> [!info] **Couette Flow:** 2 infinitely long parallel planes, distance $H$ apart > - No pressure gradient in the \(x\)-direction: $\dfrac{\partial p}{\partial x}=0$ > - Incompressible > - Steady state > - Fully developed in the \(x\)-direction: $\dfrac{\partial u_x}{\partial x}=0$ > - Planar flow: $\dfrac{\partial}{\partial z}(\cdots)=0,\quad u_z=0$ > - $g_x=0$ > - Newtonian fluid **Conservation of Mass:** $ \cancel{ \frac{d}{dt}\int_{CV}\rho\,dV } + \int_{CS}\rho\,\vec{u}_{rel}\cdot d\vec{A} =0 $ $ \int_{CS}\rho\,\vec{u}_{rel}\cdot d\vec{A} = -\rho \left[ u_x+\frac{\partial u_x}{\partial x}\left(-\frac{dx}{2}\right) \right](dy\,dz) + \rho \left[ u_x+\frac{\partial u_x}{\partial x}\left(\frac{dx}{2}\right) \right](dy\,dz) $ $ -\rho \left[ u_y+\frac{\partial u_y}{\partial y}\left(-\frac{dy}{2}\right) \right](dx\,dz) + \rho \left[ u_y+\frac{\partial u_y}{\partial y}\left(\frac{dy}{2}\right) \right](dx\,dz) $ $ = \rho\frac{\partial u_x}{\partial x}\,dx\,dy\,dz + \rho\frac{\partial u_y}{\partial y}\,dx\,dy\,dz $ $ \frac{\partial u_x}{\partial x} + \frac{\partial u_y}{\partial y} =0 $ $ \frac{\partial u_x}{\partial x}=0\therefore \frac{\partial u_y}{\partial y}=0\to u_{y}=\text{const.} $ $ u_{y}(0)=u_{y}(H)\implies u_y=0 \text{ everywhere} $ **Linear Momentum (x-dir):** $ F_{Sx}+\cancel{ F_{Bx} } = \cancel{ \frac{d}{dt}\int_{CV}u_x\rho\,dV } + \int_{CS}u_x\left(\rho\vec{u}_{rel}\cdot d\vec{A}\right) $ $ \int_{CS}u_x\left(\rho\vec{u}_{rel}\cdot d\vec{A}\right) = (u_x)(-\dot{m})+(u_x)(\dot{m}) =0 $ $ F_{Sx} = \left[ \tau_{yx} + \frac{\partial\tau_{yx}}{\partial y} \left(\frac{dy}{2}\right) \right](dx\,dz) - \left[ \tau_{yx} + \frac{\partial\tau_{yx}}{\partial y} \left(-\frac{dy}{2}\right) \right](dx\,dz) $ $ = \frac{\partial\tau_{yx}}{\partial y} (dx\,dy\,dz) =0,~\frac{\partial\tau_{yx}}{\partial y}=0 $ $ \tau_{yx} = \mu \left( \frac{\partial u_x}{\partial y} + \frac{\partial u_y}{\partial x} \right) $ $ u_{y}=0\therefore\frac{\partial u_y}{\partial x}=0 $ $ \frac{\partial}{\partial y} \left( \mu\frac{\partial u_x}{\partial y} \right) =0 $ $ \frac{\partial \mu}{\partial y}=0\text{ so } \frac{\partial^2u_x}{\partial y^2}=0 $ $ u_x=f(\cancel{ t },\cancel{ x },y,\cancel{ z })\to \frac{d^2u_{x}}{dy^2}=0 $ $ \boxed{u_x=C_1y+C_2} $ >[!info] Boundary Conditions >- No slip @ bottom wall: $u_x(y=0)=0 \Rightarrow C_2=0$ > - No slip @ top wall: $u_x(y=H)=U \Rightarrow C_1=\dfrac{U}{H}$ $ \text{ so }\boxed{u_x=U\left(\frac{y}{H}\right)} $