Navier-Stokes Equation
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PYQs
Navier-Stokes Equation For A Viscous Fluid
1) Find Navier-Stokes equation for steady laminar flow of a viscous incompressible fluid between two infinite parallel plates.
[2014, 20M]
Consider steady, incompressible, laminar flow between two infinite parallel horizontal plates as shown in the figure.
The flow is in the x - direction, hence there is no velocity component in either the y- or z- direction (i.e., v=0 and w =0 ). The steady-state continuity. equation becomes ∂u∂x=0....(1)
From Eqn. 1 , it can be concluded that the velocity u is a function of both y and z only. Since the plates are infinitely wide, it can be argued that the velocity u should not be a function of z, i.e., it must be a function of y only, u=u(y). Applying the Navier-Stokes equations, along with the assumptions that v=0,w=0 and u=u(y), yields ∂p∂x=μd2udy2 ∂p∂y=−pg ∂p∂z=0 [4] Eqn. 4 indicates that the pressure is a function of x and y. Integrate Eqn. 3 to yield Eqn. 4 indicates that the pressure is a function of x and y. Integrate Eqn. 3 to yield
p=−pgy+g1(x) Hence it can be concluded that ∂p∂ is a function of x only. Now, integrate Eqn. 2 twice with respect to y, and treat ∂p∂x as a constant (with respect to y) to give: u=12μ∂p∂xy2+c1y+c2 Applying the no-slip conditions (i.e., the fluid is “stuck” to the plates, or u=0 at y=±h ) to determine the coefficients as follows: c1=0 and c2=−h22μ∂p∂x The velocity profile becomes u=12μ∂p∂x(y2−h2) which is a parabola.
The total volumetric flow per linear depth can be obtained by integrating the velocity to give q=∫h−hudy=−2h33μ(∂p∂x) Note, q is per linear depth, which is different than Q which is the total volumetric flow rate. Also note that the flow is negative, i.e. to the left, for a positive pressure gradient, dp/dx. This is due to the gradient definition where decreasing pressure to the right is negative.
2) A thin plate of very large area is placed in a gap of height h with oils of viscosities μ′ and μ′′ on the two sides of the plate. The plate is pulled at a constant velocity V. Calculate the position of the plate so that
(i)The shear force on the sides of the two sides of the plate is equal.
(ii) The force required to drag the plate is minimum. [End effects are neglected]
[2007, 30M]
3) A steady inviscid incompressible flow has a velocity field u=fx, v=−fy, w=0, where f is a constant. Derive an expression for the pressure applied p(x,y,z) if the pressure p{0,0,0}=p0 and ˙g=−gij.
[2006, 12M]
4) The space between two infinitely long coaxial cylinder of radii a and b(b>a) respectively is filled by a homogeneous fluid of density ρ. The inner cylinder is suddenly moved with velocity v perpendicular to this axis, the outer being kept fixed. Show that the resultant pressure on a length l of inner cylinder is πρa2lb2+a2b2−a2v.
[2004, 30M]
5) Prove that (v∇2−∂∂t)∇2ψ=∂(ψ,∇2ψ)∂(x,y), where V is the kinematic viscosity of the fluid and ψ is the stream function for a two-dimensional motion of a viscous fluid.
[2002, 15M]