Ok, now look at an oscillatory flow.  We will just add an oscillatory pressure gradient.

ν=.1;
dpdx=+ .05 Cos[2 ω t];
ω= 4 π/10;
L = 1;
Wormersly = Sqrt[ L^2 ω/ν]

((∂v(t, y))/(∂t)) =- (1/ρ)  ((∂p(t))/(∂x))  + (μ/ρ)(((∂^2)v(t,y))/(∂(y^2)))

ans1=NDSolve[{D[v[t,y],t]==ν  D[  D[v[t,y] ,y],y] - dpdx ,v[0,y]==0,v[t,1]==0,v[t,0]==0},v,{t,0,20},{y,0,1}]

Plot3D[Evaluate[v[t,x]/.ans1],{t,0,10},{x,0,1}, PlotRange->All,AxesLabel->{"t","y"},PlotPoints->40]

Plot[Evaluate[(v[t,x]/.ans1)/.t->4.875],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.125],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.25],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.375],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.5],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.625],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.75],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.875],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->6],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Show[%,%%,%%%,%%%%,%%%%%,%%%%%%,%%%%%%%,%%%%%%%%,%%%%%%%%%,%%%%%%%%%%%]

Plot[1+( Cos[ω t]+.2 Cos[2 ω t-π/2]+.05Cos[4 ω t-π/2]+.02Cos[8 ω t-π/2]+.01Cos[16 ω t]),{t,0,10}]

ν=.1;
dpdx=-(1+( Cos[ω t]+.2 Cos[2 ω t-π/2]+.05Cos[4 ω t-π/2]+.02Cos[8 ω t-π/2]+.01Cos[16 ω t]));
ω= 4 π/10;
L = 1;
Wormersly = Sqrt[ L^2 ω/ν]

ans1=NDSolve[{D[v[t,y],t]==ν  D[  D[v[t,y] ,y],y] - dpdx ,v[0,y]==0,v[t,1]==0,v[t,0]==0},v,{t,0,20},{y,0,1}]

Plot3D[Evaluate[v[t,x]/.ans1],{t,0,20},{x,0,1}, PlotRange->All,AxesLabel->{"t","y"},PlotPoints->40]

Plot[Evaluate[(v[t,x]/.ans1)/.t->3.5],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->3.75],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->4.],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->4.25],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->4.5],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->4.75],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.25],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Plot[Evaluate[(v[t,x]/.ans1)/.t->5.5],{x,0,1}, PlotRange->All,AxesLabel->{"y","v"}]

Show[%,%%,%%%,%%%%,%%%%%,%%%%%%,%%%%%%%,%%%%%%%%%,%%%%%%%%]

We could change the frequency

ν=.1;
dpdx=-.1 + .05 Cos[ω t];
ω= 10 π/10;
L = 1;
Wormersly = Sqrt[ L^2 ω/ν]

ans1=NDSolve[{D[v[t,y],t]==ν  D[  D[v[t,y] ,y],y] - dpdx ,v[0,y]==0,v[t,1]==0,v[t,0]==0},v,{t,0,20},{y,0,1}]

Plot3D[Evaluate[v[t,x]/.ans1],{t,0,20},{x,0,1}, PlotRange->All,AxesLabel->{"t","y"},PlotPoints->40]