion equation $\frac{\partial c}{\partial t} = \alpha \nabla^2 c - \underline{u} \cdot \nabla c$ where $c$ is the species concentration (or temperature), $\alpha$ is the diffusivity for the species (or heat transfer coefficient), and $\underline{u}$ is the flow velocity. (a) Show that the advection-diffusion equation can be non-dimensionalized into the following form $\frac{\partial c^}{\partial t^} = \frac{1}{Pe} \nabla^{2} c^ - \underline{u^} \cdot \nabla^ c^*$ where the Peclet number, $Pe$ is defined as $Pe = LU/\alpha$. (b) Give both an example of flows with high Peclet number and low Peclet number and estimate the Peclet number for each scenario (order of magnitude will suffice).

          1. Below is a form of the advection-diffusion equation
$\frac{\partial c}{\partial t} = \alpha \nabla^2 c - \underline{u} \cdot \nabla c$
where $c$ is the species concentration (or temperature), $\alpha$ is the diffusivity for the species
(or heat transfer coefficient), and $\underline{u}$ is the flow velocity.
(a) Show that the advection-diffusion equation can be non-dimensionalized into the
following form
$\frac{\partial c^*}{\partial t^*} = \frac{1}{Pe} \nabla^{*2} c^* - \underline{u^*} \cdot \nabla^* c^*$
where the Peclet number, $Pe$ is defined as $Pe = LU/\alpha$.
(b) Give both an example of flows with high Peclet number and low Peclet number
and estimate the Peclet number for each scenario (order of magnitude will suffice).

1. Below is a form of the advection-diffusion equation
(∂ c)/(∂ t) = α∇^2 c - u·∇ c
where c is the species concentration (or temperature), α is the diffusivity for the species
(or heat transfer coefficient), and u is the flow velocity.
(a) Show that the advection-diffusion equation can be non-dimensionalized into the
following form
(∂ c^*)/(∂ t^*) = (1)/(Pe)∇^*2 c^* - u^*·∇^* c^*
where the Peclet number, Pe is defined as Pe = LU/α.
(b) Give both an example of flows with high Peclet number and low Peclet number
and estimate the Peclet number for each scenario (order of magnitude will suffice).

Added by Josefina C.

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