This　work　focuses　on　the　thermo-mechanical　modeling　of　a　size-dependent　pipe　made　of　bi-directional　functionally　graded　(FG)　materials　and　conveying　fluid　through　its　inner　wall.　The　material　properties　of　this　FG　nano-scale　pipe　along　radial　and　axial　directions　are　assumed　to　be　separately　varying　according　to　the　power-law　and　exponential　distributions.　The　governing　equations　of　the　FG　nano-scale　pipe　are　derived　according　to　nonlocal　strain　gradient　theory　and　Hamilton＇s　principle,　and　interactions　between　the　size-dependent　pipe　and　its　inner　fluid　are　revealed　via　size-dependent　constitutive　relation.　The　frequency　response　of　the　pipe　and　its　critical　flow　velocity　are　calculated　via　generalized　differential　quadrature　method.　After　validation　of　the　proposed　model　by　comparing　with　previous　works,　a　detailed　parametric　study　is　conducted　to　fully　examine　the　coupling　effect　of　material　distribution　and　various　parameters　including　small　size,　aspect　ratio,　boundary　constraint,　temperature　change　and　flow　velocity　on　the　frequency　response　of　the　FG　nano-pipe.　Simultaneously,　the　critical　flow　velocity　of　inner　fluid　under　which　the　divergence　phenomenon　appears　is　also　predicted　and　physically　explained　in　detail.　Numerical　results　show　that　bi-directional　FG　design　is　especially　beneficial　to　improve　the　stability　of　FG　nano-pipe,　which　can　provide　a　guide　to　design　advanced　micro/nanofluidic　devices　for　bio-engineering　applications.　©　2023　Elsevier　Ltd