Please check out this: Two immiscible liquids A and B are flowing in laminar flow between two parallel plates. Would there be a possibility for the velocity profiles to take this form below?
No it is not possible. At the interface (i) the velocity in the two fluids must be the same and (ii) the shear stress in the two fluids must be the same ( not the velocity gradient). So you can have a velocity gradient discontinuity at the interface. But you cannot have a local velocity maximum in each layer. Look at Eq. 1 and Eq 2 in our paper on Linear Stability of Plane Poiseuille flow of two superposed liquids. You can easily prove that cannot have a local velocity maximum in each layer. You can also play with the Mathematica demonstration at http://demonstrations.wolfram.com/PlanePoiseuilleFlowOfTwoSuperposedFluids/
to see how the parameters of the flow and layer geometry interact.
No it is not possible. At the interface (i) the velocity in the two fluids must be the same and (ii) the shear stress in the two fluids must be the same ( not the velocity gradient). So you can have a velocity gradient discontinuity at the interface. But you cannot have a local velocity maximum in each layer. Look at Eq. 1 and Eq 2 in our paper on Linear Stability of Plane Poiseuille flow of two superposed liquids. You can easily prove that cannot have a local velocity maximum in each layer. You can also play with the Mathematica demonstration at http://demonstrations.wolfram.com/PlanePoiseuilleFlowOfTwoSuperposedFluids/
to see how the parameters of the flow and layer geometry interact.
It may happen if the difference in the viscosity of the two fluids is large. From the figure, fluid 'A' is more viscous than 'B'. At the interface 'A' is acting as a source of drag for 'B'.
Mahesh it is not possible. Let me elaborate. Set up coordinate system where the interface location is y=0. Then the velocity (made dimensionless with the interface velocity) in the top layer is U1=1+a1*y +b1*y^2. The velocity in the lower layer is U2=1+ a2*y+b2*y^2, where a1=(m-n^2)/(n^2+n) , b1=-(m+n)/(n^2+n), a2=a1/m, b2=b1/m, m=mu2/mu1,n=d2/d1. The top plate is at y=1 and the bottom plate at y=-n. The above velocity profile satisfies the no-slip condition at the upper and lower plates and at the interface the shear stress is continuous and the velocity is continuous. Depending on the sign of m-n^2, du/dy=0 either in the top layer or the bottom layer. Again look at the Mathematica demonstration to see how the parameters affect the velocity profile. You do not need Mathematica to use the demonstration.
Victor is correct if one or the walls is moving so that the flow is no longer plane poiesuille flow in each layer then local maxima can occur in each layer but not if it's plane poiesuille flow in each layer with zero velocity at the bounding walls.
I missed to say that the velocity profile will have only one peak , either at the interface ( for same viscosity) or in the less viscous fluid. i agree with Brian and Victor and thank you for clarification.
You solve the rectilinear equations for each layer subject to the same pressure gradient. At the walls of the channel the no slip boundary condition holds, one for each layer. At the interface the shear stress must be continuous, the parameters of the problem are the viscosity ratio, layer thickness ratio and the flow rate ratio. It is then an algebraic problem to solve
thanks for all.
I ask Brian , is it possible that the pressure gradient for the two fluid takes different values in the general case.
i think that a mistake in this exercice r simulation). give me some remarks
For rectilinear flow with a flat interface it is impossible as the pressure gradient from one layer is impsed on the other layer.
thanks doctor Brian
I mean that each layer have different pressure gradient
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