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Question 3 U (a) Consider a steady flow of an incompressible Newtonian fluid between the two horizontal, infinite parallel plates a distance \(h\) apart. The fluid flows only in the \(x\) direction, parallel to the plates. The lower plate is fixed but the upper plate moves with constant velocity \(U\). Using the continuity equation \(\nabla \cdot \mathbf{V} = 0,\) and the Navier-Stokes equation \(\rho \frac{D\mathbf{V}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{V},\) where \(\mathbf{V}\) is the velocity of the fluid, \(t\) denotes time, \(\rho\) is the density of the fluid, \(p\) is the pressure, \(\mathbf{g}\) is the gravitational force and \(\mu\) is the viscosity of the fluid, with proper boundary conditions, determine the velocity profile of the flow. (b) A wide moving belt passes through a container of a viscous, incompressible fluid. The belt moves vertically upward with a constant velocity \(V_0\). It picks up a film of fluid of thickness \(h\) because of viscous force, whereas gravitational force tends to drain fluid down the belt. Assume that the flow is steady, laminar and fully developed, with zero shearing stress on the film surface. Using the Navier-Stokes equation, determine the expression for the average velocity of the fluid film. Belt \(h\) Fluid film (c) Consider a plane flow in which only the \(x\)- and \(y\)- components are nonzero, with viscous and gravity effects are negligible. Using a differential momentum equations for incom- pressible uniform flow, find the expression for the pressure gradient, \(\nabla p\).

          Question 3
U
(a) Consider a steady flow of an incompressible Newtonian fluid between the two horizontal,
infinite parallel plates a distance \(h\) apart. The fluid flows only in the \(x\) direction, parallel
to the plates. The lower plate is fixed but the upper plate moves with constant velocity \(U\).
Using the continuity equation
\(\nabla \cdot \mathbf{V} = 0,\)
and the Navier-Stokes equation
\(\rho \frac{D\mathbf{V}}{Dt} = -\nabla p + \rho \mathbf{g} + \mu \nabla^2 \mathbf{V},\)
where \(\mathbf{V}\) is the velocity of the fluid, \(t\) denotes time, \(\rho\) is the density of the fluid, \(p\) is
the pressure, \(\mathbf{g}\) is the gravitational force and \(\mu\) is the viscosity of the fluid, with proper
boundary conditions, determine the velocity profile of the flow.
(b) A wide moving belt passes through a container of a viscous, incompressible fluid. The
belt moves vertically upward with a constant velocity \(V_0\). It picks up a film of fluid of
thickness \(h\) because of viscous force, whereas gravitational force tends to drain fluid down
the belt. Assume that the flow is steady, laminar and fully developed, with zero shearing
stress on the film surface. Using the Navier-Stokes equation, determine the expression
for the average velocity of the fluid film.
Belt
\(h\)
Fluid film
(c) Consider a plane flow in which only the \(x\)- and \(y\)- components are nonzero, with viscous
and gravity effects are negligible. Using a differential momentum equations for incom-
pressible uniform flow, find the expression for the pressure gradient, \(\nabla p\).

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