Question

# [100 points] Consider a laminar, fully developed, steady-state flow of Newtonian fluid between two parallel and stationary discs as illustrated in the Illustration below. The flow is driven by a

pressure drop AP = P,-Po, such that guarantees the desired volumetric flow rate Q. The distance between the discs is 8. Fluid enters through a central opening of radius R, along "z"axis and then turns into the horizontal-radial flow. Ignore this entrance effect in your analysis,i.e., assume that fluid enters the space between disks at Rị uni directionally (in r direction only).Fluid leaves radially at R.. a)10Derive CONTINUTY equation pertinent to this problem. b)20 Derive MOMENTUM equations (starting from Navier-Stokes equations) relevant for fluid flowing between discs only; i.e., between radius R, and radius R. c)20Determine the BOUNDARY conditions. d)50 SOLVE the mathematical model derived in a, b and c above and obtain an algebraic expression for the velocity profile (assume very low Reynolds number). Hint: Start with complete Navier-Stokes equations and a continuity equation, and then down-select terms that are pertinent to this flow situation. (Again, ignore entrance effects, i.e., ignore the redirection of flow from vertical to horizontal; this is unidirectional flow in r direction only.Assume very low Reynolds number.)Illustration: 1 a(r u,) ¸ 1 du, du,ди,+ = 0Continuity Equation:+-arr дөdz \begin{aligned} &\text { Navier-Stokes Equations: }\\ &\rho\left[\frac{\partial u_{r}}{\partial t}+u_{r} \frac{\partial u_{r}}{\partial r}+\frac{u_{\theta}}{r} \frac{\partial u_{r}}{\partial \theta}-\frac{u_{\theta}^{2}}{r}+u_{z} \frac{\partial u_{r}}{\partial z}\right]=-\frac{\partial P}{\partial r}+\mu\left[\frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r u_{r}\right)\right)+\frac{1}{r^{2}} \frac{\partial^{2} u_{r}}{\partial \theta^{2}}-\frac{2}{r^{2}} \frac{\partial u_{\theta}}{\partial \theta}+\frac{\partial^{2} u_{r}}{\partial z^{2}}\right]+\rho g_{r} \end{aligned} \rho\left[\frac{\partial u_{\theta}}{\partial t}+u_{r} \frac{\partial u_{\theta}}{\partial r}+\frac{u_{\theta}}{r} \frac{\partial u_{\theta}}{\partial \theta}+\frac{u_{r} u_{\theta}}{r}+u_{z} \frac{\partial u_{\theta}}{\partial z}\right]=-\frac{1}{r} \frac{\partial P}{\partial \theta}+\mu\left[\frac{\partial}{\partial r}\left(\frac{1}{r} \frac{\partial}{\partial r}\left(r u_{\theta}\right)\right)+\frac{1}{r^{2}} \frac{\partial^{2} u_{\theta}}{\partial \theta^{2}}-\frac{2}{r^{2}} \frac{\partial u_{r}}{\partial \theta}+\frac{\partial^{2} u_{\theta}}{\partial z^{2}}\right]+\rho g_{\theta} \rho\left[\frac{\partial u_{z}}{\partial t}+u_{r} \frac{\partial u_{z}}{\partial r}+\frac{u_{\theta}}{r} \frac{\partial u_{z}}{\partial \theta}+u_{z} \frac{\partial u_{z}}{\partial z}\right]=-\frac{\partial P}{\partial z}+\mu\left[\frac{1}{r} \frac{\partial}{\partial r}\left(r \frac{\partial u_{z}}{\partial r}\right)+\frac{1}{r^{2}} \frac{\partial^{2} u_{z}}{\partial \theta^{2}}+\frac{\partial^{2} u_{z}}{\partial z^{2}}\right]+\rho g_{z}

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